The Complete Problem Solver

THE

© 1981 b y T H E F R A N K L I N I N S T I T U T E Published b y T H E F R A N K L I N I N S T I T U T E P R E S S Philadelphia, Pennsylvania

S M

All rights reserved. No part of this book may be reproduced in any form, for any purpose, or by any means, abstracted, or entered into any electronic or other data base without specific permission in writing from the publisher. Current printing (last digit): 5 4 3 2 Printed in the United States of America Library of Congress Cataloging in Publication Data Hayes, John R„ 1940The complete problem solver, 1. Problem solving. 2. Thought and thinking. I.Title. BF441.H33 153.4'3 81-3295 ISBN 0-89168-028-4 AACR2

ACKNOWLEDGMENTS

T h i s b o o k i s t h e r e s u l t o f m a n y y e a r s ' s e a r c h i n g for a s a t i s f a c t o r y w a y t o t e a c h a course on general p r o b l e m solving skills. By g e n e r a l p r o b l e m s o l v i n g skills I m e a n skills that can b e u s e d b y a n y o n e i n s o l v i n g p r o b l e m s that o c c u r i n e v e r y d a y life. T h e i d e a t h a t I s h o u l d try t o t e a c h a p r o b l e m s o l v i n g c o u r s e a r o s e i n a c o n v e r sation w i t h H e r b S i m o n a n d S t e v e R o s e n b e r g i n 1 9 7 4 . T h e t h r e e o f u s d e c i d e d t h a t if we really k n e w something useful about problem solving, we ought to be able to t e a c h it. S t e v e a n d I , a c c e p t i n g t h e c h a l l e n g e , p u t t o g e t h e r a n i n i t i a l v e r s i o n o f t h e c o u r s e w h i c h w e t h e n t a u g h t t o a m i x e d g r o u p o f e n g i n e e r s a n d f i n e arts s t u d e n t s a t C a r n e g i e - M e l l o n U n i v e r s i t y . O u r first v e r s i o n h a d m a n y f a u l t s , b u t s t u d e n t r e s p o n s e w a s still p o s i t i v e . I w a s s u f f i c i e n t l y e n c o u r a g e d t o c o n t i n u e . T h e c o u r s e then u n d e r w e n t a l o n g evolution as I b o r r o w e d ideas from a w i d e variety of sources and tested t h e m in the classroom. M a n y of these ideas c a m e from p e o p l e w h o , at one time or another, have taught the course with me: • S t e v e n R o s e n b e r g — w h o h e l p e d m e t o start t h e w h o l e e n t e r p r i s e , • L i n d a F l o w e r — w h o made me see the relevance of problem to the crucial skill of writing,

solving

• L e e G r e g g — w h o encouraged me to think b i g about the course, and • L y n n e R e d e r — w h o a d d e d her sophistication in the fields of m e m o r y and decision making. O t h e r ideas c a m e from p e o p l e w h o w e r e g e n e r o u s e n o u g h t o c o n t r i b u t e g u e s t lectures. Delarese Ambrose spoke about h o w to get along in small groups; E d w a r d Constant, about cultural influences in creativity; John Gatchnig, about recursive methods in problem solving; Douglas Lenat, about heuristics; D a v i d Meeker, about information retrieval; Robert N e c h e s , about listening to lectures; John P a y n e , about decision making; Herbert Simon, about series completion problems; and Richard T e a r e , about problem solving in engineering. I w o u l d like to add a special note to thanks to John P a y n e . It w a s through his e x a m p l e and e n c o u r a g e m e n t that I a d d e d the section on d e c i s i o n m a k i n g to the c o u r s e — a section w h i c h m a n y students have found extremely useful. In teaching a course on p r o b l e m solving skills, it is e x t r e m e l y important to provide students w i t h close personal supervision. T e a c h i n g p r o b l e m s o l v i n g skills is a bit like coaching: T h e instructor n e e d s to w a t c h the students in action to be sure that t h e y are p e r f o r m i n g t h e s k i l l s i n t h e r i g h t w a y . A s c o u r s e e n r o l l m e n t i n c r e a s e d , i t b e c a m e i m p o s s i b l e for m e t o g i v e t h i s s u p e r v i s i o n m y s e l f . I n s t e a d , I r e l i e d o n a group of teaching assistants—mostly u n d e r g r a d u a t e s — w h o v o l u n t e e r e d to supervise groups of about 15 students in w e e k l y sections and to m e e t w e e k l y with me to discuss section problems. T h e success of the course has d e p e n d e d v e r y h e a v i l y on t h e efforts o f t h e t e a c h i n g a s s i s t a n t s l i s t e d h e r e : Doug Bauman

Kevin Brown

Cynthia Berkowitz

Steve Ciampi

Sandra J. Bond

Aaron Clevenson

O16870

Barbara Madera Clevenson Dana Dunn

Anne Lux Bill L y d e n

Suzanne Eckert

Marilyn Mantei

Carole Elm

John Maslany

Anna-Lena Ericsson-Neches

David

Becky

Ernie Prescott

Freeland

A m y Gift

Meeker

Greg Pisocky

Richard Gorelick

Susan Robinson

Mark Hanna

Mark Segal

Ron Kander

Lisa Thaviu

Anne Karcher

Judi Vitale

Rich Kleinhample

Philip Werner

Jeanne Kravanja

Ellen Zoll

Jamie L e a c h

J e a n n e H a l p i n has t a u g h t the c o u r s e v e r y s u c c e s s f u l l y o n h e r o w n . B y d o i n g so, she has d e m o n s t r a t e d the important point that the course is g e n u i n e l y "exportable." Finally, I want to note the very important contributions of Sandra Bond. She has c o o r d i n a t e d the c o m p l e x m e c h a n i s m s of the c o u r s e : m a k i n g sure that lectures got d e l i v e r e d , that g r a d i n g got d o n e , that T A ' s w i t h difficulties got l i s t e n e d to, and t h a t h u n d r e d s o f s t u d e n t s f o u n d t h e i r w a y t o w e e k l y s e c t i o n m e e t i n g s . S'he s e r v e d a s a t e a c h i n g a s s i s t a n t for m a n y y e a r s , c o n s i s t e n t l y r e c e i v i n g e x c e l l e n t e v a l u a t i o n s from h e r students. In addition, she g u e s t l e c t u r e d on creativity in w o m e n . S h e has also contributed h e a v i l y in the preparation of this book. S h e t y p e d the manuscript, d r e w the figures, e d i t e d a n d proofread the text, a n d d i d m u c h o f the research and w r i t i n g for t h e f i n a l c h a p t e r . I n s h o r t , a g r e a t d e a l o f t h e w o r k y o u s e e i n t h i s b o o k is hers. In appreciation, I dedicate this b o o k to the m a n y m e n t i o n e d a b o v e w h o h a v e contributed to this project.

CONTENTS Introduction I. PROBLEM

i SOLVING THEORY AND PRACTICE

One. U n d e r s t a n d i n g P r o b l e m s : T h e P r o c e s s o f R e p r e s e n t a t i o n

1

Two. S e a r c h

25

Three. P r o t o c o l A n a l y s i s

51

II. M E M O R Y

AND

KNOWLEDGE

ACQUISITION

Four. T h e S t r u c t u r e o f H u m a n M e m o r y Five. U s i n g M e m o r y E f f e c t i v e l y Six. L e a r n i n g S t r a t e g i e s III. D E C I S I O N Seven.

71 93 113

MAKING

G e t t i n g the Facts Straight: M a k i n g D e c i s i o n s in A C o m p l e x

World Eight. T h e L u c k o f t h e D r a w : D e a l i n g W i t h C h a n c e i n D e c i s i o n

145

Making Nine. C o s t - B e n e f i t A n a l y s i s

161 183

IV. C R E A T I V I T Y

AND

INVENTION

Ten. C o g n i t i v e P r o c e s s e s i n C r e a t i v e A c t s Eleven. H o w S o c i a l C o n d i t i o n s A f f e c t C r e a t i v i t y

197 217

V. A P P E N D I X 1

239

VI. A P P E N D I X 2

243

VII. I N D E X

251

INTRODUCTION

T h i s b o o k has t w o purposes. It is d e s i g n e d to p r o v i d e y o u with skills that will make you a better p r o b l e m solver, and to give y o u up-to-date information about the psychology of problem solving. T h e first p u r p o s e i s c l e a r l y a p r a c t i c a l o n e , b u t I b e l i e v e t h e s e c o n d p u r p o s e i s , too. I t i s i m p o r t a n t for p e o p l e t o k n o w h o w t h e i r m i n d s w o r k . C e r t a i n l y for h u m a n istic r e a s o n s — k n o w l e d g e o f o u r h u m a n n a t u r e i s v a l u a b l e i n i t s e l f — b u t i t i s a l s o important b e c a u s e it p r o v i d e s us with a d e g r e e of flexibility w h i c h we m i g h t not otherwise have. If we can examine our o w n problem-solving processes with some degree of understanding, then we have a better chance of improving them. Further, if we have s o m e understanding of h o w p e o p l e think, we can be more effective in h e l p i n g o t h e r s . A n y o n e w h o i s t o t e a c h , o r t o tutor, o r e v e n t o h e l p a c h i l d w i t h h o m e w o r k , can b e n e f i t from k n o w l e d g e o f h o w h u m a n p r o b l e m - s o l v i n g p r o c e s s e s work and h o w they can go wrong. Early in my career as a psychologist, a student a s k e d me about my special area of interest. I told h i m that I s t u d i e d p e o p l e ' s t h i n k i n g p r o c e s s e s . " O h , t h i n k i n g ! " he s a i d , " I k n o w all a b o u t that. I ' m a m a t h m a j o r . " O f c o u r s e , h e d i d k n o w a l o t a b o u t t h i n k i n g — h e k n e w a b o u t how to do it, at l e a s t in c e r t a i n c a s e s . G i v e n a m a t h problem, he could draw on a wealth of experience to h e l p h i m find a solution. But i f h e w e r e like m o s t p e o p l e , h e w o u l d h a v e - a - v e r y difficult t i m e articulating that wealth o f e x p e r i e n c e ; h e k n e w h o w t o think b u t h e didn't k n o w h o w t o d e s c r i b e his o w n t h i n k i n g . W h e n t h e y a r e f a c e d w i t h t h e i r first t e a c h i n g t a s k , w h e t h e r i n s c h o o l or out, m a n y professionals d i s c o v e r a vast d i f f e r e n c e b e t w e e n their ability to do what they do v e r y w e l l and their ability to describe w h a t they do to others. I n this b o o k , t h e n , w e h o p e t o p r o v i d e y o u w i t h s o m e skills that w i l l h e l p y o u t o solve p r o b l e m s , b u t w e also h o p e t o p r o v i d e y o u w i t h s o m e k n o w l e d g e that will g i v e y o u greater insight into w h a t y o u are d o i n g and an i n c r e a s e d ability to understand others. W H A T IS A PROBLEM? If y o u are on one side of a river and you want to get to the other side but you don't k n o w h o w , y o u h a v e a p r o b l e m . If y o u are a s s e m b l i n g a mail-order p u r c h a s e , and the instructions l e a v e y o u c o m p l e t e l y baffled about h o w to " p u t tab A in slot B , " you h a v e a p r o b l e m . If you are writing a letter and y o u just can't find the polite w a y t o s a y , " N o , w e d o n ' t w a n t y o u t o c o m e a n d stay for a m o n t h , " y o u h a v e a problem. W h e n e v e r there i s a gap b e t w e e n w h e r e y o u are n o w a n d w h e r e y o u w a n t to b e , and y o u d o n ' t k n o w h o w to find a w a y to cross that g a p , y o u h a v e a p r o b l e m . S o l v i n g a p r o b l e m m e a n s finding an appropriate w a y to cross a gap. T h e process of finding a solution has t w o major parts: 1. R e p r e s e n t i n g the g a p — t h a t is, u n d e r s t a n d i n g t h e n a t u r e o f t h e p r o b l e m , a n d 2 . S e a r c h i n g for a m e a n s t o c r o s s it. REPRESENTING THE G A P I f p e o p l e fail t o u n d e r s t a n d t h e n a t u r e o f t h e g a p , t h e y m a y w e l l s e t off i n t h e w r o n g d i r e c t i o n t o s e a r c h for t h e s o l u t i o n . S u p p o s e y o u t o l d a f r i e n d t h a t y o u w o u l d

The C o m p l e t e Problem Solver g i v e h i m $10,000 i f h e p u t h i s e l b o w i n h i s e a r . " E a s y , " y o u r f r i e n d s a y s ; " I ' l l j u s t c u t off m y e l b o w a n d p u t i t i n a n y e a r y o u c h o o s e . " N o w y o u m a y q u e s t i o n y o u r friend's v a l u e s , b u t y o u are also pretty sure that he understands the nature of the difficulty—the g a p — t h a t the problem presents. On the other hand, if your friend said, " E a s y , I'll s t a n d o n a c h a i r , " y o u w o u l d s u s p e c t t h a t h e d i d n ' t r e a l l y u n d e r stand the nature of the difficulty. R e p r e s e n t i n g t h e g a p i s n ' t a l w a y s e a s y . I n fact, t h e m a i n d i f f i c u l t y i n m a n y problems is just the difficulty of representing the gap. C o n s i d e r the Driver's License Problem. Problem 1. The Driver's License W h e n T o m and Bill applied for their drivers' licenses, they w e r e asked their ages. Bill, w h o was a bit of a revolutionary, said they w e r e both in their twenties and that was all he was going to reveal to a bunch of bureaucrats. T h e clerk insisted on more specific information so, to smooth things over, T o m a d d e d that they both had the same birthday, and that he was four times as old as Bill was w h e n he was three times as old as Bill was w h e n he was t w i c e as old as Bill was. At this the clerk fainted and the two snatched up their licenses and disappeared. W h e n the clerk came to and realized that he w o u l d have to complete his records some w a y or other he b e g a n to do a little figuring, and before long had found out h o w old the two were. C a n you tell, too? A typical reaction to this p r o b l e m is to say, " W h a t ? " or b e a t a hasty retreat explaining, " I n e v e r w a s m u c h g o o d a t p u z z l e s . " B u t the p r o b l e m really isn't v e r y d i f f i c u l t o n c e w e f i n d a n a p p r o p r i a t e r e p r e s e n t a t i o n for it. I n C h a p t e r 1 , w e w i l l discuss processes by w h i c h we c o m e to understand the nature of a problem; we will s h o w that the w a y w e r e p r e s e n t the g a p can m a k e a n e n o r m o u s difference i n the difficulty of the p r o b l e m ; and we will p r o v i d e s o m e hints on h o w to represent problems to m a k e t h e m easier. F I N D I N G A S O L U T I O N PATH O n c e w e u n d e r s t a n d t h e n a t u r e o f a p r o b l e m , t h e r e a r e still m a n y r e a s o n s w h y w e m a y h a v e d i f f i c u l t y i n f i n d i n g a s o l u t i o n t o it. T h e p r o b l e m s b e l o w i l l u s t r a t e some of the most important reasons. Problem 2. The Loser A man once offended a fortune teller by laughing at her predictions and saying that fortune telling was all nonsense. He offended her so much, in fact, that she cast a spell on him w h i c h turned him into both a compulsive gambler and in addition a consistent loser. That was pretty mean. We w o u l d expect the spell w o u l d shortly have turned him into a miserable, impoverished wreck. Instead, he soon married a wealthy b u s i n e s s w o m a n w h o took him to the casino every day, gave him money, and smiled happily as he lost it at the roulette table. T h e y lived happily in just this way ever after. W h y was the man's wife so happy to see him lose? T h i s s t o r y p o s e s a p r o b l e m for m o s t o f u s w h e n w e first s e e it. I t w o u l d b e n o p r o b l e m i f the m a n w e r e w i n n i n g m o n e y . W e k n o w right a w a y h o w t o get from w i n n i n g m o n e y to h a p p y smiles, but to get from losing m o n e y to happy smiles is a p r o b l e m — t h e r e is a g a p that we can't i m m e d i a t e l y cross. H o w can losing m o n e y lead to happiness? In trying to bridge the gap, p e o p l e propose a variety of solutions: " P e r h a p s she is so rich that she really d o e s n ' t care a b o u t the m o n e y . " " P e r h a p s s h e i s b e c o m i n g a n u n a n d w a n t s t o g i v e all h e r m o n e y a w a y . " " P e r h a p s h e r c r a z y g r a n d f a t h e r left a w i l l w h i c h r e q u i r e d h e r t o l o s e all h e r m o n e y by 21 in order to inherit a billion." " P e r h a p s s h e is a m a s o c h i s t . "

Introduction T h e s e solutions vary in quality. T h e solution about the w o m a n b e c o m i n g a nun has t h e d i f f i c u l t y t h a t i t i g n o r e s h e r h u s b a n d . A s o l u t i o n w h i c h s e e m s t o u s b e t t e r t h a n all o f t h e s e i s t h i s : W h e n p l a y i n g r o u l e t t e , t h e m a n b e t s , s a y , o n r e d a n d l o s e s , as his spell r e q u i r e s . T h e w o m a n , h o w e v e r , bets t w i c e as m u c h on b l a c k and w i n s . In short, s h e has t u r n e d h e r h u s b a n d ' s misfortune into an a d v a n t a g e . H i s loss is their gain, and so the smiles. T h e gap is crossed. T h e p r o b l e m illustrates a very important process in p r o b l e m s o l v i n g — t h e p r o c e s s o f i n v e n t i o n . I n m a n y p r o b l e m s , lots o f a p p r o a c h e s a r e c o n c e i v a b l e — s o m e o f t h e m b e t t e r t h a n o t h e r s . T y p i c a l l y , a p e r s o n w i l l try s e v e r a l a p p r o a c h e s b e f o r e hitting on a good one. If p e o p l e can't think up any approaches, then they can't solve the p r o b l e m . Invention is an important problem-solving process, but it isn't the only process r e q u i r e d i n s o l v i n g p r o b l e m s . T h e r e are m a n y p r o b l e m s i n w h i c h i n v e n t i o n i s e a s y b u t t h e p r o b l e m i s still d i f f i c u l t .

Problem 3. The Combination Lock S u p p o s e that y o u h a v e the p r o b l e m of o p e n i n g the 10-dial c o m b i n a t i o n lock s h o w n a b o v e . P r o p o s i n g p o s s i b l e solutions is easy. T h e dial setting s h o w n m a y be a s o l u t i o n — a n d i t m a y not. T h e r e i s a total o f 1 0 o r 10,000,000,000 o r t e n b i l l i o n p o s s i b l e d i a l s e t t i n g s , a n y one o f w h i c h m a y b e t h e s o l u t i o n . T h i s i s w h e r e t h e difficulty o f the p r o b l e m l i e s — f i n d i n g the single correct c o m b i n a t i o n a m o n g ten billion possibilities. If we tried one combination e v e r y second, w o r k i n g day and n i g h t , i t w o u l d t a k e u s 3 1 7 y e a r s t o try t h e m a l l . S o m e p r o b l e m s , t h e n , l i k e t h e C o m b i n a t i o n L o c k P r o b l e m , a r e d i f f i c u l t t o s o l v e b e c a u s e w e h a v e t o s e a r c h for t h e solution a m o n g a very large n u m b e r of alternatives. 1 0

T h e f o l l o w i n g p r o b l e m is a difficult one, e v e n t h o u g h it i n v o l v e s neither invention n o r e x a m i n i n g l a r g e n u m b e r s o f a l t e r n a t i v e s . Problem 4. The Rational Investor Suppose that you have a choice b e t w e e n a safe investment w h i c h yields a sure 2 5 % return and a risky investment w h i c h gives you an e v e n chance of either tripling your m o n e y or of losing it. Which investment is best? T h e d i f f i c u l t y o f p r a c t i c a l d e c i s i o n m a k i n g d e m a n d e d b y p r o b l e m s l i k e this o n e lies in e v a l u a t i n g the alternatives. P e o p l e are often u n r e l i a b l e w h e n m a k i n g such evaluations. If they c h o o s e from the same alternatives on several different occasions, the results m a y be q u i t e inconsistent. W h i c h o f t h e t w o i n v e s t m e n t s i s b e s t d e p e n d s i n a c o m p l e x w a y o n t h e investor's financial circumstances. To evaluate the alternatives accurately, most people require explicit decision procedures such as those described in Chapter 7. S o m e p r o b l e m s are difficult b e c a u s e w e h a v e trouble r e m e m b e r i n g w h e r e w e are o n o u r p a t h t o t h e s o l u t i o n . T r y t o s o l v e P r o b l e m # 5 b e f o r e r e a d i n g f u r t h e r .

The C o m p l e t e Problem Solver

Problem 5. Cats A m o n g Pigeons Messrs. D o w n s , Heath, Field, Forest, and M a r s h — f i v e elderly pigeon f a n c i e r s — w e r e worried by the depredations of marauding cats o w n e d by five not less elderly ladies, and, hoping to get control of the cats, they married the cat owners. T h e scheme worked well for each of them so far as his o w n cat and pigeons were concerned; but it was not long before each cat had claimed a victim and each fancier had lost his favorite pigeon. Mrs. D o w n s ' cat killed the p i g e o n o w n e d by the man w h o married the owner of the cat w h i c h killed Mr. Marsh's pigeon. Mr. D o w n s ' pigeon was killed by Mrs. Heath's cat. Mr. Forest's pigeon was killed by the cat o w n e d by the lady w h o married the man w h o s e p i g e o n was killed by Mrs. F i e l d ' s cat. Who was the owner of the pigeon killed by Mrs. Forest's cat? (from Phillips, 1961)

U n l e s s y o u a r e a n e x p e r t i n s o l v i n g t h i s sort o f p r o b l e m , y o u m a y h a v e h a d s o m e difficulty in k e e p i n g track of y o u r p l a c e on y o u r w a y to the solution. Y o u m a y h a v e f o u n d y o u r s e l f a s k i n g q u e s t i o n s like, " W a s n ' t M r s . M a r s h the l a d y w h o ate the cat that m a r r i e d M r . F o r e s t ' s p i g e o n ? — O r w a s i t t h e o t h e r w a y a r o u n d ? " B e i n g a b l e t o r e m e m b e r y o u r p l a c e on the solution path is a critical p r o b l e m s o l v i n g skill. C o n s i d e r P r o b l e m # £ , b u t d o n ' t c o n s i d e r i t for v e r y l o n g . ').

Problem 6. Who's Got the E n t h a l p y ? L i q u i d water at 2 1 2 ° F and 1 atm has an internal energy (on an arbitrary basis) of 180.02 Btu/Ib . T h e specific volume of liquid water at these conditions is 0.01672 fif/lbm. What is its enthalpy? m

(from Smith and Van Ness, 1959)

P r o b l e m #6 is not a v e r y difficult p r o b l e m if y o u k n o w s o m e t h i n g about t h e r m o d y n a m i c s . I f y o u d o n ' t , h o w e v e r — i f , for e x a m p l e , y o u h a v e n ' t t h e f o g g i e s t i d e a w h a t e n t h a l p y i s — t h e n it's a n i m p o s s i b l y h a r d p r o b l e m . I p r e s e n t this p r o b l e m n o t t o m a k e y o u f e e l bacj, p u t t o d r a m a t i z e t h e e x t r e m e i m p o r t a n c e o f k n o w l e d g e i n p r o b l e m s o l v i n g . I f y o u are m i s s i n g r e l e v a n t k n o w l e d g e , a n e a s y p r o b l e m may appear difficult or impossible. If your k n o w l e d g e of math and science is w e a k , the p r o b l e m s that scientists solve m a y appear m u c h harder to y o u than they really a r e . I f t h e h u m a n i t i e s o r t h e arts a r e y o u r w e a k suit, t h e n p e o p l e w h o c a n u n d e r stand p h i l o s o p h y or w h o c a n interpret a m u s i c a l score m a y s e e m m a g i c a l l y intellig e n t t o y o u . T h e m o r a l i s t h i s : M u c h that p a s s e s for c l e v e r n e s s o r i n n a t e q u i c k n e s s /V

Introduction of m i n d actually d e p e n d s on s p e c i a l i z e d k n o w l e d g e . If y o u a c q u i r e that s p e c i a l i z e d k n o w l e d g e , y o u too m a y be a b l e to solve hard p r o b l e m s and a p p e a r c l e v e r to y o u r less learned friends. O R G A N I Z A T I O N O F THE B O O K T h e six p r o b l e m s a b o v e i l l u s t r a t e six i m p o r t a n t a s p e c t s o f h u m a n p r o b l e m solving which we will emphasize: Problem 1: Problem Problem Problem Problem Problem

Representation

2: I n v e n t i o n 3 : S e a r c h for t h e S o l u t i o n A m o n g M a n y A l t e r n a t i v e s 4: D e c i s i o n M a k i n g 5: M e m o r y 6: K n o w l e d g e

T h e book is d i v i d e d into four sections: Section I . Section II. Section III. Section IV.

Problem Solving Theory and Practice (representing problems and s e a r c h i n g for s o l u t i o n s ) M e m o r y and K n o w l e d g e Acquisition Decision Making Creativity and Invention

W h i l e t h e o r d e r i n w h i c h y o u r e a d S e c t i o n s 1 , 2 , 3 , a n d 4 i s n o t c r i t i c a l for u n d e r s t a n d i n g , I d o r e c o m m e n d that y o u r e a d S e c t i o n 1 first for a n o v e r v i e w o f t h e problem-solving process. REFERENCES Phillips,

H. My Best Puzzles in Logic and Reasoning.

N e w York:

Dover,

S m i t h , J . M . , a n d V a n N e s s , H . C . Introduction to Chemical Engineering dynamics, S e c o n d E d i t i o n . N e w Y o r k : M c G r a w - H i l l , I n c . , 1 9 5 9 .

1961. Thermo-

ONE.

UNDERSTANDING PROBLEMS: THE PROCESS OF REPRESENTATION

Usually w h e n we solve a problem, we put most of our attention on the problem a n d v e r y l i t t l e a t t e n t i o n o n o u r s e l v e s — t h a t i s , o n w h a t w e a r e doing t o s o l v e t h e problem. If we d i d attend to our o w n actions, we m i g h t notice that t h e y often occur in a c h a r a c t e r i s t i c s e q u e n c e : 1 . Finding the Problem: r e c o g n i z i n g t h a t t h e r e i s a p r o b l e m t o b e s o l v e d . 2 . Representing the Problem: u n d e r s t a n d i n g t h e n a t u r e o f t h e g a p t o b e crossed. 3 . Planning the Solution: c h o o s i n g a m e t h o d for c r o s s i n g t h e g a p . 4. Carrying O u t the Plan 5 . Evaluating the Solution: a s k i n g " H o w g o o d i s t h e r e s u l t ? " o n c e t h e p l a n i s carried out. 6 . Consolidating Gains: l e a r n i n g f r o m t h e e x p e r i e n c e o f s o l v i n g . This s e q u e n c e of actions is illustrated in the following problem. Action Finding the Problem

Problem I o b s e r v e Smith, w h o claims to be too poor to r e p a y t h e $50.00 h e o w e s m e , b u y i n g r o u n d after r o u n d o f d r i n k s for h i s f r i e n d s .

Representing the Problem

I c o n c l u d e t h a t S m i t h is n o t s u f f i c i e n t l y s e rious a b o u t r e p a y i n g his debt.

Planning the Solution

I c o n s i d e r a p o l i t e t e l e p h o n e c a l l or a n o t e reminding Smith of his indebtedness, but decide instead to ask three v e r y large friends of m i n e to call on Smith in person.

Carrying O u t the Plan

I call my friends, w h o then d e l i v e r my m e s -

Evaluating the Solution

Since

sage to Smith. Smith p a i d up rapidly w i t h o u t major

b l o o d s h e d , I r e g a r d t h e p r o b l e m as satisfactorily solved.

2

The C o m p l e t e Problem Solver

Consolidating Gains

I r e v i s e m y r u l e s for l e n d i n g m o n e y t o S m i t h and reflect on the v a l u e of having a f e w large friends.

In easy problems, we m a y go through these actions in order and without any difficulties. In hard p r o b l e m s , though, we m a y h a v e to do a great deal of backtracki n g . F o r e x a m p l e , w h e n w e e v a l u a t e w h a t w e h a v e d o n e , w e m a y d e c i d e that o u r solution i s t e r r i b l e , e.g., " A s b e s t o s b r e a d will n o t s o l v e t h e b u r n e d toast p r o b l e m ! ! " and go back to planning. Or while trying to execute a solution, we may discover something about the problem which will lead us to represent it in an entirely n e w w a y — " O h , n o w I s e e w h a t k i n d o f a p r o b l e m i t i s ! " R e t r a c i n g o f t h i s sort i s characteristic of p r o b l e m s that are c a l l e d " i l l - d e f i n e d . " We w i l l discuss t h e s e in m u c h m o r e detail later. O u r success as p r o b l e m solvers d e p e n d s on the effectiveness with w h i c h we c a n c a r r y o u t e a c h o f t h e s e six a c t i o n s . I n this c h a p t e r , w e w i l l e x a m i n e t h e n a t u r e of p r o b l e m representations a n d the processes p e o p l e u s e to form them. In addition, w e w i l l d e s c r i b e t e c h n i q u e s for i m p r o v i n g r e p r e s e n t a t i o n s s o t h a t t h e y m a k e p r o b lem solving easier. In the next chapter we will discuss planning, executing, evaluating, and consolidating. We will d e l a y the discussion of p r o b l e m finding until the final s e c t i o n o f t h e b o o k b e c a u s e t h i s t o p i c i s s o c l o s e l y r e l a t e d t o t h e t o p i c o f creativity. H O W D O PEOPLE

U N D E R S T A N D PROBLEMS?

Suppose we w e r e to spy on people as they w e r e trying to understand a n e w problem, such as the Monster Problem below. Monster Problem #1 T h r e e five-handed extra-terrestrial monsters w e r e holding three crystal globes. B e c a u s e of the quantum-mechanical peculiarities of their neighborhood, both monsters and globes come in exactly three sizes with no others permitted; small, m e d i u m , and large. T h e medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-sized globe. S i n c e this situation offended their k e e n l y d e v e l o p e d sense of symmetry, they p r o c e e d e d to transfer globes from one monster to another so that each monster w o u l d h a v e a globe proportionate to its o w n size. Monster etiquette complicated the solution of the problem since it requires: 1. that only one g l o b e may be transferred at a time, 2. that if a monster is holding two globes, only the larger of the two may be transferred, and 3. that a globe may not be transferred to a monster w h o is holding a larger globe. By what s e q u e n c e of transfers could the monsters h a v e solved this problem? We might see p e o p l e reading the problem over several times and pausing over the hard parts. We m i g h t s e e t h e m d r a w i n g sketches or w r i t i n g s y m b o l s on paper, and we might hear them mutter to themselves, something like: "Let's see . . . If a m o n s t e r is h o l d i n g t w o g l o b e s . . . W h a t d o e s this m e a n ? . . ." If we w e r e to ask p e o p l e t o " t h i n k a l o u d " a s t h e y w o r k e d o n the p r o b l e m , w e w o u l d find that their reading, sketching, and muttering reflected a whirlwind of internal activities—imaging, inferencing, decision making, and retrieving of knowledge from m e m o r y — a c t i v i t i e s w h i c h are d i r e c t e d t o w a r d " u n d e r s t a n d i n g the p r o b l e m . " I f w e look i n m o r e d e t a i l , w e w o u l d find that p e o p l e are s e l e c t i n g information and i m a g i n g o b j e c t s a n d r e l a t i o n s i n t h e p r o b l e m . F o r e x a m p l e , after r e a d i n g t h e first line of the M o n s t e r P r o b l e m , a p e r s o n m i g h t form a visual i m a g e of three b l o b s , e a c h touching a circle. T h e imagined blobs and circles, of course, correspond to the monsters and the g l o b e s , and touching in the image corresponds to the relation of h o l d i n g . T h e i m a g e s u s u a l l y reflect s o m e s e l e c t i o n o f information, e.g., the b l o b s

Understanding Problems: T h e Process of Representation

3

may h a v e no h a n d s , or the circles m a y g i v e no indication that the g l o b e s are crystaline. To understand a p r o b l e m , then, the p r o b l e m solver creates (imagines) objects and relations in his h e a d w h i c h correspond to objects a n d relations in the externally presented p r o b l e m . T h e s e internal objects and relations are the p r o b l e m solver's internal representation o f t h e p r o b l e m . S i n c e p e o p l e c r e a t e t h e m , d i f f e r e n t p e o p l e may create different internal representations of the same p r o b l e m . F r e q u e n t l y , p r o b l e m s o l v e r s w i l l m a k e a n external representation o f s o m e parts o f t h e p r o b l e m . T h e y d o t h i s b y d r a w i n g s k e t c h e s a n d d i a g r a m s o r b y w r i t i n g d o j y j c L s y m h o l s . o r e q u a t i o n s w h i c h c o r r e s p o n d t o p a r t s oFthe i n t e r n a l r e p r e s e n t a tion. S u c h external representations can be e n o r m o u s l y h e l p f u l in s o l v i n g p r o b l e m s . T h e Relation of Internal and External Representations S o m e t i m e s w e c a n s o l v e a p r o b l e m u s i n g only a n i n t e r n a l r e p r e s e n t a t i o n . F o r example, most of us c a n m u l t i p l y 17 by 23 entirely in our h e a d s and, w i t h a little effort, g e t t h e r i g h t a n s w e r . M a n y p r o b l e m s , h o w e v e r , a r e v e r y d i f f i c u l t t o s o l v e without the aid of an external representation. T h e Monster P r o b l e m and the Driver's L i c e n s e P r o b l e m in the Introduction are e x a m p l e s of s u c h p r o b l e m s . While it is possible to solve the Monster P r o b l e m entirely mentally, it is very d i f f i c u l t t o k e e p t r a c k o f w h e r e y o u a r e i n this p r o b l e m w i t h o u t a n e x t e r n a l r e p r e sentation. Y o u find y o u r s e l f asking questions like, " D i d I g i v e the small g l o b e to the big monster or didn't I ? " In the Driver's License Problem, if you don't invent and write d o w n a g o o d a l g e b r a i c notation, y o u are v e r y l i k e l y to confuse s u c h things as T o m ' s a g e now w i t h h i s a g e a t a n e a r l i e r t i m e . External representations, then, are often v e r y helpful in s o l v i n g difficult p r o b l e m s . We s h o u l d n o t e , t h o u g h , t h a t e x t e r n a l r e p r e s e n t a t i o n s can't help us at all unless we also have an internal representation of the problem. I m a g i n e t h a t we a r e p l a y i n g c h e s s . In front of us t h e c h e s s b o a r d a n d p i e c e s p r o v i d e a v e r y useful external representation o f the chess g a m e . B u t w h e n w e m a k e a m o v e , w e typically try i t i n o u r h e a d s b e f o r e m a k i n g i t o n t h e b o a r d . P l a n n i n g i s d o n e i n t e r n a l l y . Further, we couldn't m a k e m o v e s either in our heads or on the board if we didn't h a v e a n internal representation o f h o w e a c h p i e c e m o v e s . I n short, i n t e l l i g e n t p l a y w o u l d be i m p o s s i b l e w i t h o u t an internal representation. In summary: 1 . A n i n t e r n a l r e p r e s e n t a t i o n i s e s s e n t i a l for intelligent p r o b l e m s o l v i n g . I n t e r n a l representations are the m e d i u m in w h i c h we think, in the s a m e w a y that words are t h e m e d i u m i n w h i c h w e talk. W i t h o u t i n t e r n a l r e p r e s e n t a t i o n s , w e c a n ' t think through the solution of a problem, just as without words we can't speak. 2 . S o m e t i m e s a n i n t e r n a l r e p r e s e n t a t i o n i s s u f f i c i e n t for s o l v i n g . I f w e w e r e v e r y skillful, w e c o u l d p l a y " b l i n d f o l d c h e s s , " that is, w e c o u l d p l a y u s i n g o n l y our internal representation, b u t it w o u l d n ' t be easy. 3 . F o r m a n y p r o b l e m s , a n e x t e r n a l r e p r e s e n t a t i o n i s very h e l p f u l . W e w i l l e x p l o r e h o w external representations can h e l p later in this chapter. W H A T D O W E NEED T O REPRESENT I N AN INTERNAL REPRESENTATION? C o n s i d e r t h e M o n s t e r P r o b l e m d i s c u s s e d p r e v i o u s l y . I f w e are t o s o l v e this p r o b l e m , t h e r e are four p r o b l e m parts that w e n e e d t o i n c l u d e i n o u r internal representation: 1 . The G o a l — w h e r e w e w a n t the g l o b e s t o b e w h e n w e are d o n e .

Understanding Problems: T h e Process of Representation

5

a t all a b o u t h o w t h e m o n s t e r s w e r e a r r a n g e d . Y o u m a y a l s o h a v e a d d e d s h a p e s for the monsters such as those s h o w n in F i g u r e 1.

Figure 1. A Representation of t h e M o n s t e r P r o b l e m Selecting Information W h i l e y o u p r o b a b l y a d d e d s o m e d e t a i l s , y o u p r o b a b l y left o t h e r s o u t . F o r example, your i m a g e of the p r o b l e m situation p r o b a b l y didn't contain anything about "the quantum-mechanical peculiarities" of the monsters' neighborhood. V e r y likely y o u r e g a r d e d this material as "just part of the c o v e r story" a n d not really relevant t o t h e solution o f the p r o b l e m . Further, y o u m a y h a v e r e c o g n i z e d that the n u m b e r o f m o n s t e r h a n d s i s i r r e l e v a n t a n d left t h a t o u t o f y o u r r e p r e s e n t a t i o n a s well. R e l e v a n c e j u d g m e n t s s u c h as t h e s e are useful b e c a u s e t h e y a l l o w us to pare our representations d o w n t o m a n a g e a b l e size. In a study exploring relevance judgments in problem solving (Hayes, Waterman, and Robinson, 1977), the experimenter read p r o b l e m s aloud w h i c h the subjects had not heard before and asked t h e m to m a k e r e l e v a n c e j u d g m e n t s . T h e p r o b l e m s w e r e p r e s e n t e d in small p i e c e s so that the subjects c o u l d m a k e separate j u d g m e n t s a b o u t e a c h p i e c e . A t y p i c a l s e s s i o n for t h e " A l l s p o r t s P r o b l e m " p r o ceeded as follows: Experimenter (reading problem): "I w e n t to tea." Subject: " N o t r e l e v a n t . " Exp "Yesterday." Sub " R e l e v a n t — t h i s m a y b e s o m e sort o f a t i m e p r o b l e m . " Exp "With an old friend." Sub " M a y b e t h e ' o l d ' i s i m p o r t a n t i f t h i s i s a t i m e p r o b l e m . N o , I'll s a y irrelevant." Exp "Mrs. Allsports." Sub "Relevant. Probably an important person." Exp " S h e has three daughters: A m e l i a , Bella, and C e l i a . " Sub " O K , n o w w e ' r e i n t o it. It's g o i n g t o b e a b o u t t h e d a u g h t e r s . R e l e v a n t . " Exp " O n the doorstep, I met another friend." Sub "Irrelevant." E v e n o n first r e a d i n g , s u b j e c t s w e r e q u i t e a c c u r a t e i n t h e i r r e l e v a n c e j u d g ments. T h e y correctly identified m o r e than 80 p e r c e n t of the material in the text w h i c h w a s actually relevant, w h i l e rejecting m o r e than 40 p e r c e n t of the text as irrelevant. C l e a r l y , r e l e v a n c e j u d g m e n t s c a n h e l p u s t o focus o n t h e important parts of the p r o b l e m a n d thus m a k e our task of b u i l d i n g a r e p r e s e n t a t i o n easier. H o w do people make relevance judgments? T h e Hayes, Waterman, and Robinson s t u d y s u g g e s t s t h a t o n e v e r y i m p o r t a n t f a c t o r i s t h e p e r s o n ' s k n o w l e d g e o f problem types. A sophisticated problem solver recognizes many problem types, such as distance-rate-time p r o b l e m s , a g e p r o b l e m s , river-crossing p r o b l e m s , and so on. O n c e t h e p r o b l e m s o l v e r h a s i d e n t i f i e d t h e p r o b l e m t y p e , j u d g i n g w h a t i s

The C o m p l e t e Problem Solver

6

r e l e v a n t for s o l u t i o n i s m u c h e a s i e r . F o r e x a m p l e , i n a n o t h e r s t u d y ( H i n s l e y , H a y e s , a n d S i m o n , 1 9 7 7 ) , t h e e x p e r i m e n t e r h a d p r e s e n t e d o n l y t h e first t h r e e w o r d s o f t h e p r o b l e m , " A river s t e a m e r . . . " A t this p o i n t the subject said, "It's g o i n g t o b e one o f t h o s e r i v e r t h i n g s w i t h u p s t r e a m , d o w n s t r e a m , a n d still w a t e r . [ S h e w a s r i g h t ] Y o u are g o i n g to c o m p a r e times upstream and d o w n s t r e a m — o r if the t i m e is constant i t w i l l b e d i s t a n c e . " B y r e c o g n i z i n g t h e p r o b l e m t y p e , t h e s u b j e c t c o u l d p r e d i c t that any m e n t i o n of s p e e d either upstream or d o w n s t r e a m w o u l d be import a n t for s o l u t i o n , a n d t h a t t h e n a m e s o f t h e b o a t , t h e c a p t a i n , a n d t h e i r d e s t i n a t i o n w o u l d b e irrelevant. I f y o u w a n t t o b e q u i c k i n f i n d i n g t h e essential parts o f a p r o b l e m , k n o w l e d g e of problem types can be a great help. Using K n o w l e d g e to Interpret Problems W h e n we form a p r o b l e m representation, we not o n l y a d d information and d e l e t e information, w e also interpret i n f o r m a t i o n — t h a t is, w e u s e our k n o w l e d g e of the language and the w o r l d to understand p r o b l e m information. I m a g i n e the f o l l o w i n g s c e n e : A r o o m in w h i c h we find a father, a m o t h e r , a son, a n d a b a b y . T h e f a t h e r s a y s , " P e d r o , J u a n i t a i s c r y i n g . P l e a s e c h a n g e h e r . " W e i n f e r t h a t t h e father is talking to the b o y , b e c a u s e we k n o w P e d r o is a male n a m e and b e c a u s e the c o m m e n t isn't an appropriate one to address to a b a b y . We infer from our knowle d g e of p e o p l e ' s b e h a v i o r at various ages that it is the b a b y w h o is c r y i n g and in n e e d of a c h a n g e , not the mother. Further, we infer that the b a b y is f e m a l e and n a m e d Juanita, a n d that t h e family is p r o b a b l y S p a n i s h or M e x i c a n . F i n a l l y , our k n o w l e d g e o f family life s u g g e s t s that w h e n t h e father says, " P l e a s e c h a n g e h e r , " h e d o e s n o t m e a n , " T u r n h e r i n for a n e w m o d e l . " E v e n a s i m p l e situation like this o n e requires us to m a k e a n u m b e r of k n o w l e d g e - b a s e d i n f e r e n c e s i n o r d e r t o u n d e r s t a n d it. W e m a k e t h e s e i n f e r e n c e s s o naturally a n d automatically that often w e b e l i e v e that our c o n c l u s i o n s w e r e a c t u a l l y s p e l l e d o u t i n t h e p r o b l e m a n d that t h e y w e r e not i n f e r e n c e s a t all. F o r e x a m p l e , i f w e w e r e t o ask s o m e o n e , " H o w d o y o u k n o w the boy's n a m e was P e d r o ? " the r e s p o n s e m i g h t w e l l b e , " I t said s o i n the p r o b l e m . " I t didn't. T h a t was an inference b a s e d on the p r o b l e m solver's k n o w l e d g e . A t i m e w h e n w e b e c o m e a c u t e l y a w a r e that w e n e e d k n o w l e d g e t o interpret p r o b l e m s i s w h e n w e d o n ' t h a v e that k n o w l e d g e . W h e n y o u r e a d t h r o u g h the Enthalpy Problem in the introduction, you may have found yourself asking questions s u c h as, " W h a t is internal e n e r g y ? " " W h a t ' s a B T U ? " " A n d w h a t in the w o r l d is e n t h a l p y ? " Without k n o w l e d g e of concepts and relations in thermodynamics to b u i l d o n , w e c a n ' t r e p r e s e n t t h e p r o b l e m . W e d o n ' t u n d e r s t a n d t h e i n i t i a l state o r the goal, and we h a v e no idea w h a t the operators m i g h t b e . ANALOGIES A N D SCHEMAS V e r y often, w h e n we e n c o u n t e r a p r o b l e m , we r e c o g n i z e that we h a v e seen a s i m i l a r p r o b l e m b e f o r e . F o r e x a m p l e , s u p p o s e t h a t w h i l e y o u a r e d r i v i n g , y o u r car begins to lose p o w e r . It will only make 30 miles per hour on l e v e l road and slows t o a c r a w l u p h i l l s . A l s o i t h a s a t e r r i b l e t e n d e n c y t o stall w h e n t h e traffic l i g h t turns g r e e n . T h e first t i m e t h i s h a p p e n s , y o u m a y t h i n k t h a t y o u r c a r i s a b o u t t o d i e a ghastly death. If y o u h a v e b e e n through it before, though, y o u m a y recognize the s y m p t o m s of a familiar p r o b l e m . Y o u r car (you h o p e ) just n e e d s a t u n e u p . W i t h luck, n e w spark p l u g s a n d points will m a k e it healthy a n d h a p p y again. For another example, consider the following problem. T w o ancient Egyptians, A h m e d a n d G e o r g e , w e r e m e a s u r i n g a f i e l d o n t h e b a n k s o f t h e N i l e . S t a r t i n g from

Understanding Problems: T h e Process of Representation

7

one corner of the field, A h m e d w a l k e d 20 cubits south and G e o r g e w a l k e d 60 cubits w e s t . H o w far a p a r t w e r e t h e y a t t h i s p o i n t ? N o w , I'm r e a s o n a b l y sure that y o u h a v e n e v e r h e a r d this p r o b l e m b e f o r e , since I just m a d e it u p . N e v e r t h e l e s s , y o u l i k e l y r e c o g n i z e that this is a " t r i a n g l e " p r o b l e m , a n d b e c a u s e y o u h a v e a t r i a n g l e p r o b l e m schema, y o u k n o w t h a t y o u s h o u l d u s e t h e P y t h a g o r e a n t h e o r e m t o s o l v e it. T h e r e a r e m a n y f a m i l i a r p r o b l e m s c h e m a s . F o r e x a m p l e , t h e r e a r e s c h e m a s for distance-rate-time problems, triangle problems, interest p r o b l e m s , river-current problems, river-crossing problems, mixture problems, age problems, and many m o r e . A p r o b l e m s c h e m a is a p a c k a g e of information a b o u t t h e properties of a p a r t i c u l a r p r o b l e m t y p e . A s c h e m a for t r i a n g l e p r o b l e m s , for e x a m p l e , m a y i n c l u d e i n f o r m a t i o n that: 1 . T h e initial state w i l l s p e c i f y l e n g t h s o f s o m e o f t h e s i d e s o f a r i g h t - a n g l e t r i a n g l e ; 2. T h e goal w i l l be to find the l e n g t h of a n o t h e r s i d e ; a n d 3. T h e operator will i n v o l v e application of the P y t h a g o r e a n t h e o r e m . T h e r e is a variety of " o p t i m i s t " story w h i c h (inadvertently) illustrates t h e importance of our k n o w l e d g e of p r o b l e m schemas in r e p r e s e n t i n g p r o b l e m s . Optimist Story 1 An optimist put a n e w kind of furnace in his house and found that it cut his heating bills in half. D e l i g h t e d , he had another one installed, expecting that he would cut his fuel bill to zero. i Optimist Story 2 An optimist really likes his doctor except that every time he visits his office, he has to wait an hour to see him. T h e n a brilliant idea strikes him. He decides that if he takes t w o friends w i t h him to help, he should only have to wait for 20 minutes. N o w , the p e c u l i a r t h i n g a b o u t the optimists' t h i n k i n g is not that t h e y are failing to u s e k n o w l e d g e , b u t rather that t h e y are u s i n g the k n o w l e d g e inappropriately. T h e r e are m a n y situations in w h i c h it is true that if o n e of s o m e t h i n g d o e s half a j o b , then two of them will do the w h o l e job. If one can of paint covers half of the house, then t w o cans o u g h t to c o v e r the w h o l e h o u s e . T h e optimist's error is that he has a p p l i e d this s c h e m a t o h e a t i n g h o u s e s , w h e r e i t i s n o t a p p r o p r i a t e . T h e optimist in t h e s e c o n d story u s e s a s c h e m a w h i c h is p e r f e c t l y appropriate in " w o r k " problems. If one person can do a j o b in an hour, three p e o p l e ought to be able to do the j o b in 20 minutes. H o w e v e r , there are m a n y activities that can't b e h a s t e n e d b y h a v i n g s e v e r a l p e o p l e c o m b i n e t h e i r effort. T h e s e i n c l u d e w a i t i n g , f a l l i n g off c l i f f s , a n d m a t u r i n g — I f o n e b o y r e a c h e s p u b e r t y a t 1 2 , c o u l d 1 2 b o y s reach puberty at o n e ? Problem s c h e m a s are an important part of the k n o w l e d g e we u s e to solve problems. H o w e v e r , a s t h e optimist stories s h o w , w e also n e e d t o k n o w w h e n t h e s c h e m a s a r e a p p r o p r i a t e a n d w h e n t h e y a r e not. INDIVIDUAL DIFFERENCES IN PROBLEM REPRESENTATION E v e n w h e n t w o p e o p l e represent the same problem, they may w e l l not represent it in the same way. A person w h o is v e r y good at filtering out irrelevant detail may p r o d u c e a v e r y spare representation, as in F i g u r e 2. Another person w h o is not good at filtering out irrelevant detail m a y produce a c o m p l e x and ornate representation, as in F i g u r e 3. T h e r e are m o r e differences b e t w e e n representations, t h o u g h , than just the amount of detail they contain. O n e person m a y represent a p r o b l e m in visual

8

The C o m p l e t e Problem Solver

Initial State

Figure 2. A Spare Representation

Figure 3 . A n O r n a t e Representation o f t h e S a m e Problem imagery, another in s e n t e n c e s , and a third in auditory irnages. If t w o p e o p l e represent a p r o b l e m in visual images, they may w e l l not use the same images. For e x a m p l e , i n i m a g i n i n g t h e m o n s t e r s i n t h e M o n s t e r P r o b l e m , s o m e s a w t h e m arranged horizontally, s o m e vertically, and some in a circle. O u r skill in p r o b l e m s o l v i n g d e p e n d s in a v e r y i m p o r t a n t w a y on our store of problem schemas. E a c h problem schema we know gives us a very valuable advantage in s o l v i n g a w h o l e class of p r o b l e m s — a n advantage w h i c h m a y consist in k n o w i n g w h a t to p a y attention to, or h o w to represent the p r o b l e m , or h o w to search for a s o l u t i o n , o r all t h r e e . C l e a r l y t h e m o r e s c h e m a s w e k n o w , t h e b e t t e r p r e p a r e d we are as problem solvers. W h i l e our p r o b l e m skill d e p e n d s o n h o w m a n y s c h e m a s w e h a v e , i t also dep e n d s on the nature of those schemas. M c D e r m o t t and Larkin (1978) have s h o w n that n o v i c e s in p h y s i c s are more likely to h a v e s c h e m a s that are tied to concrete a s p e c t s o f t h e p r o b l e m situation, e.g., " s p r i n g p r o b l e m " s c h e m a s a n d " b a l a n c e p r o b l e m " s c h e m a s , w h e r e a s experts are m o r e l i k e l y to h a v e s c h e m a s tied to abstract p h y s i c s p r i n c i p l e s , e.g., " w o r k p r o b l e m " s c h e m a s and " m o m e n t o f inertia" schemas. In the same w a y , i n e x p e r i e n c e d math students are likely to use separate s c h e m a s for t h e f o l l o w i n g p r o b l e m s : Mr. Russo takes 3 min. less than Mr. L l o y d to pack a case w h e n each works alone. O n e day, after Mr. Russo spent 6 min. in packing a case, the boss called him

Understanding Problems: T h e Process of Representation

9

away, and Mr. L l o y d finished packing in 4 more minutes. H o w many minutes w o u l d it take Mr. Russo alone to pack a case? O n e v e g e t a b l e oil contains 6% saturated fats and a second contains 2 6 % saturated fats. In making a salad dressing h o w many ounces of the second may be added to 10 oz. of the first if the percent of saturated fats is not to e x c e e d 1 6 % ? T h e y w i l l u s e a " w o r k p r o b l e m " s c h e m a for t h e first a n d a " m i x t u r e p r o b l e m " s c h e m a for t h e s e c o n d . M o r e e x p e r i e n c e d m a t h s t u d e n t s w o u l d i n c l u d e b o t h o f these problems in a "linear e q u a t i o n s " schema. Several years ago, I did some studies of the imagery p e o p l e use to solve elementary, math p r o b l e m s (Hayes, 1973). W h e n I g a v e p e o p l e long-division problems to do in their h e a d s , I h e a r d my subjects do a lot of talking to t h e m s e l v e s : "Two-seventy-three into nine-forty-one, is two, and t w o times two-seventy-three is . . ." " A h a ! " I said to myself. " A u d i t o r y i m a g e s are important h e r e . " W h a t really surprised me t h o u g h w a s the b e h a v i o r of subjects recruited from the faculty of the modern languages department. T h e s e subjects w e r e p e o p l e w h o w e r e born in E u r o p e b u t h a d b e e n i n t h e U n i t e d S t a t e s for m a n y y e a r s a n d s p o k e e x c e l l e n t English. T h e s e subjects did a lot of talking, too, b u t in F r e n c h , Spanish, Italian, Polish, or L a t v i a n — w h a t e v e r language they spoke w h e n t h e y originally learned division. O n e p e r s o n told me that he d i d e l e m e n t a r y mathematics in Catalan, his first language, and more advanced mathematics in Spanish, the language he used i n his later s c h o o l i n g . A p p a r e n t l y m a n y p e o p l e u s e auditory i m a g e r y i n s o l v i n g arithmetic p r o b l e m s . In particular they u s e the sound of the l a n g u a g e in w h i c h they originally l e a r n e d arithmetic. Visual imagery w a s also frequently used by p e o p l e solving arithmetic problems. W h i l e d o i n g p r o b l e m s in their heads, subjects reported visual images of the digits of t h e a n s w e r a n d of marks i n d i c a t i n g b o r r o w i n g or c a n c e l l a t i o n . In one experiment, I presented subjects with the card s h o w n in F i g u r e 4, and asked them to a d d the n u m b e r s in their heads. I p u r p o s e l y did not l e a v e any space o n t h e c a r d for t h e m t o w r i t e t h e i r a n s w e r s , b u t t h e n , s i n c e t h e y w e r e d o i n g t h e p r o b l e m i n t h e i r h e a d s , t h e y d i d n ' t n e e d a n y s p a c e t o w r i t e t h e a n s w e r . I n fact, 1 2 of the 16 subjects c o m p l a i n e d that there w a s no p l a c e to write the a n s w e r . T w o of

589 + 734 Figure 4. Problem Used in H a y e s ' Mental M a t h Experiment N o t e that t h e s p a c e b e l o w t h e p r o b l e m ( w h e r e o n e w o u l d o r d i n a r i l y w r i t e t h e a n s w e r if the p r o b l e m w e r e a w r i t t e n one) has b e e n e l i m i n a t e d .

The C o m p l e t e Problem Solver

10

t h e s u b j e c t s p r o v i d e d i n t e r e s t i n g s o l u t i o n s t o this d i f f i c u l t y . O n e s a i d that s h e s u d d e n l y r e a l i z e d t h a t t h e r e w a s n o r e a s o n t h a t s h e c o u l d n ' t w r i t e t h e a n s w e r above the problem. T h e other imagined a p i e c e of paper scotchtaped to the bottom of the card. My most interesting subject was one w h o had very restricted vision and who, b e t w e e n fourth and twelfth grade, h a d r e l i e d entirely on Braille. W h e n I questioned him about the imagery he u s e d in doing arithmetic, he asked if I wanted to k n o w a b o u t the " p r i n t " i m a g e s , the Braille i m a g e s , or the colors. In my b e s t psyc h o l o g i s t f a s h i o n , I m a i n t a i n e d m y u p r i g h t p o s t u r e a n d s a i d , " T e l l m e a b o u t all o f that." In d o i n g m e n t a l arithmetic, this subject c o u l d i m a g e n u m b e r s in the familiar f o r m o f n u m e r a l s a s B r a i l l e p a t t e r n s , o r a s b a r s o f c o l o r . E a c h d i g i t h a d its o w n u n i q u e c o l o r , a s d i d t h e d a y s o f t h e w e e k ( s e e T a b l e 1). H e c o u l d r e p r e s e n t a p p o i n t m e n t t i m e s b y a l a r g e p a t c h o f c o l o r for t h e d a y , w i t h a s m a l l p a t c h i n t h e c e n t e r for t h e h o u r . T h u s , T u e s d a y a t 3:00 w a s r e p r e s e n t e d b y a l a r g e p a t c h o f a q u a w i t h a small patch of p i n k in t h e center. Another subject, w h o learned her arithmetic in H o n g K o n g on an abacus, continually m o v e d her fingers w h i l e solving arithmetic problems. As Hatano, Miyako, and Binks (1977) h a v e shown, subjects w h o use the abacus to do arithmetic h a v e difficulty d o i n g m e n t a l arithmetic if t h e y are p r e v e n t e d from u s i n g their fingers. We h a v e o b s e r v e d that p e o p l e use v e r y diverse forms of representation e v e n w h e n t h e y are solving s i m p l e problems. N u m b e r s m a y be r e p r e s e n t e d as the sound of words in one's native language. T h e y m a y be represented as visual images of print forms, Braille patterns, or colors. A n d they m a y be r e p r e s e n t e d as finger m o v e m e n t s . M o s t of our subjects u s e d t w o or more of the forms. T h u s , w h e n they are representing a p r o b l e m , p e o p l e appear to h a v e considerable c h o i c e in h o w they r e p r e s e n t it. T h i s c h o i c e i s i m p o r t a n t b e c a u s e t h e f o r m o f r e p r e s e n t a t i o n t h a t s u b jects c h o o s e can m a k e a b i g difference in the difficulty t h e y h a v e in solving problems, and in the success they have in generalizing the solutions.

T a b l e 1 . O n e Subject's C o l o r Associations Numbers 0 = black 1 = white 2 = y e l l o w (with some orange) 3 = pink 4 = blue 5 = yellow (paler than 2) 6 = dark green 7 = orange pink 8 = dark b r o w n 9 = y e l l o w (paler t h a n 5)

D a y s of the W e e k M o n d a y = dark green Tuesday = greenish white W e d n e s d a y = red orange Thursday = light red orange Friday = yellow Saturday = b r o w n Sunday = greenish white, b u t w h i t e r than T u e s d a y

Months January = greenish w h i t e February = redish yellow M a r c h = dark b r o w n i s h red April = celery green M a y = light red orange (like T h u r s d a y ) June = green (lighter than Monday) July = greener than April August = red orange (lighter than W e d n e s d a y ) September = orange with some red October = relatively pure white N o v e m b e r = dark green, but not as dark as M o n d a y D e c e m b e r = darker than September

Understanding Problems: T h e Process of Representation EXTERNAL

11

REPRESENTATIONS

I n m a n y c a s e s , a n e x t e r n a l r e p r e s e n t a t i o n i s v e r y h e l p f u l for s o l v i n g p r o b l e m s . D r a w i n g a sketch, jotting d o w n lists, w r i t i n g out e q u a t i o n s , and m a k i n g diagrams can h e l p us to r e m e m b e r information a n d to notice n e w relations in the p r o b l e m . C o n s i d e r t h e f o l l o w i n g rate p r o b l e m : A car can average 20 mph up to Pike's Peak and 60 mph back down the same road. What is the average s p e e d for the whole trip?

S o m e p e o p l e will find this p r o b l e m e a s y e n o u g h t o s o l v e i n their h e a d s . O t h e r s feel m u c h more comfortable with pencil and p a p e r — w r i t i n g d o w n relations as they o c c u r t o t h e m a n d not t r y i n g t o j u g g l e all t h e facts i n t h e i r h e a d s a t o n c e . T h e s c r a t c h sheet of such a person m i g h t look like this: total d i s t a n c e (1; average rate = total t i m e (2) d i s t a n c e up = d i s t a n c e d o w n = X (3) total d i s t a n c e = 2X ... . distance (4) t i m e = rate

20 m p h X

60 m p h X

(6) total

time = — + — 20 60 2X 2X 20 • 60 • 2 X (7) a v e r a g e r a t e = = = X^ X_ 6 0 X + 2 0 X 80X 20 60 20 • 60 2 0 - 6 0 - 2 2400 (o) = = = 30 mph 80 80 C l e a r l y this external representation is an e n o r m o u s aid to m e m o r y . T h e p r o b l e m s o l v e r c a n c o m p u t e total t i m e i n l i n e s 4 t h r o u g h 6 w i t h o u t h a v i n g t o r e m e m b e r total d i s t a n c e . I n c o m p u t i n g a v e r a g e r a t e i n l i n e s 7 a n d 8 , h e c a n a p p l y e a c h algebraic step w i t h o u t h a v i n g to r e m e m b e r the effects of p r e v i o u s steps. W o r k i n g w i t h o u t s u c h a n e x t e r n a l r e p r e s e n t a t i o n w o u l d b e v e r y d i f f i c u l t for m o s t p e o p l e . Other kinds of external representation can also be v e r y useful m e m o r y aids. F o r example, matrix r e p r e s e n t a t i o n is v e r y useful in s o l v i n g identification p r o b l e m s s u c h a s this o n e : M

s

D i c k e n s , Einstein, F r e u d , and Kant are professors of E n g l i s h , Physics, Psychology, and Philosophy (though not necessarily respectively). 1. Dickens and F r e u d w e r e in the audience w h e n the psychologist delivered his first lecture. 2. Both Einstein and the philosopher w e r e friends of the physicist. 3. T h e philosopher has attended lectures by both Kant and Dickens. 4. Dickens has never heard of Freud. Match the professors to their fields.

O u r task i s t o m a t c h t h e p r o f e s s o r s t o t h e i r f i e l d s . T o d o t h i s , w e c o n s t r u c t a matrix a s s h o w n i n F i g u r e 5 . N o w , r e a d i n g s e n t e n c e 1 , w e c o n c l u d e t h a t t h e p s y -

12

The C o m p l e t e Problem Solver

English

Physics

Psychology

Philosophy

Dickens Einstein Freud Kant Figure step: 5. A Matrix for t h e D i c k e n s , Einstein, Freud, and Kant P r o b l e m After first chologist is neither D i c k e n s nor Freud, so we put X's (indicating combinations r u l e d out) i n t w o Ebnl o n hiyns itchse t o p m a tPr isxy cohf oFliogguyr e 6 . I n t hPeh isleocsoonpdh yl i n e , g cl ikssh a s s h o w P we learn that E i n s t e i n is neither the p h i l o s o p h e r nor the physicist, and in the third lD i ni ec ,k et hnas t t h e p h i l o s o p h e r i s n e i t h e r K a n t n o r D i c k e n sX, s o w e c a n fill i n f o u r m o r e Einstein Freud

X

Kant

After

second

step: English

Physics

Dickens Einstein

Psychology

Philosophy

X

X

X

Freud

X X

Kant

After

X

third

step: English

Physics

Dickens

X

Einstein Freud Kant

Psychology

X

X X

X X

Philosophy

X

o X

Figure 6. Steps for Solving t h e D i c k e n s , Einstein, Freud, and Kant Problem

Understanding Problems: T h e Process of Representation

13

X's, as s h o w n in the m i d d l e matrix of F i g u r e 6. N o w that l e a v e s o n l y F r e u d w h o could be the philosopher, so we put an 0 in the b l o c k corresponding to F r e u d and p h i l o s o p h y , a n d X o u t t h e o t h e r a l t e r n a t i v e f i e l d s for F r e u d ( s e e r e m a i n i n g m a t r i x , F i g u r e 6). P r o c e e d i n g i n t h i s w a y ( t h o u g h y o u m a y h a v e s o m e d i f f i c u l t y w i t h t h e last f e w s t e p s ) , y o u c a n i d e n t i f y t h e f i e l d s o f all o f t h e p r o f e s s o r s . T h e matrix, like the notations in the p r e v i o u s p r o b l e m , p r o v i d e s us w i t h a great deal of h e l p in r e m e m b e r i n g the results we h a v e obtained in previous steps. Without such aids, some p r o b l e m s w o u l d be difficult or i m p o s s i b l e to solve. W h i l e external representations are clearly v e r y useful as m e m o r y aids, t h e y can h e l p in other w a y s as w e l l . As P a i g e and S i m o n (1966) h a v e s h o w n , s o m e relations in problems are easier to discover w h e n we u s e a diagram than w h e n we don't. T h i n k a b o u t t h e f o l l o w i n g p r o b l e m for a m o m e n t a n d t h e n e x a m i n e F i g u r e 7 . A car radiator contains exactly one liter of a 9 0 % alcohol-water mixture. What quantity of water will change the liter to an 8 0 % alcohol mixture?

Figure 7. Representation of t h e Radiator Problem (After Paige and Simon) F o u r r e l a t i o n s w h i c h a r e i m p o r t a n t for s o l v i n g t h e p r o b l e m c a n b e r e a d d i r e c t l y from t h e d i a g r a m . T h e s e a r e : ALCOHOL BEFORE = ALCOHOL AFTER WATER AFTER

= WATER

BEFORE

+ WATER

ADDED

TOTAL VOLUME BEFORE = ALCOHOL BEFORE TOTAL VOLUME AFTER

+ WATER BEFORE

= ALCOHOL AFTER + WATER AFTER

The C o m p l e t e Problem Solver

14

I f a n y o f t h e s e r e l a t i o n s h a d n ' t o c c u r r e d t o y o u after r e a d i n g t h e p r o b l e m text, t h e diagram c o u l d h a v e g i v e n y o u a v e r y useful hint.

N o w , s o l v e this p r o b l e m b e f o r e y o u p r o c e e d . A board was sawed into two pieces. O n e p i e c e was two-thirds as long as the whole board and was e x c e e d e d in length by the second p i e c e by 4 ft. H o w long was the board before it was cut?

D i d you notice the contradictory nature of the p r o b l e m ? Paige and Simon f o u n d that p e o p l e w h o d r a w a diagram to r e p r e s e n t this p r o b l e m c a n u s e the d i a g r a m t o d i s c o v e r its c o n t r a d i c t o r y n a t u r e . P e o p l e w h o d o n o t d r a w a d i a g r a m are likely to miss the contradiction and some may be quite happy to accept an answer o f - 1 2 f e e t for t h e l e n g t h o f t h e b o a r d ! E x t e r n a l r e p r e s e n t a t i o n s , t h e n , c a n b e e n o r m o u s l y u s e f u l . Y o u s h o u l d always consider u s i n g t h e m w h e n y o u are solving difficult p r o b l e m s .

CHANGE A N D GROWTH

I N REPRESENTATIONS

An important fact a b o u t a representation is that it can c h a n g e or d e v e l o p as we w o r k o n t h e p r o b l e m . O f t e n e n o u g h , w h e n w e start t o s o l v e a p r o b l e m , t h e r e a r e s o m e important parts that w e are v a g u e a b o u t o r w h i c h h a v e e s c a p e d u s entirely. W e m a y n o t f u l l y u n d e r s t a n d t h e w h o l e p r o b l e m u n t i l w e h a v e w o r k e d o n i t for s o m e t i m e . W h e n p e o p l e start t o s o l v e t h e M o n s t e r P r o b l e m , t h e y u s u a l l y h a v e a pretty c l e a r u n d e r s t a n d i n g of t h e initial state, the goal, a n d t h e operator. O f t e n t h o u g h , t h e y d o n ' t really u n d e r s t a n d the restrictions. As t h e y try to m a k e a m o v e , we m a y h e a r t h e m mutter, " I f t w o g l o b e s are h o l d i n g t h e s a m e monster . . . N o . T h a t ' s n o t i t . " T h e y m a y n o t a c h i e v e full u n d e r s t a n d i n g u n t i l t h e y h a v e b e e n c o r r e c t e d s e v e r a l t i m e s for m a k i n g i l l e g a l m o v e s . I m a g i n e that y o u are faced with the f o l l o w i n g matchstick p r o b l e m : G i v e n 16 matches arranged in five squares like this

m o v e just three sticks to form four squares.

W h e n y o u first r e a d t h e p r o b l e m , i t i s r e a s o n a b l e t o s u s p e c t t h a t y o u r r e p r e s e n t a t i o n o f t h e g o a l w i l l b e v e r y s t r a i g h t f o r w a r d : y o u a r e l o o k i n g for any a r r a n g e m e n t of four squares, such as t h e s e :

1

1

1,

I

1

1, e t c .

I f y o u r first s e v e r a l s o l u t i o n a t t e m p t s fail, t h o u g h , y o u m a y r e e x a m i n e t h e p r o b l e m statement to s e e if there is a n y t h i n g y o u o v e r l o o k e d . Y o u m a y n o t i c e that 16 matches w i l l m a k e exactly four squares only if e a c h match is u s e d in one and only one square. A n y arrangement in w h i c h a match serves as the side of t w o different

Understanding Problems: The Process of Representation

75

squares w o n ' t work. T h i s observation allows y o u to c h a n g e your representation of t h e g o a l . N o w y o u a r e s e a r c h i n g for a r r a n g e m e n t s l i k e t h i s :

or t h i s :

This n e w , m o r e p r e c i s e representation of the goal can h e l p y o u to avoid false l e a d s in y o u r s e a r c h for a s o l u t i o n . I observed another e x a m p l e of c h a n g e in representation firsthand w h e n a friend c h a l l e n g e d m e t o s o l v e t h e F o u r K n i g h t s P r o b l e m . T h i s p r o b l e m i n v o l v e s a 3 x 3 chess b o a r d a n d four c h e s s p i e c e s — t w o w h i t e knights and t w o b l a c k knights, arranged as s h o w n in F i g u r e 8. T h e goal is to i n t e r c h a n g e the positions of the w h i t e a n d b l a c k k n i g h t s u s i n g o n l y l e g a l k n i g h t m o v e s . F o r t h o s e w h o a r e n ' t familiar w i t h c h e s s , F i g u r e 9 s h o w s t h e l e g a l k n i g h t m o v e s . T h e k n i g h t c a n m o v e one space straight a h e a d a n d o n e s p a c e d i a g o n a l l y forward.

Figure 8. Original Position in t h e Four Knights Problem

16

The C o m p l e t e Problem Solver

Figure 9. Eight Possible Knight M o v e s Unfortunately, I had never seen the Four Knights Problem before. H o w e v e r , o n g e n e r a l p r i n c i p l e s , I s e t u p s o m e g u i d e l i n e s i n s e a r c h i n g for a s o l u t i o n . F i r s t , I d e c i d e d to work with an external representation of the p r o b l e m to h e l p me k e e p my p l a c e . I u s e d a 3 x 3 m a t r i x l i k e t h a t s h o w n in F i g u r e 8, on w h i c h I p e n c i l l e d t h e current position of e a c h p i e c e a n d e r a s e d the p r e v i o u s position. S e c o n d , I k n e w that if I m o v e d p i e c e s at random, I w o u l d h a v e trouble r e m e m b e r i n g w h i c h paths I had a l r e a d y e x p l o r e d a n d w h i c h I h a d y e t to explore. F u r t h e r , I k n e w that e v e n w h e n I r e a c h e d the goal, I m i g h t forget the path that h a d l e d t h e r e . * F o r t h e s e reasons, I m a d e m o v e s w h i c h w o u l d b e e a s y t o r e m e m b e r : I m o v e d all f o u r p i e c e s o n e s t e p c l o c k w i s e , g i v i n g me the arrangement s h o w n in F i g u r e 10. T h i s r e s u l t s u g g e s t e d a b s o l u t e l y n o t h i n g t o m e , s o I m o v e d all p i e c e s o n e m o r e step c l o c k w i s e to y i e l d the formation s h o w n in F i g u r e 1 1 . A t t h i s p o i n t , I r e c o g n i z e d t h a t I h a d r o t a t e d t h e o r i g i n a l p a t t e r n b y 90° a n d t h a t if I just r e p e a t e d w h a t I had d o n e up until n o w , the p r o b l e m w o u l d be solved! Earlier I had thought small. In my representation, there was just one o p e r a t o r — t h e single knight's m o v e . N o w , b e c a u s e of the result I o b s e r v e d in the external representation, I a d d e d a larger operator to my r e p r e s e n t a t i o n — a m a c r o - o p e r a t o r — consisting of eight knights' moves. U s i n g the macro-operator, I could solve the p r o b l e m in just t w o m o v e s rather than 16. * (Question: C o u l d I have used a better external representation?)

Understanding Problems: T h e Process of Representation

17

Figure 1 1 . Four Knights Problem Rotated T w o Steps C l o c k w i s e From O r i g i n a l Position

78

The C o m p l e t e Problem Solver

V e r y often, then, we c h a n g e our representation of a p r o b l e m w h i l e we are s o l v i n g it. I n m a n y c a s e s , t h e s e c h a n g e s a p p e a r t o b e i m p r o v e m e n t s w h i c h m a k e t h e p r o b l e m easier to solve. If y o u are h a v i n g difficulty in s o l v i n g a p r o b l e m , it m a k e s sense to consider c h a n g i n g your problem representation, A useful w a y to proceed, as P o l y a (1945) has s u g g e s t e d , is to r e e x a m i n e the p r o b l e m statement v e r y carefully. P e r h a p s w e can m a k e a n inference w h i c h w i l l h e l p u s r e p r e s e n t the goal m o r e accurately as in the Matchstick Problem. Or perhaps we can form a macro-operator a s i n t h e F o u r K n i g h t s P r o b l e m . C a r e f u l e x a m i n a t i o n o f e a c h o f t h e four p r o b l e m p a r t s — i n i t i a l state, goal, operators, and r e s t r i c t i o n s — c a n s u g g e s t w a y s to i m p r o v e our representation. REPRESENTATIONS MAKE A DIFFERENCE A p r o b l e m m a y b e d i f f i c u l t o r i m p o s s i b l e for u s t o s o l v e i n o n e r e p r e s e n t a t i o n , but m u c h easier in another. For example, consider the N i n e Dots Problem.

The Nine Dots Problem

Without raising your pencil from the paper, draw four straight lines so that each of the dots above is touched by at least one of the lines.

I f y o u d o n ' t a l r e a d y k n o w t h e p r o b l e m , try t o s o l v e i t b e f o r e p r o c e e d i n g . S o m e p e o p l e h a v e trouble b e c a u s e they h a v e a d d e d a restriction to their representation w h i c h m a k e s the p r o b l e m unsolvable.* T h e restriction is that the lines should n e v e r extend b e y o n d the square defined by the nine dots. T y p i c a l solution a t t e m p t s for s u b j e c t s a d d i n g t h i s r e s t r i c t i o n a r e s h o w n i n F i g u r e 1 2 . T h e first r e p r e s e n t a t i o n g a v e t r o u b l e b e c a u s e i t a d d e d a n e x t r a r e s t r i c t i o n t o t h e p r o b l e m . A n y r e p r e s e n t a t i o n that a d d s or d e l e t e s significant things from the initial state, f r o m t h e g o a l , f r o m t h e o p e r a t o r s , o r f r o m t h e r e s t r i c t i o n s i s v e r y l i k e l y t o g i v e u s s e r i o u s t r o u b l e . W e c a n a v o i d this t r o u b l e t o s o m e e x t e n t b y c h e c k i n g o u r representation v e r y carefully against the p r o b l e m statement b e f o r e we l a u n c h into any massive solution attempt. E v e n w h e n our representation of a problem is essentially correct, though, there are other w a y s to form a correct representation. S o m e of t h e m are easier to use than others. C o n s i d e r the following river-current problem.

A River-Current Problem You are standing by the side of a river which is flowing past you at the rate of 5 mph. You spot a raft 1 mi. upstream on which there are two boys h e l p l e s s l y adrift. T h e n you spot the b o y s ' parents 1 mi. downstream paddling upstream to save them. You know that in still water the parents can paddle at the rate of 4 mph. H o w long will it be before the parents reach the b o y s ?

* See Wickelgren (1974) for an alternative interpretation.

Understanding Problems: T h e Process of Representation

19

O n e v e r y natural w a y to r e p r e s e n t the p r o b l e m is to take the point of v i e w of the o b s e r v e r s t a n d i n g b y t h e s i d e o f t h e r i v e r . ( T h e p r o b l e m r e a l l y s e t s y o u u p t o d o this.) W e c a n c o m p u t e t h e s p e e d o f t h e b o y s w i t h r e s p e c t t o t h e o b s e r v e r ( 5 m p h downstream) and the s p e e d of the parents with r e s p e c t to the observer (5 m p h - 4 mph =1 m p h downstream). T h e difference in s p e e d b e t w e e n the b o y s and their parents i s four m i l e s p e r h o u r . T h u s , i t s h o u l d t a k e h a l f a n h o u r t o c o v e r t h e two-mile distance. An alternate and s i m p l e r w a y to r e p r e s e n t the p r o b l e m is to take the point of v i e w o f t h e b o y s o n t h e raft. I f w e t a k e t h i s p o i n t o f v i e w , w e c a n i g n o r e t h e r a t e a t w h i c h the boys a n d their parents are m o v i n g w i t h r e s p e c t to t h e observer. ( T h e o b s e r v e r r e a l l y i s i r r e l e v a n t i n t h i s p r o b l e m . ) I n a d d i t i o n , w e c a n i g n o r e t h e rate o f the current, since it is affecting t h e b o y s a n d their parents e q u a l l y . (If this s e e m s strange t o y o u , r e m e m b e r that w e r o u t i n e l y i g n o r e m o t i o n s t h a t a f f e c t b o t h p a r t i e s e q u a l l y s u c h a s t h e r o t a t i o n o f t h e e a r t h o n its axis a n d t h e m o t i o n o f t h e e a r t h a r o u n d t h e sun.) A l l t h a t i s l e f t i s t h e r a t e a t w h i c h t h e p a r e n t s a r e p a d d l i n g — f o u r miles per hour. T h u s , t h e n e w r e p r e s e n t a t i o n a l l o w s u s t o find t h e rate o f a p p r o a c h in one step, w h i l e the old representation r e q u i r e d three. WORKING

BACKWARDS

W h e n y o u h a v e t r o u b l e s o l v i n g a p r o b l e m head-on, it is often useful to try to work backwards. W o r k i n g b a c k w a r d s i n v o l v e s a simple c h a n g e in representat i o n — o r point o f v i e w — i n w h i c h y o u r n e w starting p l a c e i s the original goal, and vice versa. W o r k i n g b a c k w a r d s can be helpful b e c a u s e p r o b l e m s are often easier to solve in o n e direction than another. To s e e w h y this is so, c o n s i d e r F i g u r e 14. S u p p o s e y o u r p r o b l e m is to g e t from X to Y. Starting from X, t h e r e are five e q u a l l y p r o m i s i n g p a t h s t o e x p l o r e i n s e a r c h i n g for t h e s o l u t i o n . S t a r t i n g f r o m Y , h o w e v e r , there is j u s t o n e path t o X . In this case, t h e n , s o l v i n g b a c k w a r d s is easier than s o l v i n g f o r w a r d s . H e r e is a p r o b l e m w h i c h illustrates the usefulness of w o r k i n g b a c k w a r d s : Y o u are g i v e n f o u r b l a c k c a r d s a n d f o u r r e d c a r d s f r o m a n o r d i n a r y d e c k . Y o u h a v e t o arrange t h e m i n a s t a c k , f a c e d o w n , s o t h a t y o u c a n d e a l t h e m o u t a s f o l l o w s :

1. You p l a c e the top card on the table, face up. It is black. 2. You place the next card (now on the top of the deck) on the bottom of the deck. 3. You p l a c e the next card on the table, face up. It is red. 4. You place the next card on the bottom of the deck.

Y o u p r o c e e d i n t h i s w a y — p u t t i n g a l t e r n a t e c a r d s o n t h e t a b l e — u n t i l all t h e c a r d s are d e a l t out. W h e n y o u h a v e f i n i s h e d , t h e p a t t e r n o f c a r d s o n t h e t a b l e s h o u l d l o o k like F i g u r e 1 5 . T h i s i s a m o d e r a t e l y d i f f i c u l t p r o b l e m i f y o u try t o s o l v e i t f o r w a r d s . T h e r e a r e lots o f p o s s i b l e a r r a n g e m e n t s o f t h e c a r d s (70 o f t h e m ) , a n d o n l y o n e o f t h e a r r a n g e m e n t s i s r i g h t ! I f y o u w o r k b a c k w a r d s f r o m t h e final r e s u l t , h o w e v e r , t h e r e i s j u s t one p a t h t o t h e c o r r e c t a r r a n g e m e n t . T o w o r k b a c k w a r d s , y o u h a v e t o r e v e r s e t h e original o r d e r o f o p e r a t i o n s . P u t t h e t o p c a r d o n t h e t a b l e , p u t t h e t o p c a r d o n t h e b o t t o m , etc., b e c o m e s : P u t a c a r d f r o m t h e t a b l e o n t o p , p u t t h e b o t t o m c a r d o n t o p , etc. W h e n y o u a p p l y t h i s s e q u e n c e o f o p e r a t i o n s t o t h e c a r d s l a i d o u t o n t h e t a b l e as s h o w n a b o v e , y o u e n d up w i t h a stack of cards arranged in the correct order. Working b a c k w a r d s is useful in m a n y practical situations. F o r e x a m p l e , supp o s e that y o u w e r e w r i t i n g a p o s i t i o n p a p e r t o p e r s u a d e t h e m a n a g e m e n t o f y o u r

The C o m p l e t e Problem Solver

20

Figure 1 2 . Typical Solution Attempts for the N i n e D o t s P r o b l e m

company to accept your pet idea. Y o u might work forwards by writing a paper in w h i c h y o u p r e s e n t all t h o s e d e t a i l s a n d c o m p u t a t i o n s t h a t y o u f i n d s o f a s c i n a t i n g a n d f i r i n g i t off t o m a n a g e m e n t . O r y o u m i g h t w o r k b a c k w a r d s b y s a y i n g t o y o u r s e l f , " N o w h o l d on. W h a t I'm trying to do is to c o n v i n c e m a n a g e m e n t . If they're g o i n g t o b e c o n v i n c e d , w h a t h a s t o h a p p e n ? I'll h a v e t o m e e t t h e i r o b j e c t i o n s . N o w w h a t are those objections l i k e l y t o b e ? " B y w o r k i n g b a c k w a r d s — b y starting w i t h y o u r g o a l — y o u are more likely to write a convincing paper. HYPOTHETICAL

REASONING

C o n s i d e r this p r o b l e m : Truthtellers and Liars You are visiting a strange country in w h i c h there are just two kinds of p e o p l e — t r u t h t e l l e r s and liars. Truthtellers always tell the truth and liars always lie. You hail the first two p e o p l e you m e e t and say, "Are you truthtellers or liars?" T h e first person mumbles something you can't hear. T h e second says, " H e says he is a truthteller. He is a truthteller and so am I." C a n you trust the directions that these two may g i v e y o u ?

A g o o d w a y to s o l v e this p r o b l e m is to p r o p o s e h y p o t h e s e s . F o r e x a m p l e , y o u m i g h t s a y , " S u p p o s e t h e first g u y w a s a liar. W h a t w o u l d h e h a v e s a i d t h e n ? " T h i s i s a h y p o t h e s i s b e c a u s e , for t h e m o m e n t , y o u a r e p u r p o s e l y a s s u m i n g s o m e t h i n g

Figure 1 3 . Solution o f the N i n e D o t s Problem

Understanding Problems: T h e Process of Representation C

W

21 H —

M

Figure 1 4 . A G o o d Problem for W o r k i n g B a c k w a r d s

that w a s n ' t s t a t e d i n t h e p r o b l e m — t h a t i s , t h a t t h e first m a n w a s a liar. N o w y o u w o r k o u t t h e c o n s e q u e n c e s o f t h i s h y p o t h e s i s . " I f t h e first m a n w a s a liar, h e m u s t h a v e l i e d a b o u t h i m s e l f a n d s a i d t h a t h e w a s a t r u t h t e l l e r . " N o w y o u try t h e a l t e r native h y p o t h e s i s : " S u p p o s e he w a s a truthteller." T h e c o n s e q u e n c e of this hyp o t h e s i s i s t h a t t h e first m a n w o u l d h a v e t o t e l l t h e t r u t h a b o u t h i m s e l f a n d s a y t h a t he w a s a truthteller. T h e critical d i s c o v e r y that y o u can m a k e by trying t h e s e h y p o t h e s e s i s t h a t w h e t h e r t h e first m a n w a s a t r u t h t e l l e r o r a liar, h e m u s t h a v e said that h e w a s a t r u t h t e l l e r . T h e s e c o n d m a n , t h e n , m u s t h a v e b e e n a t r u t h t e l l e r , a n d s o w a s t h e first m a n s i n c e t h e s e c o n d m a n s a i d so. This solution m e t h o d is called hypothetical reasoning. It i n v o l v e s a d d i n g to the problem representation by making hypotheses and then w o r k i n g out the consequences of these hypotheses to learn more about the problem. Hypothetical reasoning is useful in a v e r y w i d e variety of practical situations. For example, s u p p o s e that a c o m p a n y m a n a g e r is trying to d e c i d e w h e t h e r to rent or to b u y n e w office space. S h e m i g h t consider a variety of h y p o t h e s e s in m a k i n g her d e c i s i o n , e.g., " S u p p o s e w e u s e i t for m o r e t h a n t w o y e a r s — t h e n i t w i l l b e c h e a p e r t o b u y . " Or " S u p p o s e t h e r e i s a r e c e s s i o n a n d w e h a v e t o c u t o u r staff. T h e n w e w o u l d w a n t t o b e a b l e t o g e t o u t fast, s o r e n t i n g w o u l d b e b e t t e r . " We will discuss ways to evaluate these hypotheses in the chapters on decision making. ILL-DEFINED PROBLEMS Many of the practical problems we encounter e v e r y day are ill-defined ones. T h a t is, t h e y a r e p r o b l e m s t h a t w e c a n ' t s o l v e u n l e s s w e t a k e a c t i o n t o d e f i n e t h e m

1

2

3

4

5

6

7

8

B

B

B

B

B

B

B

B

Figure 1 5 . Final Position in t h e Black/Red C a r d Problem

22

The C o m p l e t e Problem Solver

b e t t e r . T h e r e a r e t w o sorts o f a c t i o n s w e m a y h a v e t o t a k e : w e m a y h a v e t o m a k e gap-filling decisions and we m a y h a v e to j u m p into the p r o b l e m before we can fully u n d e r s t a n d it. H e r e i s a n e x a m p l e o f a p r o b l e m t h a t r e q u i r e s g a p - f i l l i n g d e c i s i o n s . J u s t a s y o u a r e a b o u t t o l e a v e y o u r h o u s e for C h r i s t m a s s h o p p i n g , y o u r s i s t e r r u s h e s u p t o y o u a n d s a y s , " I h a v e n ' t h a d t i m e t o g e t a n y t h i n g for M o t h e r . W o u l d y o u m i n d g e t t i n g s o m e t h i n g for h e r — O h , j u s t a n y t h i n g n i c e — f o r a b o u t 2 0 d o l l a r s — p l e e e s e ? " Y o u r sister has s o l v e d h e r ill-defined p r o b l e m b y m a k i n g i t y o u r ill-defined p r o b l e m . T o s o l v e this p r o b l e m , y o u w i l l h a v e t o m a k e s o m e decisions. W o u l d your mother prefer clothes? Books? Records? Suppose you decide o n r e c o r d s . Y o u still h a v e m a n y d e c i s i o n s t o m a k e . I s s h e r e a l l y t i r e d o f L i b e r a c e ? W i l l h e r p u n k r o c k p h a s e l a s t ? W h e n y o u f i n a l l y c h o o s e a gift, y o u r d e c i s i o n s w i l l h a v e contributed a great deal to the definition of this ill-defined p r o b l e m . I t i s e a s y e n o u g h t o c r e a t e a n i l l - d e f i n e d p r o b l e m . I f y o u sit d o w n a t y o u r l o c a l lunch counter and order "a milk shake, please, any flavor," you will have presented t h e c o u n t e r p e o p l e w i t h a n i l l - d e f i n e d p r o b l e m . T h e y m a y r e f u s e t o a c c e p t it, s a y i n g , " I ' m sorry, w e c a n ' t h e l p y o u u n l e s s y o u c h o o s e some f l a v o r . " O r t h e y m a y accept the problem enthusiastically, reveling in the creative freedom you have p r o v i d e d t h e m , a n d m a n u f a c t u r e for y o u y o u r first h o t p a s t r a m i m i l k s h a k e . S i n c e e a c h p r o b l e m s o l v e r m a y m a k e d i f f e r e n t " g a p - f i l l i n g " d e c i s i o n s i n solving m e J a l l T r f W a result, arrive at a different solution. T h e s e solutions m a y differ c o n s i d e r a b l y in quality d e p e n d i n g on the decisions the p r o b l e m s o l v e r has m a d e . O u r skill i n s o l v i n g i l l - d e f i n e d p r o b l e m s , t h e n , d e p e n d s in an important w a y on our ability to make good gap-filling decisions. T h e second action we may h a v e to take to define an ill-defined problem is to s j u m p i n t o t h e p r o b l e m b e f o r e w e f u l l y u n d e r s t a n d it. V e r y o f t e n t h e r e a l n a t u r e o f a p r o b l e m i s h i d d e n f r o m u s u n t i l w e a c t u a l l y t r y t o s o l v e it. F o r e x a m p l e , s u p p o s e w e h a v e t h e t a s k o f w r i t i n g a m a g a z i n e a r t i c l e for t e n - y e a r - o l d s a b o u t a b o r t i o n . T h e real nature o f this p r o b l e m p r o b a b l y w o n ' t b e a p p a r e n t t o u s until w e b e g i n writing. T h e n a multitude of difficulties will leap out at us. Will they k n o w w h a t the w o r d " f e t u s " m e a n s ? H o w can the question about w h e n a fetus b e c o m e s a h u m a n be e x p l a i n e d ? C a n t h e y understand the p r o b l e m s that l e a d a w o m a n to h a v e an abortion? O f t e n w i t h p r o b l e m s s u c h a s this one, w e can't fully define the p r o b l e m until w e a r e n e a r l y f i n i s h e d s o l v i n g it. R e a l - w o r l d p r o b l e m s are often ill-defined. F o r e x a m p l e , m a n y o f the p r o b l e m s that architects face are ill-defined. T h e architect's c l i e n t m a y desire a b u i l d i n g w i t h a certain a m o u n t of floor s p a c e at a s p e c i f i e d cost b u t m a y l e a v e m o s t of the other d e c i s i o n s a b o u t t h e b u i l d i n g ' s d e s i g n — i t s f l o o r p l a n , its a p p e a r a n c e , e t c . — u p t o the architect. H o w w e l l architects m a k e these gap-filling decisions determines in large m e a s u r e h o w g o o d t h e y are as architects. Similarly, professional p r o b l e m s f a c e d b y w r i t e r s , p a i n t e r s , c o m p u t e r p r o g r a m m e r s , c o m p o s e r s , l a w y e r s , etc., are v e r y often i l l - d e f i n e d o n e s . T o s o l v e such p r o b l e m s successfully, w e m u s t b e p r e pared to m a k e gap-filling decisions and to j u m p into the p r o b l e m s before we understand them. CONCLUSION S o m e p r o b l e m s are m u c h easier to solve in one representation than another. If y o u are h a v i n g difficulty s o l v i n g a p r o b l e m , consider c h a n g i n g t h e representation. First c h e c k to be sure your representation is correct. T h e n consider c h a n g i n g y o u r point of view: Consider hypothetical reasoning or visual imagery; consider working b a c k w a r d s , m a k i n g a gap-filling d e c i s i o n , c h a n g i n g straight lines into circles — anything that m a y y i e l d a representation that is easier to w o r k with. Y o u m a y not

Understanding Problems: T h e Process of Representation

23

find o n e , b u t i f y o u a r e a l r e a d y h a v i n g d i f f i c u l t y i n s o l v i n g t h e p r o b l e m , y o u h a v e relatively little to lose. Summary 1. Internal representations are the m e d i u m of our thought w h e n we think about a problem. 2. O u r representation of a p r o b l e m must i n c l u d e a representation of the goal, and m a y a l s o i n c l u d e r e p r e s e n t a t i o n s o f a n i n i t i a l state, o p e r a t o r s , a n d r e s t r i c t i o n s . 3. An internal representation is not a c o p y of an external situation. P e o p l e are v e r y active w h e n they form representations: T h e y add information, d e l e t e information as irrelevant, a n d interpret information. T h e internal r e p r e s e n t a t i o n is v e r y different from the external p r o b l e m situation. 4. K n o w l e d g e of l a n g u a g e a n d k n o w l e d g e of the w o r l d are b o t h important in interpreting p r o b l e m information. P r o b l e m schemas are an e s p e c i a l l y important part o f our w o r l d k n o w l e d g e for i n t e r p r e t i n g p r o b l e m i n f o r m a t i o n . 5. T h e r e are large differences in the w a y different p e o p l e r e p r e s e n t the same i n f o r m a t i o n — s o m e m a y u s e a u d i t o r y i m a g e r y ; o t h e r s , v i s u a l i m a g e r y ; still others, s e n t e n c e s , a n d so on. 6. External representations can provide considerable aid in p r o b l e m solving. In d i f f i c u l t p r o b l e m s , w e s h o u l d always c o n s i d e r u s i n g e x t e r n a l r e p r e s e n t a t i o n s . 7. T h e s e differences in representation can make an important difference in the difficulty o f the p r o b l e m . W h e n p r o b l e m solvers e n c o u n t e r difficulty, t h e y s h o u l d c o n s i d e r s e a r c h i n g for a n e w p r o b l e m r e p r e s e n t a t i o n . REFERENCES Hatano, G., M i y a k o , Y., a n d B i n k s , M . G . " P e r f o r m a n c e o f E x p e r t A b a c u s O p e r a t o r s . " Cognition, 5, 5 7 - 7 1 , 1 9 7 7 . H a y e s , J.R. " O n t h e F u n c t i o n o f V i s u a l I m a g e r y i n E l e m e n t a r y M a t h e m a t i c s . " I n Visual Information Processing, e d i t e d b y W . C h a s e . N e w Y o r k : A c a d e m i c P r e s s , Inc., 1 9 7 3 . H a y e s , J.R., W a t e r m a n , D . A . , a n d R o b i n s o n , C . S . " I d e n t i f y i n g t h e R e l e v a n t A s p e c t s of a P r o b l e m T e x t . " Cognitive Science, 1 ( 3 ) , 2 9 7 - 3 1 3 , 1 9 7 7 . H i n s l e y , D . A . , H a y e s , J.R., a n d S i m o n , H . A . " F r o m W o r d s t o E q u a t i o n s : M e a n i n g a n d R e p r e s e n t a t i o n in A l g e b r a W o r d P r o b l e m s . " In Cognitive Processes in Comprehension, e d i t e d b y P . C a r p e n t e r a n d M . Just. H i l l s d a l e , N J : L a w r e n c e E r l baum, 1977. M c D e r m o t t , J.R., a n d L a r k i n , J . H . " R e p r e s e n t i n g T e s t b o o k P h y s i c s P r o b l e m s . " Proceedings of the S e c o n d National Conference of the Canadian Society of C o m putational Studies of I n t e l l i g e n c e , T o r o n t o , 1978. Paige, J.M., and S i m o n , H.A. " C o g n i t i v e Processes in Solving A l g e b r a W o r d Probl e m s . " In Problem Solving: Research, Method and Theory, e d i t e d by B. K l e i n muntz. N e w York: John W i l e y & Sons, Inc., 1966. P o l y a , G. How to Solve It, S e c o n d E d i t i o n . P r i n c e t o n , N J : P r i n c e t o n U n i v e r s i t y Press, 1957. W i c k e l g r e n , W . A . How to Solve Problems. 1974.

S a n F r a n c i s c o : W . H . F r e e m a n &. C o . ,

TWO

SEARCH

O n c e p e o p l e h a v e arrived at a crude understanding of a p r o b l e m — a prelimin a r y r e p r e s e n t a t i o n o f its g o a l , o p e r a t o r s , e t c . — t h e y c a n l a u n c h i n t o ^ s e a r c h f o r t h e s o l u t i o n . I n t h i s c h a p t e r , w e w i l l d e s c r i b e t h a t s e a r c h p r o c e s s i n its v a r i o u s f o r m s and provide some practical guidelines w h i c h can h e l p y o u in your o w n solution searches. A s a m e t a p h o r for t h e p r o b l e m s o l v e r ' s s e a r c h for s o l u t i o n , w e i m a g i n e a p e r s o n s e a r c h i n g t h r o u g h a m a z e . T h e e n t r a n c e t o t h e m a z e i s t h e i n i t i a l state o f t h e p r o b l e m a n d its e x i t i s t h e g o a l . T h e paths i n t h e m a z e , i n c l u d i n g all its b y w a y s a n d blind alleys, c o r r e s p o n d ' " ' J ^ J ^ , _ p r o b l e m ^ s p a c e — t h a t i s , t o all t h e s e q u e n c e s o f moves available to the p r o b l e m solver. :

A factor w h i c h h a s a n i m p o r t a n t e f f e c t o n t h e d i f f i c u l t y o f a m a z e i s its s i z e . F i g u r e 1 s h o w s six " t r e e " m a z e s w h i c h v a r y i n b r a n c h i n g , l e n g t h , a n d s i z e . A tree maze i s o n e w i t h n o l o o p s . T h e size o f a m a z e i s t h e total n u m b e r o f d i f f e r e n t p a t h s from lop"T6"b6rT6m7TRe m a z e s i n t h e u p p e r p a r t o f t h e f i g u r e b r a n c h t w i c e a t e a c h choice point, w h i l e those in the l o w e r part b r a n c h three times. T h i s difference in b r a n c h i n g d o e s n ' t m a k e v e r y m u c h d i f f e r e n c e i n s i z e i f t h e m a z e i s short. I f t h e maze is long, h o w e v e r , it m a k e s an e n o r m o u s difference. N o t i c e that m a z e 6 is m o r e than 5 0 t i m e s a s b i g a s m a z e 5 . * Familiar p r o b l e m s differ a great deal in size. T h e R i v e r C r o s s i n g P r o b l e m in t h e last c h a p t e r h a s a v e r y s m a l l p r o b l e m s p a c e , w h i l e t h e C o m b i n a t i o n L o c k Problem in the Introduction has a h u g e one. N e w e l l and S i m o n (1972) estimate that t h e s i z e o f t h e p r o b l e m s p a c e for a t y p i c a l c h e s s g a m e i s 1 0 . ( T h e y a s s u m e a g a m e of 40 moves with branching of 20 alternatives per move.) It will h e l p you to understand j u s t h o w l a r g e a n u m b e r 1 0 is if y o u r e c o g n i z e that the e s t i m a t e d n u m b e r of atoms in the u n i v e r s e is o n l y 1 0 . 1 2 0

1 2 0

8 0

VARIETIES OF SEARCH W h i l e most p e o p l e are v a g u e l y familiar w i t h "trial a n d e r r o r " search, f e w can d e s c r i b e o t h e r s e a r c h m e t h o d s . I n fact, t h e r e a r e a g r e a t m a n y s e a r c h m e t h o d s . I n this c h a p t e r , w e w i l l d e s c r i b e f o u r g e n e r a l s e a r c h m e t h o d s : * If B is the degree of branching and L is length, then size = B

L

25

The C o m p l e t e Problem Solver

Figure 1. Six T r e e M a z e s

MAZE 1

branching = 2 length = 1 size = 2

1

= 2

MAZE 2

branching = 3 length = 1 size = 3

1

= 3

Search

branching = 2 length = 3 size = 2

= 8

3

MAZE 4

G branching = 3 length = 3 size = 3

3

= 27

The C o m p l e t e Problem Solver

MAZE 5 S

branching = 2 l e n g t h = 10 size = 2 = 1,024 1 0

The C o m p l e t e Problem Solver

30 1. 2. 3. 4.

Trial and Error Proximity Methods Fractionation Methods Knowledge-based Methods

U n d e r s t a n d i n g t h e s e s t r a t e g i e s a n d t h e i r s t r e n g t h s a n d w e a k n e s s e s i s e s s e n t i a l for choosing good problem solving procedures.

TRIAL-AND-ERROR

SEARCH

T h e d e f i n i n g characteristic of trial-and-error search is that the searchers either don't h a v e or don't u s e information that indicates to t h e m that a n y path is m o r e l i k e l y to l e a d to the goal than any other path. In a m a z e , trial-and-error searchers n e v e r k n o w that t h e y are on the right path until t h e y o p e n the last door and find themselves outside. T r i a l - a n d - e r r o r s e a r c h c o m e s i n t w o f o r m s — blind a n d systematic. I n b l i n d search, the searchers p i c k paths to explore blindly, without considering w h e t h e r they h a v e already e x p l o r e d the path. In systematic search, the searchers k e e p track of the paths w h i c h they have already explored and choose to explore only unexp l o r e d paths. B e c a u s e this m e t h o d avoids m u l t i p l e s e a r c h e s , systematic search is twice a s e f f i c i e n t o n t h e a v e r a g e a s b l i n d s e a r c h .

A D r a m a t i c Example S u p p o s e that y o u h a v e 50 eggs in a b a g b a l a n c e d on your head. O n e of the eggs i s b r o w n a n d all t h e r e s t a r e w h i t e . J u s t a s y o u a r e c r o s s i n g a d e e p c h a s m b y b a l a n c i n g on a fallen log, y o u hear a c o m m a n d i n g v o i c e say, " S t o p w h e r e y o u are and s h o w m e the b r o w n e g g ! " S i n c e y o u h a v e n o idea w h a t this maniac m a y do, y o u d e c i d e to pacify him. Y o u reach into the b a g with your one free h a n d and pull out a n e g g . It's w h i t e ! W h a t d o y o u d o n o w ? I f y o u p u t t h e w h i t e e g g s b a c k w i t h t h e o t h e r s a n d k e e p t r y i n g , y o u w i l l b e d o i n g r e l a t i v e l y i n e f f i c i e n t blind s e a r c h . O n t h e a v e r a g e , it w i l l take y o u 50 tries to find the b r o w n e g g a n d m a y take m u c h longer. H o w e v e r , if y o u mark the eggs y o u h a v e already e x a m i n e d in the only w a y you can, b y t o s s i n g t h e m i n t o t h e c h a s m , y o u w i l l b e d o i n g a r e l a t i v e l y e f f i c i e n t systematic search. On the a v e r a g e , it will take y o u only 25 tries to find the b r o w n e g g , a n d it w i l l n e v e r take m o r e than 50. S i n c e a little reflection s u g g e s t s h o w truly d e m e n t e d t h i s p e r s o n m u s t b e , y o u d e c i d e t o s a c r i f i c e y o u r e g g s a n d g o for t h e q u i c k e r systematic search. T r i a l - a n d - e r r o r s e a r c h c a n b e v e r y u s e f u l for f i n d i n g o n e ' s w a y i n s m a l l m a z e s a n d for s o l v i n g p r o b l e m s w i t h s m a l l p r o b l e m s p a c e s . I n t h e R i v e r C r o s s i n g P r o b l e m i n C h a p t e r I , i t i s e a s y t o s e a r c h t h r o u g h all p o s s i b l e p a t h s t o f i n d t h e s o l u t i o n . H o w e v e r , as the size of the p r o b l e m space increases, the m e t h o d b e c o m e s less and l e s s u s e f u l . T r i a l a n d e r r o r i s a r e a s o n a b l y c o m f o r t a b l e s t r a t e g y for o p e n i n g a c o m bination l o c k w i t h o n e or e v e n t w o dials. W i t h t h r e e dials (1,000 paths), it gets a b i t b o r i n g . W i t h f i v e d i a l s (100,000 p a t h s ) , w e h a v e t o b e p r e t t y d e s p e r a t e t o u s e it, a n d with 10 dials, as we pointed out in the Introduction, it is essentially useless unless y o u h a v e a spare 3 1 7 years to invest. S u p p o s e y o u w e r e p a s s i n g t h r o u g h Pittsburgh, a n d y o u w a n t e d t o call y o u r f r i e n d , B i l l M i l l e r . W h e n y o u l o o k i n t h e p h o n e b o o k , y o u f i n d t h a t t h e r e are 180 W i l l i a m M i l l e r s l i s t e d . Y o u c o u l d c a l l t h e m o n e after a n o t h e r u n t i l y o u f o u n d t h e r i g h t o n e , b u t i t w o u l d b e far m o r e s e n s i b l e t o try a n o t h e r s t r a t e g y . Y o u c o u l d try

Search

31

t o r e m e m b e r h i s m i d d l e i n i t i a l o r h i s a d d r e s s . O r y o u c o u l d try t o r e a c h a m u t u a l friend w h o m i g h t h a v e his n u m b e r . T h e m o r a l h e r e i s v e r y c l e a r . T r i a l - a n d - e r r o r s e a r c h i s not e f f i c i e n t i n l a r g e p r o b l e m s . I f y o u find y o u r s e l f d o i n g t r i a l - a n d - e r r o r s e a r c h i n a l a r g e p r o b l e m , y o u should almost certainly stop and use a better m e t h o d . S o m e t i m e s (alas!) w e d o n ' t h a v e a c h o i c e . T r i a l a n d e r r o r m a y t r u l y b e t h e o n l y m e t h o d available. I f so, y o u s h o u l d c o n s i d e r v e r y carefully w h e t h e r s o l v i n g the problem is worth the cost of the search. It may well not b e !

PROXIMITY

METHODS

Proximity search m e t h o d s are in a s e n s e just o n e step a h e a d of trial-and-error search. In proximity search, the searcher looks exactly o n e step ahead, w h i l e in trial-and-error s e a r c h , t h e s e a r c h e r d o e s n ' t l o o k a h e a d a t a l l . T h i s o n e - s t e p difference, h o w e v e r , makes an enormous difference in the effectiveness of the methods. T h e t w o p r o x i m i t y m e t h o d s w e w i l l d i s c u s s are " h i l l c l i m b i n g " a n d " m e a n s ends analysis." In b o t h of t h e s e m e t h o d s the basic q u e s t i o n is, " W h a t next step can rtake that w i l l b r i n g m e c l o s e r to t h e g o a l ? " N e i t h e r m e t h o d l o o k s b e y o n d t h a t o n e step to see w h a t difficulties lie ahead.

The Hill C l i m b i n g M e t h o d

-y~-

S u p p o s e t h a t y o u a r e l o s t a t n i g h t i n a forest. Y o u c a n ' t s e e a t h i n g . Y o u r e a s o n that i f y o u w e r e o n t h e t o p o f a h i l l y o u m i g h t b e a b l e t o s p o t t h e l i g h t o f a h o u s e or a campfire. C a n y o u do any better than trial-and-error search to find the top of a h i l l ? I n fact, y o u c a n d o m u c h b e t t e r ! T h e h i l l - c l i m b i n g m e t h o d c a n g r e a t l y r e d u c e the time y o u require to reach your goal. T h e m e t h o d works like this: J. Y o u p u t o u t a foot to t a k e a s t e p . 2. If the step is u p , y o u take it and repeat the procedure. 3. If the step is d o w n , y o u p u l l y o u r foot back, take a quarter turn c l o c k w i s e , a n d try a g a i n . 4 . I f y o u h a v e t u r n e d a full c i r c l e a n d h a v e f o u n d n o t h i n g b u t s t e p s d o w n , y o u s t o p because y o u k n o w y o u must be on the top of a hill. H a v i n g c o m p l e t e d t h e h i l l - c l i m b i n g p r o c e d u r e , y o u c a n n o w l o o k a r o u n d for s i g n s of civilization. H o w e f f i c i e n t i s t h e h i l l - c l i m b i n g m e t h o d c o m p a r e d t o trial a n d e r r o r ? V e r y e f f i c i e n t i n h i l l y c o u n t r y . F o r e x a m p l e , i f w e w e r e s e a r c h i n g for a h i l l t o p i n a n a r e a 1,000 s t e p s s q u a r e , h i l l c l i m b i n g w o u l d b e a b o u t 1,000 t i m e s m o r e e f f i c i e n t t h a n trial-and-error s e a r c h . T h e r e l a t i v e e f f i c i e n c y o f h i l l c l i m b i n g w o u l d b e e v e n g r e a t e r for l a r g e r s e a r c h a r e a s . D e s p i t e its e f f i c i e n c y , t h e h i l l - c l i m b i n g m e t h o d h a s s o m e i n h e r e n t w e a k n e s s e s . T o i l l u s t r a t e , l e t ' s p i c k u p t h e e x a m p l e w h e r e w e l e f t off. H a v i n g r e a c h e d t h e t o p o f t h e h i l l , y o u l o o k a r o u n d a n d s e e n o t h i n g . T i r e d a n d d i s c o u r a g e d , y o u fall a s l e e p o n l y t o w a k e i n full m o r n i n g l i g h t . Y o u l o o k a n d f i n d t o y o u r s u r p r i s e t h a t your hill i s e n t i r e l y s u r r o u n d e d b y m u c h taller hills. N o w y o u can u n d e r s t a n d w h y y o u d i d n ' t s e e a n y s i g n s o f c i v i l i z a t i o n l a s t n i g h t . Y o u w a n t e d t o b e o n a tall h i l l , b u t the hill-climbing m e t h o d can't tell o n e hill from another. It took y o u to the top of the nearest hill and that h a p p e n e d to be a small one. If y o u w a n t to be sure to find the h i g h e s t hill, t h e n y o u w i l l h a v e t o s u p p l e m e n t the hill c l i m b i n g m e t h o d w i t h some other m e t h o d that can c h o o s e a m o n g hills or u s e a different m e t h o d entirely.

32

T h e C o m p l e t e Problem Solver

Figure 2. Hill C l i m b e r W i t h V e r y Few Hills

A n o t h e r p r o b l e m w i t h h i l l c l i m b i n g i s t h a t i t w o r k s w e l l o n l y w h e n m o s t o f its s e a r c h a r e a s l o p e s . I f its s e a r c h a r e a i s a l m o s t e n t i r e l y flat w i t h a n o c c a s i o n a l v e r y s t e e p p e a k (see F i g u r e 2), t h e hill c l i m b e r is v e r y little b e t t e r off than t h e trial-anderror searcher. T h e hill-climbing method is important b e c a u s e it can be u s e d and is used to s o l v e a w i d e r a n g e o f p r o b l e m s . H e r e are t w o e v e r y d a y e x a m p l e s : 1. Y o u are d r i v i n g in a strange city a n d w a n t to find the d o w n t o w n area. At e a c h c o r n e r y o u s t o p a n d l o o k i n all d i r e c t i o n s . Y o u p r o c e e d i n t h e d i r e c t i o n t h a t appears busiest. In this e x a m p l e , the c h o i c e of streets corresponds to the c h o i c e of steps, and the a m o u n t o f c o m m o t i o n o n t h e s t r e e t s c o r r e s p o n d s t o t h e h e i g h t o f t h e s t e p s . Just a s the m e t h o d may take y o u to a low hilltop w h e n y o u w a n t the tallest one, it may also take y o u to a local s h o p p i n g center w h e n y o u w a n t d o w n t o w n . 2 . A s m a l l c h i l d h a s r e a d j u s t e d y o u r T V s e t s o t h a t i t s h o w s all sorts o f i n t e r e s t i n g d i a g o n a l s t r i p e s . T o r e t u r n t h e s e t t o its m o r e t r a d i t i o n a l a d j u s t m e n t , y o u b e g i n to t w i d d l e k n o b s . T h a t is, y o u c h o o s e a k n o b a n d turn it a little bit w h i l e w a t c h i n g the screen. If the pictures gets better, y o u do it again. If it gets worse, you m o v e it a little in t h e o p p o s i t e direction. W h e n it gets w o r s e w h e n y o u m o v e in e i t h e r direction, y o u l e a v e it w h e r e it is, h a v i n g found the " r i g h t " adjustment. T h e n y o u d o t h e s a m e for t h e o t h e r k n o b s — t h a t i s , y o u " h i l l c l i m b " t o b e s t a d j u s t m e n t with each knob. Since the adjustment of some knobs influences the adjustment o f o t h e r s , y o u m a y h a v e t o r e d o s o m e o f t h e k n o b s s e v e r a l t i m e s b e f o r e y o u are a b l e to get a d e c e n t picture.

After Hill C l i m b i n g , W h a t ? I n t h e p r o b l e m s w e u s e d t o i l l u s t r a t e t h e h i l l - c l i m b i n g m e t h o d , t h e i n i t i a l state a n d t h e goal differed in a single d i m e n s i o n , e.g., h e i g h t , d e n s i t y of b u s i n e s s , picture

33

Search

q u a l i t y . F u r t h e r , i n e a c h o f t h e s e p r o b l e m s , t h e o p e r a t o r s (if t h e r e w a s m o r e t h a n o n e ) all w o r k e d i n e s s e n t i a l l y t h e s a m e w a y t o c h a n g e t h a t s i n g l e d i m e n s i o n , e . g . , i n t h e T V e x a m p l e , all o f t h e k n o b s c h a n g e d p i c t u r e q u a l i t y i n r o u g h l y t h e s a m e d e g r e e . I n m a n y p r o b l e m s , t h o u g h , w e m u s t deal w i t h differences that h a v e several dimensions and w e must deal w i t h t h e m b y u s i n g s p e c i a l i z e d o p e r a t o r s — t h a t is, o p e r a t o r s that c h a n g e o n e o r a f e w o f t h e d i m e n s i o n s b u t n o t all o f t h e m . I m a g i n e t h a t y o u are a c o n t r a c t o r b u i l d i n g a h o u s e . T h e e m p t y l o t t h a t y o u h a v e r i g h t n o w differs i n m a n y d i m e n s i o n s f r o m y o u r g o a l . I t d o e s n ' t h a v e a f o u n d a t i o n , w a l l s o r roof, p l u m b i n g , w i r i n g , o r p a i n t . F u r t h e r , y o u r o p e r a t o r s — m a s o n s , carpenters, plumbers, electricians, and p a i n t e r s — a r e specialized. Masons can build foundations and walls, but they can't paint. Electricians can p u t in wiring, b u t not roofs. W h e n w e w e r e t u n i n g t h e T V s e t , i t w a s r e a s o n a b l e t o t r y k n o b s a t r a n d o m to s e e w h a t h e l p e d . In b u i l d i n g a h o u s e , t h o u g h , this strategy w o u l d be wasteful and foolish. I f the h o u s e n e e d e d painting, w e w o u l d n ' t r a n d o m l y call i n p l u m b e r s o r c a r p e n t e r s t o s e e i f t h e y h e l p e d . W e w o u l d a n a l y z e w h a t still n e e d e d t o b e d o n e a n d s e a r c h for a n a p p r o p r i a t e m e a n s t o a c c o m p l i s h it. T h i s i s j u s t w h a t t h e t e c h nique of means-ends analysis does in solving problems.

Means-Ends Analysis M e a n s - e n d s analysis ( N e w e l l and S i m o n , 1972), like hill c l i m b i n g , tries to reach the goal by taking a s e q u e n c e of steps, e a c h of w h i c h r e d u c e s the distance to t h e g o a l . T h e m e a n s - e n d s t e c h n i q u e for d o i n g t h i s , h o w e v e r , i s b o t h m o r e c o m p l e x and more powerful than the hill-climbing t e c h n i q u e . It can h a n d l e more c o m plex differences and e m p l o y s subgoals to h e l p execute each step. T h e technique involves three procedures: Procedure

Example

1. F i n d a list of differences b e t w e e n the current state a n d t h e goal. (If y o u c a n ' t f i n d a n y d i f f e r e n c e s , report that t h e p r o b l e m is solved.)

The house needs plumbing, wiring, and paint.

2 . T a k e t h e first d i f f e r e n c e a n d f i n d a n o p e r a t o r a p p r o p r i a t e for r e d u c i n g it.

T r y to hire a p l u m b e r .

If you can't find an appropriate operator, go to t h e n e x t d i f f e r e n c e .

If the plumber is out of town, try to hire an electrician.

If you run out of differences to find operators for, r e p o r t t h a t y o u c a n ' t solve the p r o b l e m .

I f e v e r y o n e i s o u t o f t o w n for a builder's convention, give u p — a t l e a s t for n o w .

3 . C o m p a r e t h e c o n d i t i o n s for a p p l y i n g t h e o p e r a t o r w i t h t h e c u r r e n t state to f i n d a d i f f e r e n c e . ( I f t h e r e is no difference, of course, just apply the operator.) If t h e r e is a d i f f e r e n c e , try to r e d u c e it.

T h e painter says he'll d o the j o b , b u t he b r o k e his ladder.

L e n d the painter your ladder.

34

The C o m p l e t e Problem Solver

Sometimes it is convenient to s h o w the relation b e t w e e n means and ends in a means-ends table such as this: foundation

walls

X

X

mason

carpenter

roof

plumbing

wiring

painting

X

X

plumber

X

electrician

X

painter

X

H e r e i s a n o t h e r e x a m p l e . I m a g i n e t h a t V l a d i s a S l o v a k e m i g r a n t i n t h e 1880's w h o i s w e l l t r a i n e d i n m e a n s - e n d s a n a l y s i s . V l a d i s t r a v e l i n g f r o m his n a t i v e v i l l a g e of Bicske, 35 miles w e s t of Budapest, to Pittsburgh, Pennsylvania, to work in the s t e e l m i l l s . T h e r e a r e s e v e r a l m e a n s o f t r a n s p o r t a t i o n a v a i l a b l e w h i c h are s u i t e d t o various purposes. T h e s e means and ends can be summarized as follows:

ship

large ocean distances

large land distances

m e d i u m distances

medium-small distances, urban areas

small distances

train

coach

trolley

X

X

walking

X

X

X

X

V l a d looks in his atlas a n d d i s c o v e r s that P i t t s b u r g h is s e p a r a t e d from B i c s k e by a v e r y large o c e a n . T h e m e a n s - e n d s table indicates that his original intention to take a s t a g e c o a c h i s o u t . O n l y a s h i p w i l l d o . T h e first t r a v e l a g e n t h e s p e a k s t o c a n a r r a n g e p a s s a g e for h i m o n a v e r y n i c e s h i p t o C a l c u t t a . V l a d r e j e c t s this offer

35

Search

b e c a u s e h i s atlas s h o w s t h a t C a l c u t t a i s e v e n f a r t h e r f r o m P i t t s b u r g h t h a n B i c s k e is. A f t e r all, h i s g o a l i s t o reduce t h e d i s t a n c e b e t w e e n h i m a n d P i t t s b u r g h . T h e s e c o n d a g e n t offers h i m p a s s a g e o n a s h i p s a i l i n g f r o m B r e m e n t o N e w Y o r k . S i n c e N e w Y o r k i s m u c h c l o s e r t o P i t t s b u r g h t h a n B i c s k e , V l a d b u y s a t i c k e t for N e w Y o r k . N o w V l a d h a s i d e n t i f i e d a n o p e r a t o r for r e d u c i n g h i s d i s t a n c e f r o m P i t t s b u r g h — t a k i n g a ship from B r e m e n . B u t he can't a p p l y it b e c a u s e the operator has a c o n d i t i o n — h e has t o b e i n B r e m e n t o sail. S o V l a d s e t s u p a s u b g o a l — t o g e t from B i c s k e t o B r e m e n . R e l y i n g o n h i s atlas a n d h i s m e a n s - e n d s t a b l e , h e d e c i d e s t o t a k e a train. F o r t u n a t e l y , h e i s a b l e t o b u y a t i c k e t for a n e x p r e s s train f r o m B u d a p e s t to B r e m e n . H e r e again, the operator i m p o s e s a c o n d i t i o n — V l a d has to g e t t o B u d a p e s t t o t a k e t h e train. S o h e sets u p a n e w s u b g o a l t o g e t f r o m B i c s k e t o Budapest. Since B i c s k e is out in the sticks, the m e a n s - e n d s table tells V l a d that he has o n l y o n e o p t i o n , t h e s t a g e c o a c h . S i n c e t h e c o a c h d o e s n ' t s t o p a t V l a d ' s h o v e l , h e has t o s e t u p o n e m o r e s u b g o a l — t o g e t t o t h e c o a c h s t o p . H i s r e l i a b l e m e a n s ends table tells him to walk.

HILL CLIMBING A N D MEANS-ENDS ANALYSIS: A C O M P A R I S O N B o t h m e t h o d s s e a r c h for a n e x t s t e p w h i c h w i l l r e d u c e t h e d i s t a n c e t o t h e g o a l . Means-ends analysis is a m o r e p o w e r f u l m e t h o d than hill c l i m b i n g b e c a u s e : 1 . i t c a n c o n s i d e r m a n y d i m e n s i o n s o f d i f f e r e n c e b e t w e e n t h e c u r r e n t state a n d t h e goal, and 2. it can set up a s e q u e n c e of subgoals to h e l p in a c c o m p l i s h i n g the next step. B e c a u s e both methods look only one step ahead, both h a v e difficulty w i t h b l i n d alleys and detour problems. For example, in solving the River Crossing Problem in C h a p t e r 1 , i t i s n e c e s s a r y a t s o m e p o i n t for t h e f a r m e r t o t a k e o n e o f h i s p o s s e s s i o n s back across the river a w a y from the goal. H i l l c l i m b e r s , m e a n s - e n d s a n a l y z e r s , a n d , not incidentally, p e o p l e , h a v e trouble w i t h p r o b l e m s o f this k i n d . In the next section, a m o n g other things, we will discuss some procedures w h i c h can h e l p us out of our difficulties w i t h detour p r o b l e m s .

FRACTIONATION

METHODS

A v e r y p o w e r f u l w a y t o s i m p l i f y t h e s e a r c h for a s o l u t i o n i s t o b r e a k t h e p r o b l e m intojaarts. I m a g i n e t h a t y o u are w o r k i n g y o u r w a y t h r o u g h a t r e e m a z e that h a s a length of ten, and has t w o alternatives at e a c h of ten c h o i c e points. In this m a z e , y o u w o u l d have to find the goal a m o n g 2 o r 1,024 a l t e r n a t i v e s . B u t n o t i c e h o w t h i s situation w o u l d c h a n g e i f s o m e o n e w h o k n e w t h e m a z e g a v e y o u t h e f o l l o w i n g h i n t . _The last p e r s o n t o s o l v e t h e m a z e l o s t h i s p u p p y h a l f w a y t h r o u g h t h e . m a z e - T h e p u p p y i s n o w s i t t i n g r i g h t i n front o f t h e d o o r t h r o u g h w h i c h t h e s o l u t i o n p a t h a n d his m a s t e r p a s s e d . N o w y o u c a n b r e a k t h e p r o b l e m i n t o t w o p a r t s . A s asuhgoal, y o u first s e a r c h for a n d r e s c u e t h e p u p p y . T h e n , w h e n y o u h a v e f o u n d it, y o u c a n s e a r c h foTtrre-exit f r o m t h e r e . B r e a k i n g t h e p r o b l e m i n t o p a r t s i s a r e a l a d v a n t a g e b e c a u s e solving the t w o s u b p r o b l e m s is easier than solving the original p r o b l e m . F i g u r e 3 illustrates t h e p o i n t . S i n c e y o u k n o w t h a t t h e p u p p y i s l o c a t e d h a l f w a y t h r o u g h t h e m a z e , t h e r e are j u s t 2 o r 3 2 p l a c e s w h e r e i t m a y b e . O n c e y o u f i n d t h e p u p p y , t h e r e are j u s t f i v e m o r e s t e p s t o r e a c h t h e g o a l , a n d , t h e r e f o r e , j u s t 3 2 a l t e r n a t i v e p a t h s t o t h e exit. T o f i n d t h e p u p p y a n d t h e n f i n d t h e g o a l w i l l r e q u i r e u s t o e x a m i n e n o more than 32 + 32 or 64 alternatives. W i t h o u t the hint, we w o u l d h a v e to e x a m i n e a m a x i m u m of 1,024 a l t e r n a t i v e s . 1 0

5

W h e r e d o e s this a d v a n t a g e c o m e f r o m ? W h e n w e c o m e t o a d o o r h a l f w a y t h r o u g h t h e m a z e a n d i t d o e s n ' t h a v e a p u p p y i n front o f it, w e k n o w t h a t w e d o n ' t

The C o m p l e t e Problem Solver

36

> 32 paths to h e r e

1,024 paths to here

32 paths

Y 1,024 paths

Figure 3. An Example of Breaking a P r o b l e m Into Parts

h a v e to e x p l o r e any of t h e 32 paths w h i c h lie b e y o n d . S i n c e there are 31 doors w i t h o u t p u p p i e s , that m e a n s that we e l i m i n a t e 31 X 32 or 992 paths. B r e a k i n g the p r o b l e m into parts, then, has r e d u c e d the a m o u n t o f search w e h a v e 1024 to do by a factor of , or 16. It is surprising w h a t o n e little s u b g o a l can do. H a v i n g 64 several subgoals w o u l d help e v e n more. C l e a r l y , i f s o m e o n e g i v e s y o u a s u b g o a l , a s i n t h e e x a m p l e a b o v e , that w i l l h e l p you to solve the problem. But in most cases you aren't g i v e n subgoals.„You have to f i n d t h e m for y o u r s e l f . H o w c a n y o u d o t h a t ? S u p p o s e that y o u are trying to solve a p r o b l e m like that r e p r e s e n t e d in F i g ure 4. Y o u are g i v e n A a n d s o m e h o w y o u h a v e to g e t to G. Y o u m a y be a b l e to find a useful s u b g o a l if y o u w o r k into the p r o b l e m either forward or b a c k w a r d . F o r exa m p l e , if y o u are a b l e to g e t from A to B, t h e n y o u can set up the s u b g o a l of getting from B t o G . * I f y o u can g e t f r o m G t o F , t h e n y o u r subgoal i s getting f r o m F t o A .

* T h e hill-climbing method and means-ends analysis break up the problem in just this way.

Search

37

I

I I U I I I L_J Il Figure 6. First Subgoal in t h e Figure 7. S e c o n d Subgoal in the T o w e r of Hanoi Problem T o w e r of Hanoi Problem N o t i c e that y o u m a y s e t u p s u b g o a l s i n this w a y w h i c h are n o t u s e f u l . F o r e x a m p l e , s u p p o s e that y o u d i s c o v e r that y o u c a n g e t t o M f r o m A , a n d s e t u p t h e s u b g o a l t o g e t from M to G. In fact, t h e r e is no p a t h f r o m M to G so t h a t t h i s s u b g o a l c a n ' t be a c h i e v e d . S u b g o a l s are so useful in p r o b l e m s o l v i n g that it is almost a l w a y s advisab l e t o s e a r c h for t h e m , e v e n t h o u g h w e s o m e t i m e s f i n d o n e s t h a t a r e n o t u s e f u l . Another w a y to find subgoals is to analyze the goal we are trying to reach. T h i s process is illustrated in the T o w e r of Hanoi P r o b l e m s h o w n in F i g u r e 5. T h e p r o b l e m i s t o m o v e t h e d i s k s f r o m t h e left p e g t o t h e r i g h t p e g w i t h t h e f o l l o w i n g restrictions: 1. only one disk can be m o v e d at a time, and 2. a large disk m u s t n e v e r be p l a c e d on a small disk.

Figure 8 . Subgoals C a n b e Used t o G u i d e t h e Problem Solver A r o u n d D e t o u r s

38

The C o m p l e t e Problem Solver

E x a m i n i n g the goal, we can ask w h a t part of the goal is the hardest to a c h i e v e . S i n c e the s e c o n d restriction constrains the small disk least, and the large disk most, w e m a y r e a s o n t h a t g e t t i n g t h e l a r g e d i s k i n p l a c e i s t h e h a r d e s t part t o a c h i e v e . T h e r e f o r e , w e s e t a s o u r s u b g o a l t h e state o f affairs s h o w n i n F i g u r e 6 , w i t h t h e o t h e r d i s k s a t u n s p e c i f i e d l o c a t i o n s . B y c o m p a r i n g t h e s u b g o a l w i t h t h e initial s t a t e , w e c a n s e e t h a t t h e i n i t i a l state c o u l d b e t r a n s f o r m e d i n t o t h e s u b g o a l i f t h e t w o smaller disks w e r e r e m o v e d . This suggests the second subgoal s h o w n in Figure 7. A r m e d w i t h these t w o subgoals, most find it quite easy to s o l v e the problem. T h e use of subgoals in the T o w e r of Hanoi P r o b l e m illustrates a second very i m p o r t a n t f u n c t i o n of s u b g o a l s : subgoals can be used to guide the problem solver around detours. F i g u r e 8 i l l u s t r a t e s t h i s p o i n t . W i t h t h e i n i t i a l state a t A a n d t h e g o a l a t G , p r o b l e m s o l v e r s u s i n g a d i s t a n c e r e d u c i n g m e t h o d are b o u n d to run into the barrier. H o w e v e r , if we p r o v i d e a subgoal, S, to lead the p r o b l e m solvers in a detour around the barrier, they can use t h e i r d i s t a n c e r e d u c i n g m e t h o d s first to g e t f r o m A to S a n d t h e n f r o m S to G. To s e e that the T o w e r of H a n o i P r o b l e m is a detour p r o b l e m , look at F i g u r e 9. T h e figure shows the c o m p l e t e solution, together with a measure of the distance f r o m t h e g o a l a t e a c h s t e p . S i n c e t h e o b j e c t i v e w a s t o g e t t h r e e p e g s off d i s k 1 a n d onto disk 3, d i s t a n c e to the goal w a s m e a s u r e d as f o l l o w s : D i s t a n c e = d i s k s on p e g 1 + d i s k s off p e g 3 As y o u can s e e , the solution r e q u i r e s y o u to take t w o steps that increase the distance to the g o a l — g o a l step 3 and step 5. T h e s e detours disappear, h o w e v e r , if you use t h e t w o s u b g o a l s , S I a n d S 2 , f o u n d b y a n a l y z i n g t h e g o a l . F i g u r e 9 s h o w s that t r a v e l i n g f r o m t h e start t o S I a n d f r o m S 2 t o t h e g o a l r e q u i r e s o n l y f o r w a r d m o v e s . T h i s situation is r e p r e s e n t e d in F i g u r e 10. T h e s o l u t i o n m e t h o d c a l l e d abstraction a l s o w o r k s b y g e n e r a t i n g s u b g o a l s . S u p p o s e that y o u are s o l v i n g a p r o b l e m w i t h several restrictions,X, Y, a n d Z . S i n c e y o u a r e h a v i n g t r o u b l e , y o u t u r n t o t h e m e t h o d o f a b s t r a c t i o n . W i t h this m e t h o d , y o u b r e a k the p r o b l e m into parts. First, y o u drop s o m e of the restrictions, say Y a n d Z , a n d s o l v e t h e r e s u l t i n g " a b s t r a c t e d " p r o b l e m . N e x t , y o u try t o fix u p t h i s s o l u t i o n , s o t h a t r e s t r i c t i o n Y i s s a t i s f i e d . F i n a l l y , y o u try t o fix t h e s o l u t i o n a g a i n s o t h a t t h e restriction Z is satisfied a n d the original p r o b l e m is solved. Let's illustrate this w i t h a p r o b l e m . S u p p o s e that we are r u n n i n g a m e e t i n g of 16 p e o p l e , m a l e s a n d f e m a l e s , from four c o m p a n i e s a n d o c c u p y i n g four positions, a s s h o w n i n F i g u r e 1 1 . O u r p r o b l e m i s t o o r g a n i z e four, f o u r - p e r s o n g r o u p s w i t h t h e restrictions that e a c h task g r o u p i n c l u d e : -Z.

a p e r s o n from e a c h c o m p a n y

2. a p e r s o n from e a c h position 3. t w o males and t w o females. S o l v i n g this p r o b l e m b y a b s t r a c t i o n , w e start b y s o l v i n g i t j u s t for r e s t r i c t i o n 1 . This solution, s h o w n in Figure 12, is very easy to achieve. N e x t w e fix u p t h e s o l u t i o n s o t h a t i t s a t i s f i e s r e s t r i c t i o n 2 . T h i s s o l u t i o n , s h o w n in Figure 13, is also easy to achieve. As we can see, h o w e v e r , the proposed solution d o e s n o t b a l a n c e all o f t h e g r o u p s b y s e x . G r o u p s 2 a n d 4 a r e b a l a n c e d b u t 1 a n d 3 are not. H o w e v e r , this solution can b e fixed u p b y e x c h a n g i n g t h e t w o p e o p l e from c o m p a n i e s A a n d C in g r o u p 1 w i t h t h o s e in g r o u p 3. Finally, and perhaps most important, we can set up useful subgoals w h e n we are dealing with familiar classes of problems. W h e n we are writing an essay, we k n o w that w e can b r e a k the task into p a r t s — r e s e a r c h , brainstorming, organization, w r i t i n g a draft, r e v i s i n g , g e t t i n g o u t s i d e c r i t i c i s m . W e k n o w t h i s b e c a u s e w e a r e

Search

39

Figure 9. Solution to the T o w e r of Hanoi Problem familiar with w r i t i n g tasks. In the s a m e w a y , in s o l v i n g a g e o m e t r y p r o b l e m , we k n o w t h a t o u r first s u b g o a l s h o u l d b e t o d r a w a f i g u r e . USING K N O W L E D G E IN PROBLEM S O L V I N G Learning I f you k n e w that y o u w e r e g o i n g t o h a v e t o run the s a m e m a z e 5 0 times, y o u would certainly make a point of trying to learn useful things about the maze. For

A , B , C , D are companies 1,2,3,4 a r e t a b l e s Figure 1 1 . Solving by Abstraction for Restriction 1

Search

F = female, M = male 1,2,3,4 a r e t a b l e s A , B , C , D are c o m p a n i e s L,M,R,S are professions Figure 1 2 . Solving by Abstraction for Restrictions 1 and 2

e x a m p l e , y o u w o u l d try t o r e c o g n i z e b l i n d - a l l e y e n t r a n c e s s o a s n o t t o s t u m b l e i n t o t h e m a g a i n . Y o u w o u l d p r o b a b l y try t o r e m e m b e r t h e m o v e j u s t b e f o r e t h e e n d , a n d o n a p a r t i c u l a r l y s u c c e s s f u l r u n y o u m i g h t w a n t t o r e m e m b e r y o u r first f e w m o v e s as well. As you gain e x p e r i e n c e with the m a z e , the old patterns will g r o w and n e w ones will be a d d e d until t h e y b l e n d together to c o v e r the w h o l e solution path. As t h e patterns g r o w , t h e n u m b e r o f e r r o r s y o u m a k e w i l l d e c r e a s e u n t i l f i n a l l y y o u c a n run the m a z e perfectly. Clearly, pattern l e a r n i n g c a n r e d u c e the difficulty of search e n o r m o u s l y . It's fortunate t h a t t h i s i s s o , for o t h e r w i s e w e w o u l d all w a s t e a g r e a t d e a l o f t i m e f i n d i n g our w a y h o m e a t n i g h t . Searching for auxiliary problems is a v e r y p o w e r f u l p r o b l e m - s o l v i n g t e c h n i q u e w h i c h i s e f f e c t i v e e x a c t l y b e c a u s e i t p r o v i d e s o p p o r t u n i t y for r e l e v a n t l e a r n ing. W h e n w e a r e h a v i n g d i f f i c u l t y w i t h a p r o b l e m , w e c a n t r y t o p o s e a r e l a t e d , e a s i e r a u x i l i a r y p r o b l e m for o u r s e l v e s . W e d o t h i s b e c a u s e b y s o l v i n g t h e e a s i e r problem, w e h o p e t o learn s o m e t h i n g that will h e l p u s s o l v e the harder p r o b l e m . For example, consider the E i g h t Dots Problem in Chapter 1. We can create an easier auxiliary p r o b l e m quite h a n d i l y i n this case j u s t b y r e d u c i n g t h e n u m b e r o f dots. S o l v e t h e E i g h t D o t s P r o b l e m b y first s o l v i n g t h i s o n e :

42

T h e C o m p l e t e Problem Solver

• _ o then this o n e :

•

• _ o

O

•

• _ o

o

and t h e n this o n e :

•

o

A s y o u s o l v e e a c h p r o b l e m , b e careful t o notice a n y t h i n g y o u learn that helps y o u to solve the next harder problem. F r e q u e n t l y you can create an easier auxiliary p r o b l e m by r e d u c i n g the number of e l e m e n t s in the p r o b l e m or the n u m b e r of constraints.

Figure 1 3 . Solution of t h e Abstraction Problem

PATTERN

MATCHING

P l a y i n g 50 g a m e s of chess or of poker isn't like r u n n i n g the s a m e m a z e 50 times. E a c h chess g a m e and e a c h poker g a m e is u n i q u e . Still, there are m a n y patterns to b e l e a r n e d w h i c h can b e useful i n later play, e.g., standard chess gambits a n d e n d - g a m e situations. T h e importance of such patterns in chess has b e e n demonstrated by de Groot and by S i m o n and his collaborators in a significant series of studies. T h e s e investigations w e r e sparked by a surprising discovery m a d e by de Groot (1965), w h o was trying to a n s w e r the question, " W h y are chess masters better than w e a k e r p l a y e r s ? "

Search

43

It was c o m m o n l y b e l i e v e d that w h e n chess masters c h o s e a m o v e , t h e y either examined more alternatives than w e a k e r players or e x a m i n e d the alternatives in greater d e p t h — t h a t is, t h e y l o o k e d further a h e a d than w e a k e r players. D e G r o o t f o u n d that n e i t h e r o f t h e s e b e l i e f s w a s t r u e . B o t h m a s t e r s a n d w e a k e r p l a y e r s typically e x a m i n e d b e t w e e n 30 and 50 m o v e s , and both searched to a depth of t w o or three m o v e s on the a v e r a g e . T h e o n l y t h i n g that s e e m e d really different a b o u t the masters' p l a y was that t h e y m a d e better m o v e s ! W h i l e c o n d u c t i n g his investigations, de G r o o t n o t i c e d a curious thing. If the c h e s s m a s t e r s l o o k e d a t a c h e s s p o s i t i o n for o n l y a f e w s e c o n d s , t h e y c o u l d r e m e m b e r it remarkably w e l l — f a r better than w e a k e r players. T h e masters' ability to r e m e m b e r chess positions w a s n ' t a general m e m o r y skill. W h e n t h e y w e r e asked to remember random arrangements of chess p i e c e s , they did no better than the w e a k e r players. Their superior ability to r e m e m b e r chess positions d e p e n d e d specifically on their superior k n o w l e d g e of chess. T h i s o b s e r v a t i o n s u g g e s t s a n e x p l a n a t i o n for t h e m a s t e r s ' s u p e r i o r a b i l i t y t o find g o o d m o v e s . T h e e x p l a n a t i o n h a s b e e n a r t i c u l a t e d b e s t b y S i m o n a n d h i s collaborators: 1 . Chess_masters c a n r e m e m b e r b r i e f l y e x p o s e d c h e s s p o s i t i o n s b e c a u s e t h e y h a v e l i v e r y large n u m b e r of chess patterns stored in memory. T h e s e patterns allow the "^master to r e m e m b e r a c h e s s p o s i t i o n as a c o m b i n a t i o n of a s m a l l n u m b e r of f a i r l y large patterns a l r e a d y familiar to t h e m . L e s s e x p e r i e n c e d c h e s s players m u s t try to r e m e m b e r chess positions as a large n u m b e r of smaller p a t t e r n s — a more difficult task. S i m o n and G i l m a r t i n (1973) tested this theory b y c o m p u t e r simulation. T h e y constructed a program, called M A P P , w h i c h r e m e m b e r e d chess board positions b y c o m p a r i n g t h e m t o p a t t e r n s s t o r e d i n its m e m o r y . G i v e n a b o u t 1,000 p a t t e r n s , M A P P did better than chess beginners but only half as w e l l as masters. On the basis of their simulation, S i m o n a n d G i l m a r t i n estimate that masters h a v e b e t w e e n 10,000 a n d 100,000 c h e s s p a t t e r n s s t o r e d i n m e m o r y . 2 . C h e s s m a s t e r s c a n f i n d b e t t e r m o v e s t h a n w e a k e r p l a y e r s for a g i v e n a m o u n t o f search b e c a u s e the c h e s s masters h a v e a larger c o l l e c t i o n of patterns stored in memory. T h e y can r e c o g n i z e better than w e a k e r players that a particular pattern o n t h e c h e s s b o a r d r e q u i r e s a p a r t i c u l a r r e s p o n s e . T h e y m a y r e c o g n i z e , for example, that a b o a r d position is similar to o n e in a g a m e t h e y are familiar with, e.g., " O h , this i s l i k e G a m e 3 o f F i s c h e r v e r s u s S p a s s k y . " T h i s r e c o g n i t i o n w i l l often s u g g e s t a m o v e s i m i l a r t o o n e i n t h e o r i g i n a l g a m e . H o w l o n g d o e s it take a chess master to learn the patterns on w h i c h his skill is b a s e d ? S i m o n a n d C h a s e (1973) estimate that to b e c o m e a grand master requires from 10,000 t o 50,000 h o u r s s t a r i n g a t c h e s s p o s i t i o n s . T h e y n o t e t h a t n o o n e h a s reached the grand master l e v e l with less than about a d e c a d e ' s intense p r e o c c u p a tion w i t h t h e g a m e . SEARCH A L G O R I T H M S S o m e t i m e s w h e n w e a r e s e a r c h i n g for t h e s o l u t i o n t o a p r o b l e m , w e u s e p r o c e d u r e s that are o f t e n u s e f u l , b u t a r e b y n o m e a n s g u a r a n t e e d t o p r o d u c e t h e s o l u t i o n o f t h e p r o b l e m . S u c h p r o c e d u r e s a r e c a l l e d search heuristics. H i l l c l i m b i n g , m e a n s e n d s a n a l y s i s , a n d l o o k i n g for a u x i l i a r y p r o b l e m s a r e all e x a m p l e s o f s e a r c h h e u r i s tics. F o r s o m e c l a s s e s o f p r o b l e m s , t h o u g h , t h e r e a r e p r o c e d u r e s w h i c h , i f c o r r e c t l y applied, are g u a r a n t e e d to y i e l d the right solution. T h e s e p r o c e d u r e s are c a l l e d algorithms. T h e long-division procedure is an example of a search algorithm. W h e n you r e c o g n i z e that y o u a r e d e a l i n g w i t h a d i v i s i o n p r o b l e m , y o u c a n a p p l y t h e m e t h o d

The C o m p l e t e Problem Solver

44

and b e sure o f g e t t i n g the a n s w e r . Y o u can s o l v e d i v i s i o n p r o b l e m s b y trial-anderror s e a r c h , b u t i t i s v e r y i n e f f i c i e n t w i t h c o m p l e x p r o b l e m s . I f I a s k e d y o u t o divide 19 into 323, y o u c o u l d carry out a systematic trial-and-error search as follows: " 1 9 x 1 = 19. 19 x 2 = 3 8 . 19 x 3 = 57,

N o , t h a t ' s n o t it. N o , t h a t ' s n o t it. etc."

Clearly the long-division algorithm is better. W e all l e a r n a n u m b e r o f s e a r c h a l g o r i t h m s i n s c h o o l . W e l e a r n a b o u t " c o m p l e t ing the s q u a r e " in algebra, "balancing equations" in chemistry, the "right-hand r u l e " in physics, the Phythagorean theorem in geometry, and so on. A s e a r c h a l g o r i t h m w h i c h y o u m a y n o t k n o w i s t h e maze algorithm. T h e m a z e a l g o r i t h m i n s u r e s t h a t w h e n y o u e n t e r a m a z e , y o u w i l l always b e a b l e t o f i n d t h e exit. T h e algorithm i s v e r y s i m p l e . A s y o u enter the m a z e , p u t y o u r h a n d o n the w a l l , and keep i t there a s y o u w a l k t h r o u g h t h e m a z e . T r y t h e a l g o r i t h m w i t h t h e m a z e s h o w n i n F i g u r e 1 4 . I m a g i n e first t h a t y o u a r e w a l k i n g t h r o u g h t h e m a z e w i t h y o u r r i g h t h a n d o n t h e w a l l . T h e n t r y i t u s i n g y o u r left h a n d . N o t i c e t h a t y o u t o o k a d i f f e r e n t p a t h t o t h e e x i t w h e n y o u u s e d y o u r left h a n d than w h e n y o u u s e d y o u r right hand. Either h a n d will work, b u t it is important not to s w i t c h h a n d s ! If y o u s w i t c h from right to left w h e n y o u get to the arrow (see F i g u r e 1 4 ) , y o u w i l l never g e t o u t o f t h e m a z e . T h e a d v a n t a g e t h a t t h e m a z e algorithm g i v e s is that it insures that y o u w o n ' t g e t c a u g h t w a n d e r i n g in circles through the maze. The split-half method is a n o t h e r s e a r c h a l g o r i t h m a n d a v e r y u s e f u l o n e .

It is

u s e d by electricians, m e c h a n i c s , electronics technicians, a n d others to locate troub l e s i n f a u l t y e q u i p m e n t . A s a n e x a m p l e o f its u s e , s u p p o s e t h a t y o u r front d o o r b e l l

Start

Figure 1 4 . A P r a c t i c e M a z e for the M a z e Algorithm

45

Search

Figure 1 5 . Using t h e Split-half Algorithm to Locate a C u r r e n t Break doesn't work. Y o u find that it isn't g e t t i n g current, e v e n t h o u g h t h e p o w e r s u p p l y in the kitchen is w o r k i n g fine. T h e r e m u s t be a break s o m e w h e r e in the 20 feet of wall b e t w e e n t h e k i t c h e n a n d t h e front door. Y o u w a n t to locate t h e b r e a k w i t h a minimum of d a m a g e to your house. U s i n g the split-half t e c h n i q u e , y o u m a k e a hole h a l f w a y b e t w e e n t h e k i t c h e n a n d t h e b e l l , a n d t e s t t h e w i r e for c u r r e n t . I f t h e r e i s current, t h e n t h e b r e a k m u s t b e b e t w e e n t h e h o l e a n d t h e front door. A p p l y i n g t h e t e c h n i q u e a g a i n , y o u m a k e a s e c o n d h o l e h a l f - w a y b e t w e e n t h e first h o l e a n d t h e b e l l . I f y o u d i d n ' t f i n d c u r r e n t o n y o u r first t e s t , t h e b r e a k m u s t b e b e t w e e n t h e h o l e and the kitchen. I n this case y o u w o u l d h a v e m a d e t h e s e c o n d h o l e h a l f w a y b e t w e e n t h e first h o l e a n d t h e k i t c h e n . I f y o u m a k e h o l e s t h a t a r e a b o u t t h r e e i n c h e s across, y o u s h o u l d n e v e r h a v e t o m a k e m o r e t h a n six o f t h e m t o f i n d t h e b r e a k , a s shown in Figure 15. SUMMARY: S E A R C H I N G

FOR S O L U T I O N S

W e h a v e d i s c u s s e d f o u r g e n e r a l m e t h o d s for s e a r c h i n g for p r o b l e m s o l u t i o n s : • Trial and error • Proximity methods • Fractionation methods, and • Knowledge-based methods. 1 . T r i a l - a n d - e r r o r s e a r c h i s u s e f u l o n l y i n s m a l l p r o b l e m s p a c e s . S y s t e m a t i c trialand-error s e a r c h i s t w i c e a s e f f i c i e n t a s b l i n d t r i a l - a n d - e r r o r s e a r c h . 2. Proximity methods attempt to solve p r o b l e m s by s e l e c t i n g a step at a time, e a c h of which reduces the distance to the goal. a. T h e h i l l - c l i m b i n g m e t h o d s e e s just o n e k i n d of d i f f e r e n c e b e t w e e n the curr e n t state a n d t h e g o a l , e . g . , h e i g h t , p i c t u r e q u a l i t y . A l l its o p e r a t o r s h a v e t h e same e f f e c t — t o c h a n g e that difference. b. Means-ends analysis can handle many kinds of differences in the same problem. It e m p l o y s s p e c i a l i z e d operators w h i c h h a v e different effects on the differences. If it runs into difficulties in a p p l y i n g an operator, it can set up a s e q u e n c e of subgoals to r e d u c e that difference. All proximity methods h a v e trouble with detour problems.

46

The C o m p l e t e Problem Solver

3. Fractionation m e t h o d s involve breaking the p r o b l e m into a s e q u e n c e of smaller parts, that,is, by setting up subgoals. Subgoals m a k e p r o b l e m s easier to solve b e c a u s e t h e y r e d u c e the amount of search r e q u i r e d to find the solution, and t h e y may g u i d e the problem solver around a detour. ~ W e can set u p subgoals b y w o r k i n g part w a y into the p r o b l e m , analyzing the goal to be a c h i e v e d , d r o p p i n g some of the p r o b l e m restrictions and solving the "abstracted" p r o b l e m , or using k n o w l e d g e of familiar p r o b l e m types. 4. K n o w l e d g e - b a s e d methods u s e information stored in the p r o b l e m solver's m e m o r y t o g u i d e t h e s e a r c h for s o l u t i o n . P r o b l e m s o l v e r s m a y s e t o u t t o a c q u i r e t h e n e e d e d k n o w l e d g e w h e n t h e y h a v e difficulty. F o r e x a m p l e , t h e y m a y solve a n auxiliary p r o b l e m to learn h o w to solve the one they are h a v i n g difficulty with. Or p r o b l e m solvers m a y use information already k n o w n to them, as w h e n a chess player uses stored chess patterns, a p u z z l e enthusiast recognizes a p r o b l e m type a n d a p p l i e s a p r o b l e m s c h e m a , or a m e c h a n i c a p p l i e s a search algorithm such as the split-half t e c h n i q u e . EXECUTING THE S O L U T I O N PLAN O n c e w e h a v e p l a n n e d a s o l u t i o n t o a p r o b l e m , w e h a v e t o c a r r y i t out. S o m e t i m e s e x e c u t i o n o f t h e p l a n i s q u i t e e a s y . I n t h e D r i v e r ' s L i c e n s e P r o b l e m , for example, the hard part is figuring out w h a t equations we w a n t to solve. Carrying out the solution is routine algebra. We n e e d only take care not to m a k e d u m b mistakes w h i c h w i l l s p o i l o u r b r i l l i a n t l y c o n c e i v e d p l a n . ( D u m b m i s t a k e s g i v e p e o p l e far m o r e trouble than t h e y m a y care to admit.) I n o t h e r c a s e s , e x e c u t i o n i s b y n o m e a n s t r i v i a l . F o r e x a m p l e , after w e h a v e p l a n n e d a n e s s a y , w e f i n d that t u r n i n g o u r p l a n i n t o s m o o t h l y f l o w i n g s e n t e n c e s o r e v e n into marginally intelligible scrawl is v e r y difficult i n d e e d . F r e q u e n t l y the plan has to be m o d i f i e d or in some cases a b a n d o n e d entirely. O u r attempts at execution send us scurrying back to the drawing board to devise a n e w plan. With luck, we will h a v e l e a r n e d e n o u g h from our disastrous attempt to e x e c u t e the old plan that our n e w plan will h a v e a better c h a n c e of success. Executing the solution provides us with a very valuable check on the adequacy of our plans. S o m e t i m e s students will look at the problems at the e n d of a chapter and d e c i d e that s i n c e t h e y k n o w h o w t o s o l v e t h e m , t h e y n e e d n ' t b o t h e r w i t h the d r u d g e r y of actually e x e c u t i n g the solutions. S o m e t i m e s the students are right, b u t s o m e t i m e s t h e y miss a n e x c e l l e n t opportunity t o d i s c o v e r that t h e y w e r e w r o n g . EVALUATING THE S O L U T I O N H o w do we k n o w w h e n we have gotten the right a n s w e r to a p r o b l e m ? If we a r e r u s h e d , w e m a y d o l i t t l e m o r e t h a n g u e s s t h a t w h a t w e h a v e looks l i k e a n a n s w e r . T h e p r o b l e m s a i d " S o l v e for X , " a n d h e r e w e h a v e X o n o n e s i d e a n d e v e r y t h i n g e l s e on t h e other side, so that's t h e a n s w e r . T h i s q u i c k g l a n c e at the a n s w e r to s e e if it has the g e n e r a l s h a p e of a solution m a y b e the b e s t w e can d o w h e n w e are rushed, b u t i t can hardly b e v i e w e d a s a n a d e q u a t e e v a l u a t i o n . I f a d r u g g i s t w e r e w e i g h i n g o u t a m e d i c a t i o n for u s , w e w o u l d not o n l y w a n t h i m to c h e c k carefully to s e e that his m e a s u r e d quantities corresponded to the quantities in the prescription. We might e v e n w a n t h i m to be sure t h a t t h e p r e s c r i p t i o n h e i s u s i n g i s t h e o n e w i t h o u r n a m e o n it! T h e c r i t i c a l q u e s t i o n i n e v a l u a t i o n i s t h i s : " D o e s t h e a n s w e r I p r o p o s e m e e t all o f t h e g o a l s a n d c o n d i t i o n s s e t b y t h e p r o b l e m ? " T h u s , after t h e effort o f f i n d i n g a solution, we m u s t turn our attention b a c k to the p r o b l e m statement and c h e c k c a r e f u l l y t o b e s u r e o u r s o l u t i o n s a t i s f i e s it. In easy problems, we may be tempted strongly to skip evaluation because the

47

Search

probability of an error s e e m s small. In s o m e cases this can be costly. T h i s can be s h o w n b y t h e f o l l o w i n g p r o b l e m s f r o m W h i m b e y a n d L o c h h e a d (1980). S o l v e t h e m before proceeding. 1. There are 3 separate, equal-size boxes, and inside each box there are 2 separate small boxes, and inside each of the small boxes there are 4 e v e n smaller boxes. H o w many boxes are there altogether? a. 24

b.

13

c. 21

d. 33

e. some other number

2. T e n full crates of walnuts w e i g h 4 1 0 lb., w h i l e an e m p t y crate w e i g h s 10 lb. H o w much do the walnuts alone w e i g h ?

a. 400 lb.

b. 390 lb.

c. 3 1 0 lb.

d. 320 lb.

e. 420 lb.

3. Three empty cereal boxes w e i g h 9 ozs. and each box holds 11 ozs. of cereal. H o w much do 2 full boxes of cereal w e i g h together?

a. 20 ozs.

b. 40 ozs.

c. 14 ozs.

d. 28 ozs.

e. 1 5 ozs.

4. Cross out the letter after the letter in the word seldom w h i c h is in the same position in the word as it is in the alphabet. 5. In how many days of the w e e k does the third letter of the day's name immediately follow the first letter of the day's name in the alphabet?

a. 1

b. 2

c. 3

d. 3

e. 5

W h i l e these p r o b l e m s are not difficult, it is easy to m a k e mistakes w h i l e solving t h e m , a n d t h e m i s t a k e s a r e r e v e a l i n g . W h i m b e y a n d L o c h h e a d (1980) p o i n t o u t t h e following c o m m o n errors: I n p r o b l e m 4 , m a n y p e o p l e cross out the " d " rather than t h e " o " i n " s e l d o m . " O n e can i m a g i n e that t h e s e p e o p l e h a v e b e c o m e s o i n v o l v e d i n the p r o c e s s e s o f finding the letter w i t h t h e s a m e position in the a l p h a b e t as in t h e w o r d that t h e y forget that t h e i r task i s t o c r o s s o u t t h e l e t t e r after it. E v a l u a t i o n c o u l d h a v e r e v e a l e d the error. In p r o b l e m 2, a f r e q u e n t error is to a n s w e r 400 p o u n d s . T h e difficulty a p p e a r s to lie in representing the p r o b l e m i n c o r r e c t l y — t h a t is, in failing to notice that there are 1 0 1 0 - p o u n d b o x e s . I n e v a l u a t i n g , t h e n , it's i m p o r t a n t t o e v a l u a t e o u r representation as w e l l as our solution. While evaluation is useful in solving simple problems, it is e v e n more useful i n solving c o m p l e x ones. S u p p o s e w e h a v e the task o f w r i t i n g a n e s s a y o n " E n d a n g e r e d S p e c i e s " for a n a u d i e n c e o f 1 0 - y e a r - o l d s . L i k e t h e p e o p l e s o l v i n g p r o b lem 4, we m a y b e c o m e so i n v o l v e d in o n e part of the p r o b l e m , e.g., e x p o u n d i n g our theories of e c o l o g y , that we c o m p l e t e l y forget that we are s u p p o s e d to w r i t e to a 10-year-old a u d i e n c e . W h i l e writing, w e m u s t c o n t i n u a l l y e v a l u a t e w h a t w e w r i t e t o b e s u r e that i t s a t i s f i e s all t h e r e q u i r e m e n t s o f o u r w r i t i n g t a s k . Some problems h a v e u n i q u e solutions. We either get t h e m right or we get t h e m wrong. Other problems h a v e a w h o l e range of solutions, some better than others. P r o b l e m s o f this sort p o s e a n e s p e c i a l l y d i f f i c u l t e v a l u a t i o n t a s k — t h e t a s k o f d e t e r m i n i n g h o w g o o d our solution is. I n t h e chapters o n d e c i s i o n m a k i n g , w e w i l l d i s c u s s t e c h n i q u e s w h i c h c a n h e l p y o u t o c a r r y o u t t h i s task. Sometimes special decision-making procedures don't help. What we m a y really n e e d is an outside opinion. We will discuss just o n e practical case h e r e — t h e case of writing. W h e n we write an essay, we often write a b o u t things of special interest to u s — t h i n g s we k n o w a great d e a l about. B e c a u s e of our special k n o w l e d g e , i t m a y b e v e r y d i f f i c u l t for u s t o p u t o u r s e l v e s i n t h e p o s i t i o n o f o u r a u d i e n c e — t o imagine what it w o u l d be like not to understand things we consider obvious. R e c o g n i z i n g this difficulty w e h a v e i n e v a l u a t i n g our o w n w r i t i n g , the s e n s i b l e

48

The C o m p l e t e Problem Solver

thing to do is to s e e k the h e l p of others. A s k i n g a friend to read aloud w h a t we h a v e written and to tell us w h a t it means can be a horrifying b u t extremely educational experience. T e s t i n g w r i t i n g o u t o n t h e a u d i e n c e for w h i c h i t i s i n t e n d e d i s a v e r y s i m p l e i d e a w i t h o b v i o u s a d v a n t a g e s , b u t v e r y f e w p e o p l e d o it. M a j o r c o r p o r a t i o n s a n d g o v e r n m e n t a l a g e n c i e s p u t a great d e a l of m o n e y a n d effort into p r o d u c i n g instruction manuals and other d o c u m e n t s to inform the public, yet rarely consider testing t h o s e d o c u m e n t s o n t h e a u d i e n c e for w h i c h t h e y w e r e i n t e n d e d . W e i n t e n d our w r i t i n g t o b e understood. T o e v a l u a t e w h e t h e r o r not i t can b e u n d e r s t o o d , w e s h o u l d r o u t i n e l y t a k e i t t o o t h e r s for c r i t i c i s m . CONSOLIDATION T h i n k of the last t i m e y o u s o l v e d a difficult p r o b l e m . W h e n t h e solution finally c a m e t o y o u , y o u r strongest e m o t i o n w a s p r o b a b l y o n e o f great relief. A t s u c h t i m e s , we are likely to say to ourselves, " W h e w ! T h a n k G o d I'm through with that!" and turn our attention to m o r e r e l a x i n g thoughts. It is just at this point, t h o u g h , that we c a n g a i n m o s t f r o m t h e p r o c e s s o f consolidation—that i s , f r o m r e f l e c t i n g o n t h e p r o b l e m - s o l v i n g e x p e r i e n c e w e h a v e j u s t b e e n t h r o u g h a n d l e a r n i n g f r o m it. W h e n w e s t r u g g l e w i t h a h a r d p r o b l e m , w e m a y start off i n a h u n d r e d w r o n g directions, e x p l o r e n u m e r o u s b l i n d alleys, miss critical c l u e s , a n d only s l o w l y c o m e t o r e c o g n i z e w h a t ' s i m p o r t a n t a b o u t t h e p r o b l e m a n d w h a t i s not. A s w e m a k e progress, w e m a y m a k e important c h a n g e s i n our representation, d e t e c t faulty assumptions or discover useful detours. After a w e e k has passed, or perhaps e v e n after a f e w h o u r s , w e w i l l b e a b l e t o r e m e m b e r v e r y l i t t l e o f t h i s c o m p l e x p r o c e s s . W e r e c o g n i z e the p r o b l e m b u t forget w h y w e found i t s o difficult. Consolidation, then, is best d o n e w h e n our m e m o r y of the problem-solving p r o c e s s i s still v e r y f r e s h i n o u r m i n d s . H o w d o w e d o i t ? T h e b a s i c q u e s t i o n t o b e a n s w e r e d is, " W h a t c a n I l e a r n f r o m t h e e x p e r i e n c e o f s o l v i n g t h i s p r o b l e m ? " T h e f o l l o w i n g m o r e specific questions m a y h e l p us to a n s w e r this o n e : 1. W h y w a s this p r o b l e m difficult? 2. 3. 4. 5. 6. 7. 8. 9.

W a s it hard to find an appropriate representation? W a s i t d i f f i c u l t for m e t o k e e p m y p l a c e i n t h e p r o b l e m ? W a s it h a r d to f i n d a s o l u t i o n m e t h o d ? Was there a detour? W h y d i d I miss critical c l u e s ? D i d I m a k e false assumptions? Should I h a v e u s e d a different representation? What mistakes did I make?

JO. D i d I m a k e i m p o r t a n t d i s c o v e r i e s a b o u t r e p r e s e n t a t i o n s , m e t h o d s , d e t o u r s ? I J. If so, h o w d i d I m a k e t h e m ? 12. A r e t h e r e o t h e r p r o b l e m s s i m i l a r t o t h i s o n e ? 13. C o u l d t h e y h a v e b e e n s o l v e d i n t h e s a m e w a y ? O f c o u r s e , d i f f e r e n t q u e s t i o n s m a y b e a p p r o p r i a t e for d i f f e r e n t p r o b l e m s . T h e particular questions are not important. W h a t is important is that y o u reflect on the p r o b l e m - s o l v i n g p r o c e s s , a s k q u e s t i o n s a b o u t it, a n d l e a r n f r o m t h e e x p e r i e n c e . I f y o u m a k e i t a h a b i t t o c o n s o l i d a t e i m m e d i a t e l y after p r o b l e m s o l u t i o n , y o u can take a d v a n t a g e of a w e a l t h of information w h i c h w o u l d o t h e r w i s e be lost to y o u .

THREE.

PROTOCOL

ANALYSIS

I n this c h a p t e r w e i n t r o d u c e t h e t e c h n i q u e o f protocol analysis—cognitive p s y c h o l o g y ' s m o s t p o w e r f u l t o o l for d e s c r i b i n g p s y c h o l o g i c a l p r o c e s s e s . F i r s t , w e will answer the question, " W h a t is a protocol?" T h e n we will w o r k through several examples to s h o w h o w protocols m a y be analyzed. W H A T IS A P R O T O C O L ? A protocol is a d e s c r i p t i o n of the activities, o r d e r e d in t i m e , in w h i c h a subject e n g a g e s w h i l e p e r f o r m i n g a task. A protocol, t h e n , is a d e s c r i p t i o n , b u t not e v e r y d e s c r i p t i o n of a task performa n c e i s a p r o t o c o l . O f t e n w e d e s c r i b e tasks m e n t i o n i n g o n l y t h e i r o u t c o m e s o r g o a l s . W e m a y s a y , for e x a m p l e , " M y G r e a t D a n e , S p o t , p e r s u a d e d m e t o g i v e h i m h i s s u p p e r . " T h i s d e s c r i p t i o n tells us that the d o g d i d o n e or m o r e things to get food, but it doesn't say what these things w e r e or in what order they occurred. T h e d e s c r i p t i o n , t h e r e f o r e , is not a p r o t o c o l . T h e d e s c r i p t i o n b e l o w is a p r o t o c o l , however. E x p e r i m e n t e r : [seated at d i n n e r table cutting into his steak] Spot: [seated directiy b e h i n d the experimenter, his chin resting on the experimenter's shoulder. Spot watches intently as the steak is b e i n g cut] Exp: Spot: Exp: Spot: Exp: Spot: Spot: Exp:

[ s k e w e r s a l a r g e p i e c e o f s t e a k w i t h h i s fork] [tail w a g s , s t o m a c h r u m b l e s o m i n o u s l y ] [ b e g i n s t o r a i s e fork t o m o u t h ] [places p a w on experimenter's arm and looks intently into experimenter's eyes] "Spot!" [removes paw, continues to watch intently] [drools into e x p e r i m e n t e r ' s shirt p o c k e t ] [abandons o w n dinner and feeds dog]

T h i s i s a p r o t o c o l b e c a u s e i t lists S p o t ' s a c t i v i t i e s a n d t h e o r d e r i n w h i c h t h e y occurred. In the same w a y , w h e n we collect protocols of p e o p l e solving p r o b l e m s , w e are n o t j u s t i n t e r e s t e d i n t h e a n s w e r s t h e y g i v e u s , b u t , m o r e i m p o r t a n d y , i n t h e sequence of things they do to get those answers. T h e y do things — such as draw diagrams, m a k e c o m p u t a t i o n s , and ask q u e s t i o n s — i n a particular order.

The C o m p l e t e Problem Solver

52

W e w i l l d e s c r i b e t h r e e k i n d s o f p r o t o c o l s h e r e : motor p r o t o c o l s , eye movement p r o t o c o l s , a n d verbal p r o t o c o l s . A l l t h r e e k i n d s h a v e p r o v e d t o b e v e r y u s e f u l for d e s c r i b i n g p s y c h o l o g i c a l p r o c e s s e s , b u t i n this c h a p t e r o u r m a i n i n t e r e s t w i l l b e i n the third k i n d — v e r b a l protocols. MOTOR

PROTOCOLS

To obtain motor protocols, we observe the obvious physical activities of our subjects—activities such as walking, picking things up, and reaching. T h e " S p o t " protocol was largely a motor protocol. T h e motor protocol b e l o w , collected by Kohler (1925, p. 174), describes the activities of Sultan, an ape, as he solved a simple problem. (March 26th): Sultan is squatting at the bars, but cannot reach the fruit, which lies outside, by means of his only available short stick. A longer stick is deposited outside the bars, about t w o metres on one side of the objective, and parallel with the grating. It can not be grasped with the hand, but it can be p u l l e d within reach by means of the small stick (see Figure 1). Sultan tries to reach the fruit with the smaller of the two sticks. N o t succeeding, he tears at a p i e c e of wire that projects from die netting of his cage, b u t that, too, is in vain. T h e n he gazes about him; (there are always in the course of these tests some long pauses, during which the animals scrutinize the w h o l e visible area). He suddenly picks up the little stick once more, goes up to the bars directly opposite to the long stick, scratches it towards him with the 'auxiliary,' seizes it, and goes with it to the point opposite the objective, which , he secures. From the moment that his eyes fall upon the long stick, his procedure forms one consecutive whole, without hiatus

K o h l e r u s e d the o b s e r v a t i o n that the solution o c c u r r e d s u d d e n l y to argue that Sultan's solution process involved "insight." Further, Kohler u s e d the observation t h a t t h e s o l u t i o n r a n off s m o o t h l y a n d c o n t i n u o u s l y t o a r g u e t h a t S u l t a n " k n e w w h a t he was d o i n g " from the m o m e n t of insight. M o t o r p r o t o c o l s a r e e s p e c i a l l y v a l u a b l e for u s e w i t h s u b j e c t s w h o h a v e l i m i t e d language abilities—for example, children and animals.

53

Protocol Analysis EYE-MOVEMENT

PROTOCOLS

An e y e - m o v e m e n t protocol is a record of the places in a scene w h e r e the s u b j e c t s fix t h e i r g a z e a s t h e y p e r f o r m a task. I n s o m e c a s e s , w e c a n o b t a i n sufficiently accurate information about e y e m o v e m e n t s just by looking at the subjects, as F a n t z (1961) has d o n e , to s e e if t h e y are l o o k i n g left or right. In other cases, we must use sophisticated equipment such as an e y e - m o v e m e n t camera (Mackworth and T h o m a s , 1962) to obtain p r e c i s e e y e positions. F i g u r e 2 s h o w s t h e e y e m o v e m e n t s o f a n e x p e r t c h e s s p l a y e r d u r i n g t h e first five seconds of examining the chess position s h o w n in F i g u r e 3 (Tichomirov and P o z n y a n s k a y a , 1966). U s i n g t h e s e data, S i m o n a n d B a r e n f e l d (1969) a r g u e d that the chess expert was examining attack-and-defense relationships among the pieces. E y e - m o v e m e n t protocols h a v e also b e e n u s e d to study tasks s u c h as r e a d i n g ( C a r p e n t e r a n d Just, 1 9 7 8 ) a n d s o l v i n g n u m b e r p u z z l e s ( W i n i k o f f , 1 9 6 6 ) . VERBAL

PROTOCOLS

In a verbal, or " t h i n k i n g a l o u d " protocol, subjects are asked to say aloud everyt h i n g t h e y t h i n k w h i l e p e r f o r m i n g t h e task. T h e y are a s k e d t o s a y e v e r y t h i n g t h a t occurs t o t h e m , n o matter h o w trivial i t m a y s e e m . E v e n w i t h s u c h e x p l i c i t instruct i o n s , h o w e v e r , s u b j e c t s m a y f o r g e t a n d fall s i l e n t — c o m p l e t e l y a b s o r b e d i n t h e task. A t s u c h t i m e s t h e e x p e r i m e n t e r w i l l s a y , " R e m e m b e r , t e l l m e e v e r y t h i n g y o u are t h i n k i n g . "

54

The C o m p l e t e Problem Solver

Figure 2. Record of Eye M o v e m e n t s for the First Five S e c o n d s Reprinted from Soviet Psychology, V o l . 5, 1 9 6 6 , by permission of the publisher, M.E. Sharpe, Inc., W h i t e Plains, NY 1 0 6 0 3 .

As examples we will s h o w t w o verbal protocols, in w h i c h subjects solve waterj u g p r o b l e m s . T h e s e r e q u i r e that the subject m e a s u r e out a s p e c i f i e d quantity of water using three jugs, as shown in Figure 4.

Protocol Analysis

55

O n e typical water-jug p r o b l e m r e q u i r e s the subject t o m e a s u r e out 3 1 quarts w h e n Jug A will hold 20 quarts; J u g B , 59 quarts; and Jug C, 4 quarts. T h e p r o b l e m can be s o l v e d in four steps as follows: 1. Fill J u g B . 2. Fill Jug A from J u g B, l e a v i n g 39 quarts in B. 3. F i l l J u g C f r o m B, l e a v i n g 35 q u a r t s in B. 4 . E m p t y C a n d fill i t a g a i n f r o m B , l e a v i n g t h e d e s i r e d q u a n t i t y , 3 1 q u a r t s , i n B . W h e n y o u a n a l y z e a p r o t o c o l i t i s h e l p f u l t o h a v e a l r e a d y d o n e t h e task y o u r s e l f . S o , b e f o r e y o u b e g i n t o a n a l y z e P r o t o c o l I , try t o s o l v e t h i s p r o b l e m : Measure 100 qts. g i v e n Jug A, holding 21 qts.; Jug B, holding 1 2 7 qts; and Jug C, 3 qts.

W h e n y o u have found the solution, solve the p r o b l e m in Protocol I, if y o u have not already d o n e so.

56

The C o m p l e t e Problem Solver

PROTOCOL I Problem: G i v e n J u g A , w h i c h c o n t a i n s 9 q t s . ; J u g B , 4 2 q t s . ; a n d J u g C , 6 qts., m e a s u r e out e x a c t l y 21 qts. Subj = Subject Subj: Exp: Subj:

(1) (2) (3) (4)

Exp = Experimenter

" U h , t h e first t h i n g t h a t ' s a p p a r e n t i s h a l f o f B i s , i s t h e u h , t h e a m o u n t that y o u w a n t . " "Uh huh." " U m , y o u c a n ' t g e t 21 from j u s t m u l t i p l y i n g up A or C.

(5) Y o u g e t 1 8 a n d 1 8 r e s p e c t i v e l y , t h a t ' s a s c l o s e a s y o u c a n g e t , (6) I g u e s s . U m , s o I'll try t o t h i n k o f t h e d i f f e r e n t c o m b i n a t i o n s t h a t (7) m i g h t . . . c o m e u p w i t h a s u r p l u s . . . o r d e f i c i t o f t h e 2 1 q u a r t s . . . and (8) n i n e a n d six a r e , 1 5 . . . i f y o u t o o k t w o n i n e s a n d t w o s i x e s , y o u ' d have 30 which (9) w o u l d l e a v e y o u . . . a n d p o u r t h e m i n t o t h e 4 2 c o n t a i n e r , y o u ' d h a v e a, (10) a n o p e n s p a c e o f a , 1 2 , w h i c h m e a n s n o t h i n g . H o w a b o u t . . . s e e . . . n o w I'm (11) trying to think of h o w close to 42 you can get with a nine and the six q u a n t i t i e s . ( 1 2 ) Y o u c a n g e t . . . I f o r g e t t h e s e v e n t a b l e . It's b e e n a l o n g t i m e since I've

Protocol Analysis

57

(13) h a d to m u l t i p l y or a n y t h i n g so y o u ' l l h a v e to g i v e me s o m e time. Um, ( 1 4 ) n i n e t i m e s f i v e i s 4 5 . . . h m , six t i m e s s e v e n i s 4 2 , 1 t h i n k . I s that right?" Exp: Subj: Exp: Subj:

Exp:

(15) (16) (17) (18)

" U h huh." " O K , s o y o u c a n , u h . . . fill B w i t h C , e v e n l y . . . " "You could..." " S o , . . . i f y o u w e r e t o t a k e . . . 3 6 . . . h m , o h , u h , six t i m e s f o u r i s 2 4 . . . and if y o u , u h , (19) W h a t I ' m t r y i n g t o g e t r i d o f i s , i s , t h r e e q u a r t s t h e r e . . . " (20) " G o o d . "

S u b j : (21) " S o i f y o u w e r e t o , u m . . . still 2 4 . I , I w a s t r y i n g t o t h i n k p o s sibly, (22) s o m e w a y , o f . . . t h e d i f f e r e n c e b e t w e e n t h e six a n d t h e n i n e i s three quarts. I

Exp:

AN

(23) w a s t r y i n g t o t h i n k o f . . . a w a y t o u h . . . o h , h o w a b o u t . . . y o u p u t t h e t h r e e , six (24) q u a r t q u a n t i t y i n t o t h e 4 2 b o t t l e , w h i c h i s 1 8 , t h e n t h e r u n off, from (25) p o u r i n g a n i n e i n t o a six w h i c h i s t h r e e a n d 1 8 i s 2 1 . " (26) " G o o d . "

EXAMPLE O F P R O T O C O L ANALYSIS

N o w , l e t ' s e x a m i n e P r o t o c o l I i n d e t a i l a n d try t o m a k e s o m e r e a s o n a b l e guesses about what the subject is doing. I n h i s first s e n t e n c e , t h e s u b j e c t m e n t i o n s s o m e t h i n g t h a t a p p e a r s t o b e i r r e l e v a n t t o s o l v i n g t h e p r o b l e m . H e m e n t i o n s t h e fact t h a t t h e d e s i r e d a m o u n t ( 2 1 quarts) i s j u s t h a l f t h e q u a n t i t y c o n t a i n e d i n J u g B . N o w , d i v i s i o n i s a v e r y u s e f u l operation in m a n y a l g e b r a p r o b l e m s . In this p r o b l e m , if we c o u l d d i v i d e Jug B in half, t h e p r o b l e m w o u l d b e s o l v e d . B u t a l a s ! T h e r e i s n o d i v i s i o n o p e r a t i o n i n water-jug p r o b l e m s . All we can do is add and subtract the quantities in Jugs A, B, and C. W h y , t h e n , d o e s the subject notice that the d e s i r e d quantity is half of B ? T h e simplest a n s w e r s e e m s to be that he is c o n f u s i n g water-jug p r o b l e m s (perhaps b e c a u s e he isn't t h o r o u g h l y familiar w i t h them) w i t h the m o r e g e n e r a l class of algebra p r o b l e m s . I f this a n s w e r i s correct, w e w o u l d e x p e c t that the subject w o u l d stop n o t i c i n g d i v i s i o n r e l a t i o n s a s h e g a i n s m o r e e x p e r i e n c e w i t h w a t e r - j u g p r o b l e m s . I n fact, t h a t i s w h a t h a p p e n e d . In line 3, the experimenter does just what the experimenter is supposed to d o — t h a t is, he is n o n c o m m i t a l . In general, the e x p e r i m e n t e r s h o u l d a n s w e r only essential questions and r e m i n d the subject to k e e p talking. F r o m lines 4 a n d 5, we c a n g u e s s that the subject has s u c c e s s i v e l y a d d e d nines t o g e t n i n e , 1 8 , 2 7 . . . a n d s i x e s t o g e t six, 1 2 , 1 8 , 2 4 . . . a n d r e a l i z e d t h a t n e i t h e r s e q u e n c e includes 2 1 . F r o m lines 6 through 10, we can s e e that the subject b e g i n s to consider combinations of nines and sixes w h i c h m a y be a d d e d together to obtain interesting sums, or subtracted from the 42-quart container to obtain interesting d i f f e r e n c e s . W h i l e c o n s i d e r i n g s u m s , t h e s u b j e c t fails t o n o t i c e t h a t t h e s u m 6 + 6 + 9 solves the problem. In lines 11 through 14, the subject tries to find out if 42 quarts can be obtained by adding nines and sixes. T h e a n s w e r is positive, b u t it doesn't h e l p the subject to find a solution. It appears to be a " b l i n d a l l e y . " In this section, the subject indicates several times that he doesn't feel confident about multiplication. In lines 15 and 17, the e x p e r i m e n t e r provides the subject with a small amount

58

The C o m p l e t e Problem Solver

o f information b y c o n f i r m i n g his u n e a s y s u s p i c i o n that six t i m e s s e v e n e q u a l s 4 2 . On occasion, the experimenter must d e c i d e w h e t h e r or not to provide information the subject requests. In this case, since the e x p e r i m e n t e r w a s really interested in water j u g p r o b l e m s rather than in arithmetic, he d e c i d e d to s u p p l y an arithmetic fact. In l i n e s 18 a n d 19, t h e s u b j e c t realizes that if he h a d a w a y to subtract three quarts from 24 quarts, he c o u l d s o l v e the p r o b l e m . In l i n e 20 the e x p e r i m e n t e r appears to slip by providing the subject with approval, w h e n he w o u l d better have r e m a i n e d s i l e n t . I n l i n e 2 1 , t h e s u b j e c t i s still t h i n k i n g o f w o r k i n g f r o m 2 4 q u a r t s . I n l i n e 2 2 , h e d i s c o v e r s a w a y t o a d d (rather t h a n s u b t r a c t ) t h r e e q u a r t s b y p o u r i n g A into C a n d c a t c h i n g t h e overflow. In lines 23 and 24, he d e c i d e s to w o r k from 18 q u a r t s r a t h e r t h a n 2 4 q u a r t s a n d t h e n (on l i n e 2 5 ) i m m e d i a t e l y s o l v e s t h e p r o b l e m . N o w , let's stand b a c k from the details of the protocol to s e e if we can characterize the w h o l e p r o b l e m - s o l v i n g p r o c e s s that the subject w e n t through. B e f o r e r e a d i n g further, r e v i e w the discussion of the protocol a n d t h e n try to characterize the p r o b l e m - s o l v i n g process yourself. O n e w a y to characterize the problem-solving process is to describe it as a s e a r c h for a n o p e r a t o r o r a c o m b i n a t i o n o f o p e r a t o r s t o s o l v e t h e p r o b l e m . ( I n t h i s case, the operators are arithmetic procedures such as division and subtraction). In F i g u r e 5 , w h e r e w e h a v e d i a g r a m m e d this search p r o c e s s , w e c a n s e e that search p r o c e e d s , generally, from s i m p l e to c o m p l e x — t h a t is, from single operators to c o m p l e x combinations of operators. U p u n t i l l i n e 1 8 , t h e s u b j e c t ' s s e a r c h for a s o l u t i o n c o u l d h a v e b e e n g u i d e d b y the p r o b l e m statement. T h a t is, b y r e a d i n g the p r o b l e m statement, the subject c o u l d h a v e d e c i d e d that w h a t w a s n e e d e d t o s o l v e the p r o b l e m w a s s o m e c o m b i n a t i o n o f a l g e b r a i c o p e r a t o r s . U p t o l i n e 1 8 , h e c o u l d s i m p l y b e t r y i n g o n e c o m b i n a t i o n after a n o t h e r . W e c a l l t h i s " f o r w a r d s e a r c h " — a s e a r c h s u g g e s t e d b y t h e p r o b l e m statem e n t alone. In lines 18 a n d 19, h o w e v e r , the subject formulates a goal on the basis

59

Protocol Analysis

of his difficulties in s o l v i n g the p r o b l e m . He notes that he hasn't b e e n able to get closer to t h e a n s w e r than three quarts a n d attempts to find an operator that w i l l subtract t h r e e quarts. T h i s goal d e p e n d s not just on the p r o b l e m statement b u t also on the subject's e x p e r i e n c e in trying to solve the p r o b l e m — t h a t is, on his distance from t h e goal. It is a form of m e a n s - e n d s a n a l y s i s in w h i c h t h e s u b j e c t attempts to find a m e a n s to the e n d of r e d u c i n g his distance from the goal. T h e w h o l e solution process then consists of 1. An initial p h a s e of forward search t h r o u g h an i n c r e a s i n g l y c o m p l e x s e q u e n c e of operators, followed by 2. A phase of means-ends analysis in w h i c h the subject s u c c e e d s in finding a solution. In analyzing a protocol, we attempt to d e s c r i b e the p s y c h o l o g i c a l processes that a s u b j e c t u s e s t o p e r f o r m a task. T o d o t h i s , i t i s u s e f u l t o b e f a m i l i a r b o t h w i t h t h e properties of the task a n d with the p r o b l e m solver's c o m p o n e n t p s y c h o l o g i c a l processes. In a n a l y z i n g t h e water-jug protocol a b o v e , k n o w i n g that t h e task required algebraic operators and that h u m a n p r o b l e m solvers often u s e processes of forward search and means-ends analysis h e l p e d us to r e c o g n i z e h o w the subject had o r g a n i z e d t h e s e p r o c e s s e s i n h i s s e a r c h for a s o l u t i o n . I n t h e s a m e w a y , w h e n w e a n a l y z e other protocols, k n o w l e d g e o f other tasks a n d o f other p s y c h o l o g i c a l properties will be useful. T h i s is not to say that we m u s t already u n d e r s t a n d a performa n c e b e f o r e w e c a n a n a l y z e it. I t i s j u s t t h a t w h e n w e d o u n d e r s t a n d s o m e t h i n g s about the performance, we can use t h e m very profitably to learn other things. In analyzing Protocol II, k n o w l e d g e of the p s y c h o l o g i c a l p h e n o m e n o n of set is very helpful.

Protocol II Problem: G i v e n J u g A , w h i c h c o n t a i n s 2 3 q t s . , J u g B , 4 9 q t s . , a n d J u g C , 3 qts., m e a s u r e out exactly 20 qts. Subj: Exp: Subj:

Exp:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

" U h , . . . I'm adding and subtracting h e r e , " "Say what you're adding and subtracting." " O . K . W e l l , 2 3 from 4 9 . . . i s . . . u h . . . r e m i n d s m e w h e n we k e e p score w h e n we play cards. 2 6 . . . oh, OK. Er it w o u l d be easy enough. You t a k e . . . u m . . . OK. H o w do we g e t this into water and j u g s n o w ? . . . b e c a u s e y o u could take a . . . w h i c h is 23, and pour it into C w h i c h is three, there we go, pour it into C w h i c h is three, and 20 w o u l d be w h a t w o u l d g o . . . right into B . " "Good."

T h e p r o b l e m u s e d i n this protocol can b e s o l v e d i n either o f t w o w a y s . I t can b e solved by the p r o c e d u r e B — A — 2 C , or it can be s o l v e d by the s i m p l e r p r o c e d u r e : A — C . Just b e f o r e s o l v i n g t h i s p r o b l e m , t h e s u b j e c t w o r k e d a s e r i e s o f s i x p r o b l e m s , all o f w h i c h r e q u i r e d t h e p r o c e d u r e B — A — 2 C for s o l u t i o n . A s a r e s u l t , w e w o u l d e x p e c t t h e s u b j e c t to s h o w a s e t to u s e t h e B - A - 2C p r o c e d u r e . As l i n e s 6 t h r o u g h 9 s h o w , the subject actually solves the p r o b l e m by the A — C p r o c e d u r e . H o w e v e r , i f w e l o o k b a c k t o l i n e s 3 a n d 4 , w e s e e t h a t t h e s u b j e c t ' s first p r o b l e m solving attempt w a s to subtract A from B. T h i s s u g g e s t s that he started to u s e the B — A — 2C p r o c e d u r e e v e n t h o u g h he d i d n ' t carry it t h r o u g h . C l e a r l y , a n a l y z i n g the protocol g i v e s u s e v i d e n c e a b o u t the subject's solution p r o c e s s that w e can't g e t just by looking at the answer.

T h e C o m p l e t e Problem Solver

60

AGE PROBLEMS T h e protocol of an age p r o b l e m illustrates again the usefulness of k n o w l e d g e o f task d e m a n d s a n d c o m p o n e n t p s y c h o l o g i c a l p r o c e s s e s for u n d e r s t a n d i n g h u m a n p e r f o r m a n c e . As in m a n y p r o b l e m s , part of the difficulty of age p r o b l e m s lies in k e e p i n g track of r e l e v a n t information. U n l e s s p r o b l e m solvers are v e r y careful in l a b e l i n g information, t h e y are likely to confuse a person's a g e at o n e time with the same person's age at a different time. For example, p r o b l e m solvers w h o have only a s i n g l e l a b e l w h i c h t h e y u s e b o t h for " M a r y ' s a g e n o w " a n d " M a r y ' s a g e t h e n " — l e t us say " M a r y " or " M " — a r e very likely to confuse Mary's age n o w with M a r y ' s a g e t h e n . T h i s is j u s t w h a t h a p p e n s in this a g e - p r o b l e m protocol. Protocol III Problem: J o h n i s n o w t w i c e a s o l d a s h e w a s w h e n M a r y w a s 3 y e a r s o l d e r than he is now, but he is only half as old as Mary is at present. N o w h o w o l d are t h e y ? Subj:

Exp: Subj:

(1 " U r n , u m , J o h n i s n o w t w i c e a s o l d a s h e w a s w h e n M a r y w a s (2. t h r e e y e a r s o l d e r t h a n h e i s n o w . B u t h e i s o n l y h a l f a s o l d a s Mary (3. A t p r e s e n t . U m , u m , t h r e e y e a r s o l d e r i s t h e o n l y n u m b e r s u p plied,

(4: s o I ' m g o i n g t o start w o r k i n g f r o m t h e r e . I g u e s s , a n d u h , . . . " (5; " W h a t a r e y o u t h i n k i n g ? " (6 " I g e t a b s o l u t e l y n o t h i n g f r o m t h o s e q u e s t i o n s . (7 " I a m d r a w i n g a b l a n k a t t h e m o m e n t , u m , I a m t r y i n g t o sort o u t John

(s; (9

(10 (11 (12 (13 (14

(is; (ie; Exp: (17 Subj: (18 Exp: (19 S u b j : (20

and Mary, I guess, here, because, the words don't lend themselv e s to b e d e c i p h e r e d into, like, t w o neat little John and M a r y c o l u m n s , um, a n d it's h a r d . O K . J o h n i s t w i c e a s o l d . . . n o w . . . t h a n w h e n h e was w h e n M a r y w a s three years older than him . . . O H w o w ! U m , but h e i s only half as old as Mary at p r e s e n t . . . so, u m , John e q u a l s , u m , n o w I'm trying t o t h i n k o f a . . . I d o n ' t k n o w w h y i t j u s t o c c u r r e d t o m e that m a y b e if I p l u g g e d John and M a r y and the information into little, little, you k n o w , J o h n e q u a l s M a r y p l u s t h r e e , o r s o m e t h i n g l i k e that, y o u know, little X , Y ? " "Uh Huh." " I don't k n o w w h a t y o u call t h e m . " "Equations."

" E q u a t i o n s . O K . G o t to w a r n y o u , I f l u n k e d m a t h in sixth grade. So um, (2i; u m , A h . . . John i s n o w t w i c e a s old a s h e w a s w h e n Mary w a s three years (22 o l d e r t h a n h e i s n o w . . . O K , s o J o h n e q u a l s . . . u m , M a r y a n d three . . . , which (23

doesn't do me any good, and he is only half as old as Mary at present,

Protocol Analysis

67

(24) s o I h a v e t o k n o w h o w m u c h J o h n i s w o r t h first, h e r e , a n d , u h , . . . Mary is (25) t h r e e y e a r s o l d e r t h a n h e i s . . . , a n d h a l f a s o l d a s M a r y a t p r e s ent. U m , (26) M a r y i s t h r e e y e a r s o l d e r t h a n h e i s n o w , b u t h e i s o n l y h a l f a s o l d as

Exp:

(27) M a r y a t p r e s e n t . . . s o h e h a s t o b e u m . . . M a r y i s t h r e e y e a r s older than (28) h e i s n o w . . . a n d M a r y i s t w i c e h i s a g e . T h e n h e ' s g o t t o b e t h r e e years (29) o l d . " (30) " A l l r i g h t . "

ANALYSIS OF THE PROBLEM T o s o l v e this p r o b l e m , i t i s n e c e s s a r y t o d i s t i n g u i s h f o u r q u a n t i t i e s — J o h n ' s a g e n o w (JN), J o h n ' s a g e t h e n (JT), M a r y ' s a g e n o w ( M N ) , a n d M a r y ' s a g e t h e n ( M T ) . T h r e e equations can b e d e r i v e d from the p r o b l e m statement. JN

= 2 JT

MT = JN + 3 JN = V2 MN

[John i s t w i c e a s o l d n o w a s h e w a s t h e n . ] [Mary t h e n w a s three years o l d e r than John is now.] [John n o w is h a l f as o l d as M a r y n o w . ]

I n addition, b e c a u s e w e k n o w that John and M a r y a g e a t the s a m e rate, w e can derive a fourth equation: JN — JT = MN — MT

[The difference b e t w e e n John's age n o w and John's age then is the same as the difference b e t w e e n Mary's age n o w and Mary's age then.]

A r m e d w i t h t h e s e e q u a t i o n s , t h o s e a d e p t a t a l g e b r a c a n d e t e r m i n e that J o h n ' s a g e n o w i s six a n d M a r y ' s a g e n o w i s 1 2 . N o w , let's s e e w h a t our subject has d o n e . F r o m lines 1 5 through 29, she frequently confuses John's and Mary's ages n o w with John's and Mary's ages then. I n fact, n o w h e r e i n t h e p r o t o c o l i s t h e r e a n y e v i d e n c e t h a t s h e e v e r m a k e s t h e now-then distinction. In lines 27 and 28, she derives t w o equations: M a r y = J o h n + 3, a n d Mary = 2 x John T h e s e c o n d o n e i s c o r r e c t , b u t t h e first i s n o t b e c a u s e i t c o n f u s e s M a r y ' s a g e t h e n with Mary's age now. T h e subject then solves these equations and concludes, incorrectly, that John's a g e n o w is three. P R O T O C O L ANALYSIS APPLIED T O W R I T I N G T h e first 2 7 l i n e s o f a w r i t i n g p r o t o c o l , a b o u t 1 0 p e r c e n t o f t h e full p r o t o c o l , are r e p r i n t e d h e r e . T h e w r i t e r w a s a v o l u n t e e r w h o k n e w that s h e w a s t o e n g a g e i n a w r i t i n g s t u d y for a b o u t a n h o u r . S h e k n e w t h a t s h e w o u l d b e a s k e d t o " t h i n k a l o u d " as she p r o d u c e d a short essay. S h e d i d not k n o w the topic, h o w e v e r , until she a r r i v e d for t h e t a p e r e c o r d i n g s e s s i o n . Protocol IV Problem: W r i t e a s h o r t e s s a y o n w h y y o u w r i t e p a p e r s a t C a r n e g i e - M e l l o n University. S:

(1) " O K , u m , t h e i s s u e i s m o t i v a t i o n a n d t h e p r o b l e m o f w r i t i n g papers. For

62

T h e C o m p l e t e Problem Solver (2) m e , m o t i v a t i o n h e r e a t C a r n e g i e - M e l l o n i s t h e a c a d e m i c p r e s s u r e a n d g r a d e s that (3) a r e i n v o l v e d , s o I'd b e t t e r p u t t h a t d o w n . . . a n d g r a d e s . . . u m , t h e y kind of (4) c o m p e l m e , t h a t ' s r e a l l y w h a t m o t i v a t i o n is, u m , k i n d o f t o i m p e l o r start or (5) a , m o m e n t u m . ( P a u s e ) O K , I s u p p o s e f r o m t h e a c a d e m i c p r e s s u r e of the grades, (6) I ' m n o t s u r e w h e t h e r , I t h i n k p e r s o n a l s a t i s f a c t i o n i s i m p o r t a n t , b u t I'm not

E: S:

(7) s u r e w h e t h e r t h a t s t e m s f r o m a c a d e m i c p r e s s u r e s a n d g r a d e s , o r whether—I (8) w o u l d s a y p e r s o n a l s a t i s f a c t i o n i s a m a j o r i s s u e . O K , u m . O h . " (9) " W h a t a r e y o u t h i n k i n g ? " (10) " I ' m t r y i n g t o t h i n k o f t h e first s e n t e n c e t o start w i t h . U m , m a y b e something (11) l i k e , p e r s o n a l satisfaction is the major m o t i v a t i n g force in the writing of m y (12) p a p e r s a n d reports. O K , I'm trying to think o f . . . O K , I w a n t to s o m e h o w g e t it into (13) the a c a d e m i c pressures n o w . U m , w e l l m a y b e not s o soon. O K . N o t o n l y do I g e t (14) satisfaction from my g r a d e s , b u t I also g e t satisfaction in turning in something (15) that is g o o d quality. So, if I'm h a p p y w h e n I write a g o o d paper, it really doesn't ( 1 6 ) m a t t e r w h a t k i n d o f g r a d e I g e t b a c k o n it, i f I ' m h a p p y w i t h it. S o , um, um, ( 1 7 ) l e t ' s s e e . U m , w h a t a r e t h e — I ' m t h i n k i n g of, I ' m t r y i n g t o r e l a t e personal (18) satisfaction b e t w e e n a c a d e m i c p r e s s u r e a n d t h e g r a d e s , b u t I'm not really sure ( 1 9 ) h o w t o d o it, h o w t o b r a n c h it. I ' m r e a l l y h a v i n g a h a r d t i m e g e t t i n g started. (20) W e l l , m a y b e I'll j u s t w r i t e a b u n c h o f i d e a s d o w n , a n d m a y b e try t o c o n n e c t t h e m after (21) I finish. O K . W h e n I feel that I've w r i t t e n a h i g h quality, a n d I p u t in paren(22) t h e s e s , p r o f e s s i o n a l , p a p e r , u m , t o b e g r a d e d , w h e n I s u b m i t it, t h e grade is not (23) a l w a y s n e c e s s a r y for t h e t e a c h e r t o h a v e t h e s a m e . O K , t h a t ' s k i n d Of

;

(24) I'll c h e c k w i t h t h a t o n e . O K , a n d — l e t ' s s e e w h a t e l s e . U m , b u t (25) o f c o u r s e , t h e r e a s o n I ' m w r i t i n g t h e p a p e r i n t h e first p l a c e i s for the grade, (26) o r t o r e l a t e t h a t b a c k . T h o s e t w o i d e a s a r e v e r y i n t e r l o c k e d — m a y b e that's not (27) t h e r i g h t t e r m . " Analysis of t h e P r o t o c o l In a n a l y z i n g this protocol fragment, we w i l l call attention to just o n e class of processes which we consider especially significant—understanding processes. W h a t we refer to h e r e as u n d e r s t a n d i n g processes are w h a t Stoehr (1967) calls

Protocol Analysis

63

"thinking," and V a n Nostrand (1976) describes as a " l e a r n i n g process." E v i d e n c e that u n d e r s t a n d i n g p r o c e s s e s a r e a t w o r k m a y b e s e e n i n c h a n g e s i n t h e w r i t e r ' s understanding of the topic as writing p r o c e e d s . In lines 1 through 3, the writer identifies motivation w i t h grades. In lines 6 and 7, she introduces the idea of p e r s o n a l s a t i s f a c t i o n b u t i s n o t a t all s u r e t h a t t h i s i s a s o u r c e o f s a t i s f a c t i o n d i s t i n c t from g r a d e s . In lines 8 t h r o u g h 1 2 , p e r s o n a l satisfaction is g i v e n a v e r y important role, b u t it is not clear h o w the writer relates it to grades. In lines 13 through 15, two sources of personal satisfaction are identified: grades a n d the production of highquality work. A relation b e t w e e n the t w o sources of satisfaction is specified in lines 15 and 16 and again in lines 21 through 23. T h e relation is that if the w o r k is of good q u a l i t y , a g r a d e i s n o t n e c e s s a r y for s a t i s f a c t i o n . I n l i n e s 2 4 a n d 2 5 , a c o n t r a d i c t o r y r e l a t i o n i s n o t e d — t h a t i s , t h a t g r a d e s are t h e i n i t i a l m o t i v a t o r for all e s s a y w r i t i n g . I n l i n e s 2 6 a n d 2 7 , t h e s e r e l a t i o n s are d e s c r i b e d a s " i n t e r l o c k e d " r a t h e r t h a n a s contradictory. T h e contradiction b e t w e e n the relations is r e c o g n i z e d later in the p r o t o c o l . I n l a r g e m e a s u r e , t h e final e s s a y i s c o n c e r n e d w i t h t h e r e s o l u t i o n o f t h i s conflict.

T h e C o m p l e t e d Essay Personal satisfaction is the major motivating force in the writing of my papers and reports. T h e emphasis on 4.0's here at C a r n e g i e - M e l l o n University causes grades to b e c o m e an instinctive motivator for myself. Acquiring good grades does, in fact, g i v e me personal satisfaction. T h e initial motivator in the outset of writing a paper is the fact that a grade will be attached to it upon completion. I feel that my role as a student requires all of my efforts to be put forth into course work, w h i c h includes the writing of papers. After I b e g i n writing a paper, the grade emphasis diminishes and a higher level of personal satisfaction takes over. W h e n I feel that I've written a high-quality or professional paper to be submitted for grading, it is not mandatory for the teacher to have the same opinion. But of course, this somewhat contradicts my earlier statement that the motivation for writing paper is to achieve the ultimate g o a l — a good grade. T h u s , the combination of the grade "initiator" and later a higher l e v e l of personal satisfaction is what motivates me to write my college papers and reports.

In the course of writing the essay, the subject's understanding of the topic c h a n g e d r a d i c a l l y . A t first, s h e b e l i e v e d h e r m o t i v a t i o n h a d a s i n g l e s o u r c e . B y t h e end of the e s s a y s h e r e c o g n i z e d that it h a d t w o s o u r c e s w h i c h w e r e related in a complex w a y . W r i t i n g in this case clearly i n v o l v e d u n d e r s t a n d i n g processes. P R O T O C O L A N A L Y S I S , M O R E GENERALLY C O N S I D E R E D As we h a v e seen, protocol analysis can be u s e d as an aid in understanding a w i d e variety of tasks from s i m p l e p r o b l e m s o l v i n g by a p e s to c o m p l e x performances such as chess p l a y i n g a n d w r i t i n g by h u m a n s . T y p i c a l l y t h o u g h , protocols are incomplete. M a n y p r o c e s s e s o c c u r d u r i n g the p e r f o r m a n c e of a task w h i c h the subject can't or d o e s n ' t report. T h e psychologist's task in a n a l y z i n g a protocol is to take t h e i n c o m p l e t e r e c o r d p r o v i d e d b y t h e p r o t o c o l , t o g e t h e r w i t h h i s k n o w l e d g e of human capabilities, a n d to infer from t h e s e a m o d e l of the u n d e r l y i n g p s y c h o l o g ical p r o c e s s e s b y w h i c h t h e s u b j e c t p e r f o r m s t h e task. A n a l y z i n g a protocol is l i k e f o l l o w i n g t h e tracks of a p o r p o i s e . O c c a s i o n a l l y , t h e p o r p o i s e r e v e a l s i t s e l f b y b r e a k i n g t h e s u r f a c e o f t h e s e a . Its b r i e f s u r f a c i n g s a r e l i k e t h e g l i m p s e s o f t h e u n d e r l y i n g m e n t a l p r o c e s s w h i c h t h e p r o t o c o l affords u s . B e -

64

T h e C o m p l e t e Problem Solver

t w e e n surfacings, the mental process, like the porpoise, runs d e e p and silent. O u r task is to infer the c o u r s e of the p r o c e s s from t h e s e b r i e f traces. S o m e h a v e s h o w n c o n c e r n that thinking-aloud protocols m a y distort or interfere with the mental processes w h i c h we want to observe. Sometimes subjects c o m p l a i n that it is hard to think a b o u t d o i n g the task a n d to talk at the same time. W h i l e t h e s e difficulties typically disappear with a little practice, t h e y force us to take the questions of distortion and interference seriously. Fortunately, Ericsson and Simon (1979) h a v e p r o v i d e d a thorough and thoughtful r e v i e w of the e v i d e n c e o n this issue. T h e y h a v e f o u n d that: J.

t h e r e i s n o e v i d e n c e t h a t t h i n k i n g - a l o u d p r o t o c o l s c o l l e c t e d u n d e r t h e sorts o f conditions w e d e s c r i b e d a b o v e distort o r interfere w i t h subjects' thinking w h e n they are performing tasks; and 2. t h i n k i n g - a l o u d protocols v e r y c l o s e l y reflect the t h o u g h t processes that o c c u r d u r i n g task p e r f o r m a n c e . H o w e v e r , t h e y a l s o c o n c l u d e t h a t , u n d e r special circumstances, t h i n k i n g - a l o u d c a n interfere with the subjects' thought processes. In the usual thinking-aloud procedure, the e x p e r i m e n t e r asks the subjects to say w h a t e v e r is on their m i n d s . If instead of t h e usual p r o c e d u r e , t h e e x p e r i m e n t e r tells t h e subjects w h a t to talk about, they m a y be forced to p a y attention to things t h e y w o u l d ordinarily h a v e i g n o r e d . S u p p o s e , for e x a m p l e , t h a t w e g a v e s u b j e c t s a n u m b e r p r o b l e m a n d a s k e d t h e m t o t e l l u s e v e r y t i m e t h e y t h o u g h t o f a n o d d n u m b e r w h i l e s o l v i n g it. O r d i n a r i l y , t h e y m i g h t p a y n o a t t e n t i o n a t all t o w h e t h e r a n u m b e r w a s o d d o r e v e n i n solving the p r o b l e m . With our instructions, not only must they pay special attention t o o d d n u m b e r s , b u t t h e y m a y a l s o d e v e l o p t h e s n e a k i n g s u s p i c i o n that o d d n u m b e r s are i m p o r t a n t for t h e s o l u t i o n w h e t h e r t h e y a r e o r n o t . We c o n c l u d e that as l o n g as we avoid telling our subjects w h a t to think about,

65

Protocol Analysis

thinking-aloud protocols can p r o v i d e us w i t h a fairly clear, u n d i s t o r t e d w i n d o w into human thinking processes. PRACTICAL

APPLICATIONS

A l t h o u g h we h a v e d e s c r i b e d protocol analysis as a research tool, it is important for y o u t o r e a l i z e t h a t y o u c a n a d a p t i t for y o u r p e r s o n a l u s e . I m a g i n e t h a t y o u a r e having trouble with a particular type of p r o b l e m either in school or at work. It might b e a w r i t i n g p r o b l e m o r a p r o b l e m i n , s a y , a c c o u n t i n g o r l a w . Y o u h a v e r e a d all t h e available instruction manuals, and you have asked p e o p l e to h e l p you. Unfortun a t e l y , t h e i r a d v i c e j u s t c o n f u s e s y o u . T h e y s a y t h i n g s l i k e , " O h , s u r e . It's e a s y . Y o u j u s t r e a r t i c u l a t e t h e f r a m e s , a d j u s t for m a r g i n a l l i q u i d i t y a t c o m p o u n d i n t e r e s t , a n d stir u n t i l s l i g h t l y t h i c k e n e d . " H o w can you get out of your d i l e m m a ? O n e v e r y effective thing y o u can do is to watch an expert solve one of these problems. An ideally cooperative expert w o u l d let you tape record a thinking-aloud protocol. H o w e v e r , y o u may have to satisfy y o u r s e l f j u s t w a t c h i n g h i m o r h e r s c r i b b l i n g o u t t h e s u c c e s s i v e p a r t s o f t h e solution. If you can, of course, you should take notes so you can r e m e m b e r the s e q u e n c e o f e v e n t s . P e r h a p s all y o u w i l l b e a b l e t o d o i s t o w r i t e a s e q u e n c e o f numbers to indicate the order in w h i c h notes w e r e scribbled on the expert's scratch pad. In any case, just w a t c h i n g another person solving a p r o b l e m can be remarkably h e l p f u l . I n fact, W h i m b e y a n d L o c h h e a d ( 1 9 8 0 ) r e c o m m e n d t h a t t h i n k i n g a l o u d b e u s e d as a routine classroom exercise to teach p r o b l e m - s o l v i n g skills. In the next section, we p r e s e n t t w o rather difficult p r o b l e m s together with a detailed description of h o w some problem solvers successfully solved them. T r y to solve each p r o b l e m before reading the solution. If y o u m a n a g e to solve the probl e m , c o m p a r e y o u r s o l u t i o n t o t h e o n e i n t h e t e x t . I f y o u c a n ' t s o l v e it, try t o understand h o w the p r o b l e m solver w h o s e solution process is d e s c r i b e d in the text a v o i d e d or o v e r c a m e the difficulties that y o u e x p e r i e n c e d . T h e Handshaking P r o b l e m * A social psychologist was interested in the custom of handshaking. He noticed that some p e o p l e are more inclined than others to shake hands w h e n they are introduced. O n e e v e n i n g w h e n he and his wife had j o i n e d four other couples at a party, he took advantage of the occasion to collect data. He asked each of the other nine p e o p l e at the party h o w many p e o p l e they had shaken hands with during the introductions. He r e c e i v e d a different answer, from zero through eight, from each of the nine people. You can assume that husbands and w i v e s don't shake hands with each other during introductions, and, of course, p e o p l e don't shake hands with themselves. G i v e n this information, find out h o w often the psychologist's wife shook hands.

M a n y p e o p l e find i t hard t o b e l i e v e that this p r o b l e m can b e s o l v e d w i t h o n l y t h e little i n f o r m a t i o n g i v e n , b u t i t c a n . T r y it. W h e n I first e n c o u n t e r e d t h i s p r o b l e m , I h a d n o i d e a h o w I w a s g o i n g t o s o l v e it. D e s p i t e t h i s , I d e c i d e d t o j u m p i n b y t r y i n g t o r e p r e s e n t w h a t I k n e w i n a systematic w a y . I c h o s e t h e matrix s h o w n in F i g u r e 6 to r e p r e s e n t t h e facts. Next, I d e c i d e d arbitrarily (gap-filling decision) to r e p r e s e n t the person w h o shook h a n d s e i g h t times as H I . E q u a l l y arbitrarily, I a s s u m e d that that p e r s o n w a s t h e h u s b a n d i n c o u p l e 1 . M y o n l y j u s t i f i c a t i o n for m a k i n g t h i s p a r t i c u l a r d e c i s i o n w a s that I h a d to m a k e some d e c i s i o n in o r d e r to p r o c e e d . I e n t e r e d X ' s in t h e first r o w a n d first c o l u m n i n F i g u r e 7 for e a c h p e r s o n H I s h o o k h a n d s w i t h . T h e z e r o s indicate that H I shook h a n d s n e i t h e r w i t h h i m s e l f nor w i t h his w i f e . A t this p o i n t , j u m p i n g i n y i e l d e d t h e first i m p o r t a n t i n s i g h t i n t o t h e p r o b l e m . * T h i s p r o b l e m was d e s c r i b e d to me by R. Bhaskar.

66

The C o m p l e t e Problem Solver

Since e v e r y o n e but Hi's wife shook hands at least once, she is the only person w h o c o u l d h a v e s h a k e n h a n d s z e r o t i m e s . S o I f i l l e d i n z e r o s for W l r o w a n d c o l u m n . N e x t I arbitrarily c h o s e H2 to be the person w h o shook hands s e v e n times. I inserted zeros in the table to indicate that H2 shook h a n d s n e i t h e r with h i m s e l f nor his wife, and I put X's in H2's row and c o l u m n to indicate that he shook hands with t h e r e m a i n i n g six p e o p l e as s h o w n in F i g u r e 8. N o w , jumping in led me to a second discovery. E v e r y o n e but H2's wife had shaken hands twice. So she was the only person w h o could h a v e shaken hands just o n c e . T h i s d i s c o v e r y s u g g e s t e d a p a t t e r n . I n t h e first t w o c o u p l e s , e i g h t w a s p a i r e d w i t h z e r o a n d s e v e n w i t h o n e . I h y p o t h e s i z e d t h a t i n t h e r e m a i n i n g c o u p l e s , six w o u l d b e p a i r e d w i t h t w o , f i v e w i t h t h r e e , a n d f o u r w i t h four. W o r k i n g o u t t h e details ( e x e c u t i n g the solution) p r o v e d that the h y p o t h e s i s w a s correct. It w a s o n l y at this p o i n t that I h a d the critical fact that I n e e d e d to s o l v e the Couple 1

Couple 1

Couple 2

Couple 3

Couple 4

Couple 5

Couple 2

Couple 3

Couple 4

Couple 5

HI

Wl

H2

W2

H3

W3

H4

W4

H5

W5

HI

0

0

X

X

X

X

X

X

X

X

Wl

0

H2

X

W2

X

H3

X

W3

X

H4

X

W4

X

H5

X

W5

X ' Figure 7. S t e p O n e in Solving t h e Handshaking Problem

Protocol Analysis

Couple 1

Couple 2

Couple 3

Couple 4

67

Couple 1

Couple 2

Couple 3

Couple 4

Couple 5

HI

Wl

H2

W2

H3

W3

H4

W4

H5

W5

HI

0

0

X

X

X

X

X

X

X

X

Wl

0

0

0

0

0

0

0

0

0

0

H2

X

0

0

0

X

X

X

X

X

X

W2

X

0

0

H3

X

0

X

W3

X

0

X

H4

X

0

X

W4

X

0

X

H5

X

0

X

W5

X

0

X

Figure 8. P r o c e e d i n g to Fill in the Matrix to Solve t h e Handshaking Problem

p r o b l e m — t h e f a c t t h a t f o u r w a s p a i r e d w i t h four. T h e s o c i a l p s y c h o l o g i s t a s k e d t h e other p e o p l e h o w often t h e y h a d s h a k e n h a n d s , and f o u n d only o n e p e r s o n w h o h a d shaken hands four times. H e , himself, must be the other person w h o shook hands four t i m e s . W e c a n c o n c l u d e t h e n t h a t h i s w i f e s h o o k h a n d s f o u r t i m e s . This p r o b l e m demonstrates as w e l l as any I k n o w the v a l u e of j u m p i n g into a p r o b l e m a n d m a k i n g g a p - f i l l i n g d e c i s i o n s i n s o l v i n g w h a t for m e a t l e a s t w a s a . very ill-defined problem. T h e Christmas T r e e P r o b l e m * Arrange 10 Christmas trees in five straight rows of four trees each. Again, try to solve the problem before reading further. W e will d e s c r i b e h o w t w o subjects s o l v e d this p r o b l e m i n similar b u t slightly different w a y s . P r o b l e m solver # 1 a p p r o a c h e d the p r o b l e m cautiously. First, she asked the e x p e r i m e n t e r if it w a s important that the trees w e r e C h r i s t m a s trees rather t h a n oaks o r m a p l e s . T h e e x p e r i m e n t e r a s s u r e d h e r t h a t t h e t y p e o f t r e e w a s n o t r e l e v a n t . N e x t , s h e w a s c o n c e r n e d t h a t t h e p r o b l e m m i g h t i n v o l v e s o m e sort o f t r i c k in representation, e.g., that t h e n u m b e r s be r e p r e s e n t e d as R o m a n n u m e r a l s , so that "five rows of four" might be five lines making up the R o m a n numeral IV as b e l o w :

IV A g a i n , the e x p e r i m e n t e r r u l e d out this possibility. Subject # 2 , less cautiously, s i m p l y a s s u m e d that the p r o b l e m i n v o l v e d no such c a t c h e s . F o r t u n a t e l y for h i m , i t d i d n ' t . A t this s t a g e i n p r o b l e m s o l v i n g , b o t h s u b j e c t s r e c o g n i z e d t h a t a r r a n g i n g t e n trees in five r o w s of four r e q u i r i n g p u t t i n g s o m e trees in m o r e than o n e row. To accomplish this, b o t h settled d o w n to arranging and rearranging dots on p a p e r (see below). * T h i s problem was d e s c r i b e d t o m e b y C . Berkenkotter.

The C o m p l e t e Problem Solver

68

• •• • • • • • ••

• • • • • •

•• • • • • • • •

B o t h s u b j e c t s s t a r t e d b y c o n s i d e r i n g r e c t a n g u l a r a r r a y s first a n d t h e n p r o c e e d e d to experiment with triangles. W h e n these early attempts failed to yield a quick solution, both subjects attempted to change the representation of the problem by r e d e f i n i n g " r o w . " B o t h a r g u e d ( i n d e p e n d e n t l y ) that a r o w of five c o u l d be v i e w e d as t w o rows of four as b e l o w : Row 1 x

After s e v e r a l u n s u c c e s s f u l tries, S u b j e c t # 1 a b a n d o n e d this approach, b u t S u b ject #2 persisted and proposed the following solution.

T h e e x p e r i m e n t e r a s k e d S u b j e c t # 2 t o s e a r c h for a m o r e e l e g a n t s o l u t i o n . B o t h s u b j e c t s t h e n r e t u r n e d t o t h e first p r o b l e m r e p r e s e n t a t i o n — t h e o n e i n v o l v i n g non-overlapping rows. After considerable experimentation, Subject #1 prod u c e d this solution.

S u b j e c t # 2 r e e x a m i n e d t h e p r o b l e m s t a t e m e n t . W h i l e p r e v i o u s l y h e h a d att e n d e d m o s t s p e c i f i c a l l y to " f i v e rows of four," n o w he s h i f t e d h i s a t t e n t i o n to "five r o w s o f f o u r . " T h i s c h a n g e i n e m p h a s i s l e d h i m t o s e a r c h for i n t e r e s t i n g f i g u r e s w i t h f i v e p a r t s . T h i s s e a r c h s o o n t u r n e d u p t h e star w h i c h s u g g e s t e d S u b j e c t * 2 ' s solution.

\

FOUR.

THE STRUCTURE O F H U M A N MEMORY

E a c h , o f u s h a s s e v e r a l m e m o r i e s w h i c h h o l d d i f f e r e n t k i n d s o f m a t e r i a l for different lengths of time. F o r e x a m p l e , we h a v e a sensory m e m o r y w h i c h can hold a n i m a g e o f 1 0 t o 1 2 l e t t e r s b u t only for a q u a r t e r o f a s e c o n d . W e h a v e a s h o r t - t e r m m e m o r y w h i c h w i l l h o l d a b o u t s e v e n n u m b e r s o r l e t t e r s for r o u g h l y 1 5 s e c o n d s , a n d we have a long-term m e m o r y w h i c h will hold great quantities of information—such as our k n o w l e d g e of a l a n g u a g e — f o r many, m a n y years. In this book, we will be c o n c e r n e d p r i m a r i l y w i t h t h e s e last t w o m e m o r i e s — t h e short-term memory, o r STM, a n d t h e long-term memory, or LTM. — THE RELATIONSHIP BETWEEN STM A N D LTM F i g u r e 1 p r o v i d e s u s w i t h a f r a m e w o r k for u n d e r s t a n d i n g h u m a n m e m o r y . M e s s a g e s , r e c e i v e d m o s t l y b y e y e o r e a r , a r e s t o r e d for v e r y s h o r t p e r i o d s o f t i m e REHEARSAL

MEMORY

INFORMATION

1

INCOMING

TION

SENSORY

VERY

< SHORT-

SHORTTERM

\

\

V

LONG-

i

/

MEMORY

TERM MEMORY

TERM

Figure 1. A F r a m e w o r k for U n d e r s t a n d i n g H u m a n M e m o r y i n s e n s o r y s t o r e s . I f w e a t t e n d t o t h e m , t h e m e s s a g e s w i l l b e t r a n s f e r r e d t o shortt e r m m e m o r y . O t h e r w i s e t h e y a r e lost. A t a p a r t y w e m a y h e a r s e v e r a l c o n v e r s a t i o n s g o i n g o n a t t h e s a m e t i m e , all o f w h i c h a r e b e i n g d u m p e d i n t o o u r s e n s o r y store. Usually, we pay attention to just one conversation, a n d are conscious of the others

72

The C o m p l e t e Problem Solver

only as a b a c k g r o u n d b u z z . T h e y n e v e r enter our short-term m e m o r y and we h a v e n o i d e a , e i t h e r a t t h e t i m e o r later, w h a t t h e y a r e a b o u t . O n c e the m e s s a g e has e n t e r e d short-term m e m o r y , w e can k e e p i t there b y r e p e a t i n g i t t o o u r s e l v e s — t h a t is, b y r e h e a r s a l . O t h e r w i s e i t w i l l b e l o s t w i t h i n a f e w seconds. Later in this chapter, we will discuss h o w m e s s a g e s are transferred to long-term m e m o r y by a process called elaborative rehearsal. SHORT-TERM

MEMORY

A g o o d w a y to understand the properties of short-term m e m o r y is to test s o m e o n e ' s m e m o r y s p a n . T a b l e 1 p r o v i d e s lists o f r a n d o m n u m b e r s y o u c a n u s e for this p u r p o s e . T a b l e 1 . R a n d o m N u m b e r s for M e m o r y Span Test Length 4 5 6 7 8 9 10 11 12

Series 1

Series 2

Series 3

08 24 4 1 6 2 3 9 0 7 8 96 3 0 7 12 8 3 5 4 3 7 5 9 8 0 42 1 6 9 2 108 5 4 6 7 4 0 2 1 9 3 8 5 7 2 9 6 1 5 3 5 8 2 7 6 0 8 1 4 9 3 6 04

12 7 9 1 4 6 0 3 7 2 3 4 95 8 4 9 2 6 0 7 6 9 7 8 3 5 1 7 53 1 8 092 92 1 5 6 8 2 4 0 5 1 2 6 8 3 0 9 7 4 6 3 5 7 0 1 8 2 4 9 3 2 0 7 6 3

8 2 6 9 14 508 2 4 1 2 7 9 6 4 1 3 2 6 1 0 6 2 9 4 8 3 8 9 0 8 13 5 8 0 1 4 2 8 6 5 1 3 9 1 2 4 7 9 0 3 8 6 8 5 3 5 3 94 263 94 1 806

First, find a w i l l i n g subject—ideally a friend w h o really doesn't care h o w the test c o m e s out. D o n ' t test c h i l d r e n o r p e o p l e w h o m i g h t b e c o n c e r n e d i f t h e y d i d n ' t d o w e l l . N e x t , e x p l a i n t o t h e s u b j e c t t h a t y o u a r e g o i n g t o r e a d lists o f n u m b e r s t o h i m a n d t h a t a s s o o n a s y o u f i n i s h r e a d i n g , h e s h o u l d try t o s a y t h e l i s t j u s t a s h e h e a r d i t — t h e s a m e n u m b e r s i n t h e s a m e o r d e r . K e e p y o u r w a t c h i n front o f y o u a n d read the n u m b e r s s t e a d i l y a n d w i t h o u t e m p h a s i s , a t t h e rate o f o n e n u m b e r p e r s e c o n d . S t a r t w i t h t h e s h o r t e s t list a n d say, " N o w I ' m g o i n g t o r e a d y o u f o u r n u m b e r s : 2 . . . 7 . . . 1 . . . 4 . " Y o u r s u b j e c t s h o u l d t h e n try t o r e p e a t t h e list. N e x t p r o c e e d to five n u m b e r s , six n u m b e r s , a n d so on. If t h e s u b j e c t has t r o u b l e w i t h a l i s t o f s e v e n o r e i g h t n u m b e r s , d o n ' t f o r c e h i m t o try all t h e lists u p t o 1 2 . T h a t w i l l only p r o d u c e m a s s i v e frustration. Stop w h e n the subject has m i s s e d t w o o r three lists in a r o w . M e m o r y s p a n s t u d i e s u s u a l l y r e q u i r e the s u b j e c t t o w o r k t h r o u g h several sets o f lists l i k e t h o s e i n T a b l e 1 . T y p i c a l l y , t h e s u b j e c t g e t s t h e v e r y s h o r t lists r i g h t 1 0 0 p e r c e n t o f t h e t i m e a n d t h e v e r y l o n g lists z e r o p e r c e n t o f t h e t i m e . T h e s u b j e c t ' s m e m o r y span is the l e n g t h of the list that he can r e p e a t correctly 50 p e r c e n t of the t i m e . A t y p i c a l m e m o r y s p a n for a d u l t s i s b e t w e e n s e v e n a n d e i g h t n u m b e r s . I f w e p e r f o r m e d the m e m o r y - s p a n test u s i n g letters, w e w o u l d get a b o u t the s a m e results a s for n u m b e r s . T h e a v e r a g e m e m o r y s p a n for l e t t e r s i s a b o u t 7 . 2 . Memory "Chunks" T h e m e m o r y s p a n for f a m i l i a r w o r d s i s a b i t l e s s t h a n t h a t f o r l e t t e r s — 5 . 8 6 . H o w e v e r , this span i s surprisingly large w h e n w e c o n s i d e r h o w m a n y letters are c o n t a i n e d i n 5.86 w o r d s . A n a v e r a g e w o r d has a b o u t five letters. W h e n p e o p l e r e m e m b e r f i v e o r six w o r d s , t h e y a r e r e m e m b e r i n g 2 5 o r 3 0 l e t t e r s — a n u m b e r far b e y o n d t h e i r m e m o r y s p a n for l e t t e r s .

73

The Structure of H u m a n M e m o r y

W h y can p e o p l e r e m e m b e r so m a n y more letters w h e n t h e y are c o m b i n e d in w o r d s t h a n w h e n t h e y a r e p r e s e n t e d s e p a r a t e l y ? T h e r e a s o n i s t h a t w e s t o r e inform a t i o n in " c h u n k s . " A c h u n k is a p a c k a g e of i n f o r m a t i o n t h a t is t r e a t e d as a u n i t . L e t t e r s a r e s o m e t i m e s t r e a t e d a s u n i t s a n d s o m e t i m e s n o t . W h e n l e t t e r s are p r e s e n t e d in an u n r e l a t e d w a y as in the l i s t X Z L P T R , e a c h letter acts as an i n d e p e n d e n t u n i t . T h u s t h i s l i s t w o u l d b e r e m e m b e r e d a s t h e six c h u n k s " X , " " Z , " " L , " " P , " " T , " and " R . " W h e n letters are c o m b i n e d into familiar w o r d s , as in the list PEACH

MAP

RIVER

FLIGHT

they don't function as individual units. T h e s e 19 letters function as four units and would be remembered as the chunks " p e a c h , " " m a p , " "river," and "flight." Such observations l e d G e o r g e M i l l e r (1956) to c o n c l u d e that the capacity of short-term m e m o r y i s 7 ± 2 chunks, n o t 7 ± 2 letters. W h e n s e v e r a l l e t t e r s a r e c o m b i n e d i n e a c h chunk, we can h o l d m a n y m o r e letters in short-term m e m o r y than w h e n there is only o n e letter per chunk. W h e r e D o C h u n k s C o m e From? A person's ability to r e m e m b e r in chunks d e p e n d s on that person's k n o w l e d g e . I f y o u w a n t e d t o r e m e m b e r a l i s t c o n t a i n i n g t h e S w a h i l i w o r d for l e m o n , ndimu, y o u w o u l d p r o b a b l y h a v e t o r e m e m b e r t h r e e c h u n k s — e n , dee, a n d moo—rather t h a n t h e s i n g l e c h u n k y o u w o u l d n e e d for lemon. T h e s y l l a b l e s en, dee, a n d moo are f a m i l i a r a s u n i t s o f p r o n u n c i a t i o n i n E n g l i s h ; t h e r e f o r e , w e c a n r e m e m b e r t h e m a s c h u n k s . T h e c o m b i n a t i o n en-dee-moo, h o w e v e r , i s n o t a f a m i l i a r u n i t for E n g l i s h s p e a k e r s , s o w e c a n ' t r e m e m b e r i t a s a c h u n k . Ndimu i s a c h u n k for S w a h i l i s p e a k e r s b e c a u s e it is as f a m i l i a r to t h e m as lemon or parsnip is to u s . E a c h character in a C h i n e s e ideograph (Figure 2) can be r e m e m b e r e d as a c h u n k to those fluent in C h i n e s e . F o r those of us w h o are not c a p a b l e of reading C h i n e s e , each character appears to be c o m p l e x e s of four to 10 lines and squiggles, each of w h i c h must be r e m e m b e r e d as a chunk. C h u n k s c o m e in m a n y shapes and sizes. W h e n e v e r we recognize a pattern in

Figure 2 . Three C h i n e s e Ideographs: " A Flight o f Butterflies"

The C o m p l e t e Problem Solver

74

information, we can break the information d o w n into c h u n k s and r e m e m b e r it m o r e easily. F o r e x a m p l e , familiar sentences can be chunks. T h e list of words in T a b l e 2 A i s h a r d t o r e m e m b e r , b u t t h e s a m e list i s e a s y t o m e m o r i z e w h e n i t i s r e o r g a n i z e d into maxims, as in T a b l e 2 B .

T a b l e 2. M a x i m s as C h u n k s A.

Words in bizarre sequence:

F o r I s e e t h e r e ' s o n e o l d b a s k e t l i k e all o f m y r e g r e t i n y o u r c o u n t r y e y e s d o n ' t fool a n I s h o o t o n e b u t t h e fool t h a t y o u h a v e p u t t o l i f e till t h e i r e g g s d o n ' t g i v e no whites. B.

T h e same words in familiar maxims:

I r e g r e t t h a t I h a v e b u t o n e l i f e t o g i v e for m y c o u n t r y . D o n ' t p u t all y o u r e g g s i n o n e b a s k e t . T h e r e ' s no fool like an old fool. D o n ' t s h o o t till y o u s e e t h e w h i t e s o f their e y e s .

G r o u p s o f letters can b e chunks e v e n i f they don't form words. D D T , I B M , R S V P , a n d U S S R are e x a m p l e s of familiar letter clusters that we treat as chunks. F a m i l i a r dates s u c h as 1066 and 1492 are e x a m p l e s of n u m b e r chunks. T e l e p h o n e n u m b e r s t h a t a r e e a s y t o c h u n k , s u c h a s 7 0 7 - 1 7 7 6 , a r e e a s y t o r e m e m b e r for t h a t reason. Recoding S o m e t i m e s m e s s a g e s can be r e c o d e d so that we can g e t m o r e information into each chunk. Smith (reported in Miller, 1956) has s h o w n a dramatic effect of recodi n g o n m e m o r y s p a n for b i n a r y d i g i t s . * S m i t h first d e t e r m i n e d t h a t h i s s u b j e c t , w h o w a s u n u s u a l l y g o o d at short-term m e m o r y tasks, c o u l d h o l d 15 b i n a r y digits in i m m e d i a t e m e m o r y . T h a t is, he c o u l d listen to a n d r e p e a t b a c k a list like this: 1-0-1-1-0-1-0-0-0-1-1-0-1-1-0 N e x t h e d r i l l e d his s u b j e c t t h o r o u g h l y i n several r e c o d i n g s c h e m e s . I n the 2 : 1 r e c o d i n g s c h e m e , t h e s u b j e c t first g r o u p e d t h e l i s t o f d i g i t s i n p a i r s a n d t h e n recoded each pair by the rule s h o w n in the box labeled " 2 : 1 R e c o d e r " in F i g u r e 3. T h e subject stored the names of the pairs rather than the original binary digits i n short-term m e m o r y . T h u s , h e h a d t o r e m e m b e r o n l y six pair n a m e s rather than 12 binary digits. W h e n asked to recall, the subject d e c o d e d the pair names and a t t e m p t e d to report the original list of b i n a r y digits. T h i s s e q u e n c e of e v e n t s is also s h o w n in F i g u r e 3. In the 3 : 1 r e c o d i n g s c h e m e , the subject g r o u p e d the list of digits into triplets a n d t h e n r e c o d e d e a c h triplet b y the rule i n the b o x l a b e l e d " 3 : 1 R e c o d e r " in F i g u r e 4. W i t h this s c h e m e , the subject stored the four n a m e s of the triplets—5-5-0-6—rather than the 12 binary digits. In addition, 4 : 1 and 5 : 1 recoding schemes w e r e also taught. W h e n t h e r e c o d i n g s c h e m e s h a d b e e n m a s t e r e d , t h e s u b j e c t ' s m e m o r y s p a n for binary digits w a s again tested. T h i s time the subject r e c o d e d the binaries as he listened to them and then d e c o d e d them again w h e n he repeated them. T h e result w a s t h a t all o f t h e r e c o d i n g s c h e m e s i m p r o v e d t h e m e m o r y s p a n for b i n a r i e s . W i t h 3 : 1 recoding, he r e m e m b e r e d about 33 binaries, and with 5 : 1 recoding, more than 40. * Binary digits are s e q u e n c e s of zeros and ones.

The Structure of H u m a n M e m o r y

75

R e c e n t l y , C h a s e a n d E r i c s s o n ( 1 9 7 9 ) r e p o r t e d m e m o r y - s p a n r e s u l t s for a s u b j e c t w h o u s e s a c o m p l e x s c h e m e for r e c o d i n g d e c i m a l d i g i t s a s t i m e s for r u n n i n g races. T h i s subject can r e m e m b e r up to 81 d e c i m a l digits. Searching for C h u n k s C h u n k i n g i s a n a c t i v i t y — i t i s s o m e t h i n g that w e d o — a l t h o u g h w e are often c o m p l e t e l y u n a w a r e t h a t w e a r e d o i n g it. C h u n k i n g i n v o l v e s n o t i c i n g , b u t s o m e t i m e s w e fail t o n o t i c e a p a t t e r n i n t h e m a t e r i a l w e a r e t r y i n g t o l e a r n . T h i s fact w a s demonstrated v e r y n i c e l y i n e x p e r i m e n t s b y B o w e r a n d Springston (1970). I n both e x p e r i m e n t s , s u b j e c t s s t u d i e d a n d i m m e d i a t e l y t r i e d t o r e c a l l lists o f l e t t e r s . T h e r e w e r e t w o sorts o f l i s t s — l i s t s l i k e T V I B M T W A U S S R — i n w h i c h t h e c h u n k s are e a s y t o f i n d — a n d lists l i k e I C B M F B I U S A O K — w h e r e t h e c h u n k s a r e a little h a r d e r t o f i n d . ( B e s u r e y o u d o f i n d t h e m o r y o u w i l l m i s s a n i m p o r t a n t point.) I n o n e s t u d y , t h e g r o u p s w e r e f o r m e d b y i n s e r t i n g p a u s e s i n s p o k e n lists. I n t h e o t h e r , t h e y w e r e f o r m e d b y c h a n g i n g t h e s i z e o r t h e c o l o r o f t h e l e t t e r s i n p r i n t e d lists. I n both of the e x p e r i m e n t s , the subjects r e m e m b e r e d m o r e letters from the easy-tofind lists t h a n f r o m t h e h a r d - t o - f i n d o n e s — a n d t h e d i f f e r e n c e s w e r e b i g — i n s o m e

The C o m p l e t e Problem Solver

76

Triplet Names DEcoded Original List Reported by Subject

1-0-1 1 - 0 - 1

Figure 4. 3:1 R e c o d i n g cases, m o r e than 4 0 p e r c e n t . T h e e x p e r i m e n t e r s w e r e a b l e t o s h o w that the subjects w e r e not m a k i n g u s e of t h e c h u n k s in the hard-to-find lists. If the subjects h a d m a d e u s e o f the c h u n k s that w e r e p r e s e n t i n the lists, t h e y w o u l d h a v e r e m e m b e r e d more. I t i s c l e a r that w e can m a k e l e a r n i n g a n d r e m e m b e r i n g easier b y s e a r c h i n g c o n s c i o u s l y for c h u n k s i n t h e m a t e r i a l w e a r e s t u d y i n g . Forgetting and Short-Term M e m o r y T h e r e are t w o w a y s w e can lose information from short-term m e m o r y : b y displacement and by decay. Displacement I m a g i n e y o u r s e l f as a n o v i c e waiter w h o b e l i e v e s that g o o d waiters k e e p orders i n s h o r t - t e r m m e m o r y . Y o u t a k e t h e o r d e r o f y o u r first c u s t o m e r : m u s s e l s , b o r s c h t , spaghetti, rice, asparagus, p i c k l e s , and cocoa. " V e r y g o o d , sir," y o u say, not really m e a n i n g it, b u t c o n f i d e n t t h a t t h e o r d e r i s s a f e l y s t o r e d i n s h o r t - t e r m m e m o r y . A s you walk away, the customer says, " A n d a side of zucchini, waiter!" N o w you're in trouble. Y o u r short-term m e m o r y holds s e v e n chunks and zucchini makes eight. Y o u r e m e m b e r the zucchini, b u t i t has p u s h e d s o m e t h i n g e l s e out o f S T M . Y o u h a v e

77

The Structure of Human M e m o r y

n o i d e a w h a t i t w a s . B u t y o u k n o w that a t s o m e p o i n t t h e c u s t o m e r i s g o i n g t o s a y , "Waiter, w h e r e is my ?" T h i s i s f o r g e t t i n g b y displacement. B e c a u s e s h o r t - t e r m m e m o r y h o l d s a l i m i t e d n u m b e r o f c h u n k s , p u t t i n g a n e w c h u n k i n w h e n i t i s full w i l l p u s h a n o l d c h u n k out. Decay I n f o r m a t i o n i s l o s t v e r y q u i c k l y f r o m S T M i f w e d o n ' t r e h e a r s e it. T h i s fact was demonstrated clearly in an e x p e r i m e n t by Peterson and Peterson (1959). T h e Petersons asked their subjects to r e m e m b e r just three letters—a triplet of cons o n a n t s s u c h a s " P T K " — f o r l e s s t h a n a m i n u t e . T h i s s e e m s l i k e a v e r y e a s y task. H o w e v e r , b e t w e e n h e a r i n g t h e letters a n d t h e test, t h e s u b j e c t h a d t o c o u n t backw a r d s b y t h r e e ' s a s fast a s p o s s i b l e . T h e i n t e r c h a n g e b e t w e e n t h e e x p e r i m e n t e r a n d the subject w e n t as follows: Experimenter: " R X B , 4 7 1 " Subject: " 4 7 1 , 468, 465, 462,

.

.

." [light s i g n a l i n g test] " R X B "

Counting backwards, if it is done rapidly, prevents rehearsal quite effectively. Y o u c a n c o n v i n c e y o u r s e l f o f t h i s b y t r y i n g it. T h e results of the P e t e r s o n s ' study are s h o w n in F i g u r e 5, and t h e y are surprisi n g . A f t e r s e v e n s e c o n d s , t h e s u b j e c t ' s c h a n c e s o f g e t t i n g t h e t r i p l e t r i g h t a r e 50/50, a n d after 1 8 s e c o n d s , o n l y o n e i n 1 0 . T h e c o n c l u s i o n i s o b v i o u s : E v e n a s i m p l e m e s s a g e in S T M will d e c a y in less than 20 seconds a n d be lost if it is not refreshed by rehearsal. Rehearsal W e c a n m a i n t a i n a m e s s a g e i n s h o r t - t e r m m e m o r y for q u i t e s o m e t i m e . A c o m m o n w a y to rehearse a m e s s a g e is to say it over and over to ourselves. S u p p o s e

Figure 5. Trigram Recall After V a r i o u s Intervals Filled W i t h B a c k w a r d Counting From The Psychology of Learning and Memory by D o u g l a s L. Hintzman, W. H. Freeman and C o m p a n y , copyright © 1 9 7 8 .

78

The C o m p l e t e Problem Solver

we w a n t to m a k e a p h o n e call in a p u b l i c place w h e r e the p h o n e b o o k is in one p l a c e a n d t h e p h o n e i s i n a n o t h e r . W e all k n o w w h a t t o d o : W e l o o k u p t h e n u m b e r a n d repeat it over and over w h i l e we race madly to the phone, avoiding people w h o m i g h t ask us a question that c o u l d w i p e the n u m b e r out of our heads. Surprisingly, y o u n g c h i l d r e n m a y not k n o w this rehearsal strategy. K e e n e y , C a n n i z z o , a n d F l a v e l l (1967) f o u n d that s o m e first-grade c h i l d r e n r e h e a r s e d a n d s o m e didn't. T h o s e w h o r e h e a r s e d d i d b e t t e r i n short-term m e m o r y tests than those w h o didn't. H o w e v e r , w h e n the non-rehearsers w e r e told t o r e h e a r s e , their performance i m p r o v e d to the level of the children w h o rehearsed on their own. W i t h uninterrupted rehearsal we can k e e p a p h o n e n u m b e r in short-term m e m o r y for a l o n g t i m e . J u s t h o w l o n g w o u l d a p p e a r t o d e p e n d o n t h e p e r s i s t e n c e w i t h w h i c h the person continues to rehearse. T h e r e is currently no G u i n n e s s World R e c o r d t i m e for h o l d i n g a t e l e p h o n e n u m b e r i n S T M , b u t p e r h a p s s o m e o n e d e s i r ing eternal fame will soon establish one. S u m m a r y : Short-term M e m o r y 1. A c h u n k is a p a c k a g e of information that we treat as a unit b e c a u s e we r e c o g n i z e it as a f a m i l i a r p a t t e r n . 2. Short-term m e m o r y can hold 7 ± 2 chunks. 3 . B y a c t i v e l y s e a r c h i n g for c h u n k s i n t h e m a t e r i a l w e w a n t t o l e a r n , w e c a n m a k e learning easier. 4 . B y c o n t i n u o u s r e h e a r s a l , w e c a n k e e p a m e s s a g e i n s h o r t - t e r m m e m o r y for l o n g periods of time. 5. Without rehearsal, information in S T M will d e c a y within 20 seconds. LONG-TERM

MEMORY

T h e r e a r e t w o d r a m a t i c w a y s i n w h i c h l o n g - t e r m m e m o r y d i f f e r s f r o m shortt e r m m e m o r y . F i r s t , w h i l e s h o r t - t e r m m e m o r i e s last for l e s s t h a n 2 0 s e c o n d s , l o n g t e r m m e m o r i e s c a n l a s t for a l i f e t i m e . A v e r y o l d m a n m a y b e a b l e t o r e c a l l w h e r e h e sat i n h i s f i r s t - g r a d e c l a s s r o o m a n d w h a t h i s t e a c h e r l o o k e d l i k e . S e c o n d , w h i l e short-term m e m o r y can h o l d only 7 ±2 chunks, the capacity of long-term m e m o r y is practically u n l i m i t e d . L o n g - t e r m m e m o r y contains our k n o w l e d g e of our l a n g u a g e and of the faces of the p e o p l e we know. It contains our k n o w l e d g e of h o w to find our w a y around the city, h o w to order in a restaurant, h o w to do algebra, a n d h o w to p e e l a banana. I c o u l d go on, b u t it s h o u l d be clear a l r e a d y that the a m o u n t of information in long-term m e m o r y is huge. T h e topic o f long-term m e m o r y can b e d i v i d e d quite naturally into three p h a s e s : first, a n encoding p h a s e i n w h i c h w e p u t i n f o r m a t i o n i n t o m e m o r y ; s e c o n d , a storage p h a s e ; a n d f i n a l l y , ^retrieval p h a s e i n w h i c h w e r e c o v e r t h e i n f o r m a t i o n w e s t o r e d ( s e e F i g u r e 6 ) . I t i s n e c e s s a r y t o u n d e r s t a n d all t h r e e o f t h e s e p h a s e s i f

ENCODING |

STORAGE

[

RETRIEVAL

Figure 6. The Three Phases of Long-term M e m o r y w e are t o u n d e r s t a n d h o w t o i m p r o v e long-term m e m o r y . W e w i l l discuss the p h a s e s i n r e v e r s e o r d e r , d e a l i n g first w i t h r e t r i e v a l , t h e n s t o r a g e , a n d l e a v i n g t h e m o s t c o m p l i c a t e d t o p i c — e n c o d i n g — f o r last. Retrieval E v e n t h o u g h w e h a v e s o m e fact, s u c h a s a n a m e o r d a t e , s t o r e d i n o u r m e m o r y , t h a t d o e s n ' t m e a n w e c a n g e t a t t h e i n f o r m a t i o n w h e n e v e r w e w a n t it. W e h a v e all

The Structure of Human M e m o r y

79

felt the frustration o f r e c o g n i z i n g p e o p l e a t parties b u t b e i n g c o m p l e t e l y u n a b l e t o d r e d g e u p t h e i r n a m e s — c e r t a i n l y n o t i n t i m e for i n t r o d u c t i o n s . O f t e n w e f e e l t h a t t h e n a m e i s " o n t h e t i p o f (our) t o n g u e . " W e m a y b e s u r e t h a t i t h a s t h r e e s y l l a b l e s and b e g i n s w i t h " B " and w e k n o w that i f s o m e o n e told u s the n a m e w e w o u l d recognize it immediately. R o g e r B r o w n a n d D a v i d M c N e i l l (1966) h a v e s t u d i e d the " t i p o f the t o n g u e " p h e n o m e n o n i n p e o p l e trying t o r e m e m b e r w o r d s . T h e y r e a d definitions o f infrequently u s e d words to subjects w h o w e r e asked to respond by writing d o w n the corresponding word. For example, they w o u l d read "a navigational instrument u s e d i n m e a s u r i n g a n g u l a r d i s t a n c e s , e s p e c i a l l y t h e a l t i t u d e o f s u n , m o o n , a n d stars at sea." S o m e of the subjects h a d no idea w h a t w o r d this defines, w h i l e s o m e others w e r e immediately able to write d o w n "sextant." T h e interesting subjects, though, w e r e the ones w h o c o u l d not i m m e d i a t e l y write d o w n the a n s w e r , b u t w h o felt that i t w a s j u s t " o n t h e t i p o f (their) t o n g u e . " T h e s e s u b j e c t s w e r e e n c o u r a g e d t o g i v e all t h e i n f o r m a t i o n t h e y c o u l d a b o u t t h e w o r d : W h a t a r e its first l e t t e r s ? H o w m a n y syllables does it h a v e ? W h a t does it r h y m e w i t h ? A b o u t half w e r e able to identify t h e first l e t t e r c o r r e c t l y , a n d a b o u t h a l f k n e w t h e n u m b e r o f s y l l a b l e s i n t h e w o r d . Many of these subjects did eventually r e m e m b e r the correct word. T h e "tip o f the t o n g u e " p h e n o m e n o n illustrates clearly that e v e n though w e have information in m e m o r y , we can't always get at it w h e n we want. Retrieval Tasks Psychologists h a v e i n v e n t e d a variety of w a y s to test m e m o r y . S o m e of these methods put the b u r d e n on the subjects to s h o w w h a t t h e y k n o w , w h i l e others reveal k n o w l e d g e that the subjects t h e m s e l v e s can't report and i n d e e d are u n a w a r e they possess. S u p p p o s e t h a t y o u a s k e d s u b j e c t s t o m e m o r i z e t h e f o l l o w i n g list: Derby Carp Jane Apple Church Trout Frank

Hut Beret Peach House Grape Hat Cod

Edna Orange Flounder George Skyscraper Fez

L e t ' s a s s u m e t h a t y o u r e a d t h i s list t o t h e m i n o r d e r , o n e w o r d a t a t i m e , a n d t h a t you read it through c o m p l e t e l y three times before you tested the subjects' memory. T h e r e are m a n y w a y s t h a t y o u c o u l d t e s t t h e s u b j e c t s ' k n o w l e d g e . W e w i l l d i s c u s s four m e t h o d s : u n c u e d r e c a l l , c u e d r e c a l l , r e c o g n i t i o n , a n d s a v i n g s . Uncued recall. T h e c r i t i c a l f e a t u r e o f u n c u e d r e c a l l t a s k s i s t h a t t h e s u b j e c t s a r e

The C o m p l e t e Problem Solver

80

required to r e p r o d u c e the material they have learned with no aids. T h e r e are t w o c o m m o n k i n d s of u n c u e d r e c a l l t a s k s : serial recall a n d free recall. I f y o u a s k t h e s u b j e c t s t o w r i t e d o w n t h e w o r d s i n the order i n w h i c h t h e y h e a r d t h e m , y o u will be testing by the m e t h o d of serial recall. If the subjects write C A R P - D E R B Y rather than D E R B Y - C A R P , they get t w o wrong. R e m e m b e r i n g the n u m b e r o f y o u r c o m b i n a t i o n l o c k i s a n e v e r y d a y e x a m p l e o f a s e r i a l r e c a l l task. Y o u h a v e t o r e m e m b e r t h e n u m b e r s and t h e i r o r d e r . I f y o u l e t t h e s u b j e c t w r i t e d o w n t h e w o r d s i n a n y o r d e r , t h e n y o u are t e s t i n g b y t h e m e t h o d o f f r e e r e c a l l . F r e e r e c a l l i s e a s i e r for t h e s u b j e c t s t h a n s e r i a l r e c a l l b e c a u s e t h e y can be scored correct e v e n though they don't r e m e m b e r the order of the words. B e i n g a s k e d to say w h o w a s at a party or m e e t i n g is a c o m m o n free recall task. T h e o r d e r i n w h i c h y o u m e n t i o n t h e p e o p l e d o e s n ' t m a t t e r . Cued recall. I f y o u g a v e t h e s u b j e c t s h i n t s w h i c h t h e y t r i e d t o r e c a l l ( R e m e m b e r a n y fish n a m e s ? D o e s a n y t h i n g r h y m e w i t h f i g ? ) , t h e n y o u w o u l d b e u s i n g t h e m e t h o d of c u e d recall. A c o m m o n e x a m p l e of a c u e d recall task is the party situation in w h i c h y o u h a v e to r e m e m b e r people's names w h e n you see them. T h e f a c e i s t h e c u e t o t h e n a m e w h i c h y o u m u s t try t o r e c a l l . A c u e d r e c a l l t a s k c a n b e m a d e harder or easier by c h a n g i n g the c u e s . T h e s e c o n d list of c u e s in T a b l e 3 w o u l d p r o b a b l y b e m o r e h e l p f u l t h a n t h e first.

T a b l e 3. C u e Lists General Category Cues living things food names clothing man-made structures

Specific Category C u e s fish fruit first n a m e s h e a d gear buildings

Rhyme Cues harp teach tank bat lurch

Categories can serve as very powerful retrieval cues. T u l v i n g and Pearlstone ( 1 9 6 6 ) d e m o n s t r a t e d t h i s i n a n e x p e r i m e n t i n w h i c h t h e s u b j e c t s l e a r n e d lists o f w o r d s w h i c h c o u l d b e g r o u p e d into categories. F o r e x a m p l e , the words " b o m b , " " t a b l e , " "carrot," " c a n n o n , " " p o t a t o , " and " c h a i r " can b e g r o u p e d into the categories " w e a p o n s , " "furniture," and " v e g e t a b l e s . " T w o g r o u p s o f s u b j e c t s s t u d i e d lists w h i c h c o n s i s t e d o f 4 8 w o r d s i n 1 2 c a t e g o ries. G r o u p 1 was a s k e d to recall as m a n y words as t h e y could. G r o u p 2 h a d the s a m e task b u t t h e y w e r e also g i v e n the n a m e s o f the categories t o h e l p t h e m . P e o p l e i n G r o u p 1 r e m e m b e r e d about 19 of the words, while people in G r o u p 2 remembered nearly 30. T h e p e o p l e in the t w o groups r e m e m b e r e d about the same n u m b e r of words in each category—2.6 out of 4 — b u t G r o u p 2 r e m e m b e r e d words in m a n y m o r e c a t e g o r i e s — 1 1 . 5 c a t e g o r i e s for G r o u p 2 , b u t o n l y 7 . 3 for G r o u p 1 . T h e a d v a n tage that h a v i n g the c a t e g o r y n a m e s g a v e to G r o u p 2, t h e n , w a s that it i n c r e a s e d the n u m b e r o f categories t h e y r e m e m b e r e d . N o t i c e t h a t i f G r o u p 1 r e m e m b e r e d a n y w o r d s in a category, t h e n t h e y r e m e m b e r e d as m a n y as G r o u p 2. T h i s suggests that r e m e m b e r i n g one w o r d in a category reminds the person of the w h o l e category, just as hearing the category n a m e does. We can imagine ourselves trying to recall the words, saying to ourselves, " C H . . . C H s o m e t h i n g — C H A I R ! Oh, yeah! And bed and table!" T h i s t e n d e n c y to r e m e m b e r things in clusters from the s a m e category w a s c l e a r l y d e m o n s t r a t e d b y B o u s f i e l d ( 1 9 5 3 ) . H e a s k e d s u b j e c t s t o s t u d y a list o f 6 0 w o r d s from the four categories: animals, n a m e s , professions, a n d v e g e t a b l e s . T h e c a t e g o r i e s w e r e s c r a m b l e d t o p r o d u c e lists l i k e t h i s : g i r a f f e , O t t o , b a k e r , t u r n i p ,

The Structure of Human M e m o r y

87

N o a h , c e l e r y , B e r n a r d , c a m e l , florist, e t c . T h e s u b j e c t s w e r e a s k e d t o r e c a l l a s m a n y w o r d s a s t h e y c o u l d i n a n y o r d e r that o c c u r r e d t o t h e m . I n t h e i r r e s p o n s e s , t h e s u b j e c t s w o u l d r e c a l l first a g r o u p o f i t e m s f r o m o n e c a t e g o r y , p a u s e , a n d t h e n r e c a l l a group of items from another category, etc. R e m e m b e r i n g an i t e m s e e m e d to c u e r e c a l l for o t h e r i t e m s i n t h e s a m e c a t e g o r y . Recognition. Y o u c o u l d a l s o t e s t y o u r s u b j e c t s ' m e m o r y for 2 0 w o r d s b y a s k i n g t h e m to p i c k t h e m out of a larger list w h i c h consists of the 20 old w o r d s m i x e d w i t h 2 0 n e w ones that w e r e n ' t o n the list t h e y studied. T h e s e n e w w o r d s are c a l l e d " f o i l s . " T h e subjects' task i s t o g o through the 4 0 w o r d s , m a r k i n g t h e m either " o l d " i f t h e y r e c o g n i z e t h e m a s p a r t o f t h e s t u d i e d list, o r " n e w " i f t h e y d o n o t . T h i s i s a r e c o g n i t i o n task, w h i c h i s g e n e r a l l y e a s i e r t h a n a r e c a l l task. T h e d i f f i c u l t y o f a r e c o g n i t i o n task v a r i e s w i t h t h e n a t u r e o f t h e f o i l s . W e w o u l d e x p e c t o u r s u b j e c t s t o d o b e t t e r i f t h e foils w e r e d i f f e r e n t f r o m t h o s e i t e m s o n t h e o r i g i n a l list. F o r e x a m p l e , i f t h e o l d l i s t w e r e m a d e u p o f n a m e s o f fruit, a n d t h e foils w e r e all I t a l i a n w o r d s , t h e r e c o g n i t i o n t a s k w o u l d b e s i m p l e . R e c o g n i t i o n t a s k s i n r e a l life c a n b e m o r e c o m p l i c a t e d : I f y o u g o t o t h e a i r p o r t to m e e t s o m e o n e y o u h a v e m e t only once before, you h a v e a difficult recognition task i n p i c k i n g t h a t p e r s o n o u t o f t h e c r o w d . I f y o u r l a u n d r y g e t s m i x e d w i t h s o m e o n e e l s e ' s a t t h e l a u n d r o m a t , y o u a l s o h a v e a r e c o g n i t i o n task. Savings. I f i n s t e a d o f t e s t i n g y o u r s u b j e c t s r i g h t after l e a r n i n g , y o u t e s t e d t h e m a w e e k or a m o n t h later, t h e y w o u l d p r o b a b l y g e t v e r y l o w scores on a n y of the tests w e h a v e d i s c u s s e d . T h e y m i g h t e v e n claim that t h e y r e m e m b e r e d a b s o l u t e l y nothing a b o u t t h e lists. E v e n so, b y u s i n g t h e savings m e t h o d y o u c o u l d p r o b a b l y s h o w that t h e y still r e t a i n e d s o m e t h i n g o f t h e i r e a r l i e r l e a r n i n g . W i t h t h e s a v i n g s m e t h o d , w e m e a s u r e how l o n g i t t a k e s t h e s u b j e c t t o r e l e a r n t h e o l d l i s t i n c o m p a r i s o n t o l e a r n i n g a c o m p a r a b l e n e w list. I f i t t a k e s t h e s u b j e c t s s e v e n r e p e t i t i o n s t o r e l e a r n t h e o l d list a n d 1 0 t o l e a r n t h e n e w , t h e n r e l e a r n i n g t i m e i s 7 0 p e r c e n t o f l e a r n i n g time, and the savings score is 30 percent. P e o p l e w h o h a v e t o r e p e a t a course are often surprised b y h o w m u c h easier the c o u r s e i s t h e s e c o n d t i m e a r o u n d . E v e n i f t h e y f l u n k e d t h e c o u r s e t h e first t i m e , b e i n g in the same Jearning situation reveals k n o w l e d g e t h e y didn't suspect they had. O n e study s h o w e d that a child m a y retain a good deal of information e v e n from very early e x p e r i e n c e s . W h e n his son w a s b e t w e e n 15 months and three years old, H. E. Burtt (1941) read passages to h i m from S o p h o c l e s in the original G r e e k . He r e p e a t e d e a c h p a s s a g e a b o u t 9 0 t i m e s . W h e n t h e c h i l d w a s 8V2, h e l e a r n e d b o t h o l d a n d n e w p a s s a g e s a n d s h o w e d a s a v i n g s s c o r e o f 2 7 p e r c e n t for t h e o l d . B y t h e t i m e the child w a s 14, the savings score d r o p p e d to e i g h t p e r c e n t , a n d at 18, to z e r o percent. T h e four r e c a l l tasks h a v e b e e n p r e s e n t e d h e r e i n i n c r e a s i n g o r d e r o f s e n s i t i v ity t o t h e s u b j e c t s ' k n o w l e d g e . F o r a g i v e n d e g r e e o f l e a r n i n g , w e w o u l d e x p e c t o u r subjects t o s c o r e w o r s t i n u n c u e d r e c a l l t a s k s , b e t t e r i n c u e d r e c a l l t a s k s , a n d still b e t t e r i n r e c o g n i t i o n t a s k s . T h e s a v i n g s m e t h o d i s t h e m o s t s e n s i t i v e o f all s i n c e i t m a y r e v e a l k n o w l e d g e e v e n w h e n t h e s u b j e c t s c o r e s z e r o o n all o f t h e o t h e r t a s k s . We w i l l m a k e u s e of the fact that s o m e retrieval tasks are easier than others w h e n we discuss m e m o r y techniques in the next chapter. S o m e m e m o r y techn i q u e s w o r k by c h a n g i n g a h a r d e r retrieval task into an e a s i e r o n e , e.g., by c h a n g i n g a n u n c u e d r e c a l l t a s k i n t o a c u e d r e c a l l task. Summary: Retrieval I . E v e n t h o u g h i n f o r m a t i o n i s s t o r e d i n o u r m e m o r y , w e c a n ' t a l w a y s r e t r i e v e it, a s the "tip of the t o n g u e " p h e n o m e n o n shows.

The C o m p l e t e Problem Solver

82 2.

Retrieval tasks vary in difficulty in the f o l l o w i n g order: U n c u e d recall

(most difficult)

C u e d recall Recognition Savings

(least difficult)

F o r a g i v e n d e g r e e of learning, we w o u l d get the b e s t score on a savings test and the w o r s t o n a n u n c u e d recall test. Storage S u p p o s e s o m e o n e g i v e s y o u a m e s s a g e that y o u w i l l h a v e to act on t o m o r r o w , such as: " W h e n y o u take the d o g to the cleaners, tell t h e m to u s e less starch." B e t w e e n t h e t i m e y o u e n c o d e t h e m e s s a g e a n d t h e t i m e y o u h a v e t o r e c a l l it, y o u w o u l d h o p e that the information w i l l remain stored, p e a c e f u l and undisturbed, in l o n g - t e r m m e m o r y . H o w e v e r , lots o f t h i n g s c a n h a p p e n t o a n u n s u s p e c t i n g m e m o r y t r a c e . I f y o u w e r e h i t o n t h e h e a d s h o r t l y after r e c e i v i n g t h e m e s s a g e , y o u m i g h t suffer f r o m r e t r o g r a d e a m n e s i a — a c o n d i t i o n i n w h i c h m e m o r i e s o f e v e n t s e x p e r i e n c e d j u s t b e f o r e t h e b l o w are lost. T h i s p h e n o m e n o n r e m i n d s u s dramatically that the m e m o r y trace has a p h y s i c a l basis w i t h i n our brains and that it can be disturbed by a gross p h y s i c a l e v e n t like a b l o w on the head. I f a n y t h i n g h a p p e n s t o o u r m e m o r y t r a c e , t h o u g h , i t i s far m o r e l i k e l y t o h a p p e n through interference than through retrograde amnesia. Interference occurs w h e n l e a r n i n g o n e t h i n g m a k e s i t d i f f i c u l t for u s t o r e m e m b e r s o m e t h i n g e l s e . P s y c h o l o g i s t s d i s t i n g u i s h b e t w e e n t w o k i n d s o f i n t e r f e r e n c e — p r o a c t i v e a n d retroactive. I n proactive interference, something learned earlier m a k e s it difficult to r e m e m b e r s o m e t h i n g l e a r n e d later. F o r e x a m p l e , i f last w e e k y o u h a d gotten t h e m e s s a g e , " W h e n y o u g e t the d o g from the cleaners, tell t h e m the starch w a s just right," this m e s s a g e m i g h t b e c o n f u s e d w i t h the current m e s s a g e a n d c a u s e proactive interference. T h e effect of proactive interference can be quite large. G r e e n b e r g and U n d e r w o o d ( 1 9 5 0 ) a s k e d p e o p l e t o l e a r n 2 0 lists o f w o r d p a i r s — o n e l i s t e v e r y t w o d a y s . F o r t h e first l i s t , t h e y f o u n d t h a t i n 2 4 h o u r s , s u b j e c t s f o r g o t a b o u t 2 0 p e r c e n t o f t h e p a i r s t h a t t h e y h a d l e a r n e d . W i t h e a c h s u c c e s s i v e list t h e y l e a r n e d , t h e y f o r g o t m o r e in 24 hours until on their t w e n t i e t h list t h e y forgot n e a r l y 80 p e r c e n t in 24 hours. E a c h s u c c e s s i v e list suffers from m o r e p r o a c t i v e i n t e r f e r e n c e than d i d t h e p r e v i o u s o n e . T h i s can m a k e the difference b e t w e e n r e m e m b e r i n g four out o f five items a n d forgetting four out of five items. In retroactive i n t e r f e r e n c e , s o m e t h i n g l e a r n e d later m a k e s it difficult to rem e m b e r s o m e t h i n g w h i c h w a s l e a r n e d e a r l i e r . T h u s , i f after y o u r first m e s s a g e about the dog, y o u r e c e i v e d a great m a n y other messages about dogs and laundries, these could cause retroactive interference w h i c h w o u l d greatly increase your chances of forgetting the original message. Retroactive interference was demonstrated in a c u e d recall experiment by B a r n e s a n d U n d e r w o o d ( 1 9 5 9 ) . T h e s u b j e c t s first l e a r n e d l i s t A - B ( s e e T a b l e 4 ) s o that w h e n t h e y w e r e g i v e n t h e A i t e m s t h e y c o u l d r e p e a t t h e B i t e m s . T h e n t h e y b e g a n t o s t u d y t h e A - C l i s t ( s e e T a b l e 4). N o t i c e t h a t t h e A i t e m s i n t h e A - B list are e x a c t l y t h e s a m e a s t h e A i t e m s i n t h e A - C list. I n l e a r n i n g t h e A - B l i s t , t h e s u b j e c t s l e a r n Z U T - U S E F U L a n d i n l e a r n i n g t h e A - C list, t h e y l e a r n Z U T - B R O W N . B e t w e e n p r a c t i c e s e s s i o n s o n t h e A - C list, t h e s u b j e c t s w e r e t e s t e d b y p r e s e n t i n g t h e A l i s t a n d a s k i n g t h e m t o g i v e both t h e B a n d t h e C i t e m s . A s p r a c t i c e o n t h e A - C l i s t c o n t i n u e d , c u e d r e c a l l for t h e B i t e m s d e c l i n e d f r o m 1 0 0 p e r c e n t t o l i t t l e m o r e t h a n 5 0 p e r c e n t a f t e r 2 0 A - C p r a c t i c e trials. C l e a r l y r e t r o a c t i v e i n t e r f e r e n c e had a v e r y m a r k e d effect.

83

The Structure of Human M e m o r y Table 4. Nonsense Syllable/Adjective Pairs Nonsense Syllable A ZUT PUM BIP SEG YAD LUS VOB KIJ WOF RUP

Adjective B useful dark early green new short helpful tough sad distant

Nonsense Syllable A ZUT PUM BIP SEG YAD LUS VOB KIJ WOF RUP

Adjective C brown humble tall happy easy frequent hot wet wide lively

C a n memories d e c a y simply through the passage of time? Earlier psychologists t h o u g h t s o . T h o r n d i k e ( 1 9 1 4 ) , for e x a m p l e , b e l i e v e d t h a t m e m o r i e s g o t w e a k e r through disuse. W h e n psychologists came to understand the effect of interference on m e m o r y , t h e y b e g a n to d o u b t that t h e y n e e d e d d e c a y to explain m e m o r y losses. M c G e o c h ( 1 9 3 2 ) , for e x a m p l e , s a i d , " T i m e , i n a n d o f itself, d o e s n o t h i n g . " M a n y m o d e r n p s y c h o l o g i s t s b e l i e v e t h a t r e t r o a c t i v e a n d p r o a c t i v e i n t e r f e r e n c e a r e suffic i e n t t o a c c o u n t for a l l o f t h e f o r g e t t i n g a t t r i b u t e d t o d e c a y . T h e i s s u e , h o w e v e r , i s still i n d o u b t , a s H i n t z m a n ( 1 9 7 8 ) p o i n t s o u t . H o w can w e r e d u c e the effects o f i n t e r f e r e n c e ? O u r m a i n w e a p o n s against i t overlearning a n d reviewing. S u p p o s e that it w o u l d take y o u 10 hours of study to learn a topic w e l l e n o u g h t o s c o r e 1 0 0 p e r c e n t o n a q u i z g i v e n i m m e d i a t e l y after s t u d y . I f y o u s t u d i e d 1 5 hours, w e w o u l d say that y o u had p u t i n 5 0 p e r c e n t o v e r l e a r n i n g , a n d i f y o u s t u d i e d 20 hours, 100 p e r c e n t o v e r l e a r n i n g . T h e effect of o v e r l e a r n i n g is to r e d u c e forgetting. K r e u g e r ( 1 9 2 9 ) s h o w e d t h a t i f m a t e r i a l w a s o v e r l e a r n e d b y 5 0 p e r c e n t o r 100 percent, it w a s better recalled over the next w e e k or t w o than if it was not overl e a r n e d . F u r t h e r , e v e n after a m o n t h , t h e s u b j e c t s s h o w a 2 0 p e r c e n t s a v i n g s s c o r e w i t h o v e r l e a r n i n g , w h i l e w i t h o u t it, t h e y s h o w n o n e . are

R e v i e w i n g information also protects it against i n t e r f e r e n c e ( U n d e r w o o d , 1964). T h e m o r e f r e q u e n t l y w e r e f e r t o i n f o r m a t i o n , t h e m o r e l i k e l y w e a r e t o r e t a i n it. T e l e p h o n e n u m b e r s w e call f r e q u e n t l y s u r v i v e interference from the m a n y other numbers we m a y dial in a typical w e e k . Clearly, if students w o u l d r e v i e w the course material frequently during the term, they c o u l d a v o i d that dreadful question, " W h e r e h a v e I h e a r d a b o u t t h a t b e f o r e ? " t h a t s o o f t e n c o m e s u p d u r i n g final e x a m s . Summary: Storage 1. M u c h 2. When speak 3. When speak

of our forgetting can be attributed to interference from other learning. t h i n g s l e a r n e d e a r l i e r i n t e r f e r e w i t h m e m o r y for t h i n g s l e a r n e d l a t e r , w e of proactive interference. t h i n g s l e a r n e d l a t e r i n t e r f e r e w i t h m e m o r y for t h i n g s l e a r n e d e a r l i e r , w e of retroactive interference.

4. T h e effects of interference can be r e d u c e d through o v e r l e a r n i n g a n d r e v i e w . Encoding A t o n e time, p s y c h o l o g i s t s b e l i e v e d that w e c o u l d transfer information into l o n g - t e r m m e m o r y s i m p l y b y r e h e a r s i n g it. N o w t h e y r e c o g n i z e t h a t i n o r d e r t o fix

The C o m p l e t e Problem Solver

84

an item in long-term memory, we must do something more complicated than just repeating it over and over. We must rework the item or elaborate on it by asking q u e s t i o n s a b o u t it, m a k i n g a s s o c i a t i o n s , c r e a t i n g i m a g e s , e t c . S i m p l e r e h e a r s a l w h i c h w i l l k e e p an item in short-term m e m o r y is c a l l e d " m a i n t e n a n c e rehearsal." Rehearsal with r e w o r k i n g and elaboration is called "elaborative rehearsal" (Craik and Watkins, 1973). In this section, 1. we will d e s c r i b e the e v i d e n c e that m a i n t e n a n c e rehearsal isn't sufficient to p u t information into long-term m e m o r y , and 2. we w i l l discuss the p r o c e s s e s w h i c h d o h e l p t o fix i n f o r m a t i o n i n l o n g - t e r m m e m o r y . T h e r e i s a d i f f i c u l t y i n s h o w i n g t h a t s i m p l e r e p e t i t i o n i s n ' t s u f f i c i e n t t o fix information in long-term m e m o r y . T h e difficulty lies in getting p e o p l e to do repetition w i t h o u t also elaborating. I f subjects s u s p e c t that t h e y m a y b e tested o n w h a t t h e y are r e p e a t i n g , t h e y m a y almost automatically d o w h a t e v e r t h e y u s u a l l y d o that h e l p s to put the material into L T M , such as form images or m a k e associations. Craik a n d W a t k i n s ( 1 9 7 3 ) w e r e a b l e t o d e v i s e a task i n w h i c h t h e s u b j e c t s a p p a r e n t l y d o n ' t elaborate on w h a t t h e y are rehearsing. T h e subjects w e r e a s k e d to listen to a list of w o r d s , similar to that s h o w n in the left-hand c o l u m n of T a b l e 5. T h e words w e r e

T a b l e 5. Craik and W a t k i n s ' Task Word List Read to Subject 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

daughter oil rifle garden grain table football anchor giraffe harp lake bike purse milk issue shadow chair nail job trip elbow

Target Word Rehearsed by Subject

garden grain grain grain grain giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe giraffe

F r o m Craik and W a t k i n s , " T h e R o l e of Rehearsal in Short-term M e m o r y , " in the Journal of Verbal Learning and Verbal Behavior, 12, 1 9 7 3 . R e p r i n t e d by permission of the authors.

read to t h e m o n e at a t i m e . T h e subjects' task w a s to report the last w o r d in the list w h i c h b e g a n w i t h t h e l e t t e r " G " . T o d o t h i s , t h e s u b j e c t s w a i t e d for a w o r d b e g i n ning with G. W h e n a target w o r d was presented, the subjects rehearsed it (since it m i g h t be the last one) until another target w o r d , if any, a p p e a r e d (see the right-hand c o l u m n i n T a b l e 5). W h e n t h e n e w t a r g e t w o r d a p p e a r e d , i t r e p l a c e d t h e o l d o n e a s the word b e i n g rehearsed. From the subjects' point of v i e w , there was no point in p u t t i n g t h e o l d t a r g e t w o r d s i n t o l o n g - t e r m m e m o r y , s i n c e t h e i r task w a s t o r e p o r t

The Structure of Human M e m o r y

85

o n l y t h e c u r r e n t t a r g e t . N o t i c e that b y v a r y i n g t h e s p a c i n g b e t w e e n t a r g e t w o r d s from z e r o t o t w e l v e w o r d s , t h e e x p e r i m e n t e r s w e r e a b l e t o control t h e a m o u n t o f rehearsal g i v e n to each word. H o w e v e r , after t h e s u b j e c t s h a d w o r k e d t h r o u g h a n u m b e r o f w o r d lists l i k e that s h o w n in T a b l e 5, the experimenters u n e x p e c t e d l y a s k e d t h e m to recall as many w o r d s a s p o s s i b l e f r o m all t h e l i s t s . T h e c r i t i c a l f i n d i n g w a s t h a t w o r d s w h i c h w e r e r e h e a r s e d for a l o n g t i m e , a s d u r i n g a 1 2 - w o r d i n t e r v a l , w e r e n ' t r e m e m b e r e d a n y b e t t e r t h a n w o r d s w h i c h w e r e r e h e a r s e d for a s h o r t t i m e , a s d u r i n g a z e r o - w o r d interval. Maintenance rehearsal appeared to h a v e no effect in fixing information in long-term m e m o r y . O t h e r studies b y C r a i k a n d W a t k i n s ( 1 9 7 3 , E x p e r i m e n t II) a n d Jacoby (1974) also support this c o n c l u s i o n . We h a v e taken the t r o u b l e to e m p h a s i z e this p o i n t b e c a u s e it is of c o n s i d e r a b l e p r a c t i c a l i m p o r t a n c e . S o m e s t u d e n t s s p e n d a g r e a t d e a l o f effort o n m a i n t e n a n c e r e h e a r s a l o f t h e i r l e s s o n s i n t h e m i s t a k e n b e l i e f t h a t i t w i l l fix i n f o r m a t i o n i n t h e i r m e m o r y . T h e s e s t u d e n t s w o u l d d o far b e t t e r b y u s i n g t h e e n c o d i n g p r o c e d u r e s described below. Elaboration I s u g g e s t e d a b o v e that in order to get information into long-term m e m o r y , we m u s t e l a b o r a t e it. H o w e v e r , I d i d n ' t s a y v e r y s p e c i f i c a l l y w h a t e l a b o r a t i o n i s . P e r haps the b e s t w a y to u n d e r s t a n d elaboration is to think of it as a p r o c e s s that forms c o n n e c t i o n s — e i t h e r w i t h i n the material to be learned, or b e t w e e n the material to be learned and other things we already know. T h e more connections the material h a s , t h e m o r e l i k e l y w e a r e t o b e a b l e t o r e m e m b e r it. W e m i g h t t h i n k o f a n elaborated m e m o r y as a satchel w i t h lots of h a n d l e s . T h e m o r e h a n d l e s it has, the e a s i e r i t i s t o g e t h o l d of. I n t h e f o l l o w i n g s e c t i o n s I w i l l d e s c r i b e s o m e t e c h n i q u e s for e l a b o r a t i n g memories. Inputs and Representations. I m a g i n e t h a t y o u a r e at a p a r t y a n d t h a t y o u a r e b e i n g i n t r o d u c e d to s o m e o n e . At that m o m e n t y o u are e x p o s e d to a great deal of i n f o r m a t i o n all a t o n c e . M u c h o f t h e i n f o r m a t i o n c o m e s f r o m t h e o u t s i d e , t h r o u g h sensory channels. T h e r e is visual information—the a p p e a r a n c e of the person's face and clothing, and the confused party s c e n e around him. T h e r e is auditory information—the sound of the person's voice, the voice of your host introducing him, and noises in t h e room. T h e r e is information from the m u s c l e s e n s e s — y o u h a v e b e e n s t a n d i n g for h o u r s a n d w o u l d v e r y m u c h l i k e t o sit d o w n — a n d t h e r e i s osmic or smell i n f o r m a t i o n — a faint fragrance of p e r f u m e a n d tobacco. In addition to this s e n s o r y information, t h e r e is s e m a n t i c i n f o r m a t i o n — t h a t is, meanings y o u d e r i v e from the sensory information that are not t h e m s e l v e s sensory i n character. F o r e x a m p l e , y o u r host says, " M e e t m y son, t h e d o c t o r . " Later, e v e n though you can't r e m e m b e r the sensory information—the inflections of the host's v o i c e o r his p a r t i c u l a r w o r d i n g — y o u still r e m e m b e r t h e m e a n i n g , t h a t h e h a s a s o n and the son is a doctor. Similarly, the sensory information of a faint fragrance m a y b e p r o c e s s e d t o p r o d u c e a q u e s t i o n o f m e a n i n g far m o r e c o m p l e x t h a n t h e s c e n t w h i c h t r i g g e r e d it: " W h y i s t h i s g u y w e a r i n g C h a n e l # 5 ? " C o r r e s p o n d i n g to the various forms of information i n p u t there are forms of internal representation. R e m e m b e r i n g a visual i m a g e of a face is m u c h l i k e l o o k i n g at a face. U s u a l l y , we c a n tell the d i f f e r e n c e b e t w e e n a s e n s o r y i m a g e a n d a m e m o r y image, b u t P e r k y (1910) s h o w e d that p e o p l e can be f o o l e d into a c c e p t i n g a real external picture as s o m e t h i n g t h e y are i m a g i n i n g . Similarly, r e m e m b e r i n g an auditory i m a g e o f a f r i e n d ' s v o i c e i s m u c h l i k e l i s t e n i n g t o a r e a l v o i c e . A p a r t i c u l a r f o r m of i n t e r n a l r e p r e s e n t a t i o n , e . g . , a v i s u a l i m a g e , is t y p i c a l l y b e t t e r for r e m e m b e r i n g s o m e k i n d s o f i n f o r m a t i o n t h a n for r e m e m b e r i n g o t h e r

86

The C o m p l e t e Problem Solver

k i n d s . V i s u a l i m a g e s a r e g o o d for r e m e m b e r i n g p i c t u r e s b u t n o t for r e m e m b e r i n g t u n e s . A u d i t o r y i m a g e s a r e g o o d for t u n e s b u t t e r r i b l e for s u n s e t s . Representation and Elaboration. O n e o b v i o u s w a y to e l a b o r a t e on a m e s s a g e i s t o u s e m o r e t h a n o n e f o r m o f r e p r e s e n t a t i o n t o c o d e it. W h e n r e a d i n g , for e x a m ple, we might construct visual images of the p e o p l e and actions described in the text in addition to our usual semantic c o d i n g of the material. T h e r e is a great deal of e v i d e n c e d e m o n s t r a t i n g that i m a g e r y is a p o w e r f u l m e m o r y aid. L e v i n (1973) t a u g h t f o u r t h - g r a d e c h i l d r e n t o i m a g e w h a t t h e y w e r e r e a d i n g a n d s h o w e d that i t h e l p e d t h e m t o r e m e m b e r t h e m a t e r i a l . P a i v i o a n d C s a p o ( 1 9 7 3 ) s h o w e d that i t i s easier to learn a list of w o r d s s u c h as " a l l i g a t o r " a n d " a c c o r d i a n " w h i c h are easy to image than words such as "ability" and "afterlife" w h i c h are hard to image. In the next chapter, we will d e s c r i b e a n u m b e r of m e m o r y t e c h n i q u e s that d e p e n d on visual imagery. W h i l e i m a g e r y can be a v e r y p o w e r f u l m e m o r y aid, not e v e r y sort of i m a g e is h e l p f u l . T o h e l p t h e m l e a r n a list o f w o r d p a i r s , B o w e r ( 1 9 7 2 ) a s k e d h i s s u b j e c t s t o form e i t h e r o f t w o k i n d s o f i m a g e s : i n t e r a c t i n g i m a g e s a n d s e p a r a t e d i m a g e s . T o form a n i n t e r a c t i n g i m a g e for t h e p a i r " d o g - b i c y c l e " t h e s u b j e c t s m i g h t p i c t u r e a d o g riding a b i c y c l e . To form a separated image, t h e y w o u l d picture a d o g on the left a n d a b i c y c l e o n t h e r i g h t i n s u c h a w a y t h a t t h e t w o o b j e c t s d i d n ' t i n t e r a c t . B o w e r f o u n d t h a t w i t h i n t e r a c t i v e i m a g e s , s u b j e c t s g i v e n t h e first h a l f o f e a c h p a i r could r e m e m b e r 71 p e r c e n t of the pairs. With separated images, they r e m e m b e r e d only 46 p e r c e n t — n o m o r e than subjects w h o s i m p l y r e p e a t e d the pairs w i t h o u t i m a g i n g . I n t e r a c t i v e i m a g e s , t h a t is, i m a g e s w h i c h f o r m c o n n e c t i o n s , a r e a u s e f u l m e m o r y aid. S e p a r a t e d i m a g e s are not. Elaboration is p o s s i b l e w i t h i n a single form of representation b e c a u s e we can u s e t h a t f o r m i n m o r e t h a n o n e w a y t o c o d e a m e s s a g e . S u p p o s e , for e x a m p l e , t h a t w e w a n t e d t o f o r m a n i m a g e for a n a b s t r a c t w o r d l i k e " a g r e e m e n t . " W e m i g h t picture two p e o p l e shaking hands, or we might picture a paper marked "agreem e n t , " o r w e m i g h t p i c t u r e t w o p e o p l e s h a k i n g h a n d s and e x c h a n g i n g a p a p e r marked "agreement." Elaborating by Answering Questions. S u p p o s e t h a t y o u a r e h a l f w a y t h r o u g h a m y s t e r y story a n d a friend w h o has already read it b e g i n s to harass y o u w i t h questions such as: D i d y o u n o t i c e h o w t h e C o u n t e s s w a s d r e s s e d after t h e m u r d e r ? D i d you figure out w h y the butler w a l k e d with a limp? D i d y o u w o n d e r w h y Sir A u b r e y a l w a y s w o r e a scarf? A s s u m i n g that y o u restrain the i m p u l s e to throw the b o o k at y o u r friend, y o u will p r o b a b l y c h a n g e t h e w a y y o u represent the story t o y o u r s e l f a s y o u read. N o w y o u w i l l n o t i c e h o w p e o p l e are d r e s s e d and h o w the b u t l e r w a l k s , a n d try t o find connections b e t w e e n these things and the murder. T h e questions, then, will cause you to elaborate your representation. Craik and T u l v i n g (1975) performed an experiment to find out h o w elaborations i n d u c e d b y q u e s t i o n s w o u l d i n f l u e n c e m e m o r y . T h e y b e l i e v e d that elaborations i n v o l v i n g s e m a n t i c c o d e s w o u l d i m p r o v e m e m o r y m o r e t h a n e l a b o r a t i o n o f visual or auditory c o d e s . T h e subjects' task w a s to a n s w e r q u e s t i o n s a b o u t a seq u e n c e o f w o r d s . T h e subjects w o u l d b e g i v e n a q u e s t i o n s u c h as, " I s the w o r d i n capital letters?" before e a c h word. T h e y a n s w e r e d " y e s " o r " n o " a n d w e n t o n t o the next question. T y p i c a l questions and responses are s h o w n in T a b l e 6. T h e typeface and rhyme questions might be e x p e c t e d to induce visual and auditory elaboration respectively, and the category and sentence questions might be e x p e c t e d to i n d u c e semantic elaboration. After the subjects w o r k e d through 40 question-word pairs, they w e r e u n e x p e c t e d l y tested. T h e test required the subjects

The Structure of Human M e m o r y

87

Table 6. Typical Q u e s t i o n s and Responses in t h e Craik and Tulving Experiment Level of processing Visual Auditory Category Sentence

Question

Yes

Is the w o r d in capital letters? D o e s the word rhyme with W E I G H T ? Is the w o r d a t y p e of fish? W o u l d t h e w o r d fit t h e s e n t e n c e : " H e met a in the street?"

TABLE crate SHARK friend

No table MARKET heaven CLOUD

F r o m Craik and T u l v i n g , " D e p t h of P r o c e s s i n g and the R e t e n t i o n of Words in E p i s o d i c M e m o r y , " in the Journal of Experimental Psychology, 1 0 4 , 1 9 7 5 . C o p y r i g h t 1 9 7 5 by the A m e r i c a n P s y c h o l o g i c a l Association. Reprinted by permission.

Table 7. Percent of W o r d s C o r r e c t l y Identified in the Craik and Tulving Experiment Question T y p e : Percent Correct:

Typeface 16

Rhyme 57

Category

Sentence

78

89

to examine a list consisting of the original w o r d s m i x e d with 40 n e w ones and to check the words they had seen before. T a b l e 7 s h o w s that the s e n t e n c e a n d category questions l e d to b e t t e r r e c o g n i tion t h a n t h e r h y m e a n d t y p e f a c e q u e s t i o n s . T h i s r e s u l t i s c o n s i s t e n t w i t h C r a i k a n d Tulving's b e l i e f that semantic elaboration w o u l d aid m e m o r y more than visual or auditory elaboration. Noticing Categories. A s w e d i s c u s s e d e a r l i e r , c a t e g o r i e s a r e a v e r y p o w e r f u l retrieval c u e . I f w e n o t i c e categories i n the material w e are learning, w e can i n c r e a s e o u r c h a n c e s o f r e m e m b e r i n g it. Jacoby (1974) s h o w e d this by arranging a situation in w h i c h o n e group of subjects w a s m o r e l i k e l y to n o t i c e categories in a list than another. He s h o w e d b o t h groups a l o n g list of w o r d s w h i c h c o n t a i n e d s u c h categories as birds, v e g e t a b l e s , etc. A s e a c h w o r d w a s s h o w n , G r o u p 1 w a s a s k e d t o s a y w h e t h e r i t w a s a m e m b e r of the same c a t e g o r y as the i m m e d i a t e l y p r e c e d i n g w o r d . G r o u p 2 w a s a s k e d to say w h e t h e r i t w a s i n t h e s a m e c a t e g o r y a s any p r e c e d i n g w o r d . T h u s , i f w o r d s l i k e "parrot" a n d " f i n c h " o c c u r r e d i n t h e list five w o r d s apart, w e w o u l d e x p e c t that Group 2 w o u l d be m o r e likely to notice the category " b i r d s " than w o u l d G r o u p 1. On a surprise recall test, G r o u p 2 r e m e m b e r e d m u c h m o r e than G r o u p 1. T h e learner is w e l l a d v i s e d to notice category relations in the material b e i n g learned. Hierarchies. M u c h o f t h e i n f o r m a t i o n w e d e a l w i t h e v e r y d a y i s o r g a n i z e d i n categories, and f r e q u e n t l y those categories are t h e m s e l v e s o r g a n i z e d in hierarchies. F o r e x a m p l e , minerals can be arranged in a hierarchically o r g a n i z e d set of categories as s h o w n in F i g u r e 7. We k n o w that attention to categories can aid learning. W h a t about attention to hierarchical organization? B o w e r , Clark, L e s g o l d , and W i n z e n z (1969) h a v e studied just such hierarchically organized material. In fact, F i g u r e 7 i s o n e o f t h e e i g h t h i e r a r c h i e s t h e y s t u d i e d . T h e s u b j e c t s i n t h e i r e x p e r i m e n t l e a r n e d e i t h e r b y s e e i n g t h e c o m p l e t e h i e r a r c h y all a t o n c e o n e a c h o f t h e four trials ( w h o l e m e t h o d ) o r b y s e e i n g l e v e l 1 a n d 2 o f t h e h i e r a r c h y o n trial 1 ; l e v e l s 1 , 2 , a n d 3 o n trial 2 ; a n d all f o u r l e v e l s o n trials 3 a n d 4 ( p r o g r e s s i v e m e t h o d ) . Further, in s o m e cases the organization of the material w a s meaningful as in Figure 7, and in others it w a s random, as in T a b l e 8. M e a n i n g f u l o r g a n i z a t i o n m a d e a b i g d i f f e r e n c e i n t h e n u m b e r o f w o r d s rec a l l e d . O n t h e first t r i a l , s u b j e c t s i n t h e m e a n i n g f u l c o n d i t i o n r e m e m b e r e d m o r e

The C o m p l e t e Problem Solver

88 LEVEL

T a b l e 8. Random O r g a n i z a t i o n of Mineral Information STONES

METALS

Precious

Alloys

Rare

Masonry

Common

limestone bronze aluminum

ruby sapphire granite platinum

copper marble steel

emerald silver lead brass

iron diamond slate gold

than three times as m a n y words as those in the random condition. Performance of subjects i n t h e m e a n i n g f u l c o n d i t i o n w a s p e r f e c t b y t h e third trial, w h i l e subjects i n t h e r a n d o m c o n d i t i o n still k n e w l e s s t h a n h a l f o f t h e w o r d s o n t h e t h i r d trial. F u r t h e r , l e a r n i n g t h e t o p t h r e e l e v e l s o f t h e h i e r a r c h y first, a s i n t h e p r o g r e s s i v e m e t h o d , h e l p e d s u b j e c t s t o l e a r n t h e w o r d s o n t h e f o u r t h l e v e l . O n t h e i r first e x p o s u r e t o the w o r d s o n t h e fourth l e v e l , subjects u s i n g the w h o l e m e t h o d l e a r n e d 60 percent of the words, w h i l e subjects using the progressive m e t h o d learned 77 percent. Attending to the hierarchical organization of the material to be learned is clearly a p o w e r f u l l e a r n i n g aid. Use o f Examples. E x a m p l e s a r e t y p i c a l l y a t t h e v e r y l o w e s t l e v e l s i n t h e h i e r a r c h i c a l s t r u c t u r e o f a t e x t o r a l e c t u r e . N o n e t h e l e s s , t h e y c a n b e v e r y i m p o r t a n t for understanding the information presented. Pollchik (1975) asked subjects to read a 100-sentence text d e s c r i b i n g s e v e n psychological d e f e n s e m e c h a n i s m s . In addition, he g a v e s o m e of the subjects t w o one-sentence e x a m p l e s illustrating e a c h of the m e c h a n i s m s . To test their k n o w l e d g e of the defense m e c h a n i s m s , subjects w e r e asked to identify the mechanism involved in each of 20 cases. Subjects w h o had had e x a m p l e s as part of their instruction correctly identified 16.4 of the 20 cases. S u b j e c t s w h o h a d n o t s e e n t h e e x a m p l e s a s p a r t o f t h e i r i n s t r u c t i o n i d e n t i f i e d o n l y 9.9 of the cases. E x a m p l e s , then, are an important aid to instruction. S u m m a r y : Encoding 1. M a i n t e n a n c e rehearsal isn't sufficient to put information into long-term m e m o r y . Elaborative rehearsal is necessary. 2. T h e r e are s e v e r a l distinct m e m o r y c o d e s : auditory, v i s u a l , semantic, etc. 3. Elaborative rehearsal m a y involve:

89

The Structure of Human M e m o r y • use of extra c o d e s , s u c h as imagery • answering questions • noticing categories • a t t e n d i n g to h i e r a r c h i c a l s t r u c t u r e • s e e i n g principles illustrated in e x a m p l e s CONTROLLING YOUR O W N

LEARNING

PROCESS

U p t o this point w e h a v e d i s c u s s e d the p r o c e s s e s o f e n c o d i n g , storage, and retrieval separately. I n this section, w e w i l l d e s c r i b e h o w p e o p l e can u s e o n e o f these processes to control another. F o r e x a m p l e , we will d e s c r i b e h o w retrieval can profitably be u s e d to control information about success and failure at e n c o d i n g . W h e n y o u are s t u d y i n g , y o u c a n m a k e d e c i s i o n s a b o u t h o w t o s p e n d y o u r s t u d y t i m e . I f y o u a r e l u c k y , y o u c a n c h o o s e t o s p r e a d t h e effort y o u s p e n d o n a s u b j e c t o u t o v e r w e e k s o r m o n t h s . I f y o u a r e n ' t l u c k y , y o u m a y f i n d y o u r s e l f c r a m m i n g all y o u r effort i n t o t h e last f e w h o u r s a v a i l a b l e b e f o r e a n e x a m o r s o m e o t h e r d e a d l i n e . Y o u can c h o o s e to e m p l o y any of a w i d e variety of strategies in learning, e.g., y o u can e n c o d e information i n m a n y w a y s . Y o u m a y c h o o s e t o s p e n d 100 p e r c e n t o f y o u r study time in r e a d i n g a n d r e r e a d i n g the material y o u are trying to learn, or y o u m a y choose to s p e n d s o m e of your time testing yourself. Your decisions about h o w to spend your time h e l p determine the efficiency of your learning. I n g e n e r a l , s p r e a d i n g y o u r s t u d y effort o u t o v e r a p e r i o d o f t i m e i s m o r e effic i e n t t h a n c r a m m i n g . T h a t i s , i f y o u d i s t r i b u t e y o u r effort, y o u c a n l e a r n a t o p i c i n less time than w i t h m a s s e d practice. F i g u r e 8 s h o w s L y o n ' s (1914) results compari n g m a s s e d a n d d i s t r i b u t e d p r a c t i c e for t a s k s w h i c h v a r y i n d i f f i c u l t y . T h e a d v a n tage of distributed practice i n c r e a s e s as the task b e c o m e s m o r e difficult.

Number of syllables in list Figure 8. Massed V e r s u s Distributed P r a c t i c e in Relation to Length of List Learned Reprinted from Foundations of Psychology, Boring, E. G., Langfeld, H. S., and W e l d , H. P. By permission of John W i l e y and Sons, © 1 9 4 8 . A s t u d y b y T h o r n d y k e a n d S t a s z (1980) o n m a p l e a r n i n g i l l u s t r a t e s t h e i m p o r t a n c e o f s t r a t e g y s e l e c t i o n . T h e s u b j e c t s w e r e g i v e n six t w o - m i n u t e s t u d y p e r i o d s in w h i c h to learn p l a c e n a m e s and spatial information from a m a p . After e a c h study period, they d r e w and l a b e l e d as m u c h of the m a p as t h e y c o u l d r e m e m b e r . T h e subjects v a r i e d c o n s i d e r a b l y i n t h e i r s u c c e s s i n t h i s l e a r n i n g t a s k a n d i n t h e strateg i e s t h e y u s e d i n p e r f o r m i n g it. T h e a u t h o r s f o u n d i m p o r t a n t d i f f e r e n c e s i n t h e w a y g o o d a n d p o o r l e a r n e r s a p p r o a c h e d t h e task. F i r s t , t h e y f o u n d t h a t g o o d l e a r n e r s e m p l o y e d b e t t e r s t r a t e g i e s for e n c o d i n g s p a t i a l i n f o r m a t i o n . T h e y f r e q u e n t l y u s e d visual imagery, pattern e n c o d i n g (e.g., " I t looks like a h o u s e " ) , a n d relational

90

The C o m p l e t e Problem Solver

e n c o d i n g ( e . g . , " T h o s e t w o s t r e e t s are p a r a l l e l " ) . S e c o n d , t h e g o o d l e a r n e r s w e r e more effective in using self-evaluation to control their learning. T h e good learners w e r e more likely than the poor learners to evaluate t h e m s e l v e s w h e r e their knowle d g e was weak. T h e poor learners w e r e more likely to spend time confirming what they did know. In evaluating themselves, the good learners w e r e more accurate (96 p e r c e n t c o r r e c t ) t h a n t h e p o o r l e a r n e r s (82 p e r c e n t c o r r e c t ) . F u r t h e r , w h e n t h e good learners d i s c o v e r e d that t h e y didn't k n o w s o m e t h i n g , t h e y w e r e more likely t o t a k e i m m e d i a t e a c t i o n t o l e a r n i t (95 p e r c e n t ) t h a n w e r e t h e p o o r l e a r n e r s (75 percent). T h e s e three differences taken together a d d up to a v e r y considerable d i f f e r e n c e i n t h e e f f e c t i v e n e s s w i t h w h i c h g o o d a n d p o o r l e a r n e r s u s e selfevaluation to control learning. I n a s e c o n d s t u d y , T h o r n d y k e a n d S t a s z (1980) s h o w e d t h a t t h e s p a t i a l c o d i n g and self-evaluation strategies w h i c h the good learners use could be taught to other subjects and that d o i n g so m a r k e d l y i m p r o v e d the subjects' m a p learning performance. T h e T h o r n d y k e and Stasz study s h o w s that self-evaluation is important, b u t h o w m u c h of your study time, if any, should y o u devote to testing yourself? Gates (1917) investigated this question and c a m e up w i t h a surprising answer. His results, s h o w n i n F i g u r e 9 , indicate that p e o p l e m a y learn b e s t w h e n t h e y s p e n d u p t o 8 0 percent of their study time in testing themselves.

20

-

10

-

_] 0

I

I

I

I

20

40

60

80

P e r c e n t a g e of study time spent in self-recitation Figure 9. Effects on Recall of Spending Various Proportions of Study T i m e in A t t e m p t i n g Retrieval Rather Than in Silent Study, for Tests G i v e n Immediately After Study From Introduction to Psychology, Seventh Edition, edited by E. R. Hilgard, R. L. Atkinson, and R. C. Atkinson. By permission of Harcourt Brace Jovanovich, Inc., © 1 9 7 9 . S u m m a r y : C o n t r o l of Learning Process P e o p l e can influence their o w n learning processes and h a v e a great deal to gain b y d o i n g so. B y J . c h o o s i n g t o distribute practice, 2 . s e l e c t i n g appropriate l e a r n i n g strategies, a n d 3. u s i n g evaluation of progress to control study, p e o p l e can significantly increase their learning effectiveness.

The Structure of H u m a n M e m o r y

91

CONCLUSION Basic research on human m e m o r y provides us with a great deal of useful information. It indicates the importance of c h u n k i n g in u s i n g our limited capacity short-term m e m o r i e s effectively. Further, it p r o v i d e s a n u m b e r of g u i d i n g princip l e s for t h e e f f i c i e n t u s e o f l o n g - t e r m m e m o r y . F o r e x a m p l e , i t h a s s h o w n t h e i m p o r t a n c e o f e l a b o r a t i o n for l e a r n i n g , t h e v a l u e o f o v e r l e a r n i n g a n d r e h e a r s a l for maintaining information in storage, and the usefulness of c u e i n g in recall. In addition, research reveals that p e o p l e can b e c o m e m o r e effective learners if t h e y learn to monitor and control their o w n learning processes. REFERENCES B a r n e s , J . M . , a n d U n d e r w o o d , B.J. " ' F a t e ' " o f F i r s t - l i s t A s s o c i a t i o n s i n T r a n s f e r T h e o r y . " Journal

of Experimental

Psychology,

58,

97-105,

1961.

Bousfield, W. A. " T h e Occurrence of Clustering in the Recall of Randomly A r r a n g e d A s s o c i a t e s . " Journal of General Psychology, 49, 2 2 9 - 2 4 0 , 1 9 5 3 . B o w e r , G . H . " M e n t a l I m a g e r y a n d A s s o c i a t i v e L e a r n i n g . " I n Cognition i n Learning and Memory, e d i t e d b y L . W . G r e g g . N e w Y o r k : W i l e y , 1 9 7 2 . Bower, G.H., Clark, M.C., Lesgold, A.M., and W i n z e n z , D. "Hierarchical Retrieval S c h e m e s in R e c a l l of C a t e g o r i z e d W o r d L i s t s . " Journal of Verbal Learning and Verbal Behavior, 8, 3 2 3 - 3 4 3 , 1 9 6 9 . Bower, G.H., and Springston, F. "Pauses as R e c o d i n g Points in Letter Series." Journal

of Experimental

Brooks, L.R. Journal

Psychology,

83,

421-430,

"Spatial and Verbal Components

of Psychology,

22,

349,368,

1970.

o f t h e A c t o f R e c a l l . " Canadian

1968.

B r o w n , R., a n d M c N e i l l , D . " T h e ' T i p o f t h e T o n g u e ' P h e n o m e n o n . " Journal o f Verbal Learning and Verbal Behavior, 5, 3 2 5 - 3 3 7 , 1 9 6 6 . Burtt, H . E . " A n E x p e r i m e n t a l S t u d y o f E a r l y C h i l d h o o d M e m o r y : F i n a l R e p o r t . " Journal of Genetic Psychology, 58, 4 3 5 - 4 3 9 , 1 9 4 1 . C h a s e , W . G . , a n d E r i c s s o n , K.A. A Mnemonic System for Digit Span: Later. U n p u b l i s h e d r e p o r t , C a r n e g i e - M e l l o n U n i v e r s i t y , 1 9 7 9 .

One

Year

Craik, F.I.M., and T u l v i n g , E. " D e p t h of P r o c e s s i n g a n d the R e t e n t i o n of W o r d s in E p i s o d i c M e m o r y . " Journal of Experimental Psychology, 104, 2 6 8 - 2 9 4 , 1 9 7 5 . Craik, F.I.M., and W a t k i n s , M.J. " T h e R o l e o f R e h e a r s a l i n Short-term M e m o r y . " Journal

of Verbal

Gates, A.I. 1917.

Learning

" R e c i t a t i o n as

and

Verbal

Behavior,

a F a c t o r in

12,

599-607,

M e m o r i z i n g . " Archives

1973. of Psychology,

40,

G r e e n b e r g , R., a n d U n d e r w o o d , B.J. " R e t e n t i o n a s a F u n c t i o n o f S t a g e o f P r a c t i c e . " Journal

of Experimental

Hintzman, D.L. The W.H. Freeman, 1978.

Psychology,

40,

Psychology

of

452-457, Learning

and

1950. Memory.

San

Francisco:

Jacoby, L.L. " T h e R o l e of M e n t a l C o n t i g u i t y in M e o r y : Registration and Retrieval E f f e c t s . " Journal of Verbal Learning and Verbal Behavior, 13, 4 8 3 - 4 9 6 , 1 9 7 4 .

FIVE.

USING

M E M O R Y EFFECTIVELY

Usually w h e n we think of m e m o r y , we think of information that is stored inside t h e h e a d . I n t h i s c h a p t e r , w e w a n t t o e m p h a s i z e that, w h i l e s o m e o f t h e i n f o r m a t i o n available to us is stored inside the head, m u c h of it is stored outside. Solving p r a c t i c a l p r o b l e m s t y p i c a l l y r e q u i r e s u s t o u s e b o t h i n t e r n a l and e x t e r n a l m e m o r y . O u r e f f e c t i v e n e s s in practical m e m o r y tasks is often d e t e r m i n e d by our s u c c e s s in managing the relation b e t w e e n internal and external m e m o r y . If the answer to our p r o b l e m i s i n a r e f e r e n c e b o o k i n t h e l i b r a r y , i t w o n ' t h e l p u s a t all u n l e s s w e h a v e s u f f i c i e n t k n o w l e d g e in our heads to be a b l e to f i n d t h e b o o k . I n t h i s c h a p t e r , w e w i l l d e s c r i b e s t r a t e g i e s for u s i n g b o t h i n t e r n a l a n d e x t e r n a l memories effectively. First, w e will describe several m n e m o n i c t e c h n i q u e s — t h a t is, p r o c e d u r e s for s t o r i n g i n f o r m a t i o n r a p i d l y i n i n t e r n a l l o n g - t e r m m e m o r y — a n d w e will p r o v i d e practice e x e r c i s e s i n their use. W h e n y o u h a v e m a s t e r e d this section, y o u s h o u l d be a b l e at o n e h e a r i n g to learn a list of 20 w o r d s p r e s e n t e d at t h e rate o f o n e w o r d e v e r y f i v e s e c o n d s . I n a d d i t i o n , y o u s h o u l d b e a b l e t o l e a r n f o r e i g n l a n g u a g e v o c a b u l a r y a t t h e rate o f o n e w o r d a m i n u t e . W e w i l l p l a c e heaviest e m p h a s i s in this chapter on m n e m o n i c t e c h n i q u e s . Next, we w i l l discuss the u s e o f e x t e r n a l s o u r c e s o f i n f o r m a t i o n , s u c h a s l i b r a r i e s a n d r e f e r e n c e b o o k s . Finally, we w i l l discuss notes and r e m i n d e r s — t h a t is, self-generated external memory aids such as lecture notes, and describe their use in solving particular memory problems. M N E M O N I C T E C H N I Q U E S : THE NEED E x t e r n a l m e m o r y a i d s a r e v e r y u s e f u l i n a w i d e v a r i e t y o f s i t u a t i o n s . T h e r e are many practical situations, h o w e v e r , in w h i c h it is either necessary or very conv e n i e n t for u s t o h a v e i n f o r m a t i o n s t o r e d i n o u r m e m o r i e s . Imagine y o u r s e l f at a cocktail party; y o u r host steers y o u to a g r o u p of p e o p l e standing together w h o cordially introduce t h e m s e l v e s . Y o u i m m e d i a t e l y w h i p out your pad and p e n c i l and write d o w n their names and distinguishing features, in o r d e r t o r e m e m b e r t h e i r n a m e s a n d f a c e s . R u m o r s i m m e d i a t e l y c i r c l e that y o u are a. a n a r c o t i c s a g e n t , b. an e m i s s a r y f r o m t h e C I A , or c. a d i v o r c e l a w y e r l o o k i n g for e v i d e n c e . O b v i o u s l y , this use of external m e m o r y d o e s not reflect good social manners and is not encouraged. T h e p r e v i o u s e x a m p l e is o n e of the most c o m m o n situations r e q u i r i n g internal

94

The C o m p l e t e Problem Solver

m e m o r y s k i l l s . I n s o c i a l a n d b u s i n e s s m e e t i n g s , w e are e x p e c t e d t o s t o r e n a m e - f a c e connections internally. T h e " W h a t w a s your n a m e a g a i n ? " routine certainly isn't flattering, a n d e v e n t u a l l y i t b e c o m e s insulting. N u m e r o u s other e x a m p l e s c a n b e cited. In speaking a foreign language, most of the vocabulary and grammar rules must be stored internally. As a practical matter, o n e can use a dictionary only rarely in conversation. T h e same applies to writing. Efficiency of writing is greatly reduced if you must consult the dictionary frequently to c h e c k spelling. Also, typing s p e e d suffers h o r r i b l y i f y o u h a v e t o m a k e f r e q u e n t r e f e r e n c e t o t h e k e y b o a r d . I n football, y o u h a v e to r e m e m b e r the plays, and on the stage, y o u must r e m e m b e r y o u r l i n e s . I n m a n y s c h o o l s i t u a t i o n s , s u c h a s t e s t t a k i n g , t h e t a s k i s e x a c t l y that o f storing information internally. S o m e students take the v i e w that there is no p u r p o s e in l e a r n i n g information c o n t a i n e d i n t e x t b o o k s b e c a u s e o n e c a n a l w a y s l o o k i t u p . T h e r e are s o m e c a s e s i n w h i c h t h i s v i e w i s j u s t i f i e d . W e don't w a n t t o m e m o r i z e t a b l e s o f l o g a r i t h m s o r t r i g o n o m e t r i c f u n c t i o n s . H o w e v e r , t h e r e are m a n y c a s e s i n w h i c h i t i s n o t j u s t i f i e d . I m a g i n e v i s i t i n g a doctor w h o has a great m e d i c a l library b u t d o e s n ' t k n o w the contents. Y o u say, " D o c , I hurt right h e r e . " D o c says, " R i g h t there, e h ! O K , I h a v e n o i d e a w h a t t h a t m e a n s , b u t I'll h i t t h e m e d i c a l b o o k s r i g h t a w a y t o s e e i f I c a n t u r n a n y t h i n g u p . I'll g i v e y o u a c a l l i n a f e w w e e k s i f w e ' r e g o i n g t o n e e d a n y t e s t s . " Finally, there are situations i n w h i c h external m e m o r y w o u l d b e better b u t w e j u s t d o n ' t h a v e p e n c i l a n d p a p e r a v a i l a b l e a t t h e m o m e n t . I t i s for s i t u a t i o n s l i k e these that w e r e c o m m e n d u s i n g the s e v e n m e m o r y aids w e are about t o describe, a s w e l l a s s u c h v a r i a t i o n s a n d c o m b i n a t i o n s o f t h e m a s y o u m a y i n v e n t for y o u r o w n special needs. In niques 1. 2. 3. 4. 5. 6. 7.

the sections w h i c h follow, we will describe seven types of m n e m o n i c techor m e m o r y aids: acronyms and acrostics rhyme-keys multiple-keys the m e t h o d of loci the keyword method the image-name technique, and chaining.

THE BASIS O F M N E M O N I C T E C H N I Q U E T h e e f f e c t i v e n e s s o f m n e m o n i c t e c h n i q u e s d e p e n d s o n four m e m o r y p r i n c i p l e s : 1. i n d e x i n g , 2. s w i t c h i n g to an easier m e m o r y task, 3. c h u n k i n g , and 4. e l a b o ration. T h e last t h r e e p r i n c i p l e s s h o u l d b e familiar t o y o u from r e a d i n g C h a p t e r 4 . S w i t c h i n g to an e a s i e r m e m o r y task refers to the s e q u e n c e of retrieval tasks o r d e r e d from m o s t t o l e a s t d i f f i c u l t : u n c u e d r e c a l l , c u e d r e c a l l , r e c o g n i t i o n , a n d s a v i n g s . T h e most usual application of this principle in m n e m o n i c s is to c h a n g e an u n c u e d recall task i n t o a c u e d r e c a l l task. I n d e x i n g , the only principle not d i s c u s s e d in C h a p t e r 4, is familiar to y o u in another context. Dictionaries contain an enormous amount of useful information in the form of w o r d s paired w i t h their definitions. T h i s information w o u l d be very h a r d t o r e t r i e v e i f t h e w o r d s w e r e a r r a n g e d a t r a n d o m . F o r t u n a t e l y , d i c t i o n a r i e s are i n d e x e d . T h a t is, t h e y are a r r a n g e d i n a n o r d e r l y w a y ( a l p h a b e t i c a l l y ) s o t h a t w e c a n q u i c k l y f i n d t h e w o r d w e are i n t e r e s t e d i n w i t h o u t a n e x h a u s t i v e s e a r c h . M o r e generally, an index is any organized set of c u e s to information. In m e m ory, a n i n d e x i s a n o r g a n i z e d s e t o f c u e s (in t h e s e n s e o f c u e d r e c a l l ) t o i n f o r m a t i o n in memory. T h e r h y m e - k e y method, discussed b e l o w , provides an excellent illustration of the a p p l i c a t i o n of i n d e x i n g in m n e m o n i c s . A s y o u r e a d a b o u t e a c h t e c h n i q u e , try t o i d e n t i f y t h e m e m o r y p r i n c i p l e s o n w h i c h it relies.

Using M e m o r y Effectively

95

A c r o n y m s and A c r o s t i c s I f y o u try t o r e m e m b e r t h e n a m e s o f t h e G r e a t L a k e s w i t h o u t h e l p (that is, a s a free recall task), y o u m a y e x p e r i e n c e s o m e difficulty. H o w e v e r , if y o u k n o w that t h e first l e t t e r s o f t h e n a m e s s p e l l t h e w o r d HOMES, y o u m a y f i n d t h e t a s k m u c h e a s i e r . HOMES i s t h e k i n d o f m e m o r y a i d c a l l e d a n acronym. A n a c r o n y m i s a w o r d o r p h r a s e m a d e e n t i r e l y o f l e t t e r s w h i c h are c u e s t o w o r d s w e w a n t t o r e m e m b e r . O f t e n t h e c u e s are t h e first l e t t e r o r first s y l l a b l e o f t h e w o r d s w e w a n t t o r e m e m b e r . O t h e r e x a m p l e s o f a c r o n y m s are s h o w n i n T a b l e 1 ( i t e m s 2 , 3 , a n d 4). T a b l e 1 . A c r o n y m s and A c r o s t i c s Memory Aid

Type

Application

Acronym

HOMES

T h e names of the Great L a k e s (Huron, Ontario, Michigan, Erie, Superior)

ITCHE (pronounced "itchy")

Acronym

W o r d s d e s i g n a t i n g the four m e m o r y principles (Indexing, Task switching, C/mnking, Elaboration)

R O Y G.

Acronym

T h e colors of the rainbow (Red, Orange, Y e l l o w , G r e e n , B l u e , Zndigo, Violet)

FACE

Acronym

T h e notes represented by the spaces of the G-clef

True virtue makes dull company.

Acrostic

A s e q u e n c e of navigation terms to be r e m e m b e r e d in order (True, Variation, Magnetic, Deviation, Compass)

Every good boy does fine.

Acrostic

T h e notes represented by the lines

On old O l y m p u s topmost top, a F i n n and G e r m a n v i e w e d a hop.

Acrostic

BIV

o f t h e G-cle'f U s e d t o r e m e m b e r t h e first l e t t e r s o f the cranial n e r v e s (Optic, Olfactory, Oculomotor, Troclear, Trigeminal, Abducens, Facial, Auditory, Glossopharyngeal, Vagus, Accessory, Hypoglossal)

A n acrostic i s a s e n t e n c e o r r h y m e i n w h i c h t h e first l e t t e r o f e a c h w o r d i s a c u e . I n t h e a c r o s t i c , " T r u e v i r t u e m e a n s d u l l c o m p a n y , " t h e first l e t t e r o f e a c h w o r d i s a cue to a s e q u e n c e of terms relating to the m a g n e t i c c o m p a s s : " t r u e , uariation, magnetic, deviation, c o m p a s s . " O t h e r e x a m p l e s of acrostics are s h o w n in T a b l e 1 (items 6 a n d 7). Exercises I n v e n t a c r o n y m s o r a c r o s t i c s for e a c h o f t h e f o l l o w i n g : 1. A d o u b l e d c o n s o n a n t is p r e c e d e d by a s h o r t v o w e l . 2 . T h e first 1 3 states o f t h e u n i o n i n o r d e r o f t h e i r a d m i s s i o n t o t h e u n i o n : D e l a ware, Pennsylvania, N e w Jersey, Georgia, Connecticut, Massachusetts, Maryland, South C a r o l i n a , N e w H a m p s h i r e , Virginia, N e w York, North C a r o l i n a , and Rhode Island. 3. T h e names of the Great Lakes in order of size: Ontario, Erie, Michigan, Huron, Superior.

96

The C o m p l e t e

Problem Solver

4. G e o l o g i c a l periods of the P a l e o z o i c era from earliest to m o s t recsent: C a m b r i a n , Ordovician, Silurian, Devonian, Mississippian, Pennsylvanian, Permian. 5 . T h e p l a n e t s i n o r d e r o f d i s t a n c e from the sun: M e r c u r y , V e n i is, Earth, M a r s , Jupiter, Saturn, Uranus, N e p t u n e , Pluto. The Rhyme-Key M e t h o d T h i s m e t h o d i s u s e f u l for r e m e m b e r i n g lists o f o r d e r e d o r u n o r a l e r e d i t e m s s u c h a s s h o p p i n g lists o r c h e c k l i s t s . T o u s e t h e m e t h o d , y o u m u s t first m i e m o r i z e t h e list of k e y words r h y m e d w i t h the s e q u e n c e of digits s h o w n in T a b l e 2.

T a b l e 2.

Digits and Rhymed Key W o r d s Rhymed K e y Wc

Digit

d

gun

one two three four five six seven eight nine ten

shoe tree door hive bricks heaven gate mine hen

:

"

S u p p o s e t h a t w e w a n t t o r e m e m b e r a s h o p p i n g list c o n s i s t i n g o f t h e f o l l o w i n g items: onions, coffee, bananas, soup, spaghetti, cola, e g g s , bread, and milk. W i t h the r h y m e - k e y m e t h o d , w e w i l l c r e a t e a n i m a g e r e l a t i n g t h e i t e m s o n t h e s h o p p i n g list t o t h e s e q u e n c e o f r h y m e d k e y w o r d s . T h u s t h e first i t e m , " o n i o n s , " m i g h t b e associated to the k e y w o r d " g u n " through an image of a gun shooting a hole in an onion or an i m a g e of a g u n shooting onions as b u l l e t s . A n y i m a g e that depicts a striking interaction b e t w e e n the r h y m e d k e y w o r d and the object to be r e m e m b e r e d w i l l d o . W h e n t h e first i m a g e i s c r e a t e d , t h e s e c o n d i t e m i s r e l a t e d t o t h e k e y w o r d r h y m e d w i t h t w o , and so on. T h u s , we m i g h t picture a shoe filled w i t h ground coffee, a b a n a n a h a n g i n g on a tree, a door inset w i t h d o z e n s of soup cans, or b e e s carrying strands of l i m p spaghetti to their h i v e , a b o y t h r o w i n g bricks at cola bottles, c l o u d s (in h e a v e n ) s h a p e d l i k e e g g s , a g a t e s e t i n t o t h e s i d e o f a l o a f o f b r e a d , a n d milk pouring from the m o u t h of a mine. I f t h e m e t h o d i s t o w o r k w e l l , y o u m u s t first h a v e T a b l e 2 t h o r o u g h l y m e m o rized, and y o u must generate vivid interactive images relating the items to be remembered to the key words. Exercises Study the r h y m e d k e y pairs s h o w n in T a b l e 2 until y o u can r e p r o d u c e t h e m r a p i d l y a n d e a s i l y . T h e n m e m o r i z e t h e f o l l o w i n g s h o p p i n g lists u s i n g t h e r h y m e key method: 1 . hat, g l o v e s , s h o e s , n e e d l e s , g l a s s e s , u m b r e l l a , p e n , gift w r a p , s c o t c h t a p e , r a z o r blades 2 . p e r c h , c h e e s e , g i n g e r r o o t , salt, o n i o n s , c a s h e w s , s a n d w i c h b a g s , s t e a k , c o o k i e s 3. thread, curtains, light bulbs, saw blades, fuses, masking tape

Using M e m o r y Effectively

97

Multiple-Key M e t h o d T h i s m e t h o d i s a v a r i a t i o n o f t h e r h y m e - k e y m e t h o d a n d i s e s p e c i a l l y u s e f u l for r e m e m b e r i n g t e l e p h o n e n u m b e r s . T h e m e t h o d r e q u i r e s a fair a m o u n t o f intial effort. T a b l e 3 , o r a s i m i l a r c h a r t o f y o u r o w n d e v i c e , m u s t b e t h o r o u g h l y m e m o r i z e d . T h e effort i n v o l v e d i n l e a r n i n g T a b l e 3 i s l e s s t h a n i t a p p e a r s a t first g l a n c e , s i n c e t h e k e y for e a c h d i g i t c a n b e l e a r n e d a s a s i n g l e i m a g e , a n d t h e f o u r w o r d s i n t h e k e y r h y m e w i t h t h e c o r r e s p o n d i n g d i g i t . T o i l l u s t r a t e , t h e f o u r k e y w o r d s for zero, or " n a u g h t , " can be stored in a single image of a hot d a u g h t e r w h o is standing on h e r n e w l y - b o u g h t yacht. Further, the last c o l u m n in T a b l e 3 is identical to the last c o l u m n i n T a b l e 2 , w h i c h y o u a l r e a d y k n o w .

Table 3. Digit

A Chart for Using t h e Multiple-Key M e t h o d

Modifier

Actor

Act

Object-Place

hot

daughter

bought

yacht

nun gnu

runs flew skis

gun shoe tree door

5 6 7

funny blue wee snoring jive quick raving

8 9

crated fine

0 (nought) 1 2 3 4

me whore wife hick Kevin Beethoven lion

tore arrives fix severs inflates dines

hive bricks heaven |

gate mine

T h e h a r d p a r t o f r e m e m b e r i n g a t e l e p h o n e n u m b e r i s r e m e m b e r i n g t h e last four d i g i t s . S o w e w i l l c o n c e n t r a t e o n t h a t first. T o r e m e m b e r t h e n u m b e r 9 6 3 - 8 7 2 5 , we w i l l take the last four digits in order a n d c o d e t h e m as a modifier, an actor, an act, a n d a n o b j e c t o r p l a c e . T h u s , 8 7 2 5 b e c o m e s " c r a t e d K e v i n f l e w t o a h i v e . " N o t i c e that t h e o r d e r o f t h e n u m b e r s i s i m p o r t a n t i n d e t e r m i n i n g t h e i m a g e . T h e number 7528 b e c o m e s "a raving g n u arrives at the gate," and the n u m b e r 5872 b e c o m e s " a j i v e B e e t h o v e n s e v e r s a s h o e . " A r t i c l e s a n d p r e p o s i t i o n s c a n b e inserted to assure the grammaticality of the phrase. T h e e x c h a n g e (the first t h r e e d i g i t s o f a t e l e p h o n e n u m b e r ) i s o f t e n e a s y t o r e m e m b e r w i t h o u t a m e m o r y aid since the n u m b e r s we call most f r e q u e n t l y — l o c a l n u m b e r s — e i t h e r h a v e t h e s a m e e x c h a n g e o r are d i v i d e d i n t o a v e r y f e w e x c h a n g e areas. I f y o u d o w a n t t o u s e a m e m o r y a i d for r e m e m b e r i n g a n e x c h a n g e , h o w e v e r , t h e m e t h o d r e q u i r e s o n e m o d i f i c a t i o n — a s y m b o l for a m i s s i n g n u m b e r . W e w i l l u s e the image of a h o l e to r e p r e s e n t a m i s s i n g n u m b e r . T h u s the n u m b e r 963 can be represented as "a fine hick skis into a h o l e . " Exercises 1 . R e c i t e t h e list o f i m a g e s for t h e d i g i t s . 2 . G e n e r a t e i m a g e s ( a n d w o r d s , w h e n p o s s i b l e ) for t h e f o l l o w i n g n u m b e r s : a. 2 8 9 3 b. 5106 c. 7 4 5 4 d. 8888 3 . G e n e r a t e i m a g e s for t h e f o l l o w i n g n u m b e r s : a. 5 5 5 - 1 2 1 2 (directory assistance—information)

98

The C o m p l e t e Problem Solver b . (292) 4 5 6 - 1 4 1 4 ( t h e W h i t e H o u s e ) c . 3 9 1 - 9 5 0 0 (the " t i m e " n u m b e r )

4 . M e m o r i z e t h i s s o c i a l s e c u r i t y n u m b e r (or y o u r o w n ) : 420-14-3170 5. M e m o r i z e TT to n i n e p l a c e s : 3.14159265 The M e t h o d of Loci T h i s m e t h o d , l i k e t h e r h y m e - k e y m e t h o d , i s e s p e c i a l l y u s e f u l for r e m e m b e r i n g u n r e l a t e d lists o f o b j e c t s . B e c a u s e i t r e t a i n s i t e m s i n a s p e c i f i c s e q u e n c e , i t i s a l s o u s e f u l for m e m o r i z i n g i t e m s that m u s t b e m i x e d i n a d e f i n i t e o r d e r , s u c h a s i n g r e d i ents in a r e c i p e , or c h e m i c a l s in a laboratory e x p e r i m e n t . T h e m e t h o d is of v e r y a n c i e n t o r i g i n a n d w a s u s e d b y R o m a n orators w h e n d e l i v e r i n g s p e e c h e s . To u s e the m e t h o d of loci, y o u must select a place such as a house or a b u i l d i n g that y o u are v e r y familiar with. N e x t c h o o s e a c o n v e n i e n t starting p l a c e s u c h as a front d o o r o r m a i n e n t r a n c e a n d i m a g i n e y o u r s e l f w a l k i n g s l o w l y t h r o u g h t h e b u i l d ing. As y o u walk, choose a s e q u e n c e of 20 distinct loci—clearly defined places like a sofa o r a r e f r i g e r a t o r — w h e r e y o u c a n i m a g i n e p u t t i n g t h e o b j e c t s y o u are t r y i n g

Figure 1. A Loci M a p

Using Memory Effectively

99

t o r e m e m b e r . T o a v o i d c o n f u s i o n , d o n o t c r o s s y o u r p a t h ; t h a t is, " w a l k " i n e a c h a r e a only o n c e . F i g u r e 1 s h o w s the h o u s e that I use and my c h o s e n path through it w i t h 2 0 l o c i . T h e first l o c u s i s t h e front d o o r ; t h e s e c o n d , a r a d i a t o r , t h e t h i r d , t h e refrigerator, etc. It is important, if the m e t h o d is to w o r k w e l l , that y o u identify a u n i q u e p a t h w i t h a w e l l - d e f i n e d s t a r t i n g p l a c e a n d a v e r y d e f i n i t e s e t o f l o c i s o that w h e n y o u are r e t r i e v i n g i n f o r m a t i o n y o u c a n r e p e a t e x a c t l y t h e s a m e s e q u e n c e o f loci that y o u u s e d w h e n storing the information. T o a p p l y t h e m e t h o d o f l o c i i n l e a r n i n g a list o f i t e m s s u c h a s t h o s e s h o w n i n T a b l e 4 , y o u m u s t a s s o c i a t e t h e i t e m s o f t h e list s u c c e s s i v e l y t o t h e s e q u e n c e o f l o c i . T h u s , s i n c e t h e first i t e m i s a n u m b r e l l a , a n d m y first l o c u s i s t h e front d o o r , I imagine the u m b r e l l a stuck into the m i d d l e of the door. I i m a g i n e the s e c o n d item, the sea gull, squatting u n c o m f o r t a b l y on the radiator; the third item, the waterm e l o n , falling out of the refrigerator, etc. T a b l e 4.

A W o r d List To Practice t h e M e t h o d of Loci 1 1 . railroad tie 12. brick 13. hippopotamus

1. u m b r e l l a 2. sea gull 3. watermelon 4. 5. 6. 7. 8. 9. 10.

14. 15. 16. 17. 18. 19. 20.

Picasso painting elephant telephone cricket waterfall a u t o m o b i l e tire snow shovel

fire dog pizza uncle candlestick spinach ivy vines

It helps in u s i n g the m e t h o d to break y o u r loci into groups of five. If y o u use a s q u a r e h o u s e for y o u r l o c i , a c o n v e n i e n t a n d n a t u r a l w a y t o d o t h i s i s t o p l a c e f i v e loci along e a c h of the four w a l l s of the h o u s e . B r e a k i n g the loci up in this w a y m a k e s i t e a s i e r t o r e t r i e v e a s i n g l e i t e m i n t h e m i d d l e o f t h e list. F o r e x a m p l e , i f y o u w a n t to retrieve item 12, you can j u m p ahead to locus 10 at one corner of the house and then " w a l k " t w o loci. Exercises 1. M a k e a d i a g r a m of y o u r loci similar to F i g u r e 1. 2. M e m o r i z e the following recipe using the m e t h o d of loci: Veal

Cutlets—Parma

Style

Beat six veal cutlets lightly to flatten them. Sprinkle with salt and fry in butter on both sides in a heavy frying pan. Sprinkle the cutlets with Vt pound c h o p p e d ham, 2-3 sprigs chopped parsley, A c u p grated parmesan cheese, and 2V2 tablespoons marsala wine. Serve as soon as the c h e e s e begins to melt. l

3.

M e m o r i z e the f o l l o w i n g lists: List 1

List 2

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

an e g g T h o m a s Jefferson a freight train a water glass the T h a m e s R i v e r cotton c a n d y a rose b u s h Ty C o b b a fountain Beethoven's Second Symphony

cheese Budapest John Milton thread a yak cadmium a door the Antarctic tennis a robin

700

The C o m p l e t e Problem Solver

List 1

List 2

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

a horse an ice cream cone philosophy Ghana a pencil a carrot the Pacific O c e a n Galileo an a u t o m o b i l e tire jello

jurisprudence a vacuum cleaner Spiro A g n e w the P y r e n e e s maple syrup a light bulb Bulgaria summer honesty Cezanne

Repair manual. I f y o u h a v e t r o u b l e w i t h t h e m e t h o d — t h a t is, i f y o u f i n d y o u r e m e m b e r less than 17 or 18 out of 20 items—try the following: 1. Recheck your loci. M a k e a n o t h e r d i a g r a m of y o u r l o c i a n d c h e c k it a g a i n s t t h e diagram y o u m a d e i n E x e r c i s e 1 . B e sure that y o u k n o w w i t h o u t any d o u b t e x a c t l y w h a t y o u r l o c i a r e a n d w h a t o r d e r t h e y c o m e in. 2. Be sure that you are generating striking interactive images. If y o u are s t o r i n g t h e item " t o w e l " at the locus "sink," don't just place the towel there neatly folded and l o o k i n g as if it b e l o n g e d . M a k e the t o w e l and the sink interact. I m a g i n e that t h e w a t e r i s t u r n e d o n a n d i t i s s o a k i n g t h e t o w e l . I f y o u are s t o r i n g t h e u m b r e l l a o n t h e s t o v e , l e t t h e s t o v e s e t fire t o t h e u m b r e l l a . T h e m e t h o d o f l o c i h a s p r o v e d i t s e l f q u i t e e f f e c t i v e i n i m p r o v i n g m e m o r y for lists. S e v e n t y - f i v e o f m y s t u d e n t s w e r e g i v e n a m e m o r y t e s t b o t h b e f o r e a n d after l e a r n i n g t h e m e t h o d o f l o c i . I n b o t h c a s e s , t h e s t u d e n t s h e a r d a list o f 2 0 i t e m s r e a d o n l y o n c e a t t h e rate o f o n e i t e m e v e r y s e v e n s e c o n d s . T h e m e d i a n score o n the pre-test w a s 7; on the post-test, 17. T h e difference w a s v e r y r e l i a b l e statistically. S o m e students h a v e e x p r e s s e d c o n c e r n that the m e t h o d o f loci w i l l b e c o m e less and less e f f e c t i v e as o n e c o n t i n u e s to u s e it to learn s u c c e s s i v e lists. T h e w o r r y is that i m a g e s c o n s t r u c t e d for r e m e m b e r i n g e a r l i e r lists w i l l b e c o m e c o n f u s e d w i t h i m a g e s for t h e c u r r e n t list, a n d that t h i s c o n f u s i o n w i l l g r o w a s t h e n u m b e r o f lists w e l e a r n b y t h i s m e t h o d i n c r e a s e s . C o n f u s i o n s o f t h i s sort s o m e t i m e s o c c u r i n u s i n g the m e t h o d , b u t I d o n ' t b e l i e v e that t h e y constitute an important limitation. To test my belief, I asked p e o p l e to use the m e t h o d several times in succession. T h e y l e a r n e d a list, w e r e t e s t e d o n it, i m m e d i a t e l y l e a r n e d a s e c o n d list, w e r e t e s t e d o n t h e s e c o n d l i s t , a n d s o o n . I r e a s o n e d t h a t l e a r n i n g t h e lists i n i m m e d i a t e s u c c e s s i o n w o u l d c a u s e m o r e c o n f u s i o n t h a n w o u l d l e a r n i n g lists w i d e l y s e p a r a t e d i n t i m e . T h u s , t h i s t e s t s h o u l d b e a s e v e r e o n e for t h e m e t h o d o f l o c i . T h e r e s u l t s are s h o w n i n T a b l e 5 . C l e a r l y the m e t h o d holds u p quite w e l l e v e n u n d e r this s e v e r e test.

T a b l e 5.

Results of Using the M e t h o d of Loci To Learn S u c c e s s i v e Lists Successive Lists L e a r n e d 1

2

3

4

5

6

1

20

20

16

19

20

19

2

19

20

17

15

18

14

3

18

20

20

18

4

20

19

20

5

16

13

16

14

14

Subject

12

101

Using M e m o r y Effectively The K e y w o r d M e t h o d

T h e k e y w o r d m e t h o d i s e s p e c i a l l y u s e f u l for l e a r n i n g f o r e i g n l a n g u a g e v o c a b u lary. W e w i l l i l l u s t r a t e t h e m e t h o d b y a p p l y i n g i t i n l e a r n i n g t h e m e a n i n g s o f t h e list o f I t a l i a n w o r d s i n T a b l e 6 . C o n s i d e r t h e first I t a l i a n w o r d , agnello. T h e I t a l i a n pronunciation [AN Y E L L ' O H ] sounds quite similar to the E n g l i s h w o r d s " A N N Y E L L O W , " o u r c h o s e n k e y w o r d s for agnello. N e x t w e w i l l r e l a t e t h e k e y w o r d s t o the E n g l i s h translation: " l a m b . " T o d o this w e can create a n i m a g e o f a girl, A n n , l e a d i n g a b r i g h t y e l l o w l a m b . T h e s e c o n d w o r d , pomidoro [ P A L M E E D O O R ' O H ] , sounds like the English words " B A L M Y D O O R " o r " B O M B H I S D O O R . " For the first k e y w o r d , w e c o u l d i m a g i n e a d o o r w a y i n a t r o p i c a l r e g i o n s u r r o u n d e d b y p a l m trees and tomato plants. F o r the s e c o n d k e y w o r d we c o u l d i m a g i n e a man's door being bombarded with exploding tomatoes. Table 6. Word in Italian

Italian M e n u W o r d s

Pronunciation

Key Word

English Meaning

ann y e l l o w b e e sticker for m o d j o e cheap ollie

lamb steak

ci pol' le

an yell' oh biss tech' ah for m o d g e j o chee pole a

pom i dor' o

palm ee door oh

b a

tomatoes

man' zo rip i e n ' o

m o n zo ree pee-a' no

beef stuffed

tor' ta

tore tah in sah lah' tah

! ldoor palmyj man sew rip p i a n o (ex) tort a it's a l o t t a ' (salad) woven my alley car s h o w fee zany melon

eggs pork, p i g

a g n e F lo bis tec' ca for m a g ' g i o

in sa l a ' ta uo' va mai al' e c a r c i o ' fi m e lan zan' e vi tel' lo pol' lo fun' ghi pes' ce bi an' co sal s i c ' c e r o s ' so

w o e vah m a h y a h ' lay car c h o h ' fe m e h lahn zahn' a vee tell' oh pole' low foon ghee pesh shay b e e yon' co sahl see chay row sow

m y

v telly polo fun g e e s e passion bee uncle sol's itchy row sow

cheese onions

cake salad

artichokes eggplant veal chicken mushrooms fish white sausage red

T h e k e y w o r d m e t h o d , then, consists of two steps: First, one selects as a keyword an E n g l i s h w o r d that r e s e m b l e s the foreign w o r d in pronunciation. As is the c a s e i n u s i n g " b o m b h i s d o o r " a s a k e y w o r d for p o m i d o r o , t h e r e s e m b l a n c e n e e d n o t b e p e r f e c t . S e c o n d , o n e c r e a t e s a s t r i k i n g i n t e r a c t i v e i m a g e r e l a t i n g t h e key-word and the E n g l i s h meaning. T h e images must be actively related, as we saw in t h e last c h a p t e r , t o p r o m o t e r e t r i e v a l . Exercise

1

I n T a b l e 6 , w e h a v e s u g g e s t e d k e y w o r d s for e a c h o f t h e 2 0 I t a l i a n w o r d s . S p e n d a b o u t 10 m i n u t e s , u s i n g the k e y w o r d m e t h o d , to learn all 20 of the Italian w o r d s . B e sure t o g e n e r a t e a n interactive i m a g e relating the k e y w o r d with the E n g l i s h meaning. This is a crucial step in the k e y w o r d method. T h e n cover the p a g e so that only the Italian w o r d s are s h o w i n g a n d test yourself.

702

The C o m p l e t e Problem Solver Table 7. Word in Arabic loo'bee ba mi'a koo'sa wa'rak a reesh' la'ban kib'bee na y e e ' ta b o o ' l e e ba'ba ga nouge' hom'mos bit' h e n ' e e bas zel'la mi'shee mal foof na mool' na mu'ra ke na'fee niss oo niss bak la'wa

Arabic Menu Words Keyword

English Meaning

blue bean bomb clues w a r rock a reach

green beans okra

lob bun kb i e table leg papa canoes home moss bit any b u z z ella mishmosh m a l e foot a mule no more c a n offer not so nice bake lava

Exercise

green squash leaves grape yogurt meat and wheat raw mixture mashed eggplant chick peas sesame sweet peas stuffed cabbage leaves butter cookie coconut cake shredded wheat cake half and half h o n e y pastry

2

S t u d y t h e A r a b i c m e n u w o r d s i n T a b l e 7 for 1 0 m i n u t e s b y t h e k e y w o r d m e t h o d (time yourself). T h e n test y o u r s e l f b y c o v e r i n g all b u t the A r a b i c w o r d s . Generating

Keywords

I n t h e t w o e x e r c i s e s a b o v e , w e h a v e p r o v i d e d k e y w o r d s for y o u . N o w , t o m a k e t h e t e c h n i q u e m o r e g e n e r a l l y u s e f u l , y o u s h o u l d p r a c t i c e g e n e r a t i n g k e y w o r d s for yourself. T a b l e 8 s h o w s 20 G e r m a n words, their pronunciations, and their m e a n ings. S p e n d 20 minutes studying these words by the k e y w o r d method, generating y o u r o w n k e y w o r d s . T h e n test y o u r s e l f b y c o v e r i n g the p a g e s o that o n l y the G e r m a n w o r d s are s h o w i n g . W h e n k e y w o r d s are p r o v i d e d , we estimate that it takes a typical student a b o u t 30 s e c o n d s p e r w o r d to m e m o r i z e a list of foreign l a n g u a g e w o r d s by the k e y w o r d m e t h o d . W h e n k e y w o r d s a r e n o t p r o v i d e d , t h e s a m e task t a k e s a b o u t 6 0 s e c o n d s p e r word. T h e m e t h o d a p p e a r s t o w o r k w e l l for m o s t s t u d e n t s . E i g h t y - f i v e o f m y s t u d e n t s w e r e g i v e n 10 m i n u t e s to learn the Italian w o r d s s h o w n in T a b l e 6 ( w h i c h t h e y had n o t s e e n b e f o r e ) , w i t h t h e k e y w o r d s p r o v i d e d . G i v e n a p o s t t e s t , 6 6 p e r c e n t g o t all 20 correct, 19 p e r c e n t got 18 or 19 correct, and 15 p e r c e n t got 17 or f e w e r correct. Worries. S o m e p e o p l e h a v e e x p r e s s e d c o n c e r n t h a t b y u s i n g t h e k e y w o r d method, the language student's m e m o r y will be cluttered with meaningless keyw o r d s w h i c h , w h i l e u s e f u l for i n i t i a l l e a r n i n g , w i l l l a t e r i n t e r f e r e w i t h t h e n a t u r a l use of l a n g u a g e . Atkinson (1975), w h o has p i o n e e r e d research on the use of this m e t h o d , has i n v e s t i g a t e d this p r o b l e m . He finds in g e n e r a l that the student's consciousness of the k e y w o r d s fades as familiarity w i t h the language increases. T h u s i t a p p e a r s t h a t t h e k e y w o r d , w h i c h i s u s e f u l for i n i t i a l l e a r n i n g , d i s a p p e a r s w i t h o u t harmful after-effects, like a b u i l d e r ' s scaffolding w h e n it is no longer n e e d e d . A second frequently expressed worry relates to pronunciation. Since the key-

Using M e m o r y Effectively

103 Table 8 .

German Words

German Word

Pronunciation

abend ei essen flugzeug friihstuck

ah-bent eye ess-en flook-tsoyk free-schteeck gahsst-hahf

egg to e a t airplane breakfast hotel

kart-off-el kel-ner mit-tahg morg-en

potato waiter noon morning

nikt rowk-en shtahtt shtrahs-e trink-en vahnn vok tsite tsy-toong tsook

not to smoke city street to drink when

gasthof kartoffel kellner mittag morgen nicht rouchen stadt strasse trinken wann wo zeit zeitung zug

English Word evening

where time newspaper train

word provides only an approximation to the correct pronunciation of the foreign word, s o m e h a v e b e e n c o n c e r n e d that students u s i n g the m e t h o d m a y learn incorrect pronunciations w h i c h w i l l later be hard to u n l e a r n . T h i s is certainly a p r o b l e m t o b e a v o i d e d i n u s i n g t h e m e t h o d . S i n g e r ( 1 9 7 6 ) h a s s h o w n , h o w e v e r , that t h e k e y w o r d m e t h o d can actually be u s e d as an aid in a v o i d i n g errors in pronunciation. F o r e x a m p l e , i n l e a r n i n g t h e F r e n c h w o r d l e pantalon, h e r s t u d e n t s w o u l d t y p i c a l l y h a v e m i s p r o n o u n c e d t h e first s y l l a b l e o f t h e w o r d b y u s i n g a s h o r t " a . " " L e p a n t a lon" looks like " p a n t , " not " p o n t , " as it should be p r o n o u n c e d . By choosing the keyword, "Pontiac," the mispronunciation was avoided. T h e k e y w o r d m e t h o d m a y also b e u s e d t o learn spelling. S u p p o s e y o u w a n t e d to l e a r n to s p e l l t h e w o r d s peasant, existence, a n d determine. F i r s t , y o u m u s t i d e n t i f y a n E n g l i s h w o r d (or w o r d s ) w h i c h c o n t a i n s t h e " h a r d - t o - s p e l l " p a r t o f t h e target w o r d . W h e r e p o s s i b l e , t h e k e y w o r d s m a y c o n t a i n t h e w h o l e w o r d . T h u s , w e g e t t h e k e y w o r d s , " P E A S A N T " for PEASANT, " T E N " for EXISTENCE, a n d " D E T E R M I N E " for DETERMINE. N e x t , y o u m u s t c o n s t r u c t a n i m a g e o r s t o r y r e l a t i n g t h e k e y w o r d s t o t h e t a r g e t w o r d . T h u s , for t h e first w o r d , w e m a y c o n s t r u c t t h e s e n t e n c e , " P e a s a n t s e a t p e a s w i t h ants o n t h e m " ; for t h e s e c o n d , " E x i s t e n c e h a s t e n m i n u s o n e l e t t e r s " ; a n d for t h e t h i r d , " T h e c o a l c o m p a n y w a n t s t o d i g a h o l e i n m y front y a r d b u t I a m d e t e r m i n e d t o d e t e r t h e m i n e . " T h i s t e c h n i q u e h a s b e e n d e s c r i b e d in d e t a i l by S h e f t e r ( 1 9 7 6 ) in h i s b o o k , Six Minutes a Day to Perfect Spelling. Exercise Use the k e y w o r d m e t h o d to learn the following spelling words: acceptable cameos courtesy extraordinary

The C o m p l e t e Problem Solver

104 jurisdiction mausoleum pigeon sovereign The I m a g e - N a m e T e c h n i q u e

T h i s p r o c e d u r e i s m o s t f r e q u e n t l y u s e d for l e a r n i n g p e o p l e ' s n a m e s , b u t i t c a n a l s o b e u s e d for a s s o c i a t i n g p a i n t i n g s w i t h t h e i r t i t l e s , or, m o r e g e n e r a l l y , i m a g e s with names. Let's suppose y o u w a n t to learn the names of some people. T h e image-name t e c h n i q u e i n v o l v e s a n a l y z i n g b o t h t h e f a c e a n d t h e n a m e , a n d t h e n s e a r c h i n g for a n y r e l a t i o n b e t w e e n t h e t w o that c a n b e f o u n d . A n a l y s i s o f t h e f a c e i n v o l v e s s e a r c h i n g for i n d i v i d u a l f e a t u r e s s u c h a s c l o s e l y - s e t e y e s , h i g h h a i r l i n e , l a r g e e a r s , o r h e a v y e y e b r o w s , o r for o v e r a l l i m p r e s s i o n s u c h a s " r e m i n d s m e o f a w e a s e l , " o r " l o o k s like a j u d g e . " T h e features that y o u attend to s h o u l d be p e r m a n e n t o n e s that are l i k e l y t o b e p r e s e n t w h e n y o u m e e t t h e p e r s o n a g a i n . C l o t h e s o r c h a n g e a b l e j e w e l r y w o u l d not h e l p y o u learn the name permanently. A n a l y s i s o f the n a m e i n v o l v e s identification o f parts, r h y m e s , a n d p u n s . A n y r e l a t i o n t h a t y o u c a n f i n d b e t w e e n t h e a n a l y s i s o f t h e f a c e a n d t h e n a m e i s fair g a m e . F i g u r e s 2 and 3 s h o w o n e subject's analysis of t w o n a m e s and faces a n d the relations he found b e t w e e n them. Exercise A p p l y the image-name technique the next time y o u m e e t someone new. T h e i m a g e - n a m e t e c h n i q u e i s h a r d e r for m o s t p e o p l e t h a n t h e t e c h n i q u e s discussed previously. Prior to learning the image-name t e c h n i q u e , 75 students in

Figure 2. Using t h e Image/Name T e c h n i q u e to Learn G r e g o r y Thurstone's N a m e

Using M e m o r y Effectively

105

Figure 3 . Using t h e I m a g e / N a m e T e c h n i q u e t o Learn D w i g h t D e n t o n ' s N a m e

m y P r o b l e m S o l v i n g c o u r s e w e r e a b l e t o r e m e m b e r a m e d i a n o f six n a m e - f a c e p a i r s in a list of 20. After l e a r n i n g the t e c h n i q u e , t h e y w e r e a b l e to r e m e m b e r a m e d i a n of 10.5 n a m e - f a c e pairs. W h i l e this i m p r o v e m e n t is less i m p r e s s i v e than the i m p r o v e m e n t w i t h the m e t h o d of loci, it is v e r y reliable statistically. Of the 75 s t u d e n t s , 6 6 s h o w e d s o m e i m p r o v e m e n t after l e a r n i n g t h e m e t h o d . Repair manual. I f y o u are h a v i n g t r o u b l e w i t h t h e i m a g e - n a m e t e c h n i q u e , t h e difficulty m a y c o m e from i n a d e q u a t e analysis of the n a m e s and faces. T h i s aspect o f t h e t e c h n i q u e r e q u i r e s c o n s i d e r a b l e effort a n d c o n c e n t r a t i o n i f i t i s t o w o r k consistently w e l l . Y o u shouldn't e x p e c t m e m o r a b l e relations b e t w e e n names and f a c e s t o s p r i n g i n t o c o n s c i o u s n e s s e a s i l y . O f t e n y o u w i l l h a v e t o d i g for t h e m . Practice helps here. A final word of wisdom. T h e t e c h n i q u e w o n ' t h e l p at all if y o u d o n ' t u s e it. To get the benefit of the t e c h n i q u e y o u have to get in the habit of a p p l y i n g it w h e n you meet new people. Chaining I n all o f t h e t e c h n i q u e s w e h a v e d i s c u s s e d s o far, t h e m e m o r y c u e a n d t h e content retrieved w e r e distinct. In the c h a i n i n g t e c h n i q u e , e a c h portion of the c o n t e n t a s i t i s r e t r i e v e d s e r v e s a s t h e c u e for t h e n e x t r e t r i e v a l . W i t h t h i s t e c h n i q u e , t h e u s e r c o n s t r u c t s a s t o r y t h a t i n c o r p o r a t e s e a c h e l e m e n t o f a list t o b e r e m e m b e r e d i n turn. T h u s , t o r e m e m b e r t h e list: f u d g e , T o r o n t o , N a p o l e o n , p i n , a n t e a t e r , t e n t , brick, f l o w e r pot, river, a n d hot d o g , we c a n create a story l i k e the f o l l o w i n g : " O n c e there was a p i e c e of fudge m a d e in Toronto. It was eaten by N a p o l e o n and gave him a pain in the stomach that felt like a pin. He p u l l e d it out and t h r e w it at an anteater w h o w a s f r i g h t e n e d a n d r a n i n t o a t e n t w h e r e i t h i t its h e a d a g a i n s t a b r i c k that w a s just inside, and k n o c k e d o v e r the flower pot on top, w h i c h rolled d o w n to the river and floated next to an o l d hot d o g . " T h i s t e c h n i q u e i s u s e f u l for t h e s a m e r a n g e o f t a s k s a s t h e m e t h o d o f l o c i , i.e., r e m e m b e r i n g o r d e r e d o r u n o r d e r e d lists o f o b j e c t s . K o h u t ( 1 9 7 6 ) c o m p a r e d t h e s e t w o m e t h o d s for m e m o r i z i n g m u l t i p l e l i s t s . H e a s k e d s u b j e c t s t o h o l d six lists i n memory at the same time, three m e m o r i z e d by the m e t h o d of loci and three by the c h a i n i n g m e t h o d . H e t e s t e d for r e c a l l o f t h e lists after all six h a d b e e n m e m o r i z e d . I n t h e t e s t s , t h e s u b j e c t w a s g i v e n t h e first w o r d o f t h e l i s t a n d a s k e d t o r e c a l l t h e

706

The C o m p l e t e Problem Solver

rest. K o h u t f o u n d t h a t t h e t w o m e t h o d s w e r e a b o u t e q u a l l y g o o d i f t h e s u b j e c t h a d to r e m e m b e r b o t h the correct position and the correct list to be scored correct. H o w e v e r , the m e t h o d of loci was better if the subject just had to r e m e m b e r the c o r r e c t p o s i t i o n o f t h e w o r d i n its list a n d n o t w h i c h l i s t i t w a s o n . T h e c h a i n i n g m e t h o d w a s b e t t e r if the subject just had to get the list right b u t not the position. W H Y D O THE M N E M O N I C T E C H N I Q U E S W O R K ? As we noted earlier, the m n e m o n i c t e c h n i q u e s d e p e n d on four m e m o r y princip l e s for t h e i r e f f e c t i v e n e s s : J. 2. 3. 4.

Indexing, S w i t c h i n g to an easier m e m o r y task, Chunking, and Elaboration.

A c r o n y m s and acrostics m a k e use of t w o of the principles. T h e acronym I T C H E , for e x a m p l e , i s a c h u n k w h i c h h e l p s u s r e m e m b e r c u e s t o t h e four m e m o r y principles. T h e s e c u e s c h a n g e w h a t w o u l d o t h e r w i s e be a free recall task into a c u e d r e c a l l task. A c r o s t i c s w o r k i n t h e s a m e w a y . T h u s , a c r o n y m s a n d a c r o s t i c s d e p e n d o n c h u n k i n g a n d o n s w i t c h i n g t o a n e a s i e r m e m o r y task. T h e r h y m e - k e y m e t h o d m a k e s u s e o f a n u m e r i c a l i n d e x — t h e n u m b e r s from one t o t e n — t o c u e h i g h l y i m a g e a b l e k e y w o r d s — g u n , shoe, etc. T h e k e y w o r d s are associated with the items to be r e m e m b e r e d by forming interactive i m a g e s — a process of elaboration. T h e multiple-key method d e p e n d s on the same principles as the rhyme-key m e t h o d , as does the m e t h o d of loci. H o w e v e r , the m e t h o d of loci uses a spatial rather than a n u m e r i c a l index. T h e k e y w o r d m e t h o d , t h e i m a g e - n a m e t e c h n i q u e , a n d t h e c h a i n i n g m e t h o d all d e p e n d o n e l a b o r a t i o n o f v a r i o u s sorts. I n t h e c h a i n i n g m e t h o d , t h e e l a b o r a t i o n m a y b e l a r g e l y s e m a n t i c — w e try t o t e l l o u r s e l v e s a m e a n i n g f u l story a b o u t i t e m s t o b e r e m e m b e r e d — a n d in the k e y w o r d m e t h o d , it m a y be largely visual, e.g., i m a g e s of tomatoes thrown at doors. T h e image-name technique encourages both auditory and visual elaboration. T a b l e 9 lists t h e s e v e n m n e m o n i c t e c h n i q u e s t o g e t h e r w i t h t h e m e m o r y p r o b l e m s t h e y are a p p l i e d to and the m e m o r y principles t h e y e m p l o y . Constructing Memory Schemes We h a v e p r e s e n t e d a n u m b e r of s c h e m e s that can be u s e d to aid m e m o r y . T h e s e s c h e m e s are e a c h useful in t h e m s e l v e s in a variety of situations. H o w e v e r , the r e a d e r w o u l d be m i s s i n g a major v a l u e of t h e s e s c h e m e s and an important r e a s o n for p r e s e n t i n g t h e m , i f h e f a i l e d t o r e a l i z e t h a t t h e y i l l u s t r a t e j u s t a f e w o f t h e m a n y p o s s i b l e m e m o r y s c h e m e s . T h e reader s h o u l d realize that h e h i m s e l f can construct others. T h e s c h e m e s w e h a v e p r e s e n t e d differ i n a n u m b e r o f w a y s . S o m e o f the s c h e m e s e m p l o y m e m o r i z e d k e y s w h i l e others r e q u i r e the u s e r t o g e n e r a t e his o w n keys. S o m e have single keys; others, multiple keys. S o m e s c h e m e s use auditory imagery, others require visual imagery, and so on. If y o u have a special m e m o r y n e e d , y o u m a y find that n o n e o f the s c h e m e s w e h a v e p r e s e n t e d i s q u i t e appropriate t o it. S u p p o s e y o u w a n t a m e m o r y s c h e m e for u s e i n p l a y i n g b r i d g e . N o n e o f t h e s c h e m e s we h a v e p r e s e n t e d is especially adapted to r e m e m b e r i n g the cards yet to be p l a y e d in a bridge hand. W h a t w o u l d y o u do to put such a s c h e m e together? A w a y t o b e g i n w o u l d b e t o try t o a n s w e r t w o q u e s t i o n s : 1. W h a t is special about the bridge situation? 2. What aspects of k n o w n m e m o r y schemes match the special aspect of the bridge situation?

Using M e m o r y Effectively Table 9.

?07

S e v e n M n e m o n i c T e c h n i q u e s , Their A p p l i c a t i o n s , and the M e m o r y Principles Involved in Their Use Application

Name

Memory Principles Used

Acronyms and Acrostics

p e r m a n e n t storage of import a n t lists a n d r u l e s , e . g . , s h o p p i n g l i s t s ; n o t u s e d for temporary storage

c h u n k i n g , task

Rhyme-Key

a r b i t r a r y lists o f 1 0 i t e m s or less (ordered or unordered)

numerical indexing, elaboration

Multiple-Key

m u l t i - d i g i t n u m b e r s , e.g., telephone numbers

numerical indexing, elaboration

Method of Loci

a r b i t r a r y lists ( o r d e r e d o r unordered); length limited b y l o c i list

spatial i n d e x i n g , elaboration

Keyword Method

vocabulary learning

elaboration

Image-Name

learning names of people, w o r k s o f art

elaboration

Chaining

a r b i t r a r y lists ( o r d e r e d o r unordered)

elaboration

T o a n s w e r t h e first q u e s t i o n , w e m a y n o t e t h a t b r i d g e h a s f o u r p l a y e r s ; t h e y are a s s o c i a t e d w i t h t h e c o m p a s s d i r e c t i o n s — n o r t h , s o u t h , e a s t , a n d w e s t ; t h e r e are t w o h a n d s w h i c h d o n ' t n e e d t o b e r e m e m b e r e d — s e l f a n d d u m m y ; s o m e c a r d s are m o r e i m p o r t a n t t h a n o t h e r s ; a n d t h e c a r d s are c h a r a c t e r i z e d b y t w o d i m e n s i o n s — s u i t a n d value. T h e s e p r o p e r t i e s o f the b r i d g e task s u g g e s t s o m e features w e m i g h t w a n t t o i n c l u d e i n a u s e f u l m e m o r y s c h e m e . T h e s p a t i a l c o d i n g o f t h e four p l a y e r s s u g g e s t s t h e u s e o f a m o d i f i e d m e t h o d o f l o c i w i t h a l o c u s for e a c h o f t h e c o m p a s s d i r e c t i o n s . T h e fact t h a t t h e c a r d s h a v e t w o d i m e n s i o n s s u g g e s t s a p a i r o f m e m o r i z e d k e y s — o n e for s u i t a n d o n e for v a l u e . S i n c e t h e v a l u e s are t h e d i g i t s o n e t h r o u g h t e n a n d Jack, Q u e e n , a n d K i n g , t h e r h y m e - k e y for d i g i t s m i g h t e a s i l y b e a u g m e n t e d t o p r o v i d e t h e k e y for v a l u e . S i n c e t h e r h y m e - k e y i s f o c u s e d o n n o u n s , p e r h a p s t h e suit k e y s h o u l d b e f o c u s e d o n a d j e c t i v e s t o a l l o w e a s y c o m b i n a t i o n s o f t h e k e y s i n images. C a n a useful s c h e m e be constructed with these properties? We don't know. We l e a v e that q u e s t i o n to the interested reader. Exercises D e s i g n a m e m o r y s c h e m e b a s e d o n a n a n a l y s i s o f t h e m e m o r y task o f o n e o f t h e following: 1. 2. 3. 4. 5. 6. 7.

chemical nomenclature biological classifications appointments geological eras resister color-coding traffic l a w s cultural history

USING EXTERNAL S O U R C E S T h e r e is an enormous amount of useful information stored in such external sources as textbooks, m a g a z i n e s , journals, n e w s p a p e r s , microfilm, and so on. T h e r e i s s o m u c h u s e f u l i n f o r m a t i o n , i n fact, that w e o f t e n h a v e c o n s i d e r a b l e t r o u b l e

708

The C o m p l e t e Problem Solver

s i f t i n g t h r o u g h w h a t i s a v a i l a b l e t o f i n d w h a t w e w a n t . F o r t u n a t e l y , t h e r e are m a n y g u i d e s o r indexes t o a v a i l a b l e i n f o r m a t i o n w h i c h c a n a i d u s i n o u r s e a r c h . K n o w i n g the right i n d e x e s is an important prerequisite to work in m a n y fields. For example, p h y s i c i a n s s h o u l d b e a b l e t o u s e t h e Cumulative Medical Index, a n d p s y c h o l o g i s t s s h o u l d k n o w h o w t o l o c a t e p s y c h o l o g i c a l s t u d i e s t h r o u g h Psychological Abstracts.

Using Indexes T h e q u e s t i o n s w e m a y n e e d t o a s k w h e n d o i n g r e s e a r c h i n t h e l i b r a r y are v e r y d i v e r s e , and so are the sources of information n e e d e d to a n s w e r t h e m . T h e card catalog is one index to these sources, but typically we n e e d m a n y more. A m o n g the most g e n e r a l l y useful i n d e x e s * are: 1. World Almanac and Book of Facts 2. Statistical Abstract of the United States 3. Webster's Third New International Dictionary Unabridged 4. Encyclopedia Americana 5. Reader's Guide to Periodic Literature 6. New York Times Index

of

the

English

Language

I f there are a n y o f t h e s e r e f e r e n c e s w h i c h y o u h a v e n e v e r u s e d , y o u should m a k e a point of finding t h e m in the library and examining their contents. O n e o f t h e m o s t v a l u a b l e r e s o u r c e s o f a l i b r a r y i s t h e staff o f r e s e a r c h l i b r a r i a n s . R e s e a r c h librarians are familiar w i t h a large n u m b e r of i n d e x e s to i n f o r m a t i o n — many unfamiliar to most p e o p l e — w h i c h allow them to answer a stupifying variety of questions. B e l o w are s o m e samples of the kinds of questions research librarians must handle and the reference sources (indexes) w h i c h m a y be used to answer them.* Q: What does A.C.R.L. mean? A: L o o k in t h e Acronyms, Initialisms, b- Abbreviations Dictionary. Q: I k n o w part of a p o e m ; h o w can I find out w h a t it is so that I can g e t a c o p y ? A: Granger's Index to Poetry ( i n d e x e d by t i t l e and first l i n e ) Q: W h a t does the s y m b o l of the pyramid on the back of the dollar bill signify? A : Collier's Encyclopedia, v . 1 1 . T h i s a p p e a r s u n d e r t h e s u b j e c t o f t h e " G r e a t Seal of the U n i t e d States." A n y attempt to locate this information u n d e r the h e a d i n g s of " d o l l a r " or " c u r r e n c y " or " m o n e y " w i l l be futile. Q: H o w many steel workers b e l o n g to labor unions? A: Directory of National Unions and Employee Associations Q: W h e r e c a n I find the address of the c o m p a n y that m a k e s my slide-o-matic windows? A: Thomas Register, v. 7 ( b r a n d n a m e s a n d i n d e x ) Q: H o w m a n y houses in the Pittsburgh area have central air-conditioning? A: U.S. Census of Housing, Census Tracts, Pittsburgh, PA. Q: W h e r e can I b u y a good map of Dekalb County, Georgia? A: 10,000 Map and Travel Publications Reference Guide T h e i m p o r t a n t p o i n t t o n o t i c e i s t h a t t h e r e are m a n y o b s c u r e i n d e x e s . T h e i n d e x most appropriate to solving your problem may be one y o u never dreamed existed, a n d a q u i c k q u e s t i o n t o y o u r r e s e a r c h l i b r a r i a n m i g h t l e a d y o u t o it.

* T h e s e lists w e r e prepared by Ms. Dorothea T h o m p s o n , Reference Librarian, Hunt Library, C a r n e g i e - M e l l o n University.

709

Using M e m o r y Effectively Exercises Find appropriate indexes to help you answer the following questions: 1. Is t h e r e a poster of t h e C o n e h e a d s in print? 2. 3. 4. 5.

What do yaks eat? On what day did Estes Kefauver die? W h o is on the Board of Directors of Marvel C o m i c s ? What is Spiro A g n e w doing n o w ? N o t e s and Reminders

P e o p l e m a k e f r e q u e n t u s e o f notes a n d r e m i n d e r s i n e v e r y d a y m e m o r y tasks. W e are f a m i l i a r w i t h t h e m i n t h e f o r m o f s h o p p i n g l i s t s , l e c t u r e n o t e s , a n d t h e proverbial string around the finger. S o m e c o m m o n p l a c e r e m i n d e r s , h o w e v e r , m a y escape our notice. F o r e x a m p l e , we m a y routinely use a finger to k e e p our place in a t e l e p h o n e b o o k o r i n o u r r e a d i n g b u t n e v e r t h i n k o f i t a s a m e m o r y a i d . W e are e v e n less l i k e l y to n o t i c e that e y e position can be u s e d as a m e m o r y aid. W h e n y o u are f o l l o w i n g a p e r s o n i n a c r o w d , i t i s e s s e n t i a l t o try t o k e e p h i m i n s i g h t . I f y o u l o s e s i g h t o f h i m for a m o m e n t , i t i s i m p o r t a n t t o l o o k for h i m c l o s e t o t h e p l a c e y o u s a w h i m last. T h e sorts o f n o t e s w e m a k e w h i l e d o i n g a r i t h m e t i c c o m p u t a t i o n s o r s o l v i n g other p r o b l e m s are of great practical importance and illustrate h o w aids to m e m o r y can also be aids to p r o b l e m solving. T h i n k h o w m u c h harder it is to multiply 2,749 b y 9,472 w h e n y o u h a v e t o d o all o f the w o r k i n y o u r h e a d than w h e n y o u c a n w r i t e d o w n the partial results as y o u work, or h o w m u c h harder it m a y be to solve a p h y s i c s o r g e o m e t r y p r o b l e m w i t h o u t first d r a w i n g a d i a g r a m . W e d i s c u s s e d the function of diagrams and other forms of representations in m o r e detail in C h a p t e r 1. I n t h e n e x t c h a p t e r , w e w i l l d i s c u s s t e c h n i q u e s for t a k i n g l e c t u r e n o t e s w h i c h c a n i m p r o v e l e a r n i n g e f f i c i e n c y for m a n y s t u d e n t s . H e r e w e w i l l d i s c u s s j u s t o n e practical application of r e m i n d e r s w h i c h can be helpful to that pitiable s e g m e n t of humanity k n o w n as " t h e a b s e n t - m i n d e d " — t h o s e p e o p l e w h o regularly arrive at t h e i r o f f i c e s t o d i s c o v e r t h a t t h e i r o f f i c e k e y s a r e a t h o m e . T h e r e r e a l l y i s a difference in the d e g r e e of absent-mindedness among p e o p l e . In one study (Wilkins and Baddeley, 1978) p e o p l e w e r e asked to p u n c h a time clock four times a day—8:30 A . M . , 1:00 P . M . , 5 : 3 0 P . M . , a n d 10:00 P . M . T h i s i s a t a s k s i m i l a r t o t h a t o f r e m e m bering to take y o u r m e d i c i n e at the right time or r e m e m b e r i n g to p u t more coins in y o u r p a r k i n g m e t e r b e f o r e t h e m e t e r m a i d d i s c o v e r s h o w l o n g it's b e e n s i n c e y o u last d i d s o . T h e s t u d y s h o w e d c o n s i s t e n t d i f f e r e n c e s a m o n g p e o p l e i n t h e d e g r e e of lateness w i t h w h i c h they p u n c h e d the clock. C a n a b s e n t - m i n d e d n e s s b e h e l p e d ? W e h o p e so. I n m a n y c a s e s , the trouble seems to result from failure to p a y attention at certain critical t i m e s . A critical time is a time just b e f o r e we take an action that is hard to r e v e r s e — l i k e t r a v e l i n g to w o r k or to Tahiti; m a i l i n g a p a c k a g e ( D i d I r e m e m b e r to take off t h e p r i c e tag?); or r e v e a l i n g i n f o r m a t i o n ( O o p s , I t h i n k t h a t was a s e c r e t ! ) . O n e w a y w e c a n h e l p ourselves to notice the critical times is to use a landmark that is c o n n e c t e d with the c r i t i c a l t i m e . F o r e x a m p l e , t h e f r o n t d o o r i s a n e x c e l l e n t l a n d m a r k t o u s e for r e m e m b e r i n g w h a t w e w a n t t o t a k e t o w o r k . I f w e w o u l d j u s t s t o p for a m o m e n t a t t h e front door t o r e v i e w w h a t w e w i l l n e e d during the day, w e c o u l d a v o i d m a n y annoying attacks o f a b s e n t - m i n d e d n e s s . T h e a c t o f s e a l i n g a p a c k a g e i s a g o o d l a n d m a r k for r e m i n d i n g us to ask q u e s t i o n s l i k e , " D i d I i n c l u d e e v e r y t h i n g I s h o u l d h a v e ? " F o r the absent-minded surgeon, s e w i n g the patient back up should trigger questions s u c h a s , " D o I k n o w w h e r e all m y s c i s s o r s a r e ? O r m y w r i s t - w a t c h ? O r m y s h o e s ? "

7/0

The C o m p l e t e Problem Solver

A b s e n t - m i n d e d n e s s m a y s t r i k e e v e n w h e n w e d o s t o p a t t h e front d o o r a n d try t o r e m e m b e r w h a t w e s h o u l d t a k e w i t h u s . It's h a r d t o r e m e m b e r everything w e s h o u l d t a k e . I n s u c h s i t u a t i o n s , checklists c a n b e v e r y u s e f u l . A c h e c k l i s t i s s i m p l y a list o f i t e m s w e m a y w a n t t o r e m e m b e r . A c h e c k l i s t for l e a v i n g t h e h o u s e m i g h t look like this: car k e y s briefcase office k e y s cats ( H a v e t h e y b e e n f e d ? ) g a r b a g e (Is it o u t ? ) wallet checkbook socks pants I f s o m e d a y y o u f o r g e t a n e w i t e m , e . g . , y o u r p e n , y o u c a n s i m p l y a d d i t t o t h e list. I f t h e c h e c k l i s t c a n b e a s s o c i a t e d w i t h a l a n d m a r k , e . g . , t a p e d t o t h e front d o o r w h e r e you will see it w h e n y o u leave, it is likely to be more effective. SUMMARY 1. In solving p r o b l e m s , we m a k e use both of internal and of external m e m o r y . 2 . M n e m o n i c t e c h n i q u e s are p r o c e d u r e s w e u s e f o r p u t t i n g i n f o r m a t i o n r a p i d l y into long-term m e m o r y . a. These techniques depend on four memory principles: • Indexing • S w i t c h i n g t o a n e a s i e r m e m o r y task • Chunking • Elaboration b. S e v e n m n e m o n i c t e c h n i q u e s , their applications, and the m e m o r y principles t h e y d e p e n d on are s u m m a r i z e d in F i g u r e 4. c . I f the a v a i l a b l e m e m o r y t e c h n i q u e s don't suit o u r n e e d s , w e s h o u l d c o n s i d e r i n v e n t i n g n e w t e c h n i q u e s that d o . 3. To find information in external m e m o r y , e.g., b o o k s , libraries, we d e p e n d on indexes. a. R e f e r e n c e librarians can often provide us w i t h an i n d e x to the information we need. 4 . N o t e s a n d r e m i n d e r s a r e e x t e r n a l m e m o r y a i d s w e c o n s t r u c t for o u r s e l v e s . a. T h e y can be used to aid us in learning and in problem solving. b. T h e y can be used to reduce absent-mindedness.

REFERENCES Acronyms, Initialisms, and Abbreviations Research Company, 1978.

Dictionary,

Atkinson, R.C. " M n e m o t e c h n i c s chology, 30, 8 2 1 - 8 2 8 , 1 9 7 5 .

Second-language

Collier's

Encyclopedia.

New

York:

in

P.F.

Collier,

Sixth

New

York:

Americana

Detroit:

Gale

L e a r n i n g . " American Psy-

1975.

Directory of National Unions and Employee Associations. Labor. B u r e a u o f L a b o r Statistics, Bulletin # 1 9 3 7 . Encyclopedia Americana.

Edition.

U.S.

Corporation,

Department

1977.

of

SIX

LEARNING

STRATEGIES

Nearly e v e r y o n e uses learning strategies, but some p e o p l e clearly make better and more extensive use of t h e m than others. Differences in the w a y p e o p l e perform i n l e a r n i n g s i t u a t i o n s , e i t h e r i n s c h o o l o r o n t h e j o b , m a y d e p e n d m o r e o n differences in their learning strategies than on differences in their ability. For example, G o l d m a n and H u d s o n (1973) f o u n d that c o l l e g e students w i t h h i g h , m i d d l e , and low grade-point averages differed in learning strategies b u t not in ability. If learning strategies are so important, w h y do we hear so little about t h e m ? T h e r e are a t l e a s t t w o a n s w e r s t o t h i s q u e s t i o n . O n e i s t h a t t h e l e a r n i n g s t r a t e g i e s w e u s e m a y b e s o familiar a n d automatic that w e are a l m o s t u n a w a r e o f t h e m . F o r e x a m p l e , w e a l l r e h e a r s e i n f o r m a t i o n w h e n w e w a n t t o r e m e m b e r it. I t s e e m s s u c h a natural t h i n g to do that we h a r d l y think of it as a strategy. In C h a p t e r 4, t h o u g h , w e saw that s o m e first-grade c h i l d r e n h a d not y e t l e a r n e d t o r e h e a r s e . W h e n t h e y were taught to rehearse, their m e m o r y performance improved. Rehearsal, then, isn't i n n a t e . I t i s a s t r a t e g y w e h a v e t o l e a r n . M o s t o f u s l e a r n e d i t s o l o n g a g o , t h o u g h , t h a t w e d o n ' t r e m e m b e r l e a r n i n g it. T h e s e c o n d r e a s o n w e d o n ' t h e a r m u c h a b o u t l e a r n i n g strategies i s that research on l e a r n i n g strategies is q u i t e n e w . T h e r e is still a great d e a l to learn a b o u t them. H o w e v e r , w e d o k n o w that instruction i n l e a r n i n g strategies c a n i m p r o v e people's ability to learn. I n this c h a p t e r , w e w i l l 1 . d i s c u s s s e v e n b a s i c l e a r n i n g s t r a t e g i e s , 2 . d e s c r i b e some practical learning systems w h i c h m a k e u s e of t h e s e basic strategies, and 3 . d e s c r i b e s o m e s p e c i a l t e c h n i q u e s for l e a r n i n g d i f f i c u l t t o p i c s .

BASIC LEARNING STRATEGIES T h e seven basic learning strategies we will discuss are: 1. T h e structuring strategy 2. 3. 4. 5. 6. 7.

T h e context strategy Monitoring Inferencing Instantiation Multiple coding Attention m a n a g e m e n t

The C o m p l e t e Problem Solver

114

F u n d a m e n t a l research u n d e r l y i n g m a n y of these strategies has already b e e n described in Chapter 4. The Structuring Strategy T h e s t r a t e g y o f i n f o r m a t i o n s t r u c t u r i n g r e q u i r e s y o u t o s e a r c h a c t i v e l y for relations in the learning material. T h e relations m a y reveal categories, hierarchies, n e t w o r k s , or other information structures that w i l l h e l p y o u to understand and r e m e m b e r the material. F o r e x a m p l e , s u p p o s e t h a t y o u a r e r e v i e w i n g for a t e s t o n C h a p t e r 5 b y l e a r n i n g t h e f o l l o w i n g list: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Elaboration Displacement C u e d recall Retroactive interference 7 ± 2 T i p of the tongue Recognition Maintenance rehearsal Proactive interference Savings Chunks U n c u e d recall

T a b l e 1 s h o w s t h e c a t e g o r y s t r u c t u r e o f t h e list, a n d F i g u r e 1 s h o w s its h i e r archical structure. T h e c h a p t e r has a t h e m e — t h e structure o f h u m a n m e m o r y — w i t h s o m e major topics u n d e r the t h e m e . T h e topics in turn h a v e subtopics u n d e r t h e m , and these subtopics m a y h a v e subtopics of their o w n . At the l o w e s t l e v e l , we typically find e x a m p l e s a n d specific studies that illustrate points m a d e in h i g h e r l e v e l s . I n g e n e r a l , i t i s a g o o d p r i n c i p l e t o list r e l a t e d t o p i c s u n d e r a s i n g l e h e a d i n g i n the hierarchy. F o r e x a m p l e , the various topics u n d e r the h e a d i n g " r e t r i e v a l " are grouped together in Chapter 4. Separating these topics w o u l d have t w o unfortunate effects: 1. It w o u l d be m o r e difficult to m a k e comparisons a n d contrasts a m o n g similar topics, e.g., the difference b e t w e e n r e c o g n i t i o n a n d recall m i g h t b e m i s s e d . 2. It w o u l d be h a r d e r to identify t h e h i e r a r c h i c a l structure of t h e text. If y o u can't find the hierarchical structure of a lecture or text, either it is poorly T a b l e 1. Items A r r a n g e d by C a t e g o r i e s Short-term Memory 7±2

Encoding

Storage

Retrieval

Elaboration

Retroactive interference

T i p of the tongue U n c u e d recall

Chunks

Proactive interference

C u e d recall

Displacement Recognition Maintenance rehearsal

Savings

c o n s t r u c t e d o r y o u h a v e n o t f u l l y u n d e r s t o o d it. I n e i t h e r c a s e , y o u a r e l i k e l y t o h a v e difficulty in r e m e m b e r i n g t h e material. U s i n g the structuring strategy shouldn't be v i e w e d simply as a matter of identifying structure. It c a n a n d s o m e t i m e s s h o u l d i n v o l v e restructuring. If a s e n t e n c e ,

7 76

The C o m p l e t e Problem Solver

Figure 3. C o m m u n i c a t i o n s N e t w o r k in an Imaginary C o m p a n y paragraph, or larger text is hard to understand b e c a u s e of poor organization, y o u m a y n e e d t o r e o r g a n i z e i t i n o r d e r t o u n d e r s t a n d it. R e a d e r s o f t e n d o this w h e n r e a d i n g v e r y difficult texts (see F l o w e r , H a y e s , and Swarts, 1979). E x a m i n i n g relations in the material to be learned can lead y o u to discover more c o m p l e x structures than hierarchies. F o r e x a m p l e , if we look at authority relations in a c o m p a n y , we w i l l find a hierarchy like that s h o w n in F i g u r e 2. H o w e v e r , if we l o o k a t c o m m u n i c a t i o n s , w e a r e v e r y l i k e l y t o f i n d a c o m p l e x network l i k e t h a t s h o w n in F i g u r e 3. T h e n e t w o r k differs from a h i e r a r c h y b e c a u s e it contains loops. F o r e x a m p l e , t h e r e is a c o m m u n i c a t i o n l o o p from the p r e s i d e n t to A to F and b a c k to the president. If there w e r e a similar authority loop, the c o m p a n y w o u l d be in b i g trouble. Rhetorical

Structure

B a r t l e t t ( 1 9 7 8 ) m a d e a n i m p o r t a n t p r a c t i c a l a p p l i c a t i o n o f t h e s t r u c t u r i n g strate g y i n t e a c h i n g students h o w t o profit from their r e a d i n g , b y h a v i n g t h e m d e s c r i b e t h e rhetorical structure o f t h e t e x t , o r h o w t h e a u t h o r o r g a n i z e d i t t o a c h i e v e its major p u r p o s e . F o u r c o m m o n rhetorical structures are d e s c r i b e d b y M e y e r a n d F r e e d l e (1978): Attribution: Attribution is used to relate a description of a person, place, thing, event, quality or abstraction of that thing w h i c h it describes. Covariance: Covariance indicates a causal relationship serving as an antecedent to another serving as the consequent. . . either the antecedent or the consequent can be in the superordinate . . . position. Response: Response has three subtypes. O n e . . . can be a question and its * answer. T h e second has arguments that are a remark and a reply to it. T h e arguments of the third are a problem and its solution. Adversative: Adversative . . . relates what did not happen to what did happen, or what does not exist to what does exist. B a r t l e t t ( 1 9 7 8 ) h a s t a u g h t n i n t h - g r a d e c h i l d r e n h o w t o i d e n t i f y r h e t o r i c a l structure as a w a y to i m p r o v e their ability to learn from their reading. In a series of five one-hour class sessions, the students learned to identify 1. the main idea of the passage, and 2. w h i c h of the four rhetorical structures the author u s e d to get the idea across.

777

Learning Strategies T h i s q u i z illustrates the skills the students w e r e e x p e c t e d to acquire:

U n d e r l i n e the sentence containing the main idea. C i r c l e the top-level organization u s e d b y the writer. 1. M a r t h a w a s w o r r i e d about h e r health. T h e doctor h a d told Martha that h e r system was overtaxed. As a result she tried to rest m o r e and to eat at regular times. She k n e w her lifestyle had to change. a) d e s c r i p t i o n c) p r o b l e m - s o l u t i o n b) before-as a result d) favored v i e w vs. opposite v i e w 2 . P o l l u t i o n i s a p r o b l e m for o u r r i v e r s . P o l l u t e d r i v e r s a r e e y e s o r e s . T h e y are also health hazards. O n e solution is to stop the d u m p i n g of industrial waste. a) description c) p r o b l e m - s o l u t i o n b) before-as a result d) favored v i e w vs. opposite v i e w 3 . O u r class r e u n i o n w a s h e l d last year. W e saw m a n y old friends there. T h e business of the m e e t i n g was kept to a m i n i m u m . We spent most of our time socializing. a) d e s c r i p t i o n c) p r o b l e m - s o l u t i o n b) before-as a result

d) favored v i e w vs. opposite v i e w

4. D e s p i t e the a r g u m e n t that s m o k i n g is harmful, m a n y c l a i m it is not so. Certainly, s m o k i n g has b e e n related to l u n g cancer, h i g h b l o o d pressure, a n d l o s s o f a p p e t i t e . B u t , for s o m e p e o p l e s m o k i n g m a y r e l i e v e t e n s i o n . a) d e s c r i p t i o n c) p r o b l e m - s o l u t i o n b) before-as a result

d) favored v i e w vs. opposite v i e w

T h e students w e r e told that finding the organization (the rhetorical structure) was the k e y to getting the author's m e s s a g e . To recall that m e s s a g e , t h e y w e r e told: Our strategy involves two steps. In reading, we find the organization the writer used. In recalling, we use the same organization. This is a strategy to improve memory. T h e strategy is called using top-level organization. Students w h o had had had not. B o t h g r o u p s w e r e reading, the subjects w e r e passage. T h e experimental

this training w e r e c o m p a r e d t o a g r o u p o f students w h o a s k e d to read passages such as that s h o w n b e l o w . After a s k e d t o w r i t e d o w n all t h e y c o u l d r e m e m b e r a b o u t t h e group r e m e m b e r e d nearly t w i c e as m u c h as the controls. T h e Early Railroads*

T w o contrasting v i e w s on the usefulness of railroads existed for early Americans. American m e n of business w e r e quick to recognize the promise of railroads, and by 1830 several companies had b e e n formed to construct railroad lines in the United States. Both the South Carolina Railroad and the Baltimore and O h i o Railroads, America's earliest lines, c o m m e n c e d operations in 1830. T h e first American locomotive engines w e r e built and d e l i v e r e d in the same year. Most of the early railroads were short l i n e s — t h e longest in the United States was also the longest in the world, 136 miles. Railway d e v e l o p m e n t p r o c e e d e d rapidly. In the first six years, more than a thousand miles of track w e r e laid, and railroads b e g a n to run trains in e l e v e n states. In the 1850's, rail service was extended to the Mississippi River. E x c e p t for the bulkiest of goods, railroads b e c a m e the most economical form of transportation within the United States. Traveling by water b e t w e e n N e w York City and Detroit in the 1850's took ten days; by rail, the same trip required only four days. Railroads w e r e not popular with everybody. Farmers complained that the noise frightened their cattle, and that sparks from the engines set their fields afire. Some physicians feared that the human body could not endure travel at speeds so high as * From Bartlett, 1978

118

The C o m p l e t e Problem Solver 30 miles an hour. Canal companies tried to k e e p railroads from building lines that might compete with canals.

S i n c e k n o w i n g t h e s t r u c t u r e o f t h e t e x t w e are r e a d i n g c a n h e l p u s t o l e a r n , i t m a k e s s e n s e t o l o o k a c t i v e l y for c l u e s t o s t r u c t u r e b e f o r e w e b e g i n r e a d i n g . I f w e p r e v i e w the text q u i c k l y by r e a d i n g i n d e x e s , h e a d i n g s , and topic s e n t e n c e s , and glancing at pictures, we can often anticipate the main topic, subtopics, and the a u t h o r ' s g e n e r a l a p p r o a c h . T h e c l u e s w e u s e t o i d e n t i f y s t r u c t u r e b e f o r e w e r e a d are c a l l e d advanced organizers ( A u s u b e l , 1 9 6 0 ) . A t t e n d i n g t o t h e m c a n m a k e l e a r n i n g a good deal easier. S i n c e c u e s t o s t r u c t u r e c a n h e l p u s w h e n w e are r e a d i n g , w e s h o u l d r e m e m b e r t o p r o v i d e t h e m for o u r r e a d e r s w h e n w e are w r i t i n g . T h i s m e a n s first, t h a t o u r w r i t i n g s h o u l d h a v e structure, and s e c o n d , that w e tell the r e a d e r w h a t i t is. T h e u s e o f h e a d i n g s , c l e a r p a r a g r a p h s , a n d t o p i c s e n t e n c e s c a n e a s e y o u r r e a d e r s ' task greatly. Also, there is n o t h i n g disgraceful about telling your r e a d e r straight out w h a t y o u are u p t o , e . g . , " F i r s t , I w i l l d i s c u s s t h e a r g u m e n t s for c a p i t a l p u n i s h m e n t , t h e n I w i l l try t o r e f u t e e a c h o f t h e m . " Y o u r r e a d e r m a y w e l l a p p r e c i a t e t h e h e l p . The C o n t e x t Strategy I n t h e s t r u c t u r e s t r a t e g y , w e s e a r c h for r e l a t i o n s w i t h i n t h e m a t e r i a l t o b e l e a r n e d . I n t h e c o n t e x t s t r a t e g y , w e s e a r c h for r e l a t i o n s b e t w e e n t h e t h i n g s t o be learned and the things we already know. R e c o g n i z i n g such relations can make l e a r n i n g a great d e a l easier. In C h a p t e r 2, we found that c h e s s masters m a k e extensive use of their chess k n o w l e d g e to h e l p them learn n e w chess positions. W h e n w e r e a d a p u z z l e , w e h a v e a m u c h e a s i e r t i m e l e a r n i n g its d e t a i l s i f w e c a n r e l a t e i t t o s i m i l a r p r o b l e m s w e h a v e s e e n b e f o r e . I f w e a l r e a d y k n o w h o w t o fix l a w n m o w e r m o t o r s , w e w i l l h a v e a n e a s i e r t i m e l e a r n i n g h o w t o fix o u t b o a r d motors. Often, the things we w a n t to learn suggest a great deal of related k n o w l e d g e . In such cases, we don't n e e d a context strategy b e c a u s e the context is obvious. I n other c a s e s , w h e n t h e p r o b l e m s e e m s strange o r w h e n little related k n o w l e d g e s u g g e s t s itself, w e m a y w a n t t o u s e t h e c o n t e x t s t r a t e g y — t h a t is, w e m a y w a n t t o s e a r c h for a n a l o g i e s o r o l d k n o w l e d g e t o h e l p u s l e a r n t h e n e w m a t e r i a l . T h i s s i t u a t i o n m a y a r i s e w h e n w e start a n e w c o u r s e o r t a k e a n e w j o b . H e r e , s o m e t i m e s p e n t s e a r c h i n g for r e l e v a n t k n o w l e d g e m a y p a y off h a n d s o m e l y . A situation in w h i c h the context strategy is frequently useful is in finding the meanings of unfamiliar words. We have a great deal of k n o w l e d g e about w o r d m e a n i n g s in t h e form of w o r d roots. If it o c c u r s to us to u s e this k n o w l e d g e , we can often guess the m e a n i n g of a n e w w o r d . A l t h o u g h m a n y o f t h e r e l e v a n t w o r d roots are L a t i n o r G r e e k , o n e d o e s n ' t n e e d to k n o w either of these languages in order to do the analyses. All of the necessary information can be found in any good E n g l i s h dictionary. In learning foreign language vocabulary, one can guess the meanings of words l i k e a r b e (tree) a n d v i n ( w i n e ) b y n o t i c i n g t h e i r r e l a t i o n s t o t h e roots o f t h e E n g l i s h w o r d s arbor and v i n e . T h e m e a n i n g s o f the E n g l i s h w o r d s , o f course, m a y b e learned in the same way. For example, we can guess the m e a n i n g of the word pericardial i f w e r e l a t e i t t o roots o f w o r d s l i k e p e r i m e t e r a n d c a r d i a c . S p e l l i n g can also b e m a d e easier b y k n o w l e d g e o f w o r d roots. F o r e x a m p l e , s p e l l i n g w o r d s l i k e pneumonia, pneumatic, psychology, a n d psychomotor is e a s i e r i f w e k n o w t h a t " p n e u m o " i s a root for l u n g a n d " p s y c h o " t h e r o o t for m i n d . Monitoring C e r t a i n l y o n e o f t h e m o s t p o w e r f u l o f t h e l e a r n i n g s t r a t e g i e s i s monitoring. T h i s is the strategy in w h i c h learners test t h e m s e l v e s to find out w h a t t h e y have and

Learning Strategies

119

h a v e n o t l e a r n e d . T h e n t h e y u s e that i n f o r m a t i o n t o g u i d e s t u d y a n d p r o m o t e learning. O n e w a y t o m o n i t o r i s t o t e s t y o u r s e l f after y o u h a v e c o m p l e t e d a u n i t o f s t u d y . F o r e x a m p l e , after r e a d i n g a n a r t i c l e , y o u c o u l d w r i t e d o w n all y o u r e m e m b e r a b o u t t h e a r t i c l e a n d t h e n c h e c k y o u r r e s p o n s e a g a i n s t t h e o r i g i n a l . A n o t h e r w a y i s t o ask y o u r s e l f q u e s t i o n s while y o u are r e a d i n g . A s k i n g a q u e s t i o n f o c u s e s y o u r a t t e n t i o n o n a p a r t o f t h e t e x t a n d m a k e s i t m o r e l i k e l y t h a t y o u w i l l l e a r n t h a t part. B y a s k i n g y o u r s e l f m a n y questions w h i l e y o u read, y o u s h o u l d b e able t o learn the text more r a p i d l y than if y o u don't ask q u e s t i o n s . As we saw in C h a p t e r 4, it m a y be useful to spend as m u c h as 80 percent of your study time in such self-testing activities. W h a t sort o f q u e s t i o n s s h o u l d y o u a s k ? I f y o u r o b j e c t i v e i s j u s t t o g e t t h e facts, the questions can be quite superficial: " W h y did R o o s e v e l t d e c i d e to run as a third p a r t y c a n d i d a t e ? " or, " W h a t w e r e t h e n a m e s o f t h e first r a i l w a y c o m p a n i e s ? " I f y o u r o b j e c t i v e i s t o e v a l u a t e t h e t e x t o r t o d e e p e n y o u r u n d e r s t a n d i n g o f it, t h e n y o u s h o u l d e m p l o y t h e inference strategy, w h i c h r e q u i r e s y o u t o a s k m o r e p e n e t r a t i n g questions. Inferencing This strategy is related to monitoring b e c a u s e it involves asking questions about the material studied, and it is related to contextualizing b e c a u s e , typically, it e x p l o r e s t h e r e l a t i o n o f o t h e r k n o w l e d g e t o t h e m a t e r i a l b e i n g l e a r n e d . Its o b j e c t i v e , h o w e v e r , i s t o g e n e r a t e i n f e r e n c e s a b o u t t h e m a t e r i a l i n o r d e r t o e v a l u a t e its truth o r i m p o r t a n c e , t o w o r k o u t its i m p l i c a t i o n s , o r t o d e e p e n u n d e r s t a n d i n g . Carrying out this strategy i n v o l v e s asking p r o b i n g questions, such as: " I s the author b i a s e d ? " or, " C a n I t h i n k o f a c o u n t e r - e x a m p l e ? " or, " H o w w o u l d t h e a r g u m e n t s e e m i f the roles w e r e r e v e r s e d ? " I n C h a p t e r 10, w e w i l l d e s c r i b e s o m e critical thinking t e c h n i q u e s that can h e l p y o u to find " p e n e t r a t i n g q u e s t i o n s . " Instantiation Often, w h e n w e are h a v i n g trouble u n d e r s t a n d i n g s o m e t h i n g that s o m e o n e i s telling us, w e w i l l say, " C a n y o u g i v e m e a n e x a m p l e o f t h a t ? " T h i s i s a n e x a m p l e o f t h e instantiation strategy. W e s a w i n C h a p t e r 4 t h a t e x a m p l e s h e l p u s t o l e a r n . In the instantiation strategy, we s e e k out e x a m p l e s that illustrate the materials we are t r y i n g t o m a s t e r . W e c a n d o t h i s i n t w o w a y s : 1. We can p a y careful attention to the e x a m p l e s that others g i v e us in lectures, in textbooks, or in a n s w e r to questions such as, " C a n y o u g i v e me an e x a m p l e of that?" and 2 . W e c a n try t o c r e a t e e x a m p l e s for o u r s e l v e s . I t i s c o m m o n for p e o p l e t o d o this w h e n t h e y are r e a d i n g a difficult text. In a r e c e n t study by F l o w e r , H a y e s , and Swarts (1980), p e o p l e w e r e a s k e d to t h i n k a l o u d as t h e y r e a d a difficult federal r e g u l a t i o n a n d t r i e d t o u n d e r s t a n d it. T h e p r o t o c o l s w e r e p e p p e r e d w i t h e x a m p l e s t h e r e a d e r s c r e a t e d for t h e m s e l v e s i n a n effort t o u n d e r s t a n d . H e r e i s a n instance. [The subject reads " i n e l i g i b l e c o n c e r n s , " and t h e n says] " S a y that if a f e l l o w has a b a r a n d h e ' s s e l l i n g m o o n s h i n e w h i c h is n o t t a x e d . . . ." Multiple C o d i n g T h e e s s e n c e of the multiple-coding strategy is to represent the information we want to r e m e m b e r in m o r e than o n e w a y . We can do this by paraphrasing, forming images, w e a v i n g the material into a story, a n d by m a n y other w a y s . F o r e x a m p l e , suppose we w a n t e d to r e m e m b e r the information in the following sentence: " B r a v e W o r m f i n i s h e d first; F i g N e w t o n , s e c o n d ; a n d K i n g K o n g , t h i r d . "

The C o m p l e t e Problem Solver

120

W e c o u l d paraphrase this as: " T h e w i n , p l a c e , and s h o w horses w e r e Brave W o r m , F i g N e w t o n , a n d K i n g K o n g . " W e c o u l d f o r m a n i m a g e o f a w o r m w i t h its c h e s t s t u c k o u t s i t t i n g o n a f i g n e w t o n h e l d b y K i n g K o n g , o r w e c o u l d c r e a t e a story a b o u t t h e first o f a l i n e o f b r a v e w o r m s w h i c h all b y i t s e l f a t e t w o f i g n e w t o n s a n d a t h i r d of King Kong. A n y of these methods of multiple coding should m a k e the information easier to recall. Attention M a n a g e m e n t Far and a w a y the most important cause of failure to learn is failure to study. If y o u haven't read an article, y o u m a y w e l l e x p e r i e n c e some difficulty in answering q u e s t i o n s a b o u t it. W e all h a v e l i m i t e d t i m e a n d w e m a y o f t e n h a v e d i f f i c u l t y i n f i n d i n g t h e t i m e to do the r e a d i n g that is e x p e c t e d of us. S o m e p e o p l e , t h o u g h , s e e m to h a v e more difficulty than others. Differences in performance on the j o b or in the classroom may d e p e n d m o r e on differences in the amount of time invested than on differences in ability. Some p e o p l e have a very poor idea of w h e r e their time goes. Y o u may be one of these. For example, do y o u k n o w h o w many hours a w e e k you spend reading to learn? D o y o u k n o w h o w many hours you spend watching T V ? D o you know h o w m u c h of your time is spent b e i n g polite to p e o p l e w h o are interrupting what y o u really w a n t to do? If y o u did know, y o u might be sufficiently horrified to do s o m e t h i n g a b o u t it. A p p e n d i x I d e s c r i b e s s o m e p r o c e d u r e s t h a t c a n h e l p w i t h t i m e management. Concentration P e r h a p s y o u are p u t t i n g i n e n o u g h t i m e o n y o u r l e a r n i n g tasks b u t y o u are w o r k i n g v e r y i n e f f i c i e n t l y . F o r e x a m p l e , y o u m a y d u t i f u l l y sit d o w n for a l o n g s t u d y session and find that y o u h a v e terrible trouble concentrating. Often w h e n w e are reading, w e discover that w e haven't b e e n p a y i n g attention. O u r e y e s w e r e scann i n g l i n e after l i n e , b u t o u r m i n d w a s far a w a y . W e h a v e n o i d e a w h a t t h e last f e w p a g e s w e r e about. T h i s can b e a v e r y a n n o y i n g p r o b l e m w h e n w e are trying t o m e e t a d e a d l i n e , s u c h a s a t e s t i n t h e m o r n i n g o r a m e e t i n g after l u n c h . W e c a n ' t s p a r e the time but we can't s e e m to concentrate either! A p p e n d i x I provides some suggestions that can h e l p y o u solve concentration p r o b l e m s . S T U D Y SYSTEMS Study systems are organized approaches to learning, especially to learning from r e a d i n g . T h e r e are m a n y different study s y s t e m s ; typically t h e y i n v o l v e s o m e r u l e o r s e q u e n c e for a p p l y i n g s o m e o f t h e b a s i c l e a r n i n g s t r a t e g i e s , a n d m a n y include p r o b l e m - s o l v i n g strategies as w e l l . Survey Q 3 R O n e of the most w i d e l y used study systems is the Survey Q 3 R system devised b y R o b i n s o n (1946). T h e five steps i n this s y s t e m — s u r v e y , q u e s t i o n , read, recite, and r e v i e w — a r e d e s c r i b e d i n R o b i n s o n ' s o w n s u m m a r y (Robinson, 1946): SURVEY

•>

1. Glance over the headings in the chapter to see the few big points which will be developed. This survey should not take more than a minute and w i l l show the three to six core ideas around w h i c h the rest of the discussion will cluster. If the chapter has a final summary paragraph this w i l l also list the ideas d e v e l o p e d in the chapter. This orientation w i l l h e l p you organize the ideas as y o u read them later.

Learning Strategies

121

QUESTION 2.

READ

RECITE

N o w b e g i n to work. Turn the first heading into a question. This will arouse your curiosity and so increase comprehension. It will bring to m i n d information already known, thus h e l p i n g you to understand that section more quickly. A n d the question w i l l make important points stand out w h i l e explanatory detail is recognized as such. T h i s turning a heading into a question can be done on the instant of reading the heading, but it demands a conscious effort on the part of the reader to make this query for which he must read to find the answer. 3. Read to answer that question, i.e., to the e n d of the first h e a d e d section. T h i s is not a passive p l o w i n g along each line, b u t an active search for the answer. 4. Having read the first section, look away from the book and try briefly to recite the answer to your question. U s e your own words and name an example. If you can do this you know what is in the book; if you can't, glance over the section again. An excellent w a y to do this reciting from memory is to jot down cue phrases in outline form on a sheet of paper. M a k e these notes very brief! NOW REPEAT STEPS 2, 3 AND 4 ON EACH SUCCEEDING HEADED SECTION. THAT IS, TURN THE NEXT HEADING INTO A QUESTION, READ TO ANSWER THAT QUESTION, AND RECITE THE ANSWER HY JOTTING DOWN CUE PHRASES IN YOUR OUTLINE. READ IN THIS WAY UNTIL THE ENTIRE LESSON IS COMPLETED.

REVIEW

5. W h e n the lesson has thus b e e n read through, look over your notes to get a bird's-eye view of the points and of their relationship and check your memory as to the content by reciting on the major subpoints under each heading. This checking of memory can be done by covering up the notes and trying to recall the main points. T h e n expose each major point and try to recall the subpoints listed under it.

Survey Q R 3 m a k e s use of three of the s e v e n strategies. T h e survey step is an example of the structuring strategy. T h e question step i n v o l v e s monitoring, b u t Robinson intends it also to facilitate structuring and to p r o v i d e context by m a k i n g contact with previous k n o w l e d g e . T h e reciting step i n v o l v e s both monitoring and structuring, a n d the r e v i e w step is s i m p l y monitoring.

D a n s e r e a u ' s M U R D E R System T h e a c r o n y m M U R D E R ( r e m e m b e r a c r o n y m s ? ) s t a n d s for t h e six parts o f D a n s e r e a u e t al.'s ( 1 9 7 9 ) s t u d y s y s t e m : M o o d , U n d e r s t a n d , R e c a l l , D i g e s t , E x p a n d , and R e v i e w . 1 . Mood. T h e first s t e p o f M U R D E R i s s e t t i n g t h e m o o d for s t u d y . D a n s e r e a u s e e s t w o major p r o b l e m s i n s e t t i n g t h e r i g h t m o o d . O n e i s c r e a t i n g a p o s i t i v e a t t i t u d e — t h a t is, s o m e h o w o v e r c o m i n g t h e fear a n d l o a t h i n g o f study. T h e second is c o p i n g w i t h distractions, e.g., y o u r r o o m m a t e practicing e l e p h a n t mating calls. B o t h o f t h e s e p r o b l e m s are attention m a n a g e m e n t p r o b l e m s , w h i c h are treated in A p p e n d i x I u n d e r the h e a d i n g " C o n c e n t r a t i o n . " 2 . Understand. I n u s i n g t h e s y s t e m , y o u a r e e n c o u r a g e d w h e n first r e a d i n g a t e x t t o m a r k a n y parts o f t h e t e x t y o u d o n ' t u n d e r s t a n d . D u r i n g t h e " D i g e s t " s t e p , y o u a t t e n d t o t h e m a r k e d p a r t s w h i c h a r e still u n c l e a r t o y o u after f u r t h e r r e a d i n g . I n learning to u s e the s y s t e m , y o u are trained to locate the p r o b l e m — a difficult w o r d , s e n t e n c e , or p a r a g r a p h — a n d to b r e a k it into parts, e.g., y o u are trained to b r e a k a d i f f i c u l t w o r d i n t o its r o o t s . I f t h e m e a n i n g i s still n o t c l e a r , y o u are e n c o u r a g e d to look at the context of the difficulty to find related information, and i f all e l s e fails, t o c o n s u l t o u t s i d e s o u r c e s , e . g . , d i c t i o n a r i e s , r e s e a r c h l i b r a r i a n s , etc. T h e " U n d e r s t a n d " a n d " D i g e s t " s t e p s , t h e n , i n v o l v e a p r o b l e m - s o l v i n g s t r a t e g y — f r a c t i o n a t i o n — a n d a l e a r n i n g s t r a t e g y — t h e context strategy. 3 . Recall. A f t e r i n i t i a l r e a d i n g , y o u a r e i n s t r u c t e d t o r e c a l l t h e i n f o r m a t i o n y o u h a v e

722

T h e C o m p l e t e Problem Solver

b e e n r e a d i n g a b o u t a n d t o t r a n s f o r m i t u s i n g o n e (or p o s s i b l y b o t h ) o f t h e f o l l o w ing substrategies: 1 . Paraphrase-imagery. T h i s i s j u s t t h e m u l t i p l e - c o d i n g s t r a t e g y d e s c r i b e d above. 2 . Networking. Y o u i d e n t i f y r e l a t i o n s i n t h e m a t e r i a l s t o b e l e a r n e d , s u c h a s " p a r t of," " e x a m p l e of," a n d " l e a d s t o . " T h i s i s a n i n s t a n c e o f t h e s t r u c t u r i n g strategy d e s c r i b e d a b o v e . 4 . Digest. T h i s s t e p h a s a l r e a d y b e e n d e s c r i b e d u n d e r " U n d e r s t a n d . " 5 . Expand. I n t h i s s t e p , y o u a s k a n d a n s w e r t h r e e k i n d s o f q u e s t i o n s . 1 . I f y o u c o u l d t a l k t o t h e a u t h o r , w h a t sorts o f q u e s t i o n s o r c r i t i c i s m s w o u l d y o u raise? 2. H o w can the material be applied? 3. H o w could y o u m a k e the material more understandable and interesting to others? Clearly, the learning strategy i n v o l v e d here is inferencing. 6 . Review. H e r e , y o u r e v i e w y o u r e r r o r s w i t h t h e i n t e n t o f f i n d i n g t h e i r c a u s e s a n d m a k i n g appropriate c h a n g e s in your study habits. T h e relevant strategy here is monitoring. T h e M U R D E R s y s t e m , t h e n , m a k e s u s e o f all o f t h e l e a r n i n g s t r a t e g i e s e x c e p t instantiation. D a n s e r e a u tested the M U R D E R system by comparing 38 college students w h o took a 1 2 - w e e k learning strategies class with 28 students w h o d i d not take the class. Learning performance was measured three times in each group: once before the c l a s s b e g a n , o n c e h a l f w a y t h r o u g h , a n d o n c e after t h e c o u r s e w a s c o m p l e t e d . T h e m e a s u r e m e n t i n v o l v e d r e a d i n g a 3,000 w o r d p a s s a g e a n d t h e n t a k i n g a t e s t o n its contents o n e w e e k later. After training, the e x p e r i m e n t a l g r o u p p e r f o r m e d b e t w e e n 14 and 18 p e r c e n t better than the untrained control group. D a n s e r e a u also t e s t e d the n e t w o r k i n g strategy s e p a r a t e l y from the w h o l e s y s t e m . H e f o u n d t h a t after six h o u r s o f t r a i n i n g , t h e e x p e r i m e n t a l g r o u p r e m e m b e r e d 34 p e r c e n t m o r e m a i n ideas than the untrained control group, b u t about the same n u m b e r of details. B. F. Jones and her colleagues (1979) have d e v e l o p e d a study system to imp r o v e t h e r e a d i n g s k i l l s o f d i s a d v a n t a g e d c h i l d r e n i n t h e fifth t h r o u g h e i g h t h g r a d e s i n C h i c a g o p u b l i c s c h o o l s . S t u d e n t s are t a u g h t a c a r e f u l l y s e q u e n c e d s e t o f l e a r n i n g s t r a t e g i e s in a mastery learning format. M a s t e r y l e a r n i n g m e a n s t h a t e a c h s t u d e n t m u s t h a v e m a s t e r e d all o f t h e s t r a t e g i e s t a u g h t s o far, a s i n d i c a t e d b y p a s s i n g a t e s t , before b e i n g a l l o w e d to study the next strategy. T h e strategies i n c l u d e m u l t i p l e c o d i n g , e.g., v i s u a l i z i n g w o r d s in a n a l o g i e s ; structuring, e.g., differentiating m a i n points a n d details; c o n t e x t u a l i z i n g , e.g., finding the m e a n i n g of an unfamiliar w o r d from other information in the text; and i n f e r e n c i n g , e.g., differentiating o p i n i o n from fact in w h a t is read. Jones e t al. find that their l e a r n i n g strategies y i e l d m u c h greater i m p r o v e m e n t in the student's r e a d i n g c o m p r e h e n s i o n scores than d o e s m o r e traditional instruction.

SPECIAL T E C H N I Q U E S FOR H A R D CASES In this section, we w i l l s h o w h o w the learning strategies can be a p p l i e d in l e a r n i n g topics w h i c h m a n y p e o p l e find difficult, e.g., zoological categories, p h y s i c s e q u a t i o n s , and cultural history. O f c o u r s e , there are s o m a n y different topics that w e can't h o p e to c o v e r t h e m all. O u r intention h e r e is s i m p l y to p r o v i d e a f e w illustra-

Learning Strategies

723

tions o f h o w t h e l e a r n i n g strategies can b e a p p l i e d . W e h o p e that t h e s e e x a m p l e s w i l l h e l p y o u t o a p p l y the l e a r n i n g strategies w h e n y o u e n c o u n t e r difficult learning situations. T h e first t o p i c w i l l i l l u s t r a t e t h e a p p l i c a t i o n o f t h e s t r u c t u r i n g a n d c o n t e x t strategies. Learning C a t e g o r i e s in Z o o l o g y S u p p o s e t h a t y o u w a n t e d (or n e e d e d ) t o l e a r n m a t e r i a l l i k e t h e z o o l o g i c a l phyla, classes, and orders with associated properties and examples, as s h o w n in T a b l e 2. A t first s i g h t , t h e s t r a n g e a n d c o m p l e x n a m e s o f t h e c a t e g o r i e s m a y s e e m a t e r r i b l e b a r r i e r t o l e a r n i n g . I n fact, i f w e u s e t h e m i n t h e r i g h t w a y , t h e n a m e s b e c o m e a k e y t o a g r e a t d e a l o f k n o w l e d g e w e a l r e a d y h a v e s t o r e d — k n o w l e d g e that c a n b e c o n n e c t e d t o t h e n e w m a t e r i a l t o h e l p u s i n l e a r n i n g it. O n t h e r i g h t - h a n d s i d e o f T a b l e 2 a r e t h e roots o f t h e c a t e g o r y n a m e s . S o m e o f t h e m , l i k e " c n i d a " a n d " e c h i n o " a r e u n f a m i l i a r t o m o s t o f u s , b u t m a n y a r e parts o f w o r d s w e a r e a l r e a d y f a m i l i a r w i t h , e . g . , p r o t o t y p e , pyrocanthus (fire t h o r n ) , arthritis, d e n t i s t , e t c . T h e s e roots s h o u l d b e e a s y t o l e a r n b e c a u s e w e c a n c o n n e c t them to k n o w l e d g e we already have. W h e n w e k n o w the m e a n i n g s o f the roots, the c a t e g o r y n a m e g i v e s u s a t h u m b nail d e s c r i p t i o n of the animals in the category. S i n c e m o l l u s m e a n s "soft," M o l l u s c a s e e m s a v e r y r e a s o n a b l e n a m e for c l a m s a n d o y s t e r s . W e c a n i m a g i n e w h a t t h e A c a n t h o c e p h a l i (thorny heads) w i l l look like, and w e can g u e s s that lobsters m i g h t well be Arthropoda. O n e o f t h e m o s t i m p o r t a n t facts a b o u t c a t e g o r i e s i n z o o l o g y i s t h a t t h e y are hierarchically organized, as s h o w n in Figure 4. We could learn the categories w i t h o u t p a y i n g attention to the hierarchy, b u t learning w o u l d be m o r e difficult and m u c h less m e a n i n g f u l . F o r e x a m p l e , a p e r s o n w h o d o e s n ' t u n d e r s t a n d the hierarc h i c a l o r g a n i z a t i o n c a n ' t f i g u r e o u t t h a t b i r d s are m o r e c l o s e l y r e l a t e d t o r e p t i l e s than to s p o n g e s . T h e structure s h o w n in T a b l e 2 reflects several important principles of zoology. For e x a m p l e , the p h y l a are o r g a n i z e d by a principle of c o m p l e x i t y . T h e simplest of t h e p h y l a s , t h e P r o t o z o a , c o n s i s t o f s i n g l e c e l l s . T h e n e x t l e v e l o f c o m p l e x i t y involves multicellular phyla like Porifera and Cnidaria w h i c h have tissues but no organs. M o r e c o m p l e x p h y l a h a v e organ systems of v a r y i n g d e g r e e s of elaborateness. A s e c o n d o r g a n i z i n g p r i n c i p l e for t h e p h y l a i s t h e p r i n c i p l e o f e n c e p h a l i z a t i o n . By encephalization, we m e a n an increasing importance of the head. As we progress from u n o r g a n i z e d b l o b s t o p e o p l e , w e f i n d a n i n c r e a s i n g p r e f e r e n c e for h e a d - f i r s t motion, with an increasing concentration of sensory systems and neural centers at the h e a d e n d . S i m p l e p h y l a like Protozoa and C n i d a r i a s h o w little encephalization. W e f i n d i n c r e a s i n g e n c e p h a l i z a t i o n w i t h i n c r e a s i n g c o m p l e x i t y a s w e m o v e from Platyhelminthes to A n n e l i d a to Chordata. O t h e r o r g a n i z i n g p r i n c i p l e s for t h e c a t e g o r i e s i n T a b l e 2 i n v o l v e c h a n g e s i n s p e c i f i c o r g a n s y s t e m s . C h a n g e s i n t h e d i g e s t i v e tract p r o v i d e o n e o f t h e m o s t f u n d a m e n t a l o f t h e o r g a n i z i n g s c h e m e s for t h e p h y l a . T h e t w o l e a s t c o m p l e x p h y l a , Protozoa and Porifera, h a v e no d i g e s t i v e tract at all. T h e m o s t c o m p l e x of the p h y l a , the Molluscs, A n n e l i d s , Arthropods, E c h i n o d e r m s , and Chordates, have w e l l d e v e l o p e d d i g e s t i v e s y s t e m s w i t h a true c o e l u m . A true c o e l u m is a space separati n g t h e d i g e s t i v e tract f r o m t h e b o d y w a l l w h i c h i s l i n e d w i t h a s p e c i a l t i s s u e c a l l e d p e r i t o n e u m . C n i d a r i a a n d P l a t y h e l m i n t h e s , w h i c h are a s t e p a b o v e t h e s p o n g e s i n c o m p l e x i t y , h a v e a d i g e s t i v e tract b u t n o c o e l u m . T h e A c a n t h o c e p h a l a , w h i c h l i e

Table 2. Zoological Categories Name

s i-H

u

S-l

1)

o

Example

Protozoa

amoeba

Porifera Cnidaria Platyhelminthes Acanthocephala Mollusca Annelida

sponge jellyfish flatworms

Arthropoda Echinodermata Chordata (Vertebrates)

clam earth w o r m spiders, crabs starfish cat, p e o p l e

Agnatha Placodermi Osteichthyes Chondrichthye s Amphibia Reptilia Aves Mammalia

lampreys (extinct) salmon sharks, rays frogs reptiles birds mammals

Monotremata Marsupialia Insectivora Dermoptera Chiroptera Edentata Proboscoidia

duck-billed platypus opossum, kangaroo shrew flying lemurs bats anteaters, sloths elephants

Meaning proto (primitive, original) + z o a (animal) p o r o s ( p a s s a g e , p o r e ) + f e r r e (to b e a r ) cnida (nettle, stinging cell) p l a t y (flat) + h e l m i n t h e s ( w o r m ) acantho (thorny) + c e p h a l u s (head) m o l l u s (soft) a n n u l u s (ring) arthro (jointed) + p o d a (legs) e c h i n o ( s p i n y ) + d e r m u s (skin) c h o r d a (string, c o r d ) a (no) + g n a t h o s (jaw) p l a c o ( p l a t e s ) + d e r m u s (skin) o s t i o ( b o n y ) + i c h t h y e s (fish) c h o n d r o ( c a r t i l a g e ) + i c h t h y e s (fish) a m p h i (on b o t h s i d e s , a r o u n d ) r e p t i l i a (from r e p e r e ; t o c r a w l ) avis (bird) m a m m a (breast) m o n o (one) + trema (hole) m a r s u p i u m (little b a g ) i n s e c t u m + v o r a r e (to d e v o u r ) d e r m u s (skin) + p t e r o n ( w i n g ) c h i r u s (hard) + p t e r o n ( w i n g ) e (without) + d e n t u m (teeth) proboscis

3

era ro

The C o m p l e t e Problem Solver

726

b e t w e e n t h e P l a t y h e l m i n t h e s a n d t h e M o l l u s c s , h a v e a p s e u d o c o e l u m — t h a t is, a space w h i c h is like a c o e l u m but w h i c h is not lined with peritoneum. I n t h e s a m e w a y t h a t t h e d i g e s t i v e s y s t e m p r o v i d e s g r o u p i n g a n d o r d e r i n g for t h e p h y l a , t h e c i r c u l a t o r y s y s t e m p r o v i d e s g r o u p i n g s a n d o r d e r i n g for t h e c l a s s e s o f vertebrates. Fish have t w o - c h a m b e r e d hearts, amphibians h a v e three-chambered hearts, and reptiles, birds, and m a m m a l s have four-chambered hearts. In general, the various organizing principles such as encephalization and complexity g r o u p order the categories in w a y s that are consistent w i t h e a c h other and with an overall principle of evolutionary development. In zoology, then, understanding the structure of the categories is closely r e l a t e d t o u n d e r s t a n d i n g t h e s c i e n c e itself. T o s e e t h a t t h e l e a r n i n g s t r a t e g i e s r e a l l y h e l p t o l e a r n z o o l o g y , c o n s i d e r this e x a m p l e . S u p p o s e y o u w e r e t o hear that the imaginary p h y l u m " P l a t y p o d a " w a s considered the immediate evolutionary descendant of the Annelida. You w o u l d k n o w i m m e d i a t e l y t h a t t h e m e m b e r s o f t h e p h y l u m are c h a r a c t e r i z e d b y flat f e e t ; that t h e y a r e m o d e r a t e l y c o m p l e x , w i t h w e l l - d e v e l o p e d o r g a n s y s t e m s , a d e f i n i t e head-to-tail orientation, and a d i g e s t i v e system w i t h a true c o e l u m . T h a t is really q u i t e a l o t of i n f o r m a t i o n to g e t f r o m a n a m e a n d a p o s i t i o n in an o r d e r i n g . T h e s e l e a r n i n g s t r a t e g i e s are v e r y i m p o r t a n t i n b i o l o g i c a l s c i e n c e s , m e d i c i n e , geology, chemistry, and m a n y other areas of k n o w l e d g e .

LEARNING

MATHEMATICAL

FORMULAS

S u p p o s e that y o u h a v e an a s s i g n m e n t that r e q u i r e s y o u to learn formulas of the sort t y p i c a l l y f o u n d i n p h y s i c s o r e n g i n e e r i n g t e x t s . T h e r e are a n u m b e r o f p r o c e d u r e s y o u c a n u s e t o m a k e t h i s task e a s i e r a n d m o r e m e a n i n g f u l . W e w i l l d e s c r i b e one w h i c h m a k e s use of the multiple c o d i n g and context strategies. Physical Interpretation of Equations O n e o f t h e m o s t p o w e r f u l p r o c e d u r e s for l e a r n i n g a n d u n d e r s t a n d i n g e q u a t i o n s is the process of physical interpretation. By physical interpretation we m e a n the process in w h i c h p e o p l e m a k e use of k n o w l e d g e of a p h y s i c a l situation, perhaps in the form of s k e t c h e s or visual i m a g e s , to h e l p t h e m u n d e r s t a n d or learn an equation. O n e w a y we can use physical interpretation is to h e l p us r e m e m b e r whether a quantity should be p l a c e d in the numerator or denominator of an equation. B e f o r e d e s c r i b i n g h o w this w o r k s , l e t m e m e n t i o n a p r o p e r t y o f e q u a t i o n s w h i c h the math haters in the a u d i e n c e m a y either h a v e forgotten or n e v e r learned i n t h e first p l a c e . C o n s i d e r t h i s a r b i t r a r y e q u a t i o n : (A • B) (D

+

+

C

E)F

H o w does X change with A, B, C, D, E, and F? T h e fact I w a n t y o u t o n o t i c e i s t h a t 1. X g e t s bigger as A, B, a n d C g e t b i g g e r because those quantities are in t h e numerator, and 2. X g e t s smaller as D, E, a n d F g e t b i g g e r because those three quantities are in the denominator. N o w , s u p p o s e t h a t y o u w a n t e d t o r e m e m b e r t h e e q u a t i o n for t h e f o r c e o f g r a v i t y b e t w e e n t w o objects. Y o u k n o w that the force, F, is e q u a l to s o m e c o m b i n a t i o n of G, and constant, M and M , the masses of the objects, and r, the distance b e t w e e n them. Y o u think it might be 1

2

727

Learning Strategies

Physical interpretation might involve imagining ourselves floating in space holding t w o l a r g e g l o b e s apart. I f e i t h e r o f t h e g l o b e s w e r e v e r y h e a v y , w e w o u l d e x p e c t t h a t it w o u l d be harder to h o l d t h e m apart than if both w e r e light. S i n c e force increases as either of the masses (M's) increases, the masses must be in the numerator. (Why?) A s w e p u s h the g l o b e s farther apart, t h e force o f attraction b e t w e e n t h e m w i l l d e c r e a s e as the force of attraction b e t w e e n t w o m a g n e t s d e c r e a s e s as we pull t h e m apart. S i n c e f o r c e d e c r e a s e s a s d i s t a n c e , r , i n c r e a s e s , r m u s t b e i n t h e d e n o m i n a t o r . <

y ) GMM2 O f t h e t h r e e e q u a t i o n s a b o v e , o n l y (b), F = , satisfies t h e s e relations. It is, i n fact, t h e c o r r e c t e q u a t i o n . A s l i g h t l y m o r e c o m p l e x e x a m p l e i n v o l v e s t h e e q u a t i o n for t h e c a p a c i t y o f a condenser: w h

?

r

4TTD

A c o n d e n s e r i s a d e v i c e for s t o r i n g a n e l e c t r i c a l c h a r g e . I t c o n s i s t s o f t w o m e t a l plates p l a c e d close to e a c h other b u t not touching, as s h o w n in F i g u r e 5. A cond e n s e r w o r k s , that is, stores electricity, m o r e easily than say, the a v e r a g e d o o r k n o b , for t w o r e a s o n s :

Figure 5. A Diagram of a C o n d e n s e r

1. Charges of the same sign repel each other and they repel more strongly the closer t h e y are. T h e r e f o r e , it is m o r e difficult to p u t a c h a r g e onto a plate w i t h a s m a l l a r e a , w h e r e t h e e l e c t r o n s are c r o w d e d , t h a n o n t o a p l a t e w i t h a l a r g e a r e a w h e r e the electrons c a n spread out.

128 2.

The C o m p l e t e Problem Solver

C h a r g e s o f o p p o s i t e s i g n attract e a c h o t h e r , a n d t h e y attract m o r e s t r o n g l y t h e c l o s e r t h e y a r e . T h e fact t h a t t h e o p p o s i t e c h a r g e s a r e c l o s e t o g e t h e r i n a c o n d e n s e r m e a n s that there is an attractive force h o l d i n g the charge on the cond e n s e r w h i c h in part counteracts the forces by w h i c h the charges of the same kind on e a c h plate r e p e l e a c h other. N o w , back to the equation: C

-

k

A

4TTD

T h e capacity of the c o n d e n s e r , C, is a measure of h o w easy it is to place a given charge on the c o n d e n s e r . A is the area of the plates and D is the distance b e t w e e n them, (k is a constant w h i c h varies with the material b e t w e e n the places and won't concern us here.) It is clear from w h a t we h a v e said a b o v e that capacity must increase as the area of the plates increases and as the distance b e t w e e n the plates decreases. A therefore must be in the numerator of the equation and D in the denominator. Physical interpretation can also h e l p us learn the values of constants in equat i o n s . C o n s i d e r t h e e q u a t i o n for t h e v o l u m e o f a s p h e r e : V = - 7T r 3

3

I s t h e c o n s t a n t 4/3w r e a s o n a b l e ? T o e s t i m a t e w h a t t h e c o n s t a n t o u g h t t o b e , i m a g i n e that the s p h e r e is i n s i d e a c u b e just 2r on a side, as s h o w n in F i g u r e 6. T h e s p h e r e

A

Figure 6. S p h e r e Inside a C u b e From T w o Angles

729

Learning Strategies

j u s t t o u c h e s t h e c u b e a t a f e w p o i n t s . T h e r e i s a fair a m o u n t o f e m p t y s p a c e b e t w e e n the s p h e r e a n d t h e c u b e . W e m i g h t estimate that h a l f o f the s p a c e i n s i d e the c u b e i s o c c u p i e d b y t h e s p h e r e . N o w , s i n c e t h e s i d e o f t h e c u b e i s 2r, its v o l u m e i s (2r) = 8r 3

3

T h e r e f o r e , we estimate that the v o l u m e of the s p h e r e is r o u g h l y h a l f of that, or 4r

3

N o w s i n c e T T i s a p p r o x i m a t e l y 3 , 4/3-n- i s a p p r o x i m a t e l y 4 , w h i c h i s a b o u t r i g h t . I n t h e s a m e w a y , c o n s i d e r i n g t h e e q u a t i o n for t h e s u r f a c e a r e a o f t h e s p h e r e , A = 47rr

2

w e c a n a s k w h e t h e r o r n o t t h e c o n s t a n t 4TT i s r e a s o n a b l e . T o e s t i m a t e t h e c o r r e c t v a l u e , i m a g i n e a s q u a r e 2 r o n a s i d e i n t o w h i c h t h e s p h e r e w i l l j u s t fit, a s s h o w n i n F i g u r e 7. T h i s square has an area of (2r) = 4r 2

2

N o w i m a g i n e trying to c o v e r the sphere w i t h the square. By noting that o n e square w o n ' t c o v e r e v e n a half of the sphere, we m i g h t estimate that it w i l l take about three squares t o c o v e r t h e s p h e r e . T h e total area, t h e n , w o u l d b e a b o u t 3 times 4r

2

T h u s , 4 7 r r is a b o u t right. 2

T h e m e t h o d o f p h y s i c a l i n t e r p r e t a t i o n w o r k s w e l l for e s t i m a t i n g m a n y c o n s t a n t s . Try to apply it in the following exercises. Exercises 1.

U s e p h y s i c a l interpretation to s h o w that the constants in the f o l l o w i n g e q u a t i o n s are a b o u t right. C = 27rr w h e r e C is t h e c i r c u m f e r e n c e of a c i r c l e a n d r is its r a d i u s . V = l/3-7rr h 2

w h e r e V i s t h e v o l u m e o f a r i g h t c i r c u l a r c o n e , r i s t h e r a d i u s o f its b a s e , a n d h is its h e i g h t .

Learning Cultural History In fields like z o o l o g y and physics w h i c h h a v e w e l l - d e v e l o p e d theories, it is relatively easy to find strong organizing principles to aid learning and c o m p r e h e n s i o n . I n t h e h u m a n i t i e s , h o w e v e r , w h e r e t h e o r i e s a r e n o t a s w e l l d e v e l o p e d , organ i z i n g principles m a y be harder to find. C o n s i d e r the task of l e a r n i n g information a b o u t c u l t u r a l h i s t o r y , t h a t i s , i n f o r m a t i o n a b o u t t h e r e l a t i o n s o f art, s c i e n c e , m u s i c , politics, etc., o v e r a p e r i o d o f t i m e . W e c a n g e t h e l p i n o r g a n i z i n g s u c h material b y relating it to artistic m o v e m e n t s s u c h as c l a s s i c i s m or r o m a n t i c i s m , to r e l i g i o u s movements, such as the Reformation or the counter-Reformation, or to political events, such as the succession of the British monarchs. Often p e o p l e s e e m to apply such organizing principles v e r y narrowly. T h u s , a person m a y k n o w that baroque music p r e c e d e d romantic m u s i c and that L o u i s X I V p r e c e d e d N a p o l e o n , b u t h e m a y h a v e n o i d e a w h a t sort o f m u s i c w a s b e i n g c o m p o s e d d u r i n g e i t h e r m a n ' s r e i g n . In general, c o l l e g e students s e e m to h a v e difficulty organizing p e o p l e and events in t i m e . T h e test s h o w n in T a b l e 3 w a s a d m i n i s t e r e d to the freshman class

HISTORY TEST: INSTRUCTIONS T h i s is a test of y o u r ability to identify the times of historically important people. T h e test sheet contains a time line e x t e n d i n g from 1 1 0 0 A D t o 1900 A D and a list of 51 n a m e s . T r y to identify the t i m e of e a c h of the p e o p l e on the list by writing the n u m b e r of the person (1 for B a c h , 2 for D a n t e , e t c . ) at a p o i n t on the time line w h i c h corresponds to the time at w h i c h the person was

T a b l e 3. History Test Results

1100 1. 2. 3. 4.

1200

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Bach Dante Louis XIV Giotto Hieronymus Bosch Marco Polo Shakespeare Napoleon Elizabeth I Gutenberg Disraeli Cromwell B e n Franklin Van Gogh Lavoisier Mozart Machiavelli

18. Rembrandt 19. Darwin 20. Jane A u s t e n 21. Wagner 22. N e w t o n 23. Scarlatti 24. C h a u c e r

Percentage of students

Absolute average error in

recognizing

years

99% 94 93 26

117 195 79

99 93 64 58 86

109 140 120 73 67 95 139 60 89

100 98 40

13 150 102

98 77 98 99 60 70

140 112

20 99 100

99 24 90

105 64 43 101 133 148 126

1800

-

an adult. As an e x a m p l e , n u m b e r 5 1 , G e o r g e W a s h i n g t o n , has already b e e n placed on the time line. A n y time w h i c h falls b e t w e e n a p e r s o n ' s t w e n tieth birthday and his death w i l l be counted as correct. If y o u do not r e c o g n i z e a n a m e , cross out that n a m e . If y o u do r e c o g n i z e a name, put it some place on the time line, e v e n if you have to guess.

1

3

0

0

25. 26. 27. 28.

Tintoretto Henry VIII Copernicus Delacroix

29. 30. 31. 32. 33. 34. 35. 36. 37.

Galileo Milton Martin Luther Gainsborough Debussy Vermeer Fielding Dostoievski Van Eyck

38. L e o n a r d o d a V i n c i 39. B e e t h o v e n 40. Hogarth 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 1400

Richard the Lionhearted Tennyson El Greco Marie Antoinette Goya Descartes Byron Calvin Jonathan Swift Thomas Aquinas George Washington 1500

20 100 91 38 96 71 98 41 65 37 36 73 45 98 99 25 94 87 70 96 60 83 77 86 82 67

155 91 163 125 175 126 120 104 141 94 95 66 210 90 132

1700

128 176 127 122 115 126 143 93 119 68 207 1600

--

132

The C o m p l e t e Problem Solver

entering Carnegie-Mellon's C o l l e g e of Humanities and Social Sciences in September, 1973. H o w d i d the humanities freshmen d o ? First, e v e r y o n e r e c o g n i z e d the n a m e of Benjamin Franklin. Furthermore, nearly three-quarters w e r e able to place him accurately on the t i m e scale. O t h e r historic.persons w h o w e r e universally recognized were Shakespeare, Henry VIII, Bach, Napoleon, N e w t o n , Darwin, and B e e t h o v e n . H o w e v e r , the a v e r a g e error in p l a c i n g p e o p l e on the t i m e scale w a s in e x c e s s o f a c e n t u r y — a n d t h i s w a s j u s t for p e o p l e r e c o g n i z e d . M o r e t h a n a q u a r t e r of the students did not recognize the names of Byron, Milton, W a g n e r , El Greco, and D e b u s s y , and m o r e than 75 p e r c e n t failed to r e c o g n i z e Scarlatti a n d Tintoretto. T h e j u d g m e n t s reveal a curious "time w a r p " in the students' perception of history. F a m o u s c o m p o s e r s as a g r o u p are p l a c e d a b o u t 140 years early a n d famous scientists are p l a c e d a b o u t 130 years early. In contrast, political a n d religious leaders are p l a c e d a b o u t 45 years late. T h e m a g n i t u d e of the w a r p suggests that there is s o m e t h i n g faulty in the s t u d e n t ' s p e r c e p t i o n o f h i s c u l t u r e . A f t e r a l l , a s l i p p a g e o f six o r s e v e n g e n e r a t i o n s makes some very surprising contemporaries. For example, the average judgments make D e b u s s y and Franklin contemporaneous, and they make N e w t o n a few years older than H e n r y V I I I . O n e a p p r o a c h to structuring a b o d y of k n o w l e d g e s u c h as this is to p r o v i d e a set of landmarks. In l e a r n i n g our w a y around a city, we t y p i c a l l y m a k e use of tall buildings, parks, or other distinctive features to k e e p us oriented to the overall plan o f the city ( L y n c h , 1960). I n the s a m e w a y , w e can p r o v i d e o u r s e l v e s w i t h landm a r k s for l e a r n i n g c u l t u r a l h i s t o r y . T h e m a t r i x s h o w n i n T a b l e 4 i s a f r a m e w o r k o f l a n d m a r k s for l e a r n i n g t h e r e l a t i o n s o f art, s c i e n c e , a n d p o l i t i c s b e t w e e n 1 4 0 0 a n d 1900. I t i s o r g a n i z e d b y t i m e a n d b y field. I n c o n s t r u c t i n g t h e matrix w e d i v i d e d t i m e i n t o 5 0 - y e a r i n t e r v a l s a n d d e f i n e d six m a j o r f i e l d s . F o r e a c h t i m e i n t e r v a l , w e tried to find a notable i n d i v i d u a l in e a c h field w h o m a d e his most important contribution d u r i n g that t i m e interval. T h u s , Mozart w a s c h o s e n as a c o m p o s e r active b e t w e e n 1 7 5 0 a n d 1800 a n d G a l i l e o , as a scientist a c t i v e b e t w e e n 1600 a n d 1650. W h i l e the 50-year interval provides only a rather c r u d e specification of time, we b e l i e v e that students u s i n g the matrix s h o u l d b e a b l e t o m a k e t i m e estimates w h i c h are c o n s i d e r a b l y m o r e a c c u r a t e t h a n t h e s t u d e n t s w e t e s t e d w h o w e r e i n e r r o r o n the average by more than a century. T h e w o r k i n v o l v e d in l e a r n i n g a matrix s u c h as that s h o w n in T a b l e 4 is c o n s i d e r a b l e . To m a k e the task of l e a r n i n g it a bit easier a n d m o r e palatable, we h a v e p r o v i d e d the list o f c o n n e c t i o n s s h o w n i n T a b l e 5.* T h e s e c o n n e c t i o n s relate pairs of p e o p l e in the matrix through an e a s y - t o - r e m e m b e r bit of biographical information. F o r e x a m p l e , k n o w i n g that Voltaire's mistress translated N e w t o n ' s w o r k s into F r e n c h m a k e s it easier to r e m e m b e r that the t i m e of Voltaire c l o s e l y f o l l o w e d that of N e w t o n . T h e m a t r i x p r o v i d e s a f r a m e w o r k for r e l a t i n g p e o p l e a n d e v e n t s i n c u l t u r a l history a n d the c o n n e c t i o n s are an aid in learning the matrix. O n c e t h e matrix has b e e n learned it can help us to organize knowledge we have w h i c h is not contained in the matrix. M o s t of us h a v e at least s o m e information that can be related to the matrix. F o r e x a m p l e , w e m a y k n o w that M a r i e A n t o i n e t t e k n e w M o z a r t a s a c h i l d o r that she w a s b e h e a d e d i n the F r e n c h R e v o l u t i o n . W e m a y k n o w that the British c h e m i s t P r i e s t l y w a s i n v o l v e d i n a c o n t r o v e r s y w i t h L a v o i s i e r . T h e s e facts w o u l d a l l o w us to identify t h e s e p e o p l e as figures of the late 1700's a n d as contemporaries of Kant and G o e t h e . Further, k n o w i n g the matrix can h e l p us to relate n e w informat i o n t o w h a t w e a l r e a d y k n o w . W h e n w e h e a r , for e x a m p l e , t h a t t h e g r e a t p l a g u e

*Because of its length, T a b l e 5 has b e e n placed at the e n d of this chapter, followed by T a b l e 6.

T a b l e 4. A History Matrix Politics (P)

R e l i g i o n (R) M u s i c (M) 1400

John D u n s t a b l e 1369-1453

1450 Josquin D e s

Philosophy Nicolas of Cusa 1401-1464 Torquemada

S c i e n c e (S) Gutenberg 1398-1468

Literature (L) Death of Chaucer 1400

1420-1498

Thomas Tallis

Martin L u t h e r

1505-1585

1483-1546

1452-1519

1451-1506

1500

1525-1594

Giordano Bruno 1547-1600

Copernicus 1473-1543 Gilbert 1540-1603

1600 Monteverde 1567-1643 1650 Henry Purcell

Rene Descartes 1596-1650 Benedict

Galileo 1564-1642 Newton

1632-1704 George Berkeley

1685-1750

1685-1753

1478-1535 Shakespeare 1564-1616 John M i l t o n 1608-1674 Restoration

Michelangelo 1475-1564 El Greco 1548-1614 Rembrandt 1606-1669 Vermeer 1632-1675

1642-1727

1700 J. S. B a c h

Thomas More

Benjamin

E m a n u e l Kant

Joan of A r c 1412-1431 Lorenzo de' Medici 1449-1492 Henry VIII 1491-1547 Elizabeth I 1533-1603 Oliver Cromwell 1599-1658 Louis X I V

Voltaire

Hogarth

1638-1715 Peter the Great

Franklin 1706-1790

1694-1754

1697-1764

1750 W . A . Mozart

Wars

Comedy

Spinoza 1659-1695

1400-1455

Vinci

1440-1521

Palestrina

Fra A n g e l i c o

Leonardo da

Columbus

Prez

1550

Art (A)

Lavoisier

Goethe

Jacques D a v i d

1672-1725 A m e r i c a n and French

1756-1791

1724-1804

1743-1794

1749-1832

1800 L. Beethoven 1770-1827 1850

Tchaikovsky

J. S. M i l l 1806-1873 Karl Marx

Faraday 1791-1867 Charles Darwin

Balzac 1799-1850 Tolstoy

1748-1825 Joseph T u r n e r 1775-1851 Vincent Van

Revolutions Napoleon 1769-1821 Q u e e n Victoria

Gogh 1840-1893 1900

1818-1883

1809-1882

1828-1910

1853-1890

1819-1901

734

The C o m p l e t e Problem Solver

which killed more than 15 percent of the population of L o n d o n occurred in 1695, w e realize that i t c o u l d h a v e k i l l e d b o t h P u r c e l l a n d N e w t o n . T h e t e c h n i q u e of u s i n g a matrix of landmarks can, of course, be a p p l i e d to other areas of history s u c h as A m e r i c a n history, m u s i c history ( T a b l e 6), a n d literary history. It can also be a p p l i e d to topics in w h i c h t i m e is not an important d i m e n s i o n , as in l e a r n i n g o n e ' s w a y a b o u t a city. SUMMARY We h a v e described s e v e n basic learning strategies, each of w h i c h can provide us c o n s i d e r a b l e h e l p in learning. T h e s e are: 1. 2. 3. 4. 5. 6. 7.

T h e structuring strategy T h e context strategy Monitoring Inferencing Instantiation Multiple coding, and Attention management.

W e h a v e d e s c r i b e d t h r e e study s y s t e m s that c o m b i n e s e v e r a l b a s i c l e a r n i n g strategies in o r g a n i z e d a p p r o a c h e s to study. T h e systems we d e s c r i b e d are: 1. T h e Survey Q 3 R system 2. Dansereau's M U R D E R system, and 3. Jones et al.'s m a s t e r y l e a r n i n g s y s t e m . Finally, we d e s c r i b e d three applications of the learning strategies to hard cases: 1. L e a r n i n g zoological categories 2. L e a r n i n g formulas in physics, and 3. L e a r n i n g cultural history.

REFERENCES Ausubel, D.P. " T h e Use of A d v a n c e Organizers in the Learning and Retention of M e a n i n g f u l V e r b a l M a t e r i a l . " Journal of Educational Psychology, 51, 2 6 7 - 2 7 2 , 1960. Bartlett,

B.J.

Top-level

Structure

as

an

Organizational

Strategy for Recall

of Class-

room Text. D o c t o r a l D i s s e r t a t i o n , A r i z o n a S t a t e U n i v e r s i t y , 1 9 7 8 . D a n s e r e a u , D . F . , M c D o n a l d , B . A . , C o l l i n s , K . W . , G a r l a n d , J., H o l l e y , C . E . , D i e k hoff, G . M . , a n d E v a n s , S . H . " E v a l u a t i o n o f a L e a r n i n g S t r a t e g y S y s t e m . " I n Cognitive and Affective Learning Strategies, e d i t e d b y H . F . O ' N e i l , Jr., a n d C D . Spielberger. N e w York: A c a d e m i c Press, Inc., 1979. F l o w e r , L . S . , H a y e s , J.R., a n d S w a r t s , H. Revising Functional Documents: The Scenario Principle. T e c h n i c a l R e p o r t N o . 1 0 , C a r n e g i e - M e l l o n D o c u m e n t D e sign Project, Pittsburgh, M a r c h , 1980. G o l d m a n , R., a n d H u d s o n , D . A . " A M u l t i v a r i a t e A n a l y s i s o f A c a d e m i c A b i l i t i e s a n d S t r a t e g i e s for S u c c e s s f u l a n d U n s u c c e s s f u l C o l l e g e S t u d e n t s i n D i f f e r e n t M a j o r F i e l d s . " Journal of Educational Psychology, 65, 3 6 4 - 3 7 0 , 1973. J o n e s , B . F . , M o n s a a s , J.A., a n d K a t i m s , M . " I m p r o v i n g R e a d i n g C o m p r e h e n s i o n : E m b e d d i n g D i v e r s e L e a r n i n g Strategies W i t h i n a Mastery L e a r n i n g Instruc-

Learning Strategies

735

tional F o r m a t . " Paper p r e s e n t e d at the A n n u a l M e e t i n g of the A m e r i c a n Educational R e s e a r c h Association, San Francisco, April, 1979. L y n c h , K. The Image of the City.

C a m b r i d g e : T e c h n o l o g y Press, 1960.

M e y e r , B . J . F . , a n d F r e e d l e , R. The Effects of Different Discourse Princeton: Educational Testing Service, 1978. R o b i n s o n , F . P . Effective Study.

N e w York:

Harper & Brothers,

Types

on Recall.

1946.

T a b l e 5 . Biographical N o t e s 1400-1450 M

John D u n s t a b l e

Nationality: Occupation: Connection:

British Composer F o u g h t w i t h t h e B r i t i s h a r m i e s i n F r a n c e a g a i n s t Joan o f Arc.

R

Cardinal Nicolas of Cusa

Nationality: Occupation:

German Roman Catholic prelate, mathematician, and philosopher.

Connection:

S u g g e s t e d that the earth m i g h t m o v e around the sun. Suggested measuring the s p e e d of falling bodies with a water c l o c k c e n t u r i e s b e f o r e Galileo d i d .

S

Johann Gutenberg

Nationality: Occupation:

German D e v e l o p e d printing press w i t h m o v a b l e type. He o p e n e d t h e w a y for w i d e s p r e a d l i t e r a c y .

L

D e a t h of Chaucer Appearance of modern English.

A

Fra Angelico

Nationality: Occupation: Connection:

Italian Painter of church frescoes. Frescoed the Florentine San Marco Monastery, which was reconstructed b y the order o f C o s i m o d e ' M e d i c i , father o f Lorenzo

de'

Medici

(+1).

P

Joan of Arc

Nationality: Occupation:

French L e d F r e n c h army to victory against the British. D e c l a r e d a w i t c h and b u r n e d at the stake.

1450-1500 M

Josquin D e s Prez

Nationality: Occupation: Connection:

French-Belgian Composer of the Renaissance. Martin Luther's f a v o r i t e c o m p o s e r ( + 1 ) .

The C o m p l e t e Problem Solver

136

T a b l e 5 . Biographical N o t e s — C o n t i n u e d R

Tomas D e Torquemada

Nationality:

Spanish

Occupation:

First Grand Inquisitor of Spain. C a u s e d Jews to be exiled from Spain. A u t h o r i z e d torture to obtain confessions. Confessor to Ferdinand and Isabella, w h o financed the voyages of Columbus.

Connection:

S

Cristoforo Columbo

Nationality: Connection:

Italian-Portuguese E x p l o r e d area n o w k n o w n as the W e s t Indies. B e l i e v e d t o h a v e i n t r o d u c e d s y p h i l i s i n t o E u r o p e . Henry VIII (+1) c o n t r a c t e d s y p h i l i s , c a u s i n g m a n y o f h i s c h i l d r e n to be stillborn.

A

Leonardo da Vinci

Nationality:

Connection:

Italian Artist, scientist, e n g i n e e r , inventor. W r o t e treatises on pers p e c t i v e , color, anatomy, and m a n y sciences. R e c e i v e d p a t r o n a g e from Lorenzo de' Medici.

P

Lorenzo de' M e d i c i

Nationality:

Italian

Occupation:

Leader of republican government in Florence. Had popular support, a n d r e f o r m e d c o u n c i l s a n d i m p r o v e d g o v e r n ment.

Occupation:

Occupation:

1500-1550

M

Thomas Tallis

Nationality:

English

Occupation:

O n e o f t h e first c o m p o s e r s t o w r i t e c h u r c h m u s i c i n E n g l i s h . C o u r t c o m p o s e r u n d e r Henry VIII, w h o n a m e d h i m " g e n t l e m a n " of C h a p e l Royal. Unsettled times due to the Reformation caused him to s w i t c h the l a n g u a g e of his c o m p o s i t i o n s , d e p e n d i n g on the

Connections:

religion of the current monarch. Was granted a monopoly o n p r i n t i n g o f m u s i c a n d m u s i c p a p e r b y Elizabeth I (+1). R

Martin Luther

Nationality:

Connection:

German O r d a i n e d priest, composer. Broke a w a y from Catholic C h u r c h , established Protestantism. O p p o s e d p u b l i c a t i o n o f t h e w o r k s o f Copernicus.

S

Nicolaus Copernicus

Occupation:

Nationality:

Polish

Occupation:

A s t r o n o m e r . W r o t e On the Revolutions of the Celestial Spheres. W a s t h e first t o s t r o n g l y p r o p o s e t h a t t h e e a r t h revolves around the sun. H e w a s 1 9 w h e n Columbus d i s c o v e r e d A m e r i c a .

Connection:

Learning Strategies

137 T a b l e 5 . Biographical N o t e s — C o n t i n u e d

L

Thomas More

Nationality: Occupation:

British A u t h o r , l a w y e r . W r o t e Utopia. H e c o n d e m n e d t h e r i g h t o f man to interpret the Scriptures freely, and thus sided w i t h the Catholic position in the Reformation. F r i e n d o f Henry VIII, u n d e r w h o m h e s e r v e d a s L o r d C h a n c e l l o r . R e f u s e d to confirm the oath of the " A c t of S u c c e s s i o n " b e c a u s e it contained a repudiation of papal law.

Connection:

A

Michelangelo

Nationality: Occupation: Connections:

Italian Painter, sculptor. Painted Sistine C h a p e l ceiling.

P

Henry VIII

Nationality: Occupation:

British K i n g o f E n g l a n d , r e s p o n s i b l e for b r i n g i n g P r o t e s t a n t i s m t o England. D e f e n d e d t h e p a p a c y a g a i n s t Luther's attacks. P r e s s u r e d P o p e C l e m e n t V I I , n e p h e w of Lorenzo de' Medici (—1) to a c c e p t h i s d i v o r c e from C a t h e r i n e .

Connections:

S c u l p t e d t h e t o m b of Lorenzo de' Medici ( — 1). As a y o u n g m a n , he c o m p e t e d w i t h da Vinci ( — 1) to p a i n t a b a t t l e scene at the Palazzo Vecchio.

1550-1600 M

Palestrina

Nationality: Occupation: Connection:

Italian Composer, master of music at the Villa d'Este. T h e music of Palestrina was in the serious, anti-secular spirit of the Counter-Reformation. His m u s i c w a s approved by the C o u n c i l of Trent, w h i c h was formulating c h u r c h d i s c i p l i n e for t h e C o u n t e r - R e f o r m a t i o n i n r e a c t i o n to Luther (—1).

R

Giordano Bruno

Nationality: Occupation:

Italian R e n e g a d e monk w h o attacked various church doctrines, including prayer. Was arrested by the Inquisition and b u r n e d at the stake. He was a philosopher w h o borrowed cosmology from C o p e r n i c u s . H a r d e n e d the C h u r c h ' s attitude toward Copernican (-1) t h e o r i e s by c o n n e c t i n g it w i t h h e r e t i c a l doctrines.

Connection:

S

William Gilbert

Nationality: Occupation:

British Physician and physicist w h o discovered basic laws of magn e t i s m a n d static e l e c t r i c i t y . P e r s o n a l p h y s i c i a n to Elizabeth I. O n e of t h e first s c i e n tists o p e n l y to s u p p o r t t h e w o r k of Copernicus (—1) a n d Giordano Bruno.

Connections:

138

T h e C o m p l e t e Problem Solver T a b l e 5 . Biographical N o t e s — C o n t i n u e d

L

William Shakespeare

Nationality: Occupation:

British

Connections:

Famous playwright and poet w h o wrote both comedy and tragedy. P l a y w r i g h t o f t h e Elizabethan e r a . W r o t e a b i o g r a p h i c a l p l a y on t h e l i f e of Henry VIH (-1).

A

El Greco

Nationality: Occupation: Connection:

Artist. B e c a m e a court painter in Spain.

Greek W a s a M a n n e r i s t painter, that is, o n e w h o paints in the manner of Michelangelo (-1).

P

Elizabeth I

Nationality: Occupation: Connection:

British Queen of England. D a u g h t e r of Henry VIII (-1), w h o , d u r i n g h e r r e i g n , strengthened the British navy, e n a b l i n g British to defeat the Spanish Armada.

1600-1650 M

Claudio Monteverde

Nationality:

Italian

Occupation:

C o m p o s e r , music director at San Marco Cathedral, V e n i c e .

R

Rene Descartes

Nationality: Occupation: Connection:

French Philospher, author, scientist. He had b e e n planning the publication of a book supporti n g t h e v i e w s o f Galileo, b u t w h e n h e l e a r n e d o f G a l i l e o ' s p e r s e c u t i o n for h i s v i e w s , D e s c a r t e s d e c i d e d a g a i n s t it.

S

Galileo Galilei

Nationality: Occupation:

Italian

Connections:

S c i e n t i s t . F o r m u l a t e d l a w s o f f a l l i n g b o d i e s a n d d i d astron o m i c a l o b s e r v a t i o n s s u p p o r t i n g t h e w o r k o f Copernicus. A d m i r e d t h e w o r k De Magnete of William Gilbert (-1). D i s c o v e r e d a principle of inertia, b u t credit is usually g i v e n t o Descartes, s i n c e G a l i l e o d i d n o t f u l l y d e v e l o p it.

L

John Milton

Nationality: Occupation: Connections:

British A u t h o r . W r o t e Paradise Lost. M e t Galileo i n I t a l y . T h e r e a r e m e n t i o n s o f G a l i l e o ' s t e l e s c o p e in Paradise Lost. S e r v e d as s e c r e t a r y to t h e C o u n c i l o f S t a t e u n t i l Cromwell c a m e i n t o p o w e r .

139

Learning Strategies T a b l e 5 . Biographical N o t e s — C o n t i n u e d A

Rembrandt van Rijn

Nationality: Occupation:

Dutch Painter. Mastered the t e c h n i q u e s of realism w h i c h had b e e n d e v e l o p i n g u p t o that point, and u s e d t h e m t o g i v e

Connection:

expression to profound human emotion. His painting, " T h e Anatomy L e s s o n of Dr. T u l p , " was e x h i b i t e d t h e y e a r Vermeer (+1) w a s b o r n .

P

Oliver Cromwell

Nationality: Occupation:

British S e r v e d as C o m m a n d e r in C h i e f of British troops and as L o r d Protector to the C o m m o n w e a l t h of E n g l a n d , Scot-

Connection:

land, and Ireland. His uncle, Thomas Henry

VIII

Cromwell,

was

C h i e f Minister to

(-1).

1650-1700 M

Henry Purcell

Nationality: Occupation:

British Court composer,

organist

and

composer

for

London

Theater. R

Benedict de Spinoza

Nationality: Occupation: Connections:

Portuguese-Dutch Philosopher and lens-maker. W a s offered a p e n s i o n by F r e n c h officers o c c u p y i n g Holl a n d o n t h e c o n d i t i o n t h a t h e d e d i c a t e a w o r k t o Louis XIV. R e f u s e d t h e offer. T e r m e d b y Voltaire (+1) a s " l e s s r e a d than famous."

S

Sir Isaac Newton

Nationality: Occupation: Connection:

British Scientist. Revolutionized physics and astronomy. Voltaire's (+1) m i s t r e s s w a s a m a t h e m a t i c i a n w h o t r a n s l a t e d N e w t o n ' s w o r k f r o m L a t i n i n t o F r e n c h . V o l t a i r e inv e n t e d t h e a p p l e story a b o u t N e w t o n .

L

Restoration C o m e d y

A

Jan Vermeer

Nationality: Occupation:

Dutch P a i n t e r w h o p e r c e i v e d r e a l i t y a s a " m o s a i c o f c o l o r e d surfaces."

P

Louis XIV

Nationality: Occupation:

French King of France. Revoked freedom of worship of the French Protestants and tried to convert t h e m by force. Voltaire (+1) w r o t e a b i o g r a p h y o f L o u i s X I V , a n d a d m i r e d h i m greatly as a y o u t h .

Connection:

740

The C o m p l e t e Problem Solver Table 5 . Biographical N o t e s — C o n t i n u e d

1700-1750 M Nationality: Occupation:

Johann Sebastian Bach German Composer organist.

to

the

court

of Prince

Leopold

of Saxony;

R

George Berkeley

Nationality: Occupation:

Irish Anglican Bishop. Wrote about the religious and economic

Connection:

p r o b l e m s of Ireland. A s s e r t e d that the scientific w o r l d v i e w encouraged atheism, w h i c h encouraged vice. W r o t e a c r i t i c i s m of Newton's (—1) d i f f e r e n t i a l c a l c u l u s .

S

Benjamin Franklin

Nationality: Occupation: Connection:

American Inventor, statesman, writer. W a s a n a v i d r e a d e r o f Voltaire.

L

Voltaire

Nationality: Occupation:

French A u t h o r of sociological and political satire.

Connections:

Wrote

a

biography

of

Peter

the

Great.

Met

George

Berkeley i n E n g l a n d . A u t h o r e d t h e m y t h o f t h e a p p l e falli n g on Newton's (—1) h e a d . A

William Hogarth

Nationality:

British

Occupation:

Painter of social satires.

P

Peter the Great

Nationality: Occupation: Connection:

Russian C z a r of Russia. O p e n e d Russia to the West.

1750-1800 M Nationality: Occupation: Connections:

Wolfgang Mozart German C o m p o s e r of the Classical Era. /. S. Bach's (-1) y o u n g e s t s o n b e c a m e a f r i e n d of t h e Mozart family and influenced the music of Mozart. M o z a r t ' s p a t r o n l a t e r b e c a m e Beethoven's (+1) p a t r o n .

R

Immanuel Kant

Nationality: Occupation: Connection:

German Philosopher. Criticized the empiricism of the British philosophers such as Berkeley (-1).

141

Learning Strategies Table 5. Biographical Notes—Continued S

Antoine Laurent Lavoisier

Nationality: Occupation: Connection:

French Chemist. C r e d i t e d with the discovery of oxygen. W a s g u i l l o t i n e d d u r i n g t h e French Revolution o n

suspi-

cion by the authorities of the revolutionary tribunal. L

Johann von Goethe

Nationality: Occupation: Connection:

German Philosopher, astrologer. S t u d i e d anatomy and optics. M e t Napoleon (+1) in E r f u r t . N a p o l e o n w a s to h i m a s y m bol of political order.

A

Jacques Louis D a v i d

Nationality: Occupation:

French Painter of m a n y famous portraits.

Connection:

Painted

P

American and French Revolutions

portrait

of Lavoisier.

1800-1850 M

L u d w i g van Beethoven

Nationality: Occupation:

German Composer. M a r k e d the e n d of the Classical Era in music and the b e g i n n i n g of the Romantic Era. B e e t h o v e n a n d Goethe (-1) m e t i n 1 8 1 2 , a n d i m m e d i a t e l y d i s l i k e d e a c h other. B e e t h o v e n t h o u g h t G o e t h e p o m p o u s and snobbish; G o e t h e thought B e e t h o v e n w i l d and un-

Connection:

ruly. A

John Stuart Mill

Nationality: Occupation:

British Philosopher. O n e of the founders of the Union of W o m e n ' s Suffrage Societies.

S

Michael Faraday

Nationality: Occupation: Connection:

British Invented the d y n a m o ; investigated electromagnetism. R e t i r e d to a h o m e g i v e n to h i m by Queen Victoria (+1).

S

Honore de Balzac

Nationality: Occupation:

French A u t h o r . B e l i e v e d t h a t e x t e r n a l c i r c u m s t a n c e s are u n i m p o r tant in d e t e r m i n i n g h o w a person's life d e v e l o p s . W r o t e a b o u t Napoleon; s a i d , " W h a t N a p o l e o n a c h i e v e d b y

Connection:

the s w o r d I shall a c h i e v e by the p e n . " A

Joseph Turner

Nationality: Occupation: Connection:

British Artist. W e n t to the L o u v r e in Paris to study Italian paintings w h i c h Napoleon c a p t u r e d a n d b r o u g h t t o P a r i s .

142

The C o m p l e t e Problem Solver Table 5 . Biographical N o t e s — C o n t i n u e d

p

Napoleon I

Nationality: Occupation: Connections:

French Emperor of European empire which he conquered. David (—1) p a i n t e d portraits of h i m as a y o u n g e m p e r o r .

1850-1900 M

Peter Ilich T c h a i k o v s k y

Nationality: Occupation:

Russian

R

Karl Marx

Nationality: Occupation:

Connection:

German Author and philosopher. W r o t e Communist Manifesto. W a s the E u r o p e a n correspondent to Horace Greeley's New York Tribune. W a s a g r e a t r e a d e r of Balzac (-1).

S

Charles Darwin

Nationality: Occupation: Connection:

British Scientist,

a r c h a e o l o g i s t , b o t a n i s t . W r o t e Origin of Species.

Karl Marx

s e n t D a r w i n an

L

L e o Tolstoy

Nationality: Occupation:

Russian A u t h o r . W r o t e War

Connections:

W r o t e a b o u t t h e i n v a s i o n o f R u s s i a b y Napoleon (-1). F e l t that a classless society w o u l d c o m e about through love and m o r a l p e r f e c t i o n r a t h e r t h a n t h e p h i l o s o p h y o f Marx.

A

Vincent van G o g h

Nationality: Occupation:

Dutch Post-impressionist painter.

P

Queen Victoria

Nationality: Occupation: Connection:

Composer. Wrote highly emotional and dramatic music. H a d several nervous breakdowns.

and

i n s c r i b e d c o p y of Das Kapital.

Peace

a n d Anna

Karenina.

British Q u e e n of England. Spent much of her reign in seclusion, l e s s e n i n g t h e i n f l u e n c e o f t h e m o n a r c h y o n B r i t i s h affairs. N a p o l e o n III w a s a frequent visitor to Victoria.

143

Learning Strategies Table 6. Music Matrix England 15501600

William Byrd 1543-1623

16001650

John Bull 1585-1640

16501700

Henry Purcell 1659-1695

17001750

17501800

Germany

Italy G. Palestrina 1525-1594

H. Schutz 1585-1672

Claudio Monteverde 1567-1643

M . A. Charpentier 1634-1704

JPachelbel 1653-1706

A. Corelli 1653-1713

John G a y 1685-1732

J. P . Rameau 1683-1764

J. S. B a c h 1685-1750

D. Scarlatti 1685-1757

William Boyce 1710-1779

C. W. Gluck 1714-1787

W. A. Mozart 1756-1792

L. Boccherini 1743-1805

H. Berlioz

L. B e e t h o v e n 1770-1827

G. Rossini

1803-1869 C. Frank 1822-1890

J. B r a h m s 1833-1897

G. Verdi 1813-1901

18001850 18501900

France

Arthur Sullivan 1842-1900

1792-1868

SEVEN.

GETTING THE FACTS STRAIGHT: MAKING

DECISIONS

IN A C O M P L E X W O R L D

W h a t s h o u l d w e h a v e for b r e a k f a s t t h i s m o r n i n g ? W h a t s h o u l d w e w e a r ? D o w e h a v e t i m e for a n o t h e r c u p o f c o f f e e ? W e c a n ' t g e t t h r o u g h t h e d a y w i t h o u t m a k i n g m a n y small d e c i s i o n s a b o u t s u c h practical questions. T h e s e d e c i s i o n s are minor o n e s , b u t s o m e d e c i s i o n s — c h o o s i n g a j o b , a c o l l e g e , a b u s i n e s s p a r t n e r — a r e extremely important. D e c i s i o n m a k i n g is a frequent and important h u m a n activity. S i n c e m o s t p e o p l e w o u l d a g r e e that this is true, it is p u z z l i n g that f e w of us reflect very often or v e r y d e e p l y about the nature of our decision-making processes, or h o w t h e y m i g h t b e i m p r o v e d . S o m e r e f l e c t s o l i t t l e that t h e y are s u r p r i s e d b y t h e i d e a that t h e y u s e d e c i s i o n - m a k i n g p r o c e s s e s a t a l l . I t n e v e r o c c u r s t o t h e m t h a t t h e r e are alternative d e c i s i o n processes and that t h e y can choose a m o n g t h e m . D e c i s i o n s a r e n o t all o f o n e k i n d . P r o c e d u r e s for m a k i n g o n e d e c i s i o n — f o r e x a m p l e , b u y i n g a h o m e — m a y b e e n t i r e l y i n a p p r o p r i a t e for m a k i n g a n o t h e r decision—for example, what poker bet to make. In this chapter and the next one we will d e s c r i b e four general t y p e s of decisions w h i c h r e q u i r e different d e c i s i o n p r o c e d u r e s : J. Decisions 2. D e c i s i o n s 3. D e c i s i o n s 4. D e c i s i o n s

under under under under

certainty risk uncertainty conflict

T h i s chapter w i l l concentrate on decisions u n d e r certainty. In Chapter 8, we w i l l d i s c u s s d e c i s i o n s u n d e r risk, uncertainty, a n d conflict.

DECISIONS U N D E R CERTAINTY I m a g i n e t h a t y o u a r e s e a r c h i n g for a n a p a r t m e n t . Y o u h a v e f o u n d four, all o f w h i c h h a v e t h e s a m e r e n t — a r e n t t h a t y o u c a n afford. Y o u r t a s k i s t o r a n k t h e a p a r t m e n t s i n t h e o r d e r o f y o u r p r e f e r e n c e f r o m 1 for b e s t t h r o u g h 4 for t h e w o r s t .

746

The C o m p l e t e Problem Solver

T h e p r o p e r t i e s t h a t y o u s h o u l d c o n s i d e r i m p o r t a n t for t h i s d e c i s i o n a r e s h o w n i n T a b l e 1. Do the problem before proceeding:

• \

T a b l e 1. Student A p a r t m e n t s

Making Decisions in a C o m p l e x World

747

N o w fill o u t t h i s c h a r t f r o m t h e i n f o r m a t i o n g i v e n i n T a b l e 2 .

M a n y p e o p l e fail t o n o t i c e t h a t t h e s e t w o p r o b l e m s are r e a l l y t h e s a m e . T h a t is, apartment Al has exactly the same properties as apartment B4 in T a b l e 2; A2 matches B3; A3 matches B l ; and A4 matches B2. N o w check to see if you have ranked the m a t c h i n g apartments in the same w a y . If half of y o u r rankings w e r e the same in the t w o p r o b l e m s , y o u did as w e l l as the a v e r a g e of a g r o u p of 20 faculty m e m b e r s a t C a r n e g i e - M e l l o n U n i v e r s i t y . T h e p u r p o s e o f w o r k i n g t h r o u g h ex-

T h e C o m p l e t e Problem Solver

148

a m p l e s s u c h a s t h e s e i s t o i l l u s t r a t e t h e s a d fact t h a t o u r d e c i s i o n - m a k i n g p e r f o r m a n c e i s t y p i c a l l y i m p e r f e c t , e v e n i n d e c i s i o n - m a k i n g t a s k s o f a r a t h e r f a m i l i a r sort.

s \

B e l o w we w i l l discuss further e v i d e n c e of the shortcomings of p e o p l e as decision m a k e r s . T h e r e i s a m p l e r o o m for i m p r o v e m e n t i n h u m a n d e c i s i o n m a k i n g . In p r e s e n t i n g p r o b l e m s in this chapter, we w i l l specify the alternatives from w h i c h the d e c i s i o n m a k e r s c h o o s e , as w e l l as the properties of the alternatives that should be c o n s i d e r e d in m a k i n g the decision, such as the size of the rooms and cleanliness, in the a b o v e p r o b l e m . H o w e v e r , in real life, d e c i s i o n s usually don't p r e s e n t t h e m s e l v e s i n s u c h n e a t form. M o r e often, the d e c i s i o n m a k e r m u s t actively search out alternatives and e v a l u a t e their properties to find those important e n o u g h to be considered in making the decision. We will b e g i n our exploration of decision processes by e x a m i n i n g five different m e t h o d s w h i c h a r e u s e f u l for m a k i n g d e c i s i o n s u n d e r c e r t a i n t y . T h e first four m e t h o d s a r e optimization m e t h o d s — t h a t is, t h e y a t t e m p t t o i d e n t i f y t h e v e r y b e s t a l t e r n a t i v e a v a i l a b l e . T h e fifth m e t h o d , satisficing, s i m p l y l o o k s for t h e first satisfactory alternative. OPTIMIZING

METHODS

\ Dominance D o m i n a n c e is the simplest of the decision procedures we will discuss. To use i t i n m a k i n g d e c i s i o n s , w e m u s t first f i n d t h e d o m i n a n c e r e l a t i o n s a m o n g t h e alternatives. O n e alternative dominates another if both of the following conditions are s a t i s f i e d : 1. It is at least as good as t h e o t h e r on all p r o p e r t i e s , a n d 2. It is b e t t e r on at least one p r o p e r t y . A n y alternative that is d o m i n a t e d by another is d r o p p e d from consideration. A n y a l t e r n a t i v e t h a t d o m i n a t e s all t h e o t h e r s i s c h o s e n a s b e s t . N o w let's a p p l y the d o m i n a n c e procedure to the apartment problem. T a b l e 3 s u m m a r i z e s the information a b o u t the four apartments in T a b l e 1. Do any of the alternatives dominate any of the others? O n l y o n e alternative dominates another in this p r o b l e m : Alternative 4 d o m i nates A l t e r n a t i v e 2. A l t e r n a t i v e 4 is as g o o d as A l t e r n a t i v e 2 in " k i t c h e n , " " g e n e r a l repair," "noise l e v e l , " "brightness," and "landlord," and it is better in "distance," " s i z e , " and " c l e a n l i n e s s . " Alternative 1 does not dominate Alternative 2 b e c a u s e , w h i l e it is better in s o m e properties, such as " d i s t a n c e , " it is w o r s e in others. Strengths

and

Weaknesses

of

the

Dominance

Method

D o m i n a n c e is not a very powerful decision-making m e t h o d b e c a u s e , as the e x a m p l e a b o v e illustrates, i t u s u a l l y d o e s n ' t e l i m i n a t e v e r y m a n y o f the alternatives. T h e a d v a n t a g e of the m e t h o d is that p e o p l e can a g r e e a b o u t w h i c h alternat i v e s are d o m i n a n t . T h e y c a n d o t h i s , e v e n t h o u g h t h e y m a y d i f f e r a b o u t w h a t p r o p e r t i e s a r e m o s t i m p o r t a n t for m a k i n g t h e d e c i s i o n . T h u s , J o n e s m a y f e e l that t h e kitchen is the m o s t important property of an apartment w h i l e his w i f e b e l i e v e s that distance from w o r k is m o s t important. D e s p i t e t h e s e differences, t h e y w i l l agree that Alternative 4 d o m i n a t e s Alternative 2. We can rely on results obtained by the d o m i n a n c e m e t h o d b e c a u s e , e v e n if we c h a n g e our minds about h o w important the various properties are, the d o m i n a n c e relations w i l l be u n c h a n g e d . T h e d o m i n a n c e m e t h o d i s e a s y t o a p p l y , a n d its r e s u l t s a r e r e l i a b l e . T h e r e f o r e , it can be of v a l u e w h e n u s e d to s c r e e n s o m e alternatives from consideration b e f o r e other d e c i s i o n m e t h o d s are a p p l i e d to the p r o b l e m . A n y alternative that is d o m i -

\.

T a b l e 3 . Alternatives 1

2

3

4

Distance in Minutes

15 Min

60 M i n

20 M i n

45 Min

Size of Rooms

Average

Cramped

Comfortable

Small

Kitchen

N e w stove, etc.

Stove, etc. in good condition

Stove, etc. old but useable

Stove, etc. in good condition

General Repair

Needs no Repair

Needs no Repair

Needs one Week work

Needs no Repair

Cleanliness

Needs Vacuuming

Needs Vacuuming

Ready to Move in

Ready to Move in

Noise Level

Frequently Noisy

Often Quiet

Sometimes Noisy

Often Quiet

Brightness

Always needs artificial light

Very Bright

Fairly Bright

Very Bright

Landlord

Indifferent

Cordial

Very Friendly

Cordial

757

Making D e c i s i o n s in a C o m p l e x World

c o m p a r e d Alternatives 1 and 3 on the second most important p r o p e r t y — r o o m s i z e — a n d c h o s e n Alternative 3, since it has larger rooms than Alternative 1. Notice w h a t w o u l d happen if the order of importance of the properties w e r e r e v e r s e d , t h a t is, i f l a n d l o r d w e r e t h e m o s t i m p o r t a n t p r o p e r t y , b r i g h t n e s s n e x t , a n d s o o n . A l t e r n a t i v e s 2 a n d 4 w o u l d b e t i e d for t h e b e s t l a n d l o r d . T h e s e t w o a l t e r n a t i v e s are also t i e d on b r i g h t n e s s a n d n o i s e l e v e l . T h e tie is finally b r o k e n in favor of Alternative 4 by the difference in cleanliness. C h a n g i n g the order of importance of the properties d o e s not a l w a y s c h a n g e the alternative c h o s e n as best, b u t in this c a s e it clearly has. Exercises U s e the lexicographic m e t h o d to identify the best student in the previous exercise. A s s u m e that the order of importance of the c o u r s e s from most to least is: 1. C h e m i s t r y , P h y s i c s , C a l c u l u s , Art, E n g l i s h 2. Art, E n g l i s h , C a l c u l u s , P h y s i c s , C h e m i s t r y Strengths

and

Weaknesses

of

the

Lexicographic

Method

T h e lexicographic method is most appropriate w h e n one of the properties o u t w e i g h s all o f t h e o t h e r s i n i m p o r t a n c e . Its m a j o r s t r e n g t h u n d e r t h e s e c i r c u m stances is that it is q u i c k and easy to apply. T h e m e t h o d is least appropriate w h e n the properties are r o u g h l y e q u a l in importance. U n d e r these circumstances the m e t h o d m a y lead us to c h o o s e an alternative w h i c h has a slight advantage in the most important property, e v e n t h o u g h that advantage is o u t w e i g h e d by b i g disadvantages in other properties. T h i s happens b e c a u s e the lexicographic m e t h o d typic a l l y i g n o r e s all b u t t h e m o s t i m p o r t a n t p r o p e r t y .

Additive Weighting T h e a d d i t i v e - w e i g h t i n g m e t h o d t a k e s all o f t h e p r o p e r t i e s i n t o a c c o u n t b u t does not give t h e m equal weight. T h e more important properties receive heavy weights and the less important ones lighter weights. T o u s e t h e a d d i t i v e - w e i g h t i n g m e t h o d , o n e m u s t h a v e n u m b e r s b o t h for w e i g h t s o f t h e p r o p e r t i e s a n d for t h e v a l u e s o f t h e p r o p e r t i e s . F o r e x a m p l e , i n T a b l e 4 w e h a v e p r o v i d e d n u m e r i c a l w e i g h t s for t h e p r o p e r t i e s i n t h e r i g h t - h a n d c o l u m n : 7 for d i s t a n c e , 4 for s i z e o f r o o m s , e t c . W e h a v e a l s o p r o v i d e d n u m b e r s f r o m 1 t o 5 for t h e v a r i o u s v a l u e s o f e a c h o f t h e p r o p e r t i e s . T h e s e a r e s h o w n i n T a b l e 4 i n p a r e n t h e s e s t o t h e r i g h t o f e a c h p r o p e r t y v a l u e . T h e n u m e r i c a l v a l u e 4 i s ass i g n e d for t h e 1 5 - m i n u t e d i s t a n c e i n d i c a t i n g a r e l a t i v e l y h i g h o r " g o o d " v a l u e . T h e numerical v a l u e 1 is a s s i g n e d to the 60-minute distance, indicating a l o w or " p o o r " v a l u e . T h e n u m b e r s for t h e w e i g h t s a n d v a l u e s o f t h e p r o p e r t i e s a r e i n t e n d e d t o reflect the importance of the properties and property v a l u e s to the d e c i s i o n maker. T y p i c a l l y , d e c i s i o n m a k e r s w i l l differ s o m e w h a t i n the i m p o r t a n c e t h e y p l a c e o n the various properties and v a l u e . F o r e x a m p l e , a tenant w h o is h a n d y w i t h tools m a y p u t a r e l a t i v e l y l o w v a l u e o n general repair, w h e r e a s o n e w h o i s all t h u m b s m a y find it m u c h m o r e important. In the s a m e w a y , a p e r s o n m a y not care about the landlord's attitude as l o n g as he isn't actively hostile. T h u s , he m i g h t w a n t to assign t h e s a m e n u m e r i c a l v a l u e to indifferent a n d cordial. To make a decision by the additive-weighting method, multiply numerical v a l u e s o f t h e p r o p e r t i e s b y t h e w e i g h t s o f t h e p r o p e r t i e s for e a c h a l t e r n a t i v e . T h e n choose the alternative with the largest sum as " b e s t . " In T a b l e 4, the products of v a l u e a n d w e i g h t a r e s h o w n i n t h e l o w e r p a r t o f e a c h c e l l . T h e p r o d u c t for d i s t a n c e for A l t e r n a t i v e 1 i s t h e w e i g h t 7 m u l t i p l i e d b y t h e v a l u e 4 , o r 2 8 . A l t e r n a t i v e 1 h a s

T h e C o m p l e t e Problem Solver

754

the largest sum of products, 75, and is therefore c h o s e n as the best alternative by the a d d i t i v e - w e i g h t i n g m e t h o d . Exercises Apply the additive-weighting procedure to the following problems: J . W h i c h w o u l d b e t h e b e s t a p a r t m e n t i n T a b l e 3 i f all o f t h e w e i g h t s w e r e e q u a l ? 2 . I f t h e w e i g h t s for t h e s u b j e c t s i n E x e r c i s e 1 w e r e C h e m i s t r y 4 , P h y s i c s 4 , C a l culus 5, Art 2, and English 3, w h o w o u l d be the best student? 3 . I f t h e w e i g h t s for t h e s u b j e c t s w e r e C h e m i s t r y 2 , P h y s i c s 2 , C a l c u l u s 4 , A r t 5 , and English 4, w h o w o u l d be the best student? Strengths

and

Weaknesses

of

the

Additive

Weighting

Method

T h e a d d i t i v e - w e i g h t i n g m e t h o d t a k e s all o f t h e p r o p e r t i e s i n t o a c c o u n t i n m a k i n g the d e c i s i o n , b u t it doesn't take the interactions of the properties into account. W h e n w e say that t w o properties interact, w e m e a n that the importance w e assign to a value of one property d e p e n d s on the value w h i c h other properties happen to have. For example, the importance we place on the friendliness of the landlord m a y d e p e n d on the condition of the apartment. If the apartment needs a great deal of repair, it m a y be v e r y important that the landlord is friendly. T h e a d d i t i v e - w e i g h t i n g m e t h o d can lead t o inappropriate decisions b y ignoring these interactions, just as the lexicographic m e t h o d can lead to inappropriate d e c i s i o n s b y i g n o r i n g t h e l e s s i m p o r t a n t p r o p e r t i e s . H o w e v e r , i g n o r i n g t h e s e interactions is not really a v e r y serious p r o b l e m in most cases. T h i s fact is illustrated in the following e x a m p l e , suggested by Y n t e m a and Torgerson's (1961) analysis of decisions. S u p p o s e that y o u are a scoutmaster a n d that y o u h a v e the task of a w a r d i n g merit b a d g e s t o y o u r n u m e r o u s c h a r g e s a t t h e e n d o f t h e y e a r . T h e r e a l i t i e s o f life h a v e forced y o u to c o n c e n t r a t e on j u s t three of the 10 b o y scout virtues this year: helpfulness, cleanliness, and honesty. D u r i n g the year, you have given each scout a grade f r o m 1 ( l o w ) t o 7 ( h i g h ) for h i s p e r f o r m a n c e o n e a c h o f t h e s e t h r e e v i r t u e s . T h e problem, now, is w h o gets the badges. Y o u a n d t h e a s s i s t a n t s c o u t m a s t e r a g r e e t h a t t h e t h r e e v i r t u e s are e q u a l l y important. T h e assistant s u g g e s t s that i t w o u l d b e e a s y t o m a k e the d e c i s i o n b y a d d i t i v e w e i g h t i n g . " A f t e r a l l , " h e p o i n t s o u t , " i f t h e w e i g h t s a r e all e q u a l , w e c a n l e t t h e m all b e 1 . " T h e n M e r i t , M , w i l l j u s t b e e q u a l t o 1 • helpfulness score + 1 • cleanliness score + 1 • honesty score or M = helpfulness + cleanliness + honesty, or M = h e + cl + h o H e i s d e l i g h t e d , b u t y o u are w o r r i e d . Y o u k n o w y o u r t r o o p , a n d y o u k n o w t h a t t h e v i r t u e s i n t e r a c t w i t h e a c h o t h e r . F o r e x a m p l e , t h e r e w a s t h a t d i r t y little k i d , S n y d e r , w h o w a s r e a l l y h e l p f u l i n t h e l i b r a r y , b u t h e g o t t h e b o o k s s o d i r t y that t h e y had to be b u r n e d . H e l p f u l n e s s and honesty interact too. R e m e m b e r Gionelli, w h o h e l p e d lots o f o l d m e n a c r o s s t h e s t r e e t a n d t h e n p i c k e d t h e i r p o c k e t s ? I n y o u r w i s d o m y o u k n o w that the merit contributed b y e a c h virtue d e p e n d s o n the values of the other virtues. Y o u k n o w that true merit is m e a s u r e d by this formula: True Merit = (He • Cl) + (Cl • Ho) + (Ho • He)

Making Decisions in a C o m p l e x World

155

N o w that's a lot different from M = He + Cl + Ho! N e v e r t h e l e s s , y o u r assistant s e e m s quite u n t r o u b l e d w h e n y o u tell h i m these h a r d facts o f l i f e . " I t ' s t r u e , " h e s a y s , " t h o s e v i r t u e s r e a l l y i n t e r a c t v e r y s t r o n g l y , b u t that d o e s n ' t m e a n that the a d d i t i v e - w e i g h t i n g s c h e m e is g o i n g to c a u s e us to m a k e lots o f m i s t a k e s . W h a t w e are t r y i n g t o d o i s t o o r d e r t h e s c o u t s a c c o r d i n g t o t h e i r true merit. We make a mistake only w h e n a higher merit scout gets a l o w e r additivew e i g h t i n g s c o r e t h a n a l o w e r m e r i t s c o u t . " T h a t is, if M > M x

2

(the a d d i t i v e - w e i g h t i n g s c o r e s )

but ( T r u e M e r i t ) > ( T r u e M e r i t ^ 2

T o r g e r s o n and Y n t e m a a r g u e that in ordinary c i r c u m s t a n c e s this d o e s n ' t happen very much. S u p p o s e we w e r e to select t w o scouts at random and suppose f u r t h e r that t h e s c o u t s s p r e a d t h e m s e l v e s o u t e v e n l y o n t h e v i r t u e s c a l e . T h a t is, a s c o u t is j u s t as l i k e l y to s c o r e a 1 on c l e a n l i n e s s or h o n e s t y as 2 or 4 or 7. If t h e t r u e merit of one scout is 6 units greater than the other (which h a p p e n s 86 p e r c e n t of the time), T o r g e r s o n and Y n t e m a calculate that the additive w e i g h t i n g score, M, will a l s o b e g r e a t e r for t h e m o r e v i r t u o u s s c o u t i n 9 9 . 5 p e r c e n t o f c a s e s . What about the r e m a i n i n g 14 percent of cases in w h i c h the difference in true merit is less than 6? T h e a d d i t i v e - w e i g h t i n g procedure can't do w o r s e than chance, so it s h o u l d g e t as a m i n i m u m an additional s e v e n p e r c e n t of t h e d e c i s i o n s right. Thus, e v e n though it ignores interactions, the additive-weighting procedure makes the right c h o i c e in at least 93 p e r c e n t (86% + 7%) of cases in a p r o b l e m w i t h strong interaction. F u r t h e r , it t e n d s not to m a k e b a d errors. T h a t is, in cases w h e r e t h e r e is a b i g difference in true value b e t w e e n the alternatives, the additive-weighting m e t h o d almost a l w a y s m a k e s the right d e c i s i o n . W e can c o n c l u d e that the additiveweighting method, e v e n though it ignores interactions, is a reasonably good decis i o n m e t h o d i n g e n e r a l . I n p a r t i c u l a r , i t s h o u l d d o a g o o d j o b for u s i n a s s i g n i n g merit b a d g e s . T h e major d r a w b a c k of the a d d i t i v e - w e i g h t i n g m e t h o d is that it is time c o n s u m i n g . O b t a i n i n g t h e n u m b e r s for t h e w e i g h t s a n d v a l u e s o f t h e p r o p e r t i e s i s p r o b a b l y the most difficult part of the m e t h o d . In some cases, these n u m b e r s can be d e r i v e d from o b j e c t i v e c o n s i d e r a t i o n s , s u c h a s t h e p o s t e d p r i c e s o f cars o r t h e c a l o r i c c o n tent of various foods. In other cases, t h e y must be o b t a i n e d t h r o u g h subjective e s t i m a t i o n . I n u s i n g t h e a d d i t i v e - w e i g h t i n g m e t h o d for A p a r t m e n t s 1 , 2 , 3 , a n d 4 (Table 4), w e h a d t o m a k e e i g h t estimates o f property w e i g h t s and 4 0 estimates o f property values. O n c e the necessary numbers have b e e n obtained, a few minutes worth of m u l t i p l y i n g a n d a d d i n g still m u s t b e d o n e t o f i n d t h e b e s t c h o i c e . A l l t h i n g s c o n sidered, the additive-weighting method may require b e t w e e n half an hour and an hour to complete. T h i s is likely more time than y o u w o u l d w a n t to s p e n d choosing your dessert i n the l u n c h line. H o w e v e r , i t m a y w e l l b e w o r t h s p e n d i n g that a m o u n t of time if the d e c i s i o n is an important or e x p e n s i v e one, such as the c h o i c e of a graduate school or a n e w car.

Effectiveness Indices In s o m e c a s e s , it is important to take into a c c o u n t the interactions w h i c h the a d d i t i v e - w e i g h t i n g m o d e l ignores. T h i s m a y h a p p e n e i t h e r b e c a u s e the interact i o n s are e s p e c i a l l y s t r o n g o r b e c a u s e e r r o r s i n d e c i s i o n s a r e v e r y c o s t l y , o r b o t h . B o t h are t r u e i n c e r t a i n m e d i c a l s i t u a t i o n s i n w h i c h t h e p a t i e n t m u s t r e c e i v e t w o kinds of m e d i c a t i o n simultaneously. S o m e pairs of drugs h a v e effects in combina-

756

T h e C o m p l e t e Problem Solver

t i o n w h i c h n e i t h e r o f t h e m h a s a l o n e . T h a t is, t h e d r u g s i n t e r a c t w i t h e a c h o t h e r . I n s o m e c a s e s , t h e s e i n t e r a c t i o n s are l i f e - t h r e a t e n i n g a n d m u s t b e t a k e n i n t o a c c o u n t i n c h o o s i n g a p p r o p r i a t e d o s a g e l e v e l s . F o r c a s e s o f t h i s sort, w e n e e d a d e c i s i o n m e t h o d m o r e c o m p l e x than additive w e i g h t i n g , w h i c h will take the interactions i n t o a c c o u n t . W e w i l l u s e t h e t e r m effectiveness index t o d e s i g n a t e all s u c h " m o r e c o m p l e x " methods. Often such methods will involve a physical, chemical, or engin e e r i n g m o d e l b a s e d o n e x t e n s i v e a n a l y s i s o f t h e s i t u a t i o n a b o u t w h i c h w e are m a k i n g decisions. D e s i g n i n g and implementing such a m e t h o d may be very expensive indeed. We w i l l not ask the reader to construct any effectiveness indices. O u r purpose in d i s c u s s i n g effectiveness indices is simply to point out that w h e n it is very important to get the b e s t decisions, it is possible, though often e x p e n s i v e , to do better than the m o d e l s we discussed above. SATISFYING: A NON-OPTIMIZING APPROACH D e c i s i o n - m a k i n g s i t u a t i o n s t h a t w e m e e t i n d a i l y e x p e r i e n c e are o f t e n a g o o d deal m e s s i e r than the i d e a l i z e d situations we h a v e d e s c r i b e d a b o v e . Perhaps the m o s t i m p o r t a n t c o m p l i c a t i o n w e e n c o u n t e r i s that, r a t h e r t h a n b e i n g p r e s e n t e d w i t h a p r e d e f i n e d l i s t t o c h o o s e f r o m , w e m a y b e f o r c e d t o s e a r c h for a l t e r n a t i v e s . F o r e x a m p l e , w h e n a t e a c h e r a s s i g n s a " f r e e " t h e m e , t h e s t u d e n t ' s task o f d e c i d i n g w h a t t o w r i t e a b o u t w i l l r e q u i r e t h a t h e g e n e r a t e a list o f p o s s i b l e t o p i c s . S i m i l a r l y , w h e n a c o l l e g e w a n t s to f i n d a n e w d e a n , a s e a r c h c o m m i t t e e is a p p o i n t e d to i d e n t i f y a list of c a n d i d a t e s to be c o n s i d e r e d . G e n e r a t i n g alternatives, examining their properties, and choosing among the a l t e r n a t i v e s a r e all a c t i v i t i e s t h a t m a y a d d c o n s i d e r a b l e c o s t t o t h e d e c i s i o n - m a k i n g p r o c e s s . I n t h e p r o b l e m o f s e a r c h i n g for a d e a n , t h e s e c o s t s i n c l u d e t h e s a l a r i e s o f t h e s e a r c h - c o m m i t t e e m e m b e r s a n d t h e t r a v e l c o s t s o f c a n d i d a t e s i n v i t e d from o t h e r c i t i e s . I f t h e list o f c a n d i d a t e s t h a t t h e s e a r c h c o m m i t t e e g e n e r a t e s i s v e r y l a r g e , t h e costs m a y also be large. If the search c o m m i t t e e takes five years and generates 100,000 n a m e s , c l e a r l y d e c i s i o n m a k i n g i s g o i n g t o b e v e r y e x p e n s i v e . T h e d e c i s i o n - m a k i n g t e c h n i q u e d e s c r i b e d b e l o w can be helpful in practical situations w h e r e such costs must be considered. T h e d e c i s i o n m e t h o d s w e h a v e d i s c u s s e d s o far a r e all o p t i m i z i n g m e t h o d s , t h a t is, m e t h o d s d e s i g n e d t o f i n d t h e b e s t a v a i l a b l e a l t e r n a t i v e . I n this s e c t i o n , w e w i l l d i s c u s s q u i t e a d i f f e r e n t a p p r o a c h t o d e c i s i o n m a k i n g c a l l e d satisficing, first d e s c r i b e d b y S i m o n ( 1 9 5 5 ) . T h i s m e t h o d i s not d e s i g n e d t o i d e n t i f y t h e b e s t a l t e r n a t i v e . R a t h e r it is d e s i g n e d to f i n d t h e first satisfactory a l t e r n a t i v e . T h e satisficing m e t h o d requires the decision maker to identify the worst value h e i s w i l l i n g t o a c c e p t for e a c h o f t h e a t t r i b u t e s . H e t h e n c o n s i d e r s all o f t h e a l t e r n a t i v e s i n o r d e r , r e j e c t i n g a n y a l t e r n a t i v e s w h i c h fall b e l o w t h e m i n i m a l v a l u e s o f t h e a t t r i b u t e s , a n d a c c e p t i n g t h e first a l t e r n a t i v e w h i c h m e e t s a l l o f t h e m i n i m a l values. C o n s i d e r i n g T a b l e 3 , s u p p o s e that the d e c i s i o n m a k e r ' s m i n i m a l v a l u e s are: distance: room size: kitchen:

45 minutes small useable

repair: cleanliness: noise: brightness: landlord:

t w o w e e k s ' repair needs some cleaning sometimes noisy fairly b r i g h t indifferent

A l t e r n a t i v e 1 fails t o m e e t t h e m i n i m u m v a l u e s for n o i s e l e v e l a n d b r i g h t n e s s ,

757

Making Decisions in a C o m p l e x World

a n d A l t e r n a t i v e 2 fails t o m e e t t h e m i n i m u m v a l u e for d i s t a n c e a n d r o o m s i z e . A l t e r n a t i v e 3 , h o w e v e r , m e e t s all o f t h e m i n i m a l r e q u i r e m e n t s a n d i s t h e r e f o r e chosen. N o t i c e that Alternative 4 also m e e t s the m i n i m a l r e q u i r e m e n t s b u t w a s not c h o s e n o r e v e n c o n s i d e r e d b e c a u s e s e a r c h w a s t e r m i n a t e d w h e n t h e first satisfactory a l t e r n a t i v e w a s f o u n d . Exercises 1. In the example above, what w o u l d be the result if the decision maker insisted in addition to the other m i n i m u m requirements that the apartment n e e d no repair? T h a t the stove be n e w ? 2. U s e the satisficing m e t h o d w i t h the data on p a g e 150 to a . F i n d a g o o d a l l - a r o u n d s t u d e n t — o n e w h o c a n s c o r e 8 5 i n all s u b j e c t s . b. Find a good science s t u d e n t — o n e w h o can score 90 or better in math and science. Strengths

and

Weaknesses

of

the

Satisficing

Method

T h e satisficing m e t h o d is particularly useful w h e n we have to choose a m o n g a very large n u m b e r of alternatives and it is not essential to find the best. For examp l e , w h e n w e a r e c h o o s i n g a d o z e n a p p l e s f r o m a s u p e r m a r k e t b i n , w e d o n ' t try t o find t h e 1 2 b e s t a p p l e s . W e c a n ' t b e b o t h e r e d w i t h e x a m i n i n g a n d c o m p a r i n g all o f

Table 5. Decision Making Methods Method

Type

U s e this method:

Domi-

optimizing

for p r e l i m i nary screeni n g o f alternatives

Lexicography

optimizing

Additive Weighting

Cost of computation required

Number of alternatives examined

low

all

w h e n attrib u t e s are v e r y different in weight

very low

all

optimizing

w h e n it is important to find t h e b e s t alternative

high

all

Effectiveness Index

optimizing

w h e n it is very i m p o r tant to g e t b e s t alternative

very high

all

Satisficing

non-optimizing

w h e n the cost of examining the w h o l e set of alternatives is very high

very low

some

nance

158

The C o m p l e t e Problem Solver

t h e a p p l e s i n t h e b i n . W e s i m p l y p i c k t h e first 1 2 a c c e p t a b l e a p p l e s a n d g o o n t o u s e t h e s a t i s f i c i n g m e t h o d for s e l e c t i n g p e p p e r o n i , k i w i fruit, o r w h a t e v e r e l s e w e m a y n e e d . S i m i l a r l y , in the d e a n - s e a r c h task, it is e c o n o m i c a l to stop t h e search as soon as a s a t i s f a c t o r y c a n d i d a t e is f o u n d . T h e p r i m a r y a d v a n t a g e o f t h e s a t i s f i c i n g m e t h o d i s t h a t i t c a n y i e l d a satisfact o r y d e c i s i o n w i t h o u t r e q u i r i n g u s t o e x a m i n e all o f t h e a l t e r n a t i v e s . T h u s , c o m pared to optimization methods, it can greatly reduce the cost of search in decision making. T h e s a t i s f i c i n g m e t h o d m a y n o t y i e l d a d e c i s i o n a t all i f w e s e t o u r s t a n d a r d s t o o h i g h . F o r e x a m p l e , i n w r i t i n g w e m a y f i n d o u r s e l v e s b l o c k e d b e c a u s e w e are s e a r c h i n g for t h e p e r f e c t w o r d t o e x p r e s s o u r m e a n i n g w h e n n o n e m a y e x i s t . D i o g e n e s s e e m s t o h a v e e n c o u n t e r e d t h i s sort o f d i f f i c u l t y i n h i s s e a r c h for a n h o n e s t man. As a practical matter w h a t usually h a p p e n s w h e n we run into difficulty in f i n d i n g a satisfactory a p a r t m e n t or car or e m p l o y e e is that we c h a n g e our m i n d s a b o u t w h a t is m i n i m a l l y a c c e p t a b l e and set a n e w but, alas, l o w e r standard. T a b l e 5 summarizes the properties of the decision-making procedures described above. EXPERIMENTAL STUDIES OF D E C I S I O N M A K I N G

UNDER CERTAINTY

D o formal d e c i s i o n m e t h o d s h e l p ? T h e correct c h o i c e i n a n y decision situation d e p e n d s on the decision m a k e r s ' individual values. If t w o p e o p l e m a k e different c h o i c e s in the s a m e situation, it d o e s n ' t m e a n that o n e of t h e m is w r o n g ; it m a y just b e that t h e y h a v e different v a l u e s . T h i s m e a n s that w e can't tell h o w g o o d p e o p l e ' s d e c i s i o n - m a k i n g p r o c e s s e s are b y the c h o i c e t h e y m a k e . H o w t h e n can w e measure i m p r o v e m e n t in decision-making skills? No matter w h a t p e o p l e ' s v a l u e s are, if t h e y use g o o d d e c i s i o n methods, they should tend to agree with themselves w h e n they make the same decision twice. To t e s t for i m p r o v e m e n t i n d e c i s i o n m a k i n g , t h e n , w e m e a s u r e d s e l f - a g r e e m e n t o f s t u d e n t s m a k i n g a p a r t m e n t d e c i s i o n s ( s e e T a b l e s 1 a n d 2 ) b o t h b e f o r e a n d after t h e y h a d s t u d i e d the m e t h o d s d e s c r i b e d i n this chapter. O f the 7 1 students tested, 4 9 w e r e more consistent on posttest than pretest, 14 w e r e less consistent, and eight w e r e u n c h a n g e d . T h e s e results indicate a very reliable i m p r o v e m e n t in consistency for t h e g r o u p a s a r e s u l t o f t r a i n i n g i n f o r m a l d e c i s i o n m e t h o d s . I N F O R M A T I O N - P R O C E S S I N G LIMITS IN D E C I S I O N M A K I N G Often w h e n we are m a k i n g an important d e c i s i o n , such as w h a t c o l l e g e to a t t e n d o r w h e r e t o s p e n d o u r v a c a t i o n , w e m a k e a n effort t o g e t a s m u c h i n f o r m a t i o n a s p o s s i b l e a b o u t e a c h o f t h e a l t e r n a t i v e s . I m p l i c i t i n t h i s effort i s o u r a s s u m p t i o n that m o r e r e l e v a n t information a b o u t the alternatives w i l l e n a b l e us to m a k e better d e c i s i o n s . A study by H a y e s (1962) indicates that this a s s u m p t i o n isn't necessarily t r u e i f t h e d e c i s i o n m a k e r i s not u s i n g a f o r m a l d e c i s i o n p r o c e d u r e . H a y e s s t u d i e d m i l i t a r y p e r s o n n e l m a k i n g d e c i s i o n s a b o u t a s i m u l a t e d aird e f e n s e situation. T h e alternatives w e r e characterized b y e i g h t e q u a l l y important relevant properties. For some decisions, information was available about only two o f t h e p r o p e r t i e s . F o r o t h e r d e c i s i o n s , i t w a s a v a i l a b l e for four, six, o r all e i g h t o f t h e properties. In all cases, the correctness of a d e c i s i o n w a s s c o r e d on the basis of the "real-world situation" m e a s u r e d b y all e i g h t properties, w h e t h e r the decision m a k e r h a d s e e n all of t h e m or not. T h e surprising result w a s that the d e c i s i o n m a k e r s m a d e j u s t a s g o o d decisions w h e n t h e y w e r e g i v e n t w o r e l e v a n t facts a s w h e n t h e y w e r e g i v e n four, six, o r e i g h t . T h e s u b j e c t s m u s t h a v e b e e n d e r i v i n g u s e f u l i n f o r m a t i o n f r o m t h e e x t r a facts, t h o u g h , b e c a u s e w h e n i r r e l e v a n t facts r a t h e r t h a n r e l e v a n t o n e s w e r e a d d e d , t h e

Making Decisions in a C o m p l e x World

159

d e c i s i o n s b e c a m e d i s t i n c t l y w o r s e . T h e s e r e s u l t s s u g g e s t t h a t e x t r a r e l e v a n t facts both h e l p a n d hurt. T h e y add extra information, b u t t h e y also confuse the decision maker. C o m p a r i n g several alternatives simultaneously on four or more properties a p p e a r s t o b e a v e r y d i f f i c u l t task for m o s t p e o p l e t o d o i n t h e i r h e a d s . D e c i s i o n m e t h o d s s u c h a s l e x i c o g r a p h y a n d a d d i t i v e w e i g h t i n g are u s e f u l b e c a u s e t h e y a l l o w p e o p l e t o s u b s t i t u t e r e l i a b l e o b j e c t i v e p r o c e d u r e s for u n r e l i a b l e s u b j e c t i v e o n e s . REFERENCES H a y e s , J.R. Human Data Processing Limits in Decision Making. T e c h n i c a l D o c u mentary Report No. ESD-TDR-62-48. Bedford, MA: Operational Applications Laboratory, July, 1962. S i m o n , H.A. "A B e h a v i o r a l nomics, 6 9 , 9 9 - 1 1 8 , 1 9 5 5 .

M o d e l o f R a t i o n a l C h o i c e . " Quarterly Journal o f Eco-

Yntema, D.B., and Torgerson, W . S . Man-Computer Cooperation in Requiring Common Sense. I R E T r a n s a c t i o n s , T G H F E , H F E - 2 , 20-26,

Decisions 1961.

EIGHT.

THE LUCK OF THE D R A W : DEALING W I T H C H A N C E IN DECISION M A K I N G

I n m a k i n g d e c i s i o n s u n d e r c e r t a i n t y , o u r task i s t o f i g u r e o u t w h i c h o f t h e alternatives w e like best. W h e n w e m a k e a choice, w e a s s u m e that w e will get the alternative w e w a n t . I f w e tell the car sales dealer, " I ' l l take the y e l l o w M a z d a , " w e e x p e c t t o g e t it. I f t h e y d e l i v e r a b l u e p a i s l e y F i a t i n s t e a d , w e w o u l d l i k e l y b e v e r y upset.

T^UT,

I

ORDERED

A

YELLOUJ A \ A Z P / * '.

162

T h e C o m p l e t e Problem Solver

I n t h i s c h a p t e r , w e w i l l d i s c u s s d e c i s i o n s o f q u i t e a d i f f e r e n t sort: W h a t y o u w a n t may b e w h a t y o u g e t , b u t t h e n a g a i n i t m a y not. T h a t i s , w e w i l l s t u d y decisions in w h i c h some e v e n t not u n d e r your control intervenes b e t w e e n your c h o i c e a n d its o u t c o m e . C o n s i d e r m y d e c i s i o n t o r u n for P r e s i d e n t , for e x a m p l e . I t i s t r u e that t h e campaign w o u l d cost m a n y millions of dollars, but the j o b is really a nice one. It p r o v i d e s free h o u s i n g and, u n l i k e m a n y other j o b s , a l l o w s o n e t o m a k e State o f the Union addresses and be C o m m a n d e r - i n - C h i e f of the armed forces. N o w , e v e n t h o u g h I'd like the j o b , I've d e c i d e d not t o i n v e s t m y millions b e c a u s e I k n o w that this is not a d e c i s i o n u n d e r certainty. E v e n if I w e r e to invest m y m i l l i o n s , I w o u l d n o t b e certain o f g e t t i n g t h e j o b . B e t w e e n m y i n v e s t m e n t a n d t h e j o b l i e s a n e l e c t i o n w h i c h m i g h t n o t n e c e s s a r i l y g o m y w a y . I n fact, s o m e f o r m e r f r i e n d s s u g g e s t e d t h a t m y c h a n c e s are s o p o o r t h a t I w o u l d i n e f f e c t b e g i v i n g m y money away. My decision then d e p e n d s not just on finding the alternative I w a n t most, but also on the e v e n t s intervening b e t w e e n c h o i c e and outcome. I n t e r v e n i n g e v e n t s m a y b e o f t h r e e d i f f e r e n t sorts: c h a n c e e v e n t s t o w h i c h w e can assign a probability, such as the flipping of a coin; c h a n c e events to w h i c h w e c a n ' t a s s i g n a p r o b a b i l i t y , s u c h a s t h e e r u p t i o n o f M t . St. H e l e n ; a n d e v e n t s under the control of an o p p o n e n t w h o is trying to beat us, such as the moves of our o p p o n e n t i n a c h e s s g a m e . T h e s e t h r e e sorts o f i n t e r v e n i n g e v e n t s c o r r e s p o n d t o t h e t h r e e t y p e s o f d e c i s i o n s w e w i l l d i s c u s s i n this c h a p t e r : d e c i s i o n s u n d e r risk, decisions u n d e r uncertainty, and decisions u n d e r conflict. H e r e are e x a m p l e s of each of these three types of decisions: Risk G a m b l i n g d e c i s i o n s are t y p i c a l o f d e c i s i o n s u n d e r risk. I n r o u l e t t e , for e x a m p l e , the g a m b l e r c h o o s e s a n u m b e r on w h i c h to bet, say, 17. T h e n a c h a n c e e v e n t , the position at w h i c h a metal ball c o m e s to rest on the roulette w h e e l , d e t e r m i n e s t h e o u t c o m e o f the g a m b l e r ' s c h o i c e . I f the ball stops o n 17, h e w i n s ; o t h e r w i s e , h e loses. A n e s s e n t i a l f e a t u r e o f d e c i s i o n s u n d e r risk i s t h a t w e c a n c a l c u l a t e a p r o b a b i l ity for t h e e f f e c t o f t h e c h a n c e e v e n t . I n r o u l e t t e , for e x a m p l e , w e k n o w t h a t t h e r e are 3 8 p l a c e s a t w h i c h t h e b a l l i s e q u a l l y l i k e l y t o s t o p . T h e p r o b a b i l i t y o f g e t t i n g 1 7 , t h e n , is j u s t '/ssUncertainty L i k e d e c i s i o n s u n d e r risk, d e c i s i o n s u n d e r u n c e r t a i n t y i n v o l v e a c h a n c e factor. T h e u n i q u e feature of d e c i s i o n s u n d e r uncertainty is that we can't calculate a p r o b a b i l i t y for t h e e f f e c t o f t h e c h a n c e e v e n t . B u y i n g a C h r i s t m a s p r e s e n t for A u n t E m m a , for e x a m p l e , i s a d e c i s i o n u n d e r u n c e r t a i n t y , i f h e r t a s t e s ( t h e c h a n c e factor) are u n k n o w n t o u s . F o r most of us, d e c i d i n g w h e t h e r to go on a picnic w h e n the w e a t h e r looks t h r e a t e n i n g i s m a k i n g a d e c i s i o n u n d e r u n c e r t a i n t y . W e k n o w t h a t i t m a y rain, b u t w e don't k n o w e n o u g h a b o u t w e a t h e r forecasting t o calculate the probability that i t w i l l rain. W e can c h a n g e the d e c i s i o n from o n e u n d e r u n c e r t a i n t y t o o n e u n d e r risk i f w e a s k t h e w e a t h e r b u r e a u t o t e l l u s t h e p r o b a b i l i t y o f r a i n . Conflict D e c i s i o n s u n d e r c o n f l i c t are c o m m o n i n c o m p e t i t i v e g a m e s , l i k e c h e s s a n d tennis, and in business and war. W h e n y o u choose a m o v e in chess, or a strategy

Dealing With C h a n c e in Decision Making

763

i n t e n n i s , y o u k n o w t h a t y o u r o p p o n e n t w i l l d o h i s b e s t t o c o u n t e r y o u r m o v e o r foil your strategy. E a c h d e c i s i o n t y p e requires a different approach. In decisions u n d e r certainty, t h e m a i n d i f f i c u l t y i s d e c i d i n g w h i c h a l t e r n a t i v e i s b e s t . I n d e c i s i o n s u n d e r risk a n d uncertainty, the b i g p r o b l e m is d e a l i n g w i t h the effects of c h a n c e ; and in decisions u n d e r conflict, it is taking account of the hostile action of our opponent. T h e r e l a t i o n s b e t w e e n c h o i c e a n d o u t c o m e for all f o u r d e c i s i o n t y p e s are d i a g r a m m e d in F i g u r e 1.

D E C I S I O N S UNDER RISK* S u p p o s e that y o u h a v e several h u n d r e d spare dollars in y o u r p o c k e t and that y o u are g i v e n t h e c h o i c e o f p l a y i n g t h e f o l l o w i n g t w o g a m e s a s m a n y t i m e s a s y o u l i k e . B o t h g a m e s i n v o l v e t o s s i n g a fair c o i n . Game 1 : Y o u w i n $2.00 w h e t h e r t h e c o i n c o m e s u p h e a d s o r t a i l s . Game 2 : Y o u w i n $ 1 0 . 0 0 i f t h e c o i n c o m e s u p h e a d s a n d l o s e $5.00 i f i t c o m e s up tails. W h i c h game should y o u prefer to play? O n e w i d e l y r e c o m m e n d e d t e c h n i q u e for m a k i n g r i s k y d e c i s i o n s l i k e this o n e i s t o c h o o s e t h e a c t i o n w h i c h h a s t h e g r e a t e s t expected value. T h e e x p e c t e d v a l u e of an action is the average payoff value we can expect if we repeat the action many times. T h e a v e r a g e p a y o f f i n G a m e 1 i s e a s y t o c o m p u t e . S i n c e y o u w i n $2.00 w h e t h e r t h e c o i n c o m e s u p h e a d s o r t a i l s , t h e a v e r a g e p a y o f f h a s t o b e $2.00. I n G a m e 2 , w h e r e t h e p a y o f f s a r e n ' t e q u a l , w e c a n c o m p u t e t h e a v e r a g e payoff, o r e x p e c t e d value, using the following formula: 1.

E x p e c t e d V a l u e = a v e r a g e p a y o f f = p r o b a b i l i t y of a h e a d X p a y o f f for h e a d s + p r o b a b i l i t y of a tail X p a y o f f for tails

or s y m b o l i c a l l y

Since the e x p e c t e d value of G a m e 2 is greater than the e x p e c t e d value of G a m e 1, we should choose G a m e 2 in order to maximize our e x p e c t e d value. * Appendix II provides an elementary introduction to probability. If you are not familiar with the concept of probability, read A p p e n d i x II before proceeding.

764

The C o m p l e t e Problem Solver

765

Dealing With C h a n c e in Decision Making A n o t h e r Example

S u p p o s e t h a t y o u a r e t h e p r o d u c e r for t h e l o c a l t h e a t e r c o m p a n y . L o n g e x p e r i e n c e has t a u g h t y o u that m u s i c a l s b r i n g in a lot of m o n e y , $10,000, if t h e y s u c c e e d , b u t 8 0 % o f t h e t i m e t h e y fail a n d , d u e t o h i g h p r o d u c t i o n c o s t s , l o s e a n a v e r a g e o f $2,000. C o m e d i e s , o n t h e o t h e r h a n d , b r i n g i n m u c h l e s s w h e n t h e y s u c c e e d , $3,000, b u t t h e y s u c c e e d m o r e f r e q u e n t l y , 5 0 % o f t h e t i m e , a n d l o s e l e s s — $ 1 , 0 0 0 — w h e n t h e y f l o p . Y o u a r e p l a n n i n g a d e v e l o p m e n t a l p r o g r a m for y o u r theater, and must d e c i d e either on a series of musicals or a series of c o m e d i e s . W h i c h c h o i c e w i l l m a x i m i z e e x p e c t e d v a l u e for y o u r t h e a t e r ? EV ( m u s i c a l ) = = = EV ( c o m e d y ) = = =

0.2 x $10,000 + 0.8 x - $ 2 , 0 0 0 $2,000 - $ 1 , 6 0 0 + $400 0.5 x $3,000 + 0.5 x - $ 1 , 0 0 0 $ 1 , 5 0 0 - $500 + $1,000

To maximize e x p e c t e d value, you should choose the series of c o m e d i e s . In m a n y situations, we have to deal with more than t w o possible outcomes. For e x a m p l e , w i t h a s i n g l e d i e , w e c a n roll a n y n u m b e r f r o m o n e t h r o u g h six, a n d w i t h t w o d i c e , any n u m b e r from t w o through t w e l v e . T o c o m p u t e e x p e c t e d v a l u e s i n such cases, we have to expand Equation 1 as follows: 2.

EV = P(l)-V(l)+P(2)-V(2)+P(3)-V(3)+ . . . . +P(N)-V(N)

T h a t is, in a situation w i t h N possible o u t c o m e s , the e x p e c t e d v a l u e (EV) is simply t h e p r o b a b i l i t y o f e a c h o u t c o m e t i m e s t h e v a l u e o f t h a t o u t c o m e a d d e d u p o v e r all of the possible outcomes. W e c a n u s e E q u a t i o n 2 t o c o m p u t e t h e e x p e c t e d v a l u e for G a m e 3 . Game 3 .

R o l l i n g a s i n g l e d i e , y o u w i n $6.00 i f y o u g e t a o n e , $5.00 i f y o u g e t a t w o , etc.

For this g a m e , the probability of e a c h o u t c o m e is one-sixth, since e a c h of the six n u m b e r s h a s e q u a l p r o b a b i l i t y a n d t h e e x p e c t e d v a l u e i s EV

(Game

3)

=

V ($6.00)+V ($5.00)+V ($4.00)+V ($3.00)+V ($2.00)+V ($1.00) 6

6

6

6

6

6

_ ($6.00 + $5.00 + $4.00 + $3.00 + $2.00 + $1.00) =

$21.00 6

= $3.50 Since the e x p e c t e d value of G a m e 3 is greater than the e x p e c t e d values of either G a m e 1 or G a m e 2, y o u should prefer to play G a m e 3 to the other g a m e s in order to maximize expected value. Exercises F i n d t h e e x p e c t e d v a l u e for t h e f o l l o w i n g g a m e s : 1 . I f a fair c o i n c o m e s u p h e a d s , y o u w i n $ 1 . 0 0 ; o t h e r w i s e y o u l o s e $1.00. 2 . I f y o u r o l l o n e d i e a n d g e t a six y o u w i n $5.00; o t h e r w i s e y o u l o s e $1.00. 3 . O n a r o u l e t t e w h e e l t h e r e are 3 6 n u m b e r e d p o s i t i o n s a n d t w o h o u s e p o s i t i o n s . I f t h e b a l l s t o p s o n y o u r n u m b e r , y o u g e t y o u r d o l l a r b a c k p l u s $35.00. O t h e r w i s e y o u lose y o u r dollar.

The C o m p l e t e Problem Solver

166

4 . Y o u g e t $6.00 i f y o u r o l l a s e v e n o n a p a i r o f d i c e a n d y o u l o s e $ 1 . 0 0 i f y o u r o l l any other number. * 5 . Y o u g e t $ 3 5 . 0 0 i f y o u r o l l a 1 2 o n a p a i r o f d i c e a n d y o u l o s e $ 1 . 0 0 i f y o u roll a n y other number. DECISIONS

UNDER UNCERTAINTY Sailing,

swimming,

or

tennis?

S u p p o s e that y o u h a v e t o m a k e plans o n e s u m m e r e v e n i n g w h e t h e r t o g o sailing or s w i m m i n g the next day. T h e w e a t h e r forecast is " s u n n y " b u t says nothing about w i n d s . Y o u h a v e no i d e a w h e t h e r it w i l l be w i n d y or not. Preparations h a v e t o b e m a d e n o w , s o y o u can't w a i t till t o m o r r o w t o s e e w h a t the w i n d s w i l l b e l i k e . Y o u k n o w t h a t i f t h e w i n d s are g o o d , s a i l i n g w i l l b e e x c e l l e n t . I f it's n o t w i n d y , though, sailing will be hot and dull. On the other hand, s w i m m i n g will be chilly and u n c o m f o r t a b l e if it is w i n d y , b u t p e r f e c t if it isn't. To h e l p w i t h the d e c i s i o n y o u p r e p a r e a t a b l e l i k e t h a t s h o w n i n T a b l e 1 . T h e a l t e r n a t i v e a c t i o n s t h a t w e are t r y i n g t o d e c i d e a m o n g a r e s h o w n i n t h e r o w s o f t h e t a b l e a n d t h e u n c e r t a i n states o f nature w h i c h affect the o u t c o m e s of t h e s e actions are s h o w n as c o l u m n s . T h e values of the various o u t c o m e s are indicated by the n u m b e r s in the cells. T a b l e 1. V a l u e s of Activities in T w o Kinds of W e a t h e r

Sailing Swimming Tennis

Windy

Calm

Row Minima

10 -2 -3

-5 8 4

-5 _2** -3

Filtering o u t Alternatives W h a t e v e r strategy we d e c i d e to use in approaching decision problems, it is wise to make a habit of determining if any of the alternatives is dominated by any other alternative a n d c o u l d therefore be e l i m i n a t e d . F r o m T a b l e 1 we can s e e that s w i m m i n g d o m i n a t e s t e n n i s b e c a u s e for e v e r y state o f n a t u r e t h e o u t c o m e for s w i m m i n g i s b e t t e r t h a n t h e o u t c o m e for t e n n i s . FOUR

STRATEGIES

W e w i l l d e s c r i b e f o u r s t r a t e g i e s for m a k i n g d e c i s i o n s u n d e r u n c e r t a i n t y : t h e mini-max strategy, the maxi-max strategy, the H u r w i c z strategy, and the mini-max regret strategy. We w i l l use e a c h of these strategies to arrive at a decision in the sailing-swimming dilemma. The Mini-max Strategy T h i s s t r a t e g y , first d e s c r i b e d b y v o n N e u m a n n ( s e e V o n N e u m a n n a n d M o r g e r i stern, 1944), is a v e r y c o n s e r v a t i v e , p e s s i m i s t i c strategy w h i c h a s s u m e s that whatever action we choose, nature is against us and w i l l cause the worst possible outcome. T h u s , if we d e c i d e to go sailing, the strategy m a k e s the g l o o m y assumption that the w i n d s w i l l b e c a l m . O n the other hand, i f w e d e c i d e t o g o s w i m m i n g , then the strategy a s s u m e s that the w e a t h e r w i l l b e w i n d y . T h e v a l u e s o f t h e s e w o r s t o u t c o m e s , the r o w m i n i m a , are s h o w n in the right-hand c o l u m n of T a b l e 1. * W h e n rolling two dice, the outcomes from 2 through 12 are not equally likely. If you don't know how to find out what the probabilities are, consult the A p p e n d i x on Probabilities. **Best choice.

767

Dealing With C h a n c e in Decision Making

T h e m i n i - m a x s t r a t e g y c a l l s for c h o o s i n g t h e a c t i o n t h a t g i v e s u s t h e b e s t (largest) o f t h e s e m i n i m a . T h a t i s , i t c h o o s e s t h e a c t i o n w h o s e w o r s t p o s s i b l e o u t c o m e is not as b a d as the worst possible outcomes of the other actions. T h u s , since t h e w o r s t p o s s i b l e o u t c o m e for s a i l i n g i s — 5 a n d t h e w o r s t p o s s i b l e o u t c o m e for s w i m m i n g i s - 2 , the mini-max strategy chooses s w i m m i n g . ( R e m e m b e r , tennis w a s eliminated because it was dominated by swimming.) T h e mini-max strategy has the n i c e property that it guarantees an o u t c o m e w h i c h i s n o w o r s e t h a n t h e m i n i m u m v a l u e for t h e a c t i o n . T h e o u t c o m e m a y b e b e t t e r than that m i n i m u m , b u t it w i l l certainly be no w o r s e . H o w e v e r , this strategy, w h i c h focuses on p r e v e n t i n g disaster, has the unfortunate p r o p e r t y that it m a y e l i m i n a t e the b e s t o u t c o m e s from consideration.

The Maxi-max Strategy A more adventurous approach to making decisions is to use the maxi-max strategy. T h i s is an optimistic strategy w h i c h a s s u m e s that nature w i l l cooperate w i t h u s t o p r o v i d e t h e b e s t p o s s i b l e o u t c o m e for t h e a c t i o n w e c h o o s e . T h e v a l u e s of these b e s t p o s s i b l e o u t c o m e s , the row maxima, are s h o w n in the right-hand c o l u m n of T a b l e 2. T h e maxi-max strategy chooses the action w h i c h y i e l d s the best of the b e s t p o s s i b l e o u t c o m e s . In this case, it c h o o s e s sailing. T h i s strategy has the nice property of guaranteeing you a chance of obtaining the best possible outcome. H o w e v e r , it does not d e f e n d y o u against the possibility that y o u m a y obtain the worst p o s s i b l e o u t c o m e , as d o e s the mini-max strategy.

T a b l e 2. V a l u e s of T w o Activities in T w o Kinds of W e a t h e r

Sailing Swimming

Windy

Calm

10 -2

-5 8

Row Maxima 10* 8

The H u r w i c z Strategy T h i s strategy ( H u r w i c z , 1953) allows a c o m p r o m i s e — d e p e n d i n g on h o w optimistic y o u f e e l — b e t w e e n the v e r y pessimistic mini-max strategy and the v e r y optimistic maxi-max strategy. To use the H u r w i c z strategy, select a v a l u e b e t w e e n 0 a n d 1 for t h e c o e f f i c i e n t o f o p t i m i s m , A . L o w v a l u e s o f A , s u c h a s 0.1 o r 0.2, are pessimistic a n d reflect a b e l i e f that b a d things are m o r e l i k e l y to h a p p e n . H i g h v a l u e s , s u c h a s 0.8 o r 0.9, a r e o p t i m i s t i c . L e t ' s a s s u m e t h a t w e a r e b e i n g s l i g h t l y p e s s i m i s t i c a n d c h o o s e A = 0.4. F i n d both t h e r o w m i n i m a a n d t h e r o w m a x i m a a s shown in T a b l e 3.

T a b l e 3 . C o m p u t a t i o n s for the H u r w i c z Strategy

Sailing Swimming

Windy

Calm

R o w Max.

10 -2

-5 8

10 8

For each row we compute A ( M a x . ) + (1 - A ) M i n . *Best choice.

Row Min. -5 -2

Weighted Min. & Max. 1 2*

168

The C o m p l e t e Problem Solver

T h u s for s a i l i n g w e g e t

= =

.4(10) + 4 - 3 1.0

.6(-5)

For swimming we get .4(8) + . 6 ( - 2 ) = 3.2 - 1.2 = 2.0 N o w choose the activity w h i c h yields the maximum of these c o m p u t e d quantities. In this case, the c h o i c e is s w i m m i n g . N o t i c e that w h e n A is z e r o , the H u r w i c z strategy is the s a m e as the mini-max strategy; w h e n A is 1, the H u r w i c z strategy is the same as the maxi-max strategy. Another

Example

T h r e e b r o t h e r s — M a n n y , M o e , a n d J a c k — o w n a b a k e r y . S i n c e t h e b a k e r y has b e e n m a k i n g a profit, t h e brothers d e c i d e to i n v e s t their surplus. F o u r i n v e s t m e n t possibilities occur to them: 1. 2. 3. 4.

E x p a n d i n g facilities to increase bread production A d d i n g a l i n e of pastries I n v e s t i n g in the stock market Starting a b r e a d d e l i v e r y service.

T h e brothers r e c o g n i z e that the e x p e c t e d return on their i n v e s t m e n t w i l l d e p e n d on t h e state o f t h e e c o n o m y . T a b l e 4 s u m m a r i z e s t h e i r e s t i m a t e s o f t h e p e r c e n t r e t u r n t o b e e x p e c t e d for e a c h a l t e r n a t i v e u n d e r v a r i o u s s t a t e s o f t h e e c o n o m y . T a b l e 4. P e r c e n t Return on Four Investments in T h r e e States of t h e Economy Better More bread Pastries Stocks Delivery

Same

Worse

Max.

20 30

10 12

2 -10

20

8

6 4

4

8

-20

10

30*

Min. 2 -10

4*

Hurwicz 11* 10

6

T h e brothers can agree to eliminate the delivery service alternative since it is dominated by t w o of the other alternatives. H o w e v e r , they can't agree on w h i c h of the r e m a i n i n g t h r e e alternatives i s b e s t b e c a u s e M a n n y i s v e r y c o n s e r v a t i v e , Jack is w i l d l y optimistic, a n d M o e is a little of each. S i n c e M a n n y is c o n s e r v a t i v e , he a p p l i e s t h e m i n i - m a x s t r a t e g y a n d d e c i d e s t h a t t h e safe a n d s a n e t h i n g t o d o i s t o invest in stocks. Jack, w i t h visions of a giant industry d a n c i n g in his h e a d , a p p l i e s the maxi-max strategy a n d d e c i d e s that m a k i n g pastries is the route to w e a l t h and p o w e r . M o e , w h o s e o p t i m i s m c o e f f i c i e n t i s 0.5, a p p l i e s t h e H u r w i c z s t r a t e g y a n d decides that their best option is to increase bread production. Difficulties w i t h Expected V a l u e s While maximizing expected values seems to be a very reasonable w a y to make d e c i s i o n s i n m a n y c a s e s , t h e r e are s o m e s i t u a t i o n s i n w h i c h i t y i e l d s a n s w e r s t h a t v i o l a t e o u r i n t u i t i o n s , a n s w e r s t h a t j u s t s e e m w r o n g . F o r e x a m p l e , s u p p o s e that y o u h a d the opportunity t o c h o o s e b e t w e e n t h e f o l l o w i n g offers: e i t h e r y o u w i l l r e c e i v e o n e m i l l i o n d o l l a r s for c e r t a i n or, d e p e n d i n g o n t h e f l i p o f a fair c o i n , y o u w i l l *Best choice.

Dealing With C h a n c e in Decision Making

769

r e c e i v e four m i l l i o n dollars or nothing. In this situation, I w o u l d m u c h prefer to take t h e offer o f o n e m i l l i o n d o l l a r s for c e r t a i n e v e n t h o u g h t h e e x p e c t e d v a l u e o f t h e o t h e r offer i s t w i c e a s g r e a t (the r e a d e r s h o u l d c h e c k t h e d i f f e r e n c e i n e x p e c t e d v a l u e , if it isn't clear.) M a n y others h a v e indicated that t h e y w o u l d m a k e the same c h o i c e I h a v e m a d e in this hypothetical situation. L e t ' s c o n s i d e r a m o r e p r a c t i c a l sort o f d e c i s i o n — w h e t h e r o r n o t t o b u y i n s u r a n c e . N o w , w e all k n o w that insurance c o m p a n i e s m a k e m o n e y . T h e e x p e c t e d v a l u e o f a n i n s u r a n c e p o l i c y , t h e n , i s p o s i t i v e for t h e i n s u r a n c e c o m p a n y , b u t n e g a t i v e for t h e i n s u r e d p e r s o n . Y e t m a n y p e o p l e , p e r h a p s m o s t , c h o o s e t o b u y i n s u r a n c e , and that d e c i s i o n can't be r e g a r d e d as an error. P e o p l e are w i l l i n g to accept a small but certain l o s s — t h e p a y m e n t of the insurance p r e m i u m — t o avoid t h e risk o f a v e r y l a r g e l o s s d u e t o a c c i d e n t o r d e a t h , e v e n t h o u g h t h e e x p e c t e d v a l u e of the insurance policy is negative. Finally, let's c o n s i d e r the p r o b l e m o f lotteries: H e r e again, w e k n o w that people w h o run them do so to earn money. T h o u g h the e x p e c t e d value of b u y i n g a l o t t e r y t i c k e t i s n e g a t i v e , m a n y p e o p l e b u y t h e m w e e k after w e e k . W h y d o e s t h e e x p e c t e d v a l u e t e c h n i q u e fail i n t h e s e c a s e s ? E x p e c t e d v a l u e s are averages o f v a l u e s . T h e y are v e r y a p p r o p r i a t e w h e n w e are t r y i n g t o b a l a n c e values that are c l o s e together, e.g., the c h a n c e of l o s i n g five dollars versus the c h a n c e o f w i n n i n g t e n . A v e r a g e s are m u c h l e s s a p p r o p r i a t e , t h o u g h , w h e n w e b a l a n c e v a l u e s t h a t are v e r y d i f f e r e n t , s u c h a s t h e c o s t o f a m o d e s t i n s u r a n c e p a y m e n t v e r s u s t h e risk of b e i n g i m p o v e r i s h e d by a serious car a c c i d e n t , or the dollar price of a lottery ticket versus a prize w h i c h w o u l d a l l o w us to quit our dull old job and m o v e to Tahiti. In the four-million-dollar gamble, you w e r e asked to balance the chance of nothing versus the certainty of a fortune. M o s t choose the fortune. But suppose y o u w e r e g u a r a n t e e d that the g a m b l e w o u l d be offered to y o u 20 t i m e s in a row. In this c a s e , m o s t p e o p l e s w i t c h b a c k t o m a x i m i z i n g e x p e c t e d v a l u e s a n d g o for t h e f o u r m i l l i o n d o l l a r s e a c h t i m e . W h y ? B e c a u s e w i t h 2 0 c h a n c e s , t h e r i s k o f b e i n g left w i t h nothing practically goes a w a y — a b s o r b e d by the law of large numbers. N o w the c h o i c e i s b e t w e e n two l i f e - c h a n g i n g a l t e r n a t i v e s a n d t h e d i f f e r e n c e b e t w e e n t h e m is subjectively m u c h smaller than the difference b e t w e e n one million and zero. A f t e r a l l , i f i t i s h a r d for u s t o i m a g i n e w h a t i t w o u l d b e l i k e t o h a v e a m i l l i o n , i t i s e v e n harder t o i m a g i n e the difference b e t w e e n o n e m i l l i o n a n d four million. T h e decision strategy d e s c r i b e d in the next section m a y be better d e s i g n e d t h a n m a x i m i z i n g e x p e c t e d v a l u e t o h a n d l e d e c i s i o n s o f t h i s sort. Minimizing M a x i m u m Regret A d e c i s i o n p r o c e d u r e w h i c h w o r k s v e r y w e l l for t h o s e c a s e s i n w h i c h e x p e c t e d v a l u e fails i s t h e s t r a t e g y o f m i n i m i z i n g m a x i m u m r e g r e t ( M M R ) . H e r e r e g r e t m e a n s pretty m u c h w h a t y o u w o u l d e x p e c t . I m a g i n e that y o u h a v e d e c i d e d t o g o sailing a n d t h a t t h e w i n d i s c a l m . T h e r e y o u are a t s e a — h o t , s w e a t y , a n d r e g r e t t i n g t h a t y o u didn't go s w i m m i n g . We can take the difference b e t w e e n the value of the outcome you actually obtained and the m a x i m u m value you could have obtained if y o u had c h o s e n a different alternative as a m e a s u r e of regret. We can form a regret matrix for t h e s a i l i n g - s w i m m i n g d e c i s i o n b y c o m p u t i n g r e g r e t for e a c h p o s s i b l e o u t c o m e , a s s h o w n i n T a b l e 5 . T h e m a x i m u m r e g r e t for e a c h a c t i o n i s s h o w n i n t h e r i g h t - h a n d c o l u m n . I f w e are t o m i n i m i z e m a x i m u m r e g r e t i n t h i s c a s e , w e m u s t c h o o s e s w i m m i n g . T a b l e 6 s h o w s t h e r e g r e t m a t r i x for t h e b a k e r y p r o b l e m . I n c r e a s i n g bread production is the action w h i c h m i n i m i z e s m a x i m u m regret. T h e M M R strategy can b e applied t o t w o problems w h i c h presented some d i f f i c u l t i e s for t h e e x p e c t e d v a l u e s t r a t e g y . T h e s e a r e t h e I n s u r a n c e a n d t h e L o t t e r y p r o b l e m s . T a b l e 7 s h o w s t h e p a y o f f a n d r e g r e t m a t r i c e s for a n A u t o I n s u r a n c e

770

The C o m p l e t e Problem Solver Table 5. Payoff and Regret M a t r i c e s for the Sailing-Swimming Problem Payoff

Regret

Matrix

Matrix

Windy

Calm

Windy

Calm

Maximum Regret

10 -2

-5 8

0 12

13 0

13 12*

Sailing Swimming

T a b l e 6. Payoff and Regret Matrices for the Bakery Problem Payoff

Matrix

Regret

Economy

Matrix

Economy

Better

Same

Worse

Better

Same

20 30 8

10 12 6

2 -10 4

10 0 22

2 0

2 14

6

0

Bread Pastries Stocks

Maximum Regret

Worse

10* 14 22

T a b l e 7. Payoff and Regret Matrices for an A u t o Insurance Problem Payoff

Insure Don't Insure

Matrix

Regret

No Accident

Accident

-5 0

-5 -100

No Accident 5 0

Matrix

Accident

Maximum Regret 5* 95

0 95

T a b l e 8. Payoff and Regret Matrices for the Lottery Problem Payoff

Matrix

State of W o r l d Lose Bet Don't Bet

-10 0

Win 100 0

Regret

Matrix

State of W o r l d Lose

Win

10 0

0 100

Maximum Regret 10* 100

P r o b l e m , a n d T a b l e 8 for t h e L o t t e r y P r o b l e m . U s i n g t h e M M R s t r a t e g y w e w o u l d d e c i d e b o t h to b u y insurance and to b e t on the lottery. Exercises U s e t h e H u r w i c z s t r a t e g y w i t h A v a l u e s o f 0.0, 0.2, 0.4, 0.6, 0.8, a n d 1.0, a n d m i n i - m a x r e g r e t t o f i n d t h e b e s t a l t e r n a t i v e s for e a c h o f t h e f o l l o w i n g p a y o f f m a trices: *Best choice.

Dealing With C h a n c e in Decision Making 1.

State

JUDGMENTAL

171

of Nature

PROBABILITIES

W e h a v e m a d e a sharp distinction b e t w e e n d e c i s i o n s u n d e r risk, i n w h i c h w e can c o m p u t e probabilities, and decisions u n d e r uncertainty, in w h i c h we cannot. Actually there is a broad range of cases b e t w e e n these t w o extremes in w h i c h , while we can't calculate probabilities precisely, we can estimate them with varying degrees of accuracy. We will call these probability estimates "judgmental probabilities." The A c c u r a c y of Judgmental Probabilities P e o p l e can be quite good at making probability estimates in certain simple situations. F o r e x a m p l e , i n a n e x p e r i m e n t b y R o b i n s o n (1964) w h e r e p e o p l e w e r e e x p o s e d to a s e q u e n c e of lights w h i c h flashed on the left a n d the right, t h e y w e r e v e r y a c c u r a t e i n e s t i m a t i n g t h e p r o b a b i l i t y t h a t a f l a s h w o u l d b e o n t h e left o r t h e right. A b o u t two-thirds of t h e subjects' j u d g m e n t s fell w i t h i n ± . 10 of the true proportions. Further, g i v e n long experience, p e o p l e can m a k e good probability estimates e v e n in c o m p l e x situations. Peterson et al (1972) and M u r p h y and W i n k l e r (1974) s h o w e d that e x p e r i e n c e d m e t e o r o l o g i s t s w e r e v e r y g o o d a t estimati n g t h e p r o b a b i l i t y t h a t t h e t e m p e r a t u r e w o u l d fall w i t h i n a s p e c i f i e d r a n g e . W h i l e p e o p l e in the situations d e s c r i b e d a b o v e w e r e v e r y accurate in estimati n g p r o b a b i l i t i e s , w e s h o u l d e m p h a s i z e t h a t p e o p l e ' s p r o b a b i l i t y e s t i m a t e s are n o t u n i v e r s a l l y g o o d . T h e i r s u c c e s s d e p e n d s o n h a v i n g a p p r o p r i a t e information—obtained either through immediate perception or long experience. When p e o p l e are a s k e d to m a k e j u d g m e n t s a b o u t topics in w h i c h t h e y are not expert, they m a y do v e r y poorly. F o r e x a m p l e , A l p e r t a n d Raiffa (1969) a s k e d subjects questions o f t h e f o l l o w i n g k i n d : " H o w m a n y f o r e i g n cars w e r e i m p o r t e d i n t o t h e U n i t e d States in 1968? a.

M a k e a h i g h estimate s u c h that y o u feel there is o n l y a o n e p e r c e n t probability t h e t r u e a n s w e r w o u l d e x c e e d y o u r e s t i m a t e .

772

The C o m p l e t e Problem Solver

b. M a k e a l o w estimate s u c h that y o u feel there is o n l y a o n e p e r c e n t probab i l i t y t h a t t h e t r u e a n s w e r w o u l d b e b e l o w this e s t i m a t e . " T h e y f o u n d that in 40 to 50 p e r c e n t of c a s e s , the true a n s w e r fell o u t s i d e of the r a n g e o f v a l u e s s p e c i f i e d b y t h e s u b j e c t . I n short, p e o p l e w e r e t e r r i b l e a t m a k i n g t h e s e judgments. G a m b l e r ' s Fallacy E v e n i n s i m p l e s i t u a t i o n s , t h o u g h , p e o p l e are s u b j e c t t o s y s t e m a t i c errors i n e s t i m a t i n g p r o b a b i l i t i e s . R e c e n t l y a local sports c o m m e n t a t o r a r g u e d that the Oakland Raiders w e r e especially likely to w i n the u p c o m i n g S u p e r b o w l g a m e because o f " t h e l a w o f a v e r a g e s . " H e e m p h a s i z e d r e p e a t e d l y a n d w i t h c o n s i d e r a b l e assurance that since t h e y had lost a string of S u p e r b o w l g a m e s in the recent past, it w a s , by the l a w of a v e r a g e s , their turn to w i n . M a n y b e l i e v e that this l a w of a v e r a g e s is a s t a t i s t i c a l t r u t h . I n fact, t h i s s o - c a l l e d l a w i s a v e r y c o m m o n i l l u s i o n — s o c o m m o n that statisticians h a v e n a m e d it the " G a m b l e r ' s F a l l a c y . " L e t ' s t a k e a v e r y c l e a r c a s e . S u p p o s e w e h a v e a fair c o i n — t h a t is, a c o i n w i t h a 0.5 p r o b a b i l i t y o f c o m i n g u p h e a d s a n d a 0.5 p r o b a b i l i t y o f c o m i n g u p tails. I n a r e a s o n a b l e m o o d , w h i c h i s m o s t o f t h e t i m e , w e w o u l d all a g r e e that t h e c o i n h a s n o m e m o r y a t a l l . I t i s j u s t a m e t a l d i s k . I f w e c o u l d i m a g i n e s o m e w a y t o a s k it, " O n y o u r l a s t f l i p , d i d y o u c o m e u p h e a d s o r t a i l s ? " w e w o u l d h a v e t o b e l i e v e its a n s w e r w o u l d b e , " W h a t flip? I don't r e m e m b e r any flip!!" Y e t strange things h a p p e n to us w h e n w e a c t u a l l y b e g i n t o f l i p t h e c o i n . I m a g i n e that t h e first t w o f l i p s are h e a d s , a n d t h e n t h e third flip is a h e a d , too. We b e g i n to s u s p e c t that the fourth flip w i l l be a tail, b u t n o — a n o t h e r h e a d ! W h a t ' s h a p p e n i n g ? ! S u r e l y the next flip w i l l be a tail, o r a t l e a s t w e f e e l t h a t t h e p r o b a b i l i t y i s v e r y h i g h . A f t e r a l l , a c c o r d i n g t o t h e l a w o f a v e r a g e s it's t i m e for a tail t o t u r n u p . W e h a v e f o r g o t t e n , o f c o u r s e , that t h e c o i n h a s n o m e m o r y . I t k n o w s n o t h i n g o f t h e p a s t a n d i s b e h a v i n g o n t h e fifth f l i p j u s t a s i t d i d o n t h e first. I n b o t h c a s e s , t h e p r o b a b i l i t y o f a tail i s 0.5. T o a v o i d t h e g a m b l e r ' s fallacy, w e s h o u l d forget the past, j u s t a s the c o i n d o e s , a n d treat e a c h f l i p a s a n e v e n t i n d e p e n d e n t o f all p r e v i o u s f l i p s . L i n d s a y a n d N o r m a n (1972) point out that y o u can use the g a m b l e r ' s fallacy to earn money: In horse racing, this psychological tendency [the gambler's fallacy] suggests that you ought to b e t on the favorite w h e n e v e r favorites have b e e n consistently w i n n i n g in the previous few races. T h e assumption is that the other betters will operate according to the gambler's fallacy: T h e y w i l l assume it less and less likely that yet another favorite will win. This tendency w o u l d make the odds deviate from the objective probabilities and thus provides the opportunity for a good bet. (p. 548) THE DIFFICULTY OF REVISING AN O P I N I O N A n o t h e r d i f f i c u l t y i n u s i n g p r o b a b i l i t i e s a r i s e s b e c a u s e p e o p l e are o f t e n n o t very effective in using n e w information to modify old opinions. T h i s ineffectiveness is illustrated clearly in a study by Phillips and E d w a r d s (1966). Phillips and E d w a r d s s h o w e d their subjects a s e q u e n c e of red and w h i t e poker chips d r a w n from a b a g . T h e subjects k n e w that the b a g c o n t a i n e d either 70 red and 3 0 w h i t e c h i p s , o r 3 0 r e d a n d 7 0 w h i t e c h i p s . F u r t h e r , t h e y w e r e t o l d that t h e p r o b a b i l i t y t h a t t h e b a g w a s p r e d o m i n a n t l y r e d w a s e q u a l t o t h e p r o b a b i l i t y that i t w a s p r e d o m i n a n t l y w h i t e . T h e subjects' task w a s to m a k e an estimate of the probab i l i t y t h a t t h e b a g w a s p r e d o m i n a n t l y r e d o r p r e d o m i n a n t l y w h i t e , after s e e i n g e a c h s u c c e s s i v e c h i p . L e t ' s a s s u m e t h a t t h e c h i p s are b e i n g d r a w n f r o m a b a g o f m o s t l y red chips. Before s e e i n g any chips, the subject estimates that the probability is a b o u t 0.50 t h a t t h e b a g c o n t a i n s m o s t l y r e d c h i p s . T y p i c a l l y , a s h e s e e s m o r e a n d m o r e c h i p s , he increases his estimate of the probability that the b a g is mostly red.

Dealing With C h a n c e in Decision Making

773

H o w e v e r , h e d o e s n ' t i n c r e a s e i t n e a r l y a s fast a s h e o u g h t t o o n t h e b a s i s o f t h e e v i d e n c e . E d w a r d s e s t i m a t e s that s u b j e c t s g e t b e t w e e n o n e - h a l f a n d one-fifth o f t h e a v a i l a b l e information from e a c h c h i p . W h a t d o e s this m e a n in terms of accuracy of p r o b a b i l i t y e s t i m a t e s ? I n t h i s s i t u a t i o n , i f s u b j e c t s d r e w e i g h t r e d c h i p s a n d four w h i t e o n e s , t h e y should e s t i m a t e t h e p r o b a b i l i t y o f a m o s t l y r e d b a g a t 0.964. I n fact, R a i f f a ( 1 9 6 8 ) f o u n d t h a t h i s statistics s t u d e n t s c l u s t e r e d t h e i r e s t i m a t e s a r o u n d 0 . 7 0 — n e a r l y 3 0 p e r c e n t too l o w . P e o p l e a r e n ' t a l w a y s i n e f f e c t i v e i n u s i n g n e w i n f o r m a t i o n ( s e e , for e x a m p l e , E d w a r d s , 1 9 6 8 ) . H o w e v e r , t h e r e are e n o u g h s i t u a t i o n s i n w h i c h t h e y are i n e f f e c t i v e t o s u g g e s t t h a t s o m e a i d s for r e v i s i n g j u d g m e n t s c o u l d b e v e r y h e l p f u l t o d e c i s i o n makers. T h e aid w h i c h we will suggest is Bayes' T h e o r e m . Bayes' T h e o r e m * B a y e s ' T h e o r e m i s a p r o c e d u r e for r e v i s i n g o p i n i o n s o n t h e b a s i s o f n e w e v i d e n c e . S u p p o s e t h a t w e h o l d s o m e o p i n i o n s ( h y p o t h e s e s ) a b o u t t h e state o f t h e w o r l d . F o r e x a m p l e , i f w e w e r e m e d i c a l researchers w e m i g h t b e l i e v e that there i s a 75 p e r c e n t c h a n c e that a certain d i s e a s e is c a u s e d by organism A ( H y p o t h e s i s 1) a n d a 2 5 p e r c e n t c h a n c e t h a t i t i s n o t ( H y p o t h e s i s 2). N e x t , w e f i n d s o m e n e w e v i d e n c e a b o u t our h y p o t h e s e s . F o r e x a m p l e , we m i g h t r e c e i v e a report that a case of the disease has b e e n o b s e r v e d w h i c h does not appear to i n v o l v e organism A. We n o w w a n t to revise our opinions on the basis of the n e w e v i d e n c e . In particular, we want to reduce the probability of Hypothesis 1 and increase the probability of H y p o t h e s i s 2. T o use B a y e s ' T h e o r e m t o d o this, w e n e e d one more set o f probabilities. F o r e a c h h y p o t h e s i s , w e n e e d t o k n o w the probability that the n e w e v i d e n c e w o u l d h a v e b e e n o b t a i n e d i f that h y p o t h e s i s w e r e true. I n the m e d i c a l r e s e a r c h e x a m p l e , we will n e e d to k n o w two probabilities: we will n e e d to k n o w the probability of o b s e r v i n g a case of the d i s e a s e w i t h o u t o b s e r v i n g o r g a n i s m A — f i r s t , a s s u m i n g that H y p o t h e s i s 1 is true, and s e c o n d , a s s u m i n g that H y p o t h e s i s 2 is true. We w i l l call these probabilities P ( E | H 1 ) and P(E|H2), w h i c h should be read, "the probability of the e v i d e n c e g i v e n that HI is t r u e , " and, " t h e probability of the e v i d e n c e g i v e n that H 2 i s t r u e . " ( T h e v e r t i c a l b a r i s r e a d a s " g i v e n " o r " g i v e n t h a t . " ) A c c o r d i n g to B a y e s ' T h e o r e m , the n e w probability we ought to assign to HI g i v e n the n e w e v i d e n c e is P(H1|E)

=

P(E|H1) x P(H1) P(E|H1) x P(H1)

+

P(E|H2) x P(H2)

w h e r e P ( H 1 ) a n d P ( H 2 ) are o u r o r i g i n a l e s t i m a t e s o f t h e p r o b a b i l i t i e s o f H I a n d H 2 . N o w , in the m e d i c a l research e x a m p l e , a s s u m e that there is a small probability, say, .05, t h a t t h e d i s e a s e c o u l d b e o b s e r v e d w i t h o u t f i n d i n g t r a c e s o f o r g a n i s m A e v e n i f H y p o t h e s i s 1 i s t r u e . T h a t is, P ( E | H 1 ) = .05. A s s u m e f u r t h e r t h a t t h e r e i s a l a r g e p r o b a b i l i t y , s a y , 0.90, t h a t t h e d i s e a s e c o u l d b e o b s e r v e d w i t h o u t f i n d i n g o r g a n i s m A i f H y p o t h e s i s 2 i s t r u e , t h a t is, P ( E | H 2 ) = .90. B y B a y e s ' T h e o r e m , o u r n e w v a l u e for t h e p r o b a b i l i t y o f H I i s P(Hl|E)

=

(.05) (.75) (.05) (.75)

+

(.90) (.25)

.0375 .0375 +

.225

= 0.143 * M a n y students find the sections on Bayes' T h e o r e m difficult. T h e y may be skipped without spoiling the continuity of the discussion.

174

The C o m p l e t e Problem Solver

I n other w o r d s , w h e n w e take the n e w e v i d e n c e into account, our 7 5 p e r c e n t confidence in Hypothesis 1 should be r e d u c e d drastically to 14 percent according to Bayes' Theorem. W h y use such a c o m p l e x procedure to make a simple decision? T h e reason is that a s the P h i l l i p s a n d E d w a r d s study s h o w e d , p e o p l e don't m a k e v e r y good u s e o f n e w i n f o r m a t i o n t o m o d i f y t h e i r p r e v i o u s o p i n i o n s — t h e y are v e r y c o n s e r v a t i v e a b o u t c h a n g i n g t h e i r m i n d s . C l e a r l y , p e o p l e n e e d s o m e h e l p i f t h e y are t o m a k e adequate use of n e w information. T h e computations i n v o l v e d in using B a y e s ' T h e o rem, w h i l e t h e y w o u l d be a n u i s a n c e in m a k i n g a trivial d e c i s i o n , are really w e l l w o r t h the effort if t h e d e c i s i o n is an important o n e . The Fido Caper S u p p o s e t h a t y o u l e f t y o u r d o g F i d o a t h o m e t o g u a r d y o u r h o u s e s o that burglars w o u l d not b r e a k in a n d steal the 1 0 - p o u n d roast that is defrosting on the c o u n t e r . W h e n y o u g e t b a c k t h e l o c k s are all i n g o o d o r d e r , s o y o u k n o w t h a t n o burglar has e n t e r e d . H o w e v e r , the roast is g o n e . N e e d l e s s to say, F i d o is a p r i m e suspect. On the basis of past experience, t w o sessions w i t h the dog psychiatrist, and a c e r t a i n shifty l o o k i n h i s e y e , y o u j u d g e t h e p r o b a b i l i t y i s 0.95 t h a t F i d o d i d it. H o w e v e r , b e f o r e forcing F i d o t o role p l a y a s the m i s s i n g roast, y o u d e c i d e t o c o l l e c t o n e f u r t h e r p i e c e o f e v i d e n c e . Y o u p r e p a r e h i s o r d i n a r y d i n n e r a n d offer i t t o h i m . T o y o u r s u r p r i s e , h e g o b b l e s i t u p t o t h e last c r u m b . H a r d l y w h a t y o u w o u l d e x p e c t of t h e t h i e f w h o j u s t m a d e a 1 0 - p o u n d roast disappear. Y o u e s t i m a t e that the probab i l i t y t h a t F i d o w o u l d d o t h i s i f h e h a d i n fact e a t e n t h e r o a s t i s o n l y 0.02. N o r m a l l y , t h o u g h , h e h a s a g o o d a p p e t i t e a n d e a t s h i s d i n n e r w i t h a p r o b a b i l i t y o f 0.99. H o w are y o u t o r e v i s e y o u r e a r l i e r s u s p i c i o n s g i v e n t h e e v i d e n c e o f t h e r e a d i l y e a t e n dinner? Clearly, Bayes' T h e o r e m can come to the rescue as follows: T h e probability that F i d o i s g u i l t y g i v e n t h a t h e j u s t a t e h i s d i n n e r i s

Dealing With C h a n c e in D e c i s i o n Making

0.0190 =

+

175

0.0495

0.0190

~ 0.0685 = .28 T h i n g s l o o k e d v e r y b a d for F i d o b e f o r e t h e d i n n e r e x p e r i m e n t . H o w e v e r , w i t h the aid of B a y e s ' T h e o r e m , we w e r e able to take the results of the d i n n e r experim e n t into account and c o n c l u d e that F i d o w a s p r o b a b l y innocent. A n y o n e w h o loves dogs can see the value of Bayes' Theorem. The Antique Problem O n e d a y w a l k i n g past an antique store, y o u spot w h a t y o u are certain, w e l l , 80 p e r c e n t certain, is an antique chair w o r t h about $100. On the other hand, y o u r e c o g n i z e that there is a 20 p e r c e n t c h a n c e that it is w o r t h l e s s . Y o u k n o w that the o w n e r of the s h o p is r e a s o n a b l y c o m p e t e n t in d i s t i n g u i s h i n g a n t i q u e s from junk. If i t i s a n a n t i q u e , t h e r e i s o n l y o n e c h a n c e i n 1 0 t h a t h e w i l l fail t o r e c o g n i z e it. O n t h e o t h e r h a n d , i f i t i s j u n k , t h e r e i s a v e r y h i g h p r o b a b i l i t y , 0.98, t h a t h e w i l l recognize it as junk. Y o u e n t e r t h e s t o r e a n d s a y , " H o w m u c h for t h a t c h a i r ? " T h e o w n e r s a y s , " F i v e dollars." C l e a r l y , he thinks it is j u n k . U s i n g this information, d e t e r m i n e A. a n e w (Bayesian) estimate of the probability that the chair is an a n t i q u e , and B. w h e t h e r y o u s h o u l d b u y it or not. Answer t o A . H y p o t h e s i s H I i s t h a t t h e c h a i r i s a n a n t i q u e . H y p o t h e s i s H 2 i s that t h e c h a i r i s j u n k . T h e n e w e v i d e n c e , E , i s t h a t t h e a n t i q u e d e a l e r t h i n k s it's junk. Y o u r i n i t i a l o p i n i o n s are P ( H 1 ) = 0.8 a n d P ( H 2 ) = 0.2 T h e revised probability is P(H1|E) -

0

1

0

*

0

8

°

The C o m p l e t e Problem Solver

776

T h e n e w e v i d e n c e , t h e n , s h o u l d r e d u c e y o u r c o n f i d e n c e that the chair i s a n antique. Answer to B: To buy or not to buy? N o w e v e n t h o u g h y o u a r e l e s s c o n f i d e n t t h a t t h e c h a i r i s a n a n t i q u e , i t c o u l d still b e a g o o d b u y . T h a t i s , t h e risk m a y b e w o r t h the cost. L e t ' s c o m p u t e the e x p e c t e d value of the purchase. If it is an antique ( P = 0.29), t h e n y o u g a i n $95 ($100 c h a i r m i n u s t h e $ 5 p u r c h a s e p r i c e ) . I f it's j u n k ( P = 0 . 7 1 ) , y o u l o s e $5. T h e e x p e c t e d v a l u e o f t h e p u r c h a s e , t h e n , i s E V = (95) (.29) - (5) (.71) = $27.55 - $3.55 = $24.00 I f y o u c a n s c r a p e t o g e t h e r t h e f i v e d o l l a r s , t h e n y o u o u g h t t o t a k e t h e risk. H o w t o Understand B a y e s ' T h e o r e m Just b e f o r e the d i n n e r e x p e r i m e n t there w e r e four p o s s i b l e o u t c o m e s w e w o u l d i m a g i n e , as s h o w n in T a b l e 9. B e c a u s e t h e s e are the only possibilities, the probabilities of t h e s e four alternatives sum to 1. Table 9. Four Possible O u t c o m e s of the Dinner Experiment Fido guilty and eats d i n n e r P = .0095

Fido innocent and eats dinner

Fido guilty and doesn't eat dinner P = .9405

Fido innocent and doesn't eat dinner P = .0005

P = .0495

X =

0590

2 = .9410 2 = 1

Since F i d o did eat his dinner, the dinner e x p e r i m e n t eliminated t w o of these four alternatives. T h e e l i m i n a t e d alternatives are the l o w e r t w o i n T a b l e 9 . S i n c e o n e o r t h e o t h e r o f t h e r e m a i n i n g t w o a l t e r n a t i v e s must o c c u r , t h e p r o b a b i l i t i e s o f these t w o a l t e r n a t i v e s m u s t n o w s u m t o 1 . T h a t is, t h e t w o a l t e r n a t i v e s w h o s e p r o b a b i l i t i e s s u m m e d t o .0590 b e f o r e t h e d i n n e r e x p e r i m e n t m u s t s u m t o 1 after t h e dinner experiment. T h e probabilities of these alternatives must be increased as a result of the n e w e v i d e n c e . Bayes' T h e o r e m is just a w a y of increasing the probabili t i e s s o t h a t t h e n e w v a l u e o f e a c h p r o b a b i l i t y i s p r o p o r t i o n a l t o its o r i g i n a l v a l u e . Exercises 1. S u p p o s e F i d o had refused his dinner. First, guess the probability that F i d o w a s the roast thief. N e x t c o m p u t e t h e p r o b a b i l i t y u s i n g B a y e s ' t h e o r e m . 2 . S u p p o s e t h a t a b a g h a s a 0.5 c h a n c e o f c o n t a i n i n g 7 0 g r e e n c h i p s a n d 3 0 w h i t e c h i p s , a n d a 0.5 c h a n c e o f c o n t a i n i n g 7 0 w h i t e c h i p s a n d 3 0 g r e e n c h i p s . A g r e e n c h i p is d r a w n from the b a g at random and then replaced. W h a t is the probability n o w that the b a g is p r e d o m i n a n t l y g r e e n ? 3. A s e c o n d g r e e n c h i p is d r a w n at random. W h a t is the n e w probability that the b a g is predominantly green? DECISIONS

UNDER CONFLICT

W h e n we play a g a m e against nature, as in the sailing-swimming decision, we can v i e w the mini-max strategy as a conservative one, b e c a u s e we don't really b e l i e v e t h a t n a t u r e i s t r y i n g t o a r r a n g e t h e w o r s t o u t c o m e s for u s . W h e n w e p l a y a

Dealing With C h a n c e in Decision Making

777

g a m e a g a i n s t a h u m a n o p p o n e n t , h o w e v e r , t h e m i n i - m a x s t r a t e g y i s n ' t a t all c o n s e r v a t i v e . W e know o u r o p p o n e n t i s a c t i v e l y t r y i n g t o d o u s in. T h e m i n i - m a x s t r a t e g y i s t h e o n l y o n e w e w i l l u s e for a n a l y z i n g d e c i s i o n s u n d e r c o n f l i c t . A Two-Person Zero-Sum G a m e G i o v a n n i a n d H a n s , t h e o w n e r s o f a t o w n ' s o n l y t w o r e s t a u r a n t s , are n a t u r a l l y i n c o m p e t i t i o n w i t h e a c h o t h e r . E a c h c a n afford t o try a n y o n e o f a s e t o f s t r a t e g i e s t o attract c u s t o m e r s a w a y f r o m t h e o t h e r ( s e e T a b l e 10). F o r G i o v a n n i t h e r e i s G l , a free glass of red w i n e ; G 2 , waiters w h o sing V e r d i ; and G 3 , half-price pizza. For H a n s , there i s H I , c h e a p b e e r ; H 2 , waiters w h o y o d e l ; H 3 , free sauerkraut; and H 4 , P u m p k i n S t r e u d e l N i g h t . T a b l e 1 0 , t h e p a y o f f m a t r i x for G i o v a n n i , s h o w s t h e p e r c e n t o f t h e total m a r k e t t h a t G i o v a n n i c a n w i n f r o m H a n s w h e n e a c h c h o o s e s o n e o f t h e a v a i l a b l e s t r a t e g i e s . P o s i t i v e v a l u e s are g o o d for G i o v a n n i , a n d n e g a t i v e v a l u e s are g o o d for H a n s . W e a s s u m e t h a t w h a t G i o v a n n i g a i n s , H a n s l o s e s , a n d v i c e

v e r s a . T h a t is, t h e s u m o f G i o v a n n i ' s g a i n s (or l o s s e s ) a n d H a n s ' l o s s e s (or g a i n s ) i s zero. S u c h a g a m e , in w h i c h the sum of gains and losses a d d e d up o v e r all players is z e r o , is c a l l e d a zero-sum game. N o t i c e i n T a b l e 1 0 that H a n s ' strategy H 4 , P u m p k i n S t r e u d e l N i g h t (which h o l d s its o w n o n l y a g a i n s t s i n g i n g w a i t e r s ) i s d o m i n a t e d b y H I a n d H 3 , w h i c h h a v e l o w e r s c o r e s ( b e t t e r for H a n s ) fqr e a c h o f G i o v a n n i ' s s t r a t e g i e s . H 4 i s t h e r e f o r e

778

T h e C o m p l e t e Problem Solver

d r o p p e d . T h e s t r a t e g i e s w h i c h r e m a i n after w e h a v e s c r e e n e d for d o m i n a n c e a r e shown in Table 11. Table 1 1 . Reduced Payoff Matrix HI

H2

H3

Row Min.

I n T a b l e 1 1 , t h e r o w m i n i m a a r e t h e v a l u e s o f t h e w o r s t o u t c o m e s (from G i o v a n n i ' s p o i n t o f v i e w ) for e a c h o f h i s s t r a t e g i e s . T h e c o l u m n m a x i m a are t h e v a l u e s o f t h e w o r s t o u t c o m e s (from H a n s ' p o i n t o f v i e w ) for e a c h o f his s t r a t e g i e s . T h e m i n i - m a x d e c i s i o n for G i o v a n n i i s G 3 , h a l f - p r i c e p i z z a , a n d for H a n s , H I , c h e a p beer. If G i o v a n n i adopts strategy G 3 , he w i l l pick up at least one percent of the m a r k e t a n d h e m a y g e t m o r e i f H a n s d e v i a t e s from h i s m i n i - m a x s t r a t e g y . I f H a n s adopts strategy H I , he w i l l lose no more than one p e r c e n t of the market and he may l o s e l e s s i f G i o v a n n i d e v i a t e s f r o m h i s m i n i - m a x s t r a t e g y . I n t h i s c a s e , i t i s b e s t for e a c h participant to a d o p t a single p u r e strategy. G i o v a n n i s h o u l d a l w a y s use G 3 , a n d H a n s s h o u l d a l w a y s u s e H I . T h e r e are m a n y c a s e s , t h o u g h , i n w h i c h t h e p a r t i c i p a n t s s h o u l d a d o p t mixed strategies. T h a t is, p a r t o f t h e t i m e t h e y s h o u l d t a k e one action and part of the time another. M i x e d or Pure Strategies? T h e r e is a s i m p l e test to d e t e r m i n e w h e t h e r a p u r e or a m i x e d strategy is best. I f t h e m a x i m u m o f t h e r o w m i n i m a (the m a x i - m i n ) e q u a l s t h e m i n i m u m o f t h e c o l u m n m a x i m a (the mini-max), t h e n a p u r e strategy is best. In this situation, e a c h p l a y e r d o e s b e s t p l a y i n g the mini-max strategy. If we play our mini-max strategy, our o p p o n e n t s w i l l a l w a y s lose by d e v i a t i n g from their mini-max strategy. In the same way, if our opponents play their mini-max strategies, we will always lose by d e v i a t i n g f r o m o u r m i n i - m a x s t r a t e g y . I n t h e p a y o f f m a t r i x for G i o v a n n i a n d H a n s (see T a b l e 11), the maxi-min and the mini-max both e q u a l 1. T h u s , pure strategies are a p p r o p r i a t e for t h i s c a s e a n d t h e y g u a r a n t e e t h a t G i o v a n n i w i l l w i n a t l e a s t 1 (the m a x i - m i n ) a n d t h a t H a n s w i l l l o s e n o m o r e t h a n 1 (the m i n i - m a x ) . For the payoff matrix in this g a m e , A l e x s h o u l d gain m o r e than 1 (the mini-max). values will depend on how

T a b l e 12, the mini-max is 1 but the maxi-min is 0. In n o l e s s t h a n 0 (the m a x i - m i n ) , a n d B e r t s h o u l d l o s e n o Just w h e r e t h e g a i n s a n d l o s s e s w i l l fall b e t w e e n t h e s e Alex and Bert play. T h e mini-max decision procedure

Table 1 2 . Payoff Matrix in the Alex-Bert Conflict Bert's Strategies Bl a i

_A2 C o l u m n Max. Payoff Difference

-

1

B2

Row Min. -

3

1

0

1*

3

2

3

-

_, ^ rayon Difference 3-(-D=4

i

0*

1-0

=1 2,-5

Dealing With C h a n c e in Decision Making

779

w e u s e d a b o v e w o u l d select pure strategies f o r e a c h p a r t i c i p a n t — A l t e r n a t i v e A 2 f o r A l e x and A l t e r n a t i v e B l f o r Bert. B u t notice that t h e s e strategies don't h a v e the nice properties that the strategies chosen for G i o v a n n i and Hans had. If Alex always c h o o s e s A 2 , t h e n Bert is soon g o i n g to s w i t c h from Bl to B2 so as to lose less w i t h e a c h trial. T o k e e p B e r t f r o m d o i n g t h i s , A l e x s h o u l d s o m e t i m e s u s e A l a n d s o c k B e r t w i t h a l o s s o f 3 p o i n t s . T h a t is, A l e x s h o u l d u s e a m i x e d s t r a t e g y . In u s i n g a m i x e d strategy, though, just h o w often s h o u l d A l e x play Al and h o w often s h o u l d he p l a y A 2 ? T h e r e is a fairly s i m p l e p r o c e d u r e he c a n u s e to find the b e s t p r o p o r t i o n s for t h e v a r i o u s s t r a t e g i e s . F i r s t , h e s h o u l d f i n d t h e m a g n i t u d e o f t h e d i f f e r e n c e i n p a y o f f w i t h i n e a c h r o w o f t h e p a y o f f m a t r i x . T h e s e d i f f e r e n c e s are 4 for t h e first r o w , 1 for t h e s e c o n d , a n d t h e i r s u m i s 5 . U s i n g t h e s e n u m b e r s , A l e x can n o w do the following calculations:

and c o n c l u d e that h e s h o u l d u s e Strategy A l one-fifth o f t h e t i m e a n d Strategy A 2 four-fifths o f t h e t i m e . I n t h e s a m e w a y , B e r t c a n u s e t h e d i f f e r e n c e s i n p a y o f f i n t h e c o l u m n s to find the proportions in w h i c h he should mix his strategies. Bert calculates as follows:

a n d c o n c l u d e s that h e s h o u l d u s e B l three-fifths o f t h e t i m e a n d B 2 two-fifths o f the time. E a c h p l a y e r s h o u l d c o n c e a l h i s i n t e n d e d a c t i o n o n e a c h trial s o t h a t h e d o e s n ' t forewarn his o p p o n e n t of his i n t e n d e d m o v e . O n e w a y to do this is to let some random process like the throw of a die d e t e r m i n e w h a t one w i l l do next. I f e a c h p l a y e r p l a y s t h e o p t i m a l s t r a t e g y d e f i n e d b y t h e m e t h o d w e j u s t outl i n e d , t h e n t h e v a l u e o f t h e g a m e (the a m o u n t t h a t A l e x c a n e x p e c t t o w i n o r t h e average per game) is easy to determine. T h e probability that the p a y o f f i s - 1 i s j u s t the probability that A l e x plays A l a n d B e r t p l a y s B l , t h a t i s , 1/5 x 3/5 = 3/25. M a k i n g t h e s e c a l c u l a t i o n s for all t h e c e l l s of the payoff matrix we can find the e x p e c t e d v a l u e of the g a m e as follows:

=

3 - — + 25

6 — + 25

12 — = 25

15 — = 25

0.60

I n this c a s e , t h e v a l u e i s 0 . 6 — a v a l u e , a s w e w o u l d e x p e c t , b e t w e e n 0 , t h e r o w m a x , and 1, the column min. An interesting feature of t h e s e strategies is that the v a l u e of the g a m e w i l l remain the s a m e as l o n g as either of the p l a y e r s h o l d s to his optimal strategy.

180

The C o m p l e t e Problem Solver

A G U I D E FOR THE BEWILDERED DECISION MAKER In this and the p r e v i o u s chapter, we h a v e d e s c r i b e d a n u m b e r of decision p r o c e d u r e s a p p l i c a b l e in a variety of situations. F i g u r e 2, a d e c i s i o n tree, can h e l p y o u find the d e c i s i o n p r o c e d u r e that is appropriate to a specific situation. By answering the following questions, you can work through the decision tree: 1. Is this a d e c i s i o n u n d e r certainty? 2. 3. 4. 5.

D o e s it i n v o l v e costly search? Is t h i s a d e c i s i o n u n d e r c o n f l i c t ? C a n you estimate the relevant probabilities with reasonable accuracy? D o e s the decision involve catastrophic outcomes?

Dealing With C h a n c e in Decision Making

181

Exercises I d e n t i f y o p t i m a l s t r a t e g i e s for e a c h p l a y e r i n t h e f o l l o w i n g t w o - p e r s o n g a m e s .

REFERENCES A l p e r t , M . , a n d Raiffa, H. A Progress Report on the Training of Probability Assessors. U n p u b l i s h e d m a n u s c r i p t , H a r v a r d U n i v e r s i t y , 1 9 6 9 . Edwards,

W.

"Conservatism

sentation o f Human Judgment,

in

Human

Information

e d i t e d by B.

Processing."

Kleinmuntz.

In

N e w York:

Formal

Repre-

John W i l e y

& Sons, Inc., 17-52, 1968. H u r w i c z , L . " W h a t H a s H a p p e n e d t o t h e T h e o r y o f G a m e s ? " American Economic Review Supplement, 4 3 , 3 9 8 - 4 0 5 , 1 9 5 3 . Lindsay, P.H., and Norman,

D . L . Human Information Processing.

N e w York: Aca-

d e m i c Press, Inc., 1972. Murphy, A . H . , and Winkler, R.L. " C r e d i b l e Interval T e m p e r a t u r e Forecasting: S o m e E x p e r i m e n t a l R e s u l t s . " Monthly Weather Review,

102, 784-794,

1974.

P e t e r s o n , C . R . , S n a p p e r , K.J., a n d M u r p h y , A . H . " C r e d i b l e I n t e r v a l T e m p e r a t u r e F o r e c a s t s . " Bulletin of the American Meteorological Society, 5 3 , 9 6 6 - 9 7 0 , 1 9 7 2 . Phillips, L.D., and E d w a r d s , W. "Conservatism in a Simple Probability Inference T a s k . " Journal of Experimental Psychology, 7 2 , 3 4 6 - 3 5 7 , 1966.

NINE

COST-BENEFIT

ANALYSIS

S h o u l d saccharin b e b a n n e d ? S h o u l d pollution standards b e stricter? A r e nuclear p o w e r plants safe? S h o u l d e v e r y o n e b e v a c c i n a t e d against m e a s l e s ? A r e auto seat belts w o r t h the cost? G o v e r n m e n t a g e n c i e s a n d b u s i n e s s e s are continually faced w i t h q u e s t i o n s s u c h as t h e s e . In this chapter, we w i l l d e s c r i b e a d e c i s i o n tool widely used to help answer such questions: Cost-Benefit Analysis. Understanding cost-benefit analysis will: J. 2.

h e l p y o u to e v a l u a t e g o v e r n m e n t a l decisions w h i c h affect y o u , a n d p r o v i d e y o u w i t h a d e c i s i o n - m a k i n g tool w h i c h y o u can u s e to m a k e b u s i n e s s a n d personal decisions.

C o s t - b e n e f i t analysis is b a s e d on a v e r y s i m p l e i d e a — t h e i d e a that we s h o u l d t a k e a n a c t i o n o n l y after w e h a v e c o n s i d e r e d all o f its c o s t s a n d b e n e f i t s . " C e r t a i n l y everyone k n o w s t h a t , " y o u m a y m u t t e r . W h i l e t h e i d e a i s v e r y s i m p l e , i t i s s u r p r i s i n g h o w o f t e n p e o p l e fail t o a p p l y it. W h e n w e b u y a n e w p u p p y , w e m a y fail t o c o n s i d e r t h e w e a r a n d t e a r o n t h e r u g o r t h e cat. W h e n a g r o c e r c u t s t h e p r i c e o f strawberries, h e m a y not c o n s i d e r the possibility that b l u e b e r r y sales w i l l b e r e d u c e d as a r e s u l t . S u p p o s e that y o u w e r e t r y i n g t o d e c i d e a m o n g s e v e r a l alternative actions, e.g., t o a d d a p o r c h t o y o u r h o u s e , t o i n s u l a t e it, t o t a k e a v a c a t i o n i n A r u b a , o r t o d o nothing. To c o n d u c t a cost-benefit analysis, y o u should perform the following steps: A. For each of the alternative actions, 1. identify all the important sources of costs and b e n e f i t s ; 2. estimate the v a l u e s of the costs a n d benefits; 3. estimate the probabilities of obtaining the costs a n d benefits; and 4. compare the e x p e c t e d v a l u e s of the costs and benefits. B. C h o o s e t h e a c t i o n for w h i c h the expected value of the benefits m i n u s the expected value of the costs is g r e a t e s t . E a c h o f t h e s e s t e p s h a s its o w n u n i q u e p i t f a l l s i n t o w h i c h t h e u n w a r y c o s t - b e n e f i t a n a l y z e r m a y fall. W e w i l l d i s c u s s t h e p i t f a l l s b e l o w , b u t first w e w i l l p r e s e n t a highly simplified e x a m p l e to illustrate the m e c h a n i c s of cost-benefit analysis. S u p p o s e t h a t y o u h a v e f i n a l l y f o u n d a p o t e n t i a l c u s t o m e r for y o u r $200 c a r after m o n t h s a n d m o n t h s o f t r y i n g . Y o u a r e d r i v i n g i t t o y o u r c u s t o m e r ' s h o u s e for inspection w h e n it b e g i n s to m a k e a dreadful n o i s e — " P o c k e t a - q u e e p ! Pocketaq u e e p ! " A t this p o i n t y o u are a p p r o a c h i n g the o n l y repair shop i n t o w n — 183

T h e C o m p l e t e Problem Solver

184

" D i s h o n e s t F r a n k ' s G a r a g e and M o r t u a r y . " Y o u k n o w that F r a n k is a g o o d m e c h a n i c w h o c a n fix t h e t r o u b l e . S h o u l d y o u s t o p for r e p a i r s o r g o o n ? T o m a k e t h e decision y o u a n a l y z e the costs a n d benefits as follows: Action

1:

Stop

at

Frank's

Costs. Y o u e s t i m a t e t h a t F r a n k i s m o s t l i k e l y t o c h a r g e y o u $ 5 0 ( p r o b a b i l i t y = .80), b u t t h a t t h e b i l l m a y b e $ 1 0 0 ( p r o b a b i l i t y = .20). T h e e x p e c t e d v a l u e o f t h e costs, then, is EV (costs) = (.80) ($50) + (.20) ($100) = $40 + $20 = $60 Benefits. Y o u j u d g e t h a t t h e r e are t h r e e p o s s i b l e o u t c o m e s o f y o u r s a l e s efforts: 1. T h e person w i l l be so horrified by the appearance of y o u r car that he w i l l d e c i d e not to b u y it w i t h o u t e v e n l i s t e n i n g to the e n g i n e . In this case, the b e n e f i t is zero. Y o u e s t i m a t e t h a t t h e p r o b a b i l i t y o f t h i s o u t c o m e i s 0.50. 2 . T h e p e r s o n w i l l b e s o e n t r a n c e d b y the a p p e a r a n c e o f the car that h e w i l l b u y i t n o m a t t e r w h a t t h e e n g i n e s o u n d s l i k e . I n t h i s c a s e , t h e b e n e f i t i s $200. Y o u estimate that the p r o b a b i l i t y of this o u t c o m e is 0.10. 3 . T h e p e r s o n f i n d s t h e c a r a c c e p t a b l e o n l y i f t h e e n g i n e s o u n d s g o o d . I n this c a s e , t h a n k s t o F r a n k , t h e b e n e f i t i s $200. Y o u e s t i m a t e t h a t t h e p r o b a b i l i t y o f t h i s o u t c o m e i s 0.40. T h e e x p e c t e d value of the benefits is EV ( b e n e f i t s ) = (.50) (0) + (.10) ($200) + (.40) ($200) = $100 E V ( b e n e f i t s ) - E V (costs) = $40. Action

2:

Costs.

Don't

Stop

at

Frank's

Zero

Benefits. T h e b e n e f i t s o f t h i s a c t i o n a r e t h e s a m e a s t h e b e n e f i t s for A c t i o n 1 , w i t h one exception. If the b u y e r d e c i d e s to listen to the e n g i n e , y o u will lose the s a l e . I n t h i s c a s e , t h e b e n e f i t s w i l l b e z e r o r a t h e r t h a n $200. T h e e x p e c t e d value of the benefits is ( E V ) b e n e f i t s = (.50) (0) + (.10) ($200) + (.40) (0) = $20 E V ( b e n e f i t s ) - E V (costs) = $20. S i n c e t h e e x p e c t e d v a l u e o f t h e b e n e f i t s m i n u s t h e c o s t s i s g r e a t e r for A c t i o n 1 t h a n A c t i o n 2 , y o u d e c i d e t o s t o p for r e p a i r s .

PROBLEMS IN F I N D I N G ALL OF THE I M P O R T A N T C O S T S A N D BENEFITS S o m e actions h a v e e n o r m o u s l y w i d e s p r e a d effects w h i c h are q u i t e difficult to track d o w n . F o r e x a m p l e , b u i l d i n g a d a m m a y h a v e effects s u c h as i m p r o v i n g flood control and transportation, increasing recreational facilities, and providing electric p o w e r and irrigation. T h e s e effects should result in r e d u c e d cost of p o w e r and food a n d i n c r e a s e d t o u r i s m a n d i n d u s t r y for t h e r e g i o n . T h e s e b e n e f i t s w i l l l i k e l y b e a c c o m p a n i e d b y a n i n c r e a s e i n traffic, i n l a w e n f o r c e m e n t p r o b l e m s , i n p o l l u t i o n , a n d i n t h e c o s t o f s u m m e r h o m e s . I d e n t i f y i n g all t h e c o s t s a n d b e n e f i t s i n a c o m p l e x c a s e s u c h a s t h i s m a y p r o v e v e r y d i f f i c u l t . B e f o r e p r o c e e d i n g , try t o t h i n k o f a l l o f the other p o s s i b l e costs and benefits in this case.

Cost Benefit Analysis

185

D i d y o u think of the potential increase in v a l u e of land in areas formerly subject to f l o o d i n g ? D i d y o u think of the possibility that t e l e p h o n e s e r v i c e in the area m a y b e o v e r l o a d e d ? D i d y o u t h i n k o f fire a n d h e a l t h s e r v i c e s ? A C a s e Study Saccharin: To Ban or Not to Ban? In March of 1977, the C o m m i s s i o n e r of the F o o d and D r u g Administration created a minor public uproar w h e n he proposed b a n n i n g the sale of saccharin. T h e C o m m i s s i o n e r took this action b e c a u s e he j u d g e d that saccharin p o s e d a small b u t d e f i n i t e h e a l t h h a z a r d for h u m a n s . H e e s t i m a t e d t h a t t h e l i f e t i m e u s e o f o n e s a c c h a r i n - s w e e t e n e d soft d r i n k p e r d a y m i g h t c a u s e a s m a n y a s 1,200 c a s e s o f b l a d d e r cancer p e r y e a r in the U n i t e d States. T h e law under w h i c h the C o m m i s s i o n e r was acting is very specific. T h e Federal F o o d , D r u g , and C o s m e t i c Act w h i c h regulates food additives contains the so-called D e l a n e y anti-cancer c l a u s e (21 U S C § 348). T h i s c l a u s e states t h a t " . . . n o a d d i t i v e s h a l l b e d e e m e d t o b e safe i f i t i s f o u n d t o i n d u c e c a n c e r w h e n i n g e s t e d by m a n or a n i m a l " A recently c o m p l e t e d C a n a d i a n study (1974) p r o v i d e d the C o m m i s s i o n e r w i t h r e a s o n a b l y c o n v i n c i n g e v i d e n c e t h a t s a c c h a r i n d o e s i n d u c e c a n c e r i n rats. T h e C o m m i s s i o n e r ' s d u t y w a s c l e a r . H e had t o b a n s a c c h a r i n . H o w e v e r , t h i s w a s n o t t h e w h o l e story. A s t h e C o m m i s s i o n e r s a i d (Federal Register, A p r i l 1 5 , 1 9 7 7 ) , Press reports of the announcement of the F D A ' s intention to withdraw approval of saccharin as an ingredient in foods and beverages have g i v e n the impression that the Commissioner is acting reluctantly, based exclusively on the D e l a n e y anticancer clause of the Federal Food, Drug, and C o s m e t i c A c t . . . and, further, that the agency's action was triggered solely by the findings of the Canadian study. Neither impression is accurate.

He cited a n u m b e r of other studies, in addition to the Canadian one, w h i c h had i n d e p e n d e n t l y s u g g e s t e d that saccharin c a u s e d cancer. In addition, he stated a r a t i o n a l e for b a n n i n g e v e n m i l d c a n c e r - c a u s i n g a g e n t s s u c h a s s a c c h a r i n : Although the risk from consumption of saccharin is small compared to that of other health hazards, e.g., cigarette smoking, saccharin is only one of a potentially large number of hazards present in our environment. T h e Commissioner b e l i e v e s that reduction of prolonged, general exposure to a number of w e a k l y carcinogenic substances in our environment as they are discovered may be essential to reduce the total incidence of cancer. (Federal Register, April 15, 1977)

T h e proposed ban received a considerable amount of unfavorable comment. Marvin Eisenstadt, executive president of C u m b e r l a n d Packing Corporation, the nation's largest p r o d u c e r of l o w calorie s w e e t e n e r s , c h a r g e d that the " o u t r a g e o u s and harmful action" w a s b a s e d on "flimsy scientific e v i d e n c e . " Public opinion polls s h o w e d that a majority of c o n s u m e r s b e l i e v e d that the a n i m a l tests w e r e either i r r e l e v a n t o r n o t c o n c l u s i v e . (As w e w i l l s e e , t h e p u b l i c w a s w r o n g o n b o t h c o u n t s , but not necessarily w r o n g in their opposition to the ban.) Criticism of the scientific e v i d e n c e varied in sophistication. On the simplem i n d e d e n d w a s t h e p o p u l a r j o k e , " W h a t t h e s t u d i e s s h o w i s t h a t rats s h o u l d n ' t d r i n k d i e t c o l a . " T h e u g l y fact i s t h a t a g e n t s t h a t c a u s e c a n c e r i n rats a r e v e r y l i k e l y to cause it in h u m a n s , too. A more sophisticated criticism c o n c e r n e d the saccharin d o s a g e s i n v o l v e d i n t h e s t u d y . T h e a m o u n t o f s a c c h a r i n t h e rats w e r e g i v e n w a s , t a k i n g t h e i r b o d y w e i g h t i n t o a c c o u n t , a b o u t 1,000 t i m e s g r e a t e r t h a n t h e a m o u n t a h u m a n w o u l d t y p i c a l l y u s e . T h i s c r i t i c i s m s e e m s i m p r e s s i v e o n first h e a r i n g . H o w e v e r , i t l o s e s m u c h o f its f o r c e w h e n o n e c o n s i d e r s t h e r a t i o n a l e o f a n i m a l

786

T h e C o m p l e t e Problem Solver

s t u d i e s . F i r s t , t h e l i f e s p a n o f a rat ( t w o y e a r s ) i s m u c h s h o r t e r t h a n t h a t o f a h u m a n (75 years). If we are i n t e r e s t e d in lifespan effects, t h e n it is r e a s o n a b l e to u s e h i g h e r d o s a g e s i n t h e rat w h o i s e x p o s e d o v e r a m u c h s h o r t e r t i m e t h a n t h e h u m a n . F u r t h e r , i n a h u m a n p o p u l a t i o n , a n i n c i d e n c e o f o n e c a s e o f c a n c e r i n 1,000 o r i n 10,000 w o u l d b e c o n s i d e r e d i m p o r t a n t . A n i m a l r e s e a r c h e r s t y p i c a l l y c o n d u c t their e x p e r i m e n t s w i t h o n l y 50 or 100 animals. If a n i m a l r e s e a r c h e r s u s e d a dosage o f s a c c h a r i n w h i c h p r o d u c e d o n l y o n e c a s e i n 1,000 o r o n e c a s e i n 10,000, t h e r e i s a v e r y g o o d c h a n c e t h a t n o n e o f t h e i r o n e h u n d r e d rats w o u l d s h o w a n y s i g n s o f t h e disease. To have a reasonable chance of observing an important cancer-producing effect, the researchers m u s t e i t h e r u s e a v e r y large n u m b e r of animals or increase the effect by using a h i g h dosage. T h e researchers' d e c i s i o n to use h i g h dosages is entirely justified. Most of the criticism of the F D A ruling was directed at the validity of the e v i d e n c e that saccharin c a u s e d cancer. T h i s criticism appears t o h a v e b e e n o v e r l y severe. R e c e n t studies h a v e c o n f i r m e d the c o n c l u s i o n that saccharin d o e s c a u s e b l a d d e r cancer. D o e s i t follow t h e n that w e s h o u l d b a n saccharin? W h a t d o y o u think? We k n o w that u s i n g saccharin has a c o s t — n a m e l y an increased i n c i d e n c e of as m u c h a s 1,200 c a s e s o f b l a d d e r c a n c e r a y e a r . B u t w h a t a b o u t t h e b e n e f i t s o f saccharin? T h e r e are t w o major uses of saccharin: 1. to h e l p diabetics k e e p to their sugar-free diets, a n d 2 . t o h e l p dieters k e e p their w e i g h t d o w n . W e w i l l discuss only the second benefit. E x c e s s w e i g h t m a y s e e m a trivial health p r o b l e m c o m p a r e d to cancer, b u t in this c a s e , i t m a y n o t b e t r i v i a l a t a l l . E x c e s s w e i g h t h a s i m p o r t a n t i m p l i c a t i o n s for health and life. P e o p l e w h o are o v e r w e i g h t die y o u n g e r than other p e o p l e . Bernard C o h e n ( 1 9 7 8 ) h a s c o m p a r e d t h e b e n e f i t s o f s a c c h a r i n i n c o n t r o l l i n g w e i g h t w i t h its costs i n c a u s i n g c a n c e r . U s i n g t h e F D A data, C o h e n c a l c u l a t e d that d u e t o c a n c e r risk, e a c h soft d r i n k r e d u c e d o n e ' s l i f e e x p e c t a n c y b y a b o u t n i n e s e c o n d s . T o p u t this i n p e r s p e c t i v e , e a c h c i g a r e t t e r e d u c e s l i f e e x p e c t a n c y b y a b o u t 1 2 m i n u t e s , o r about 80 times as m u c h . D r i n k i n g a diet drink e v e r y day over a w h o l e lifetime w o u l d r e d u c e life e x p e c t a n c y b y a b o u t t w o d a y s . N o w , life i n s u r a n c e statistics s h o w that e v e r y p o u n d of e x c e s s w e i g h t a p e r s o n c a r r i e s d e c r e a s e s life e x p e c t a n c y b y 2 9 d a y s . S u b s t i t u t i n g o n e s a c c h a r i n - s w e e t e n e d d r i n k for a s u g a r - s w e e t e n e d d r i n k e v e r y d a y w o u l d r e d u c e w e i g h t e n o u g h t o c a u s e a 200-day increase in life e x p e c t a n c y — 1 0 0 times as large as the t w o - d a y d e c r e a s e d u e t o c a n c e r risk. T h u s , e v e n w i t h o u t c o n s i d e r i n g t h e b e n e f i t s o f s u g a r - f r e e d r i n k s for d i a b e t i c s , i t a p p e a r s t h a t t h e b e n e f i t s o f s a c c h a r i n a r e m u c h g r e a t e r t h a n its costs.* W h a t c a n w e l e a r n f r o m a c a s e s u c h a s t h i s ? A t least t w o t h i n g s . 1.

A law, like the D e l a n e y anti-cancer clause, w h i c h focuses narrowly on avoiding one particular k i nd of cost, m a y be more harmful than helpful. By failing to look a t all t h e c o s t s a n d b e n e f i t s , i t m a y k i l l m o r e p e o p l e t h a n i t s a v e s . T h i s p o i n t b e c o m e s d r a m a t i c a l l y e v i d e n t i n t h e c a s e o f V i t a m i n A , a s u b s t a n c e necessary t o life. V i t a m i n A has b e e n s h o w n to h a v e m i l d c a n c e r - c a u s i n g effects. If we tried to stamp out V i t a m i n A b e c a u s e of this particular cost, we w o u l d do vast harm.

* W e may want to question C o h e n ' s analysis of the healthful effects of diet drinks. For example, we might want proof that overweight people actually do reduce their intake of calories as a result of using diet soft drinks. T h e r e are e n o u g h reports of people who diet by eating large quantities of diet foods in addition to their regular meals to make us question this assumption. Whether C o h e n ' s analysis is accurate or not, it serves to alert us that health costs due to excess w e i g h t may be considerable, and should be accurately evaluated and taken into account in making the saccharin decision.

Cost Benefit Analysis

787

T h e point applies not just to laws, like the D e l a n e y clause, but to decision procedures generally. W h e n e v e r we focus narrowly on avoiding a particular cost and don't take all the important costs and benefits into account, we m a y m a k e very bad decisions. 2.

A s e c o n d l e s s o n t o b e d e r i v e d f r o m t h e s a c c h a r i n c a s e i s t h a t p e o p l e o f t e n fail t o l o o k for a l l t h e c o s t s a n d b e n e f i t s . T h e p r o p o s e d b a n f o c u s e d e n t i r e l y o n t h e c a n c e r issue, as d i d objections to the ban. T h i s w r i t e r w a s a p p a l l e d to see his favorite n e w s commentators looking v e r y serious about the controversy, s c r u n c h i n g u p t h e i r e y e b r o w s , s a y i n g , " W e l l , j u s t h o w g o o d i s t h e e v i d e n c e that saccharin causes c a n c e r ? " and c o m p l e t e l y missing the point that there might be costs as w e l l as benefits in b a n n i n g saccharin. D e l a n e y d i d n ' t c o n s i d e r all t h e c o s t s a n d b e n e f i t s o f s a c c h a r i n w h e n h e w a s w r i t i n g his law, the F D A C o m m i s s i o n e r d i d n ' t c o n s i d e r t h e m w h e n h e p r o p o s e d his ban, and the p u b l i c d i d n ' t c o n s i d e r t h e m w h e n t h e y criticized the ban. Cost-benefit analysis may seem a very elementary and obvious idea, but people surely have failed to use it in cases w h e r e it was appropriate.

In the saccharin case, applying the technique of cost-benefit analysis w o u l d have broadened people's perspective on the problem by leading them to search a c t i v e l y for c o s t s a n d b e n e f i t s . I t w o u l d b y n o m e a n s h a v e g u a r a n t e e d that t h e y w o u l d find t h e m all, h o w e v e r .

PROBLEMS IN ESTIMATING PROBABILITIES We have already discussed some of the problems caused w h e n p e o p l e make s u b j e c t i v e j u d g m e n t s o f p r o b a b i l i t y , a s w i t h t h e g a m b l e r ' s f a l l a c y ( C h a p t e r 8). Measuring objective probabilities can cause problems as w e l l . T h i s is particularly true w h e n w e a r e d e a l i n g w i t h n e w t e c h n o l o g y . I f w e w a n t e d t o m e a s u r e t h e p r o b a b i l i t y o f f a i l u r e o f a 1 9 7 0 ' s a u t o m a t i c t r a n s m i s s i o n , for e x a m p l e , w e c o u l d c o u n t t h e n u m b e r o f u n i t s t h a t f a i l e d after o n e y e a r , after t w o y e a r s , a n d after three years, and c o m p u t e the appropriate probabilities. F o r a n e w transmission just d e v e l o p e d this year, t h o u g h , we don't h a v e the data to m a k e these probability measurements. C u s t o m e r s might like to have them n o w , b u t we simply can't deliver them. W h e n n u c l e a r r e a c t o r s w e r e first d e v e l o p e d , e n g i n e e r s w a n t e d t o k n o w t h e probabilities that various materials such as steel and concrete w o u l d be w e a k e n e d by intense radiation. M a n y years of e x p e r i e n c e w i t h the n e w t e c h n o l o g y w e r e required before the answers could be provided. O t h e r c a s e s i n w h i c h i t i s d i f f i c u l t t o g e t a d e q u a t e p r o b a b i l i t y e s t i m a t e s are situations i n v o l v i n g large risks w h i c h h a v e a v e r y small p r o b a b i l i t y of occurring, such as airplane crashes a n d oil spills. S u p p o s e that y o u are c o n s i d e r i n g b u y i n g a h o u s e j u s t b e l o w a n e w d a m . I t i s i m p o r t a n t for y o u t o k n o w w h e t h e r t h e p r o b a b i l i t y that t h e d a m w i l l b r e a k i n t h e n e x t 1 0 y e a r s i s o n e i n a h u n d r e d , o n e i n t e n t h o u s a n d , one in a million or less. Y o u ask an e n g i n e e r w h o says, " W e l l , that's a n e w kind of d a m , a n d o n l y 2 0 o f t h e m are 1 0 y e a r s o l d . S i n c e n o n e h a s b r o k e n y e t , I s u p p o s e that m e a n s that t h e p r o b a b i l i t y o f f a i l u r e i s b e t w e e n z e r o a n d o n e i n 2 0 , o r f i v e p e r c e n t . " Unfortunately, this is n o w h e r e near the accuracy y o u n e e d to m a k e your decision. A great deal of e x p e r i e n c e is n e e d e d in m a k i n g d e c i s i o n s of this kind, and somet i m e s that e x p e r i e n c e j u s t i s n ' t a v a i l a b l e . In-evaluating n u c l e a r p o w e r , a major cost to be c o n s i d e r e d is the possibility of a d i s a s t r o u s n u c l e a r e x p l o s i o n . T o e v a l u a t e t h i s c o s t , a n a l y s t s m u s t try t o i d e n t i f y all o f t h e w a y s i n w h i c h a n a c c i d e n t c o u l d o c c u r . T o d o t h i s , t h e y m a y start w i t h t h e s t a n d a r d o p e r a t i n g p r o c e d u r e o f t h e p l a n t a n d ask w h a t c a n g o w r o n g i n t h e p r o c e -

788

The C o m p l e t e Problem Solver

dure. F o r e x a m p l e , t h e y m a y ask, " W h a t w o u l d h a p p e n if the reactor heat control p u m p s h o u l d f a i l ? " or, " W h a t w o u l d h a p p e n i f t h e r e a c t o r h e a t - c o n t r o l p u m p and the reactor temperature-warning system should both fail?" By w o r k i n g through the potential failures and combinations of failures in the standard operation of the plant, the analysts can arrive at an estimate of the likelihood of an accident. T h i s is a d i f f i c u l t task e v e n i f w e a s s u m e t h a t t h e o p e r a t o r s a r e f o l l o w i n g s t a n d a r d p r o c e d u r e s . I f t h e y a r e n o t , t h e d i f f i c u l t i e s are e n o r m o u s l y i n c r e a s e d . F o r e x a m p l e , v i o l a t i o n o f s t a n d a r d p r o c e d u r e w a s a n i m p o r t a n t f a c t o r i n t h e fire a t t h e B r o w n ' s F e r r y N u c l e a r P o w e r P l a n t . T h i s fire n a r r o w l y m i s s e d c a u s i n g " m a n y c a s u a l t i e s a n d radiation contamination of a large part of A l a b a m a a n d T e n n e s s e e " ( C o m e y , 1975). T e c h n i c i a n s i n t h e p l a n t h a d b e e n s e a r c h i n g for air l e a k s . O n e i n g e n i o u s t e c h n i c i a n was using a candle to aid the search, grossly violating standard operating procedure, b u t i m p r o v i n g t h e e f f i c i e n c y o f t h e s e a r c h . T h e c a n d l e c a u s e d t h e f i r e , a n d t h e fire, together w i t h a f e w other malfunctions, almost c a u s e d a major n u c l e a r disaster. Clearly violations of standard procedure can lead to accidents. Unfortunately, it's v e r y h a r d for a n a n a l y s t t o p r e d i c t all o f t h e i n g e n i o u s w a y s i n w h i c h p e o p l e w i l l violate standard procedure. T a k i n g the saccharin and the B r o w n ' s F e r r y cases together, w e can s e e that cost-benefit analysis can help in complex decisions, but we should recognize

Cost Benefit Analysis

189

c l e a r l y t h a t i t i s s u b j e c t t o error. Just b e c a u s e s o m e o n e has d o n e a c o s t - b e n e f i t a n a l y s i s d o e s n ' t g u a r a n t e e that t h e y h a v e d o n e i t r i g h t . THE PROBLEM OF C O M P A R I N G DIFFERENT SORTS OF VALUES O n e o f t h e d r a w b a c k s o f c o s t - b e n e f i t a n a l y s i s i s that i t i s s o m e t i m e s d i f f i c u l t t o c o m p a r e v a l u e s o f d i f f e r e n t sorts. F o r m a n y t h i n g s , m o n e y p r o v i d e s a c o n v e n i e n t s c a l e for c o m p a r i s o n . S i n c e i t i s o u r m e d i u m o f e x c h a n g e , m a n y t h i n g s are a s s i g n e d m o n e y v a l u e s o r p r i c e s . T h e s e p r i c e s g i v e u s a w a y o f d e c i d i n g h o w m a n y " X ' s " are worth a " Y " . M o n e y values allow us to "compare apples and oranges" in everyday commerce. T h e r e i s still a " c o m p a r i n g a p p l e s a n d o r a n g e s " p r o b l e m , h o w e v e r , b e c a u s e t h e r e are m a n y i m p o r t a n t t h i n g s for w h i c h w e d o n o t o r d i n a r i l y s e t a p r i c e . F o r e x a m p l e , a n e w d a m m a y p r o v i d e f a c i l i t i e s for w a t e r sports a s o n e o f its b e n e f i t s . H o w d o w e c o m p a r e t h e p l e a s u r e p e o p l e g e t from w a t e r sports w i t h t h e d o l l a r c o s t of the d a m ? A n e w s u b w a y system may provide a quieter, less c r o w d e d ride than the old. H o w do we c o m p a r e the increased comfort of the riders w i t h the dollar cost of the n e w system? An intensive care unit m a y save some lives w h i c h otherwise m i g h t h a v e b e e n lost. H o w d o w e c o m p a r e t h e v a l u e o f t h e s e l i v e s w i t h t h e c o s t o f the intensive care unit? I n this last c a s e , s o m e p e o p l e w i l l state flatly t h a t y o u c a n ' t p u t a v a l u e o n a l i f e — c e r t a i n l y n o t a m o n e y v a l u e . T h e r e are t w o t h i n g s s u c h a s t a t e m e n t m i g h t m e a n . F i r s t , i t m i g h t m e a n t h a t t h e r e r e a l l y i s n o w a y t o c o m p a r e t h e v a l u e o f a life t o t h e v a l u e of, s a y , t e n d o l l a r s . B y this i n t e r p r e t a t i o n , i f w e a s k t h e q u e s t i o n , " I s Smith worth ten dollars?" the a n s w e r must b e , "I don't k n o w and there is no w a y t o d e c i d e . " I t h i n k t h i s a n s w e r i s u n a c c e p t a b l e . M o s t o f u s w o u l d a g r e e r e a d i l y that a life i s w o r t h m u c h m o r e t h a n t e n d o l l a r s , e v e n t h o u g h w e all k n o w a f e w e x c e p tions. T h e o t h e r i n t e r p r e t a t i o n i s that a life i s w o r t h m o r e t h a n a n y a m o u n t o f m o n e y we can i m a g i n e . W h i l e this interpretation has a certain attractiveness, it is surely false. C o n s i d e r t h e f o l l o w i n g d i f f i c u l t p r o b l e m . S u p p o s e t h a t t h e life o f a c h i l d d e p e n d e d o n a v e r y e x p e n s i v e k i d n e y m a c h i n e — s o e x p e n s i v e , i n fact, that i t r e q u i r e d all o f t h e n a t i o n ' s r e s o u r c e s t o k e e p i t r u n n i n g . G i v e n a v o t e , w o u l d y o u v o t e t o d e v o t e all o f t h e n a t i o n ' s r e s o u r c e s t o s a v i n g this c h i l d ' s l i f e ? I f y o u d i d , y o u w o u l d b e v o t i n g for i m p o v e r i s h m e n t , s u f f e r i n g , a n d d e a t h for a g r e a t m a n y o t h e r p e o p l e . M o n e y r u n s h o s p i t a l s , m o n e y g r o w s c r o p s , m o n e y p a y s for i m m u n i z a t i o n s a n d p r e n a t a l c a r e . M o n e y r e p r e s e n t s r e s o u r c e s a n d t h e u g l y fact i s that r e s o u r c e s are limited. If we s p e n d a million dollars to save Smith, we m a y not h a v e the million dollars that w o u l d b e r e q u i r e d t o s a v e C i n c i n n a t i . W h a t c a n w e c o n c l u d e f r o m t h e k i d n e y - m a c h i n e e x a m p l e ? W e are f o r c e d t o t h e difficult c o n c l u s i o n t h a t w e c a n ' t sensibly s p e n d u n l i m i t e d a m o u n t s t o s a v e a particular p e r s o n ' s l i f e . T o d o s o w o u l d c o s t m a n y m o r e l i v e s t h a n i t w o u l d s a v e . W e l l , then, h o w m u c h should we spend to save a life? We k n o w we should spend more than t e n d o l l a r s a n d w e k n o w t h a t w e s h o u l d n ' t s p e n d s o m u c h t h a t w e c a u s e t h e d e a t h s o f lots o f o t h e r p e o p l e , b u t h o w m u c h should w e s p e n d ? T h e r e i s n o c l e a r a n s w e r t o this q u e s t i o n . N o n e t h e l e s s , t o r e v e a l t h e n a t u r e o f t h e p r o b l e m a little m o r e c l e a r l y , l e t ' s e x a m i n e t h e k i d n e y - m a c h i n e p r o b l e m i n a smaller c o n t e x t — t h e context of a single family. I m a g i n e a family w i t h five children. O n e of the five w i l l die if he d o e s n ' t h a v e the k i d n e y m a c h i n e . S h o u l d the parents p r o v i d e t h e m o n e y for t h e m a c h i n e i f i t m e a n s that 1. T h e o t h e r four w i l l d i e ? Yes ( ) No ( ) or 2 . T h e o t h e r four w i l l l i v e i n a b j e c t p o v e r t y w i t h g r e a t l y d e c r e a s e d life e x p e c tancies? Yes ( ) No ( )

790

The C o m p l e t e Problem Solver or 3 . T h e o t h e r four w i l l l i v e i n a b j e c t p o v e r t y w i t h a s t r o n g p r o b a b i l i t y t h a t t h e y will b e c o m e criminals, perhaps murderers? Yes ( ) No ( ) or 4 . T h e o t h e r four, w h o are b l i n d , w i l l h a v e t o g i v e u p r e m e d i a l s u r g e r y ? Yes ( ) No ( ) or 5. T h e other four w i l l h a v e to drop out of h i g h school b e f o r e graduation to take menial jobs? Yes ( ) No ( ) or 6. T h e other four, t h o u g h p r o m i s i n g students, w i l l h a v e no c h a n c e to go to college? Yes ( ) No ( ) or 7. T h e f a m i l y w o n ' t be a b l e to b u y a p i a n o or a c a r ? Yes ( ) No ( )

T h e s e are h o r r i b l e q u e s t i o n s t o ask. W e a s k t h e m o n l y b e c a u s e t h e y r e v e a l s o m e thing of the nature of a very important problem. Most people will answer " n o " to t h e first q u e s t i o n a n d " y e s " t o t h e last, b u t m a y d i f f e r o n t h e i r a n s w e r s t o t h e questions in b e t w e e n . In order to answer these questions we have to compare human values of very d i f f e r e n t sorts. W e h a v e t o c o m p a r e c e r t a i n d e a t h w i t h a p r o b a b i l i t y o f d e a t h , o r w i t h life-long suffering, or w i t h lost opportunities. P e o p l e differ from one another in t h e s e d e c i s i o n s , p e r h a p s b e c a u s e their v a l u e s differ, or p e r h a p s t h e y aren't q u i t e sure at a n y g i v e n m o m e n t w h a t their v a l u e s are in t h e s e difficult comparisons. L o w - i n c o m e p e o p l e m a y place more importance on curing blindness and less on graduating from h i g h school than do h i g h - i n c o m e p e o p l e . P e o p l e s e e m to place a h i g h e r v a l u e o n l i v e s i f t h e i n d i v i d u a l s are i d e n t i f i e d t h a n i f t h e y a r e a n o n y m o u s . As Z e c h h a u s e r (1975) points out, we are v e r y u n l i k e l y to s e n d astronauts on onew a y trips into s p a c e e v e n t h o u g h i t w o u l d greatly c u t costs. W e are m u c h m o r e l i k e l y t o r e d u c e e x p e n s e s for h i g h w a y s a f e t y w h e r e t h e v i c t i m s , w h i l e m o r e n u m e r ous, die in comforting anonymity. W h a t can we c o n c l u d e a b o u t c o m p a r i n g v a l u e s s u c h as life, happiness, and m o n e y ? S u c h c o m p a r i s o n s are v e r y d i f f i c u l t . P e o p l e w i l l n o t a g r e e w i t h e a c h o t h e r a n d t h e y w i l l n o t a g r e e w i t h t h e m s e l v e s a t d i f f e r e n t t i m e s a n d i n d i f f e r e n t situations. H o w e v e r , w e cannot avoid such comparisons. W e m a k e them frequently w h e t h e r w e d o s o i n full c o n s c i o u s n e s s o r not. G o v e r n m e n t s d o s o w h e n t h e y establish p u r e food a n d d r u g l a w s , or set airline safety standards. Individuals m a k e these comparisons w h e n they d e c i d e w h e t h e r or not to ride a motorcycle or to enter a high-risk profession such as police work. Cost-benefit analysis w o n ' t make the d i f f i c u l t y o f t h e s e v a l u e c o m p a r i s o n s g o a w a y . H o w e v e r , t h e f r a m e w o r k o f costbenefit analysis can h e l p us as individuals and as a society to be clearer and more systematic a b o u t w h a t w e are d o i n g w h e n w e m a k e t h e s e comparisons.

D e a l i n g w i t h V a l u e C o m p a r i s o n s in Practice T h e r e are s o m e s i t u a t i o n s i n w h i c h w e c a n g e t a r o u n d d i f f i c u l t v a l u e c o m p a r i sons by u s i n g a d o m i n a n c e rule. F o r e x a m p l e , c o n s i d e r a d i a b e t e s detection prog r a m . T h e c o s t s o f s u c h a p r o g r a m i n c l u d e f i n a n c i a l c o s t s — s u c h a s w a g e s for p h y s i c i a n s a n d t e c h n i c i a n s , testing materials, carfare a n d w a g e s lost by the persons t e s t e d — a n d h u m a n comfort c o s t s — s u c h as the a n n o y a n c e of w a i t i n g in the office. Similarly, the benefits of the program include the financial benefits of avoiding the e x p e n s e s of hospitalization and lost w a g e s , and the h u m a n comfort benefits of a v o i d i n g b l i n d n e s s and pain. W e can a p p l y the d o m i n a n c e rule i n this d e c i s i o n a s follows: If the b e n e f i t s are greater than the costs in one of the t w o areas a n d no

191

Cost Benefit Analysis

w o r s e than the costs in the other, t h e n the program s h o u l d be u n d e r t a k e n . T h u s , if the financial costs a n d benefits w e r e the s a m e , b u t the h u m a n comfort benefits w e r e g r e a t e r t h a n t h e c o s t s , t h e n t h e total b e n e f i t s w o u l d b e g r e a t e r t h a n t h e c o s t s . S u p p o s e it costs as m u c h in dollars to i m m u n i z e p e o p l e against a disease as it d o e s t o t r e a t t h e m for t h e d i s e a s e . S i n c e i t i s b e t t e r i n h u m a n t e r m s for p e o p l e t o a v o i d t h e d i s e a s e t h a n t o h a v e it, t h e total b e n e f i t s o f i m m u n i z a t i o n w o u l d e x c e e d its c o s t s . The D i r e c t A p p r o a c h t o V a l u e C o m p a r i s o n A n o t h e r w a y t o d e a l w i t h v a l u e c o m p a r i s o n i s t h e d i r e c t a p p r o a c h . W e c a n ask p e o p l e t o t e l l u s h o w m u c h t h e y w o u l d b e w i l l i n g t o p a y for t h e p l e a s u r e o f g o i n g f i s h i n g , o r for r e l i e f f r o m h a y f e v e r , o r for a l e s s r i s k y j o b . C o n s i d e r t h e f o l l o w i n g decision. S u p p o s e y o u are offered t w o s u m m e r j o b s as b a n k guard at different b r a n c h e s o f t h e s a m e b a n k . O n e j o b i n t h e s u b u r b a n b r a n c h w h e r e t h e r e i s z e r o risk t o y o u r life p a y s $5,000 for t h e s u m m e r . T h e o t h e r j o b i s i n a b u s y u r b a n a r e a a n d i n v o l v e s a o n e p e r c e n t r i s k t o y o u r life o v e r t h e s u m m e r . H o w m u c h w o u l d t h e j o b in the urban b r a n c h h a v e to pay to be as attractive as the s u b u r b a n j o b ? W o u l d an extra $ 1 0 0 b e e n o u g h ? H o w a b o u t a n e x t r a m i l l i o n ? W h i l e m e a s u r e m e n t s s u c h a s t h e s e are n o t v e r y p r e c i s e , t h e y d o g i v e u s s o m e i d e a o f h o w t h e i n d i v i d u a l c o m pares very different v a l u e s , and allow us to p r o c e e d w i t h a cost-benefit analysis. (For a more d e t a i l e d discussion of these m e t h o d s , see Z e c h h a u s e r , 1975.) PROBLEMS IN C O M P A R I N G C O S T S A N D BENEFITS RECEIVED AT DIFFERENT TIMES In d o i n g a cost-benefit analysis, we f r e q u e n t l y h a v e to c o m p a r e costs and benefits r e c e i v e d at different times. F o r e x a m p l e , suppose that on January 1 of year z e r o t h e J o n e s C o m p a n y i n v e s t s $3,200 i n a m a c h i n e w h i c h e a r n s $1,000 a y e a r , p a y a b l e o n J a n u a r y 1 o f y e a r s 1 , 2 , 3 , a n d 4 , a n d t h e n i t w e a r s o u t . A t first g l a n c e i t a p p e a r s t h a t t h e b e n e f i t s o u t w e i g h t h e c o s t s by $800 ( T a b l e 1 ) , but this isn't right! T h e benefits r e c e i v e d in years 2, 3, and 4 are not directly c o m p a r a b l e to the costs i n y e a r 1 . T o u n d e r s t a n d w h y t h e y a r e n o t d i r e c t l y c o m p a r a b l e a n d h o w t h e y can b e c o m p a r e d , w e w i l l h a v e t o d i s c u s s i n t e r e s t a n d its r e l a t i o n t o t h e p r e s e n t a n d future v a l u e o f m o n e y . C o m p u t i n g Interest Interest—Simple

and

Compound

S u p p o s e that y o u put $100 in the b a n k at 6% interest and then e v e r y year y o u r e c e i v e a c h e c k for $6.00. T h i s i s c a l l e d " s i m p l e " i n t e r e s t . I f y o u i n v e s t P d o l l a r s a t i % i n t e r e s t for n y e a r s , t h e t o t a l a m o u n t o f m o n e y y o u w i l l h a v e after n y e a r s , F , i s . F u t u r e v a l u e = P r i n c i p a l + ( P r i n c i p a l x i n t e r e s t x n u m b e r of y e a r s )

792

The C o m p l e t e Problem Solver

T h i s may also be written as F = P ( l + in) If we invest $100 at 6% simple interest, in 10 years we will have F = P + Pin = $ 1 0 0 + $ 1 0 0 x (.06) x 1 0 = $ 1 0 0 + $60 = $160 S u p p o s e o n t h e o t h e r h a n d y o u i n v e s t $ 1 0 0 i n t h e b a n k a t 6 % i n t e r e s t , b u t rather than r e c e i v i n g an interest c h e c k each year, y o u have the interest a d d e d to the amount already in the bank. T h u s , the interest w i l l also collect interest. T h i s is c a l l e d " c o m p o u n d " i n t e r e s t . I f y o u i n v e s t P d o l l a r s a t i % i n t e r e s t for n y e a r s , t h e total a m o u n t o f m o n e y y o u w i l l h a v e after n y e a r s , F , a t c o m p o u n d i n t e r e s t i s 2.

F = P(l + i)

n

If we i n v e s t $100 at 6% c o m p o u n d interest, in 10 years we will h a v e F = P(l + i) = 1 0 0 (1 + . 0 6 ) = $179.08 n

1 0

T a b l e 2 c o m p a r e s t h e v a l u e o f F for s i m p l e a n d c o m p o u n d i n t e r e s t o v e r 2 0 years. C l e a r l y c o m p o u n d interest has an increasing a d v a n t a g e o v e r s i m p l e interest a s t i m e p a s s e s . W e w i l l b e c o n c e r n e d primarily w i t h c o m p o u n d interest i n this discussion. Table 2

193

Cost Benefit Analysis

w o r k b a c k w a r d s , t h a t i s , w e are g i v e n t h e f u t u r e v a l u e a n d w e w a n t t o c o m p u t e t h e p r e s e n t v a l u e f r o m it. E q u a t i o n 3 s h o w s h o w t o d o t h i s . I t i s d e r i v e d d i r e c t l y f r o m E q u a t i o n 2.

N o w b a c k to the Jones C o m p a n y . U s i n g E q u a t i o n 3 and assuming an interest r a t e o f 1 0 % , w e c a n c o m p u t e t h e p r e s e n t v a l u e (that i s , t h e v a l u e o n Jan. 1 , y e a r 0 ) of $1,000 r e c e i v e d on Jan. 1, in years 1, 2, 3, a n d 4. A t h o u s a n d dollars r e c e i v e d on Jan. 1 , y e a r 1 , i s t h e f u t u r e v a l u e o f t h e u n k n o w n p r e s e n t v a l u e , P a n d after o n e year's interest. T h u s l 9

T h e s u m o f t h e s e v a l u e s i s $ 3 , 1 6 9 . 7 7 , w h i c h i s l e s s t h a n $3,200! T h u s , t h e present value o f t h e e a r n i n g s t o b e r e c e i v e d o v e r four y e a r s i s a c t u a l l y l e s s t h a n t h e p r e s e n t value of the investment in the machine. T h e Jones C o m p a n y w o u l d do better to invest their m o n e y at 10 percent and forget about b u y i n g the machine.

SUMMARY C o s t - b e n e f i t analysis is a d e c i s i o n p r o c e d u r e in w h i c h we c o m p a r e the exp e c t e d costs a n d benefits of alternative actions. It i n v o l v e s the f o l l o w i n g steps: A. For each of the alternative actions: 1 . i d e n t i f y all t h e i m p o r t a n t s o u r c e s o f c o s t s a n d b e n e f i t s ; 2. estimate the v a l u e s of the costs and benefits; 3. estimate the probabilities of obtaining the costs a n d benefits; and 4. c o m p a r e the e x p e c t e d v a l u e s of the costs a n d benefits. B. C h o o s e t h e a c t i o n for w h i c h the expected value of the benefits m i n u s the expected value of the costs is g r e a t e s t . W h e n w e d o cost-benefit analysis, w e m a y e n c o u n t e r a n u m b e r o f practical problems. T h e s e include: 1 . d i f f i c u l t i e s i n i d e n t i f y i n g all o f t h e i m p o r t a n t c o s t s a n d b e n e f i t s ; 2. difficulties in estimating probabilities (especially w h e n dealing with n e w technology); and 3 . d i f f i c u l t i e s i n c o m p a r i n g v a l u e s o f d i f f e r e n t sorts, e . g . , t h e v a l u e o f a j o b opportunity v e r s u s the v a l u e of a life.

194

T h e C o m p l e t e Problem Solver

SAMPLE PROBLEMS Problem #1 In an a r t i c l e in t h e Journal of the American Medical Association, K i r k l a n d ( 1 9 7 7 ) , q u e s t i o n e d t h e u s e f u l n e s s o f s c r e e n i n g p a t i e n t s o v e r 6 5 for s y p h i l i s . H e said: T h e economics of hospital care have now made the scrutiny of all expenses, no matter h o w small, necessary. At Emory University Hospital, screening for syphilis has had the sanctity of tradition and official requirements. At the time of this study, an 'admissions package' g i v e n to each n e w patient by the admissions office included a requisition for an automated reagin test (ART) [a test for syphilis].

K i r k l a n d r e v i e w e d t h e folders of 300 patients 65 or older, a n d f o u n d that o n l y one case of syphilis had b e e n d e t e c t e d and treated. T h e patient w a s a 71-year-old w o m a n admitted to the hospital with serious heart trouble. He continued: Therefore, at $3.50 per A R T , a minimum of $1,050 was spent to detect and treat that one patient. It is highly questionable whether or not that treatment will have any effect on her longevity or w e l l - b e i n g ; she probably just got upset at b e i n g told what it was for. It therefore s e e m e d wasteful to continue routine screening in this age group, and such screening was discontinued at this hospital as of January, 1975. It seems improbable that this m o v e will constitute a major public health hazard or will substantially affect the health of the individual patients.

W h a t costs a n d b e n e f i t s w e r e c o n s i d e r e d i n this c a s e ? Problem # 2 S u p p o s e t h a t y o u p u t $500 i n t h e b a n k a t 6 % simple i n t e r e s t . H o w m u c h w i l l you have in 10 years? H o w m u c h will you have if the interest is compound? Problem #3 W h a t w o u l d t h e a n s w e r s b e i n p r o b l e m # 2 i f t h e interest rate w e r e 9 % ? Problem #4 H o w m u c h m o n e y w o u l d you have to have put in the bank 10 years ago to have $1,000 n o w ? A s s u m e 8 % c o m p o u n d i n t e r e s t . Problem #5 A s s u m i n g a 1 0 % interest rate, w h a t is t h e future v a l u e 5 years from n o w of $1,000 i n v e s t e d t o d a y ? W h a t i s t h e f u t u r e v a l u e 5 y e a r s f r o m n o w o f a $1,000 i n v e s t m e n t i f w e i n v e s t $500 o f i t t o d a y and $500 a y e a r f r o m t o d a y ? Problem #6 Y o u r c o m p a n y has an opportunity to b u y a d u p l i c a t i n g m a c h i n e that w i l l last t h r o u g h 5 y e a r s o f u s e . I t c o s t s $20,000 a n d w i l l b r i n g i n $5,000 i n b u s i n e s s p a y a b l e a t t h e e n d o f e a c h y e a r . T a k i n g t h e c o s t s a n d b e n e f i t s into a c c o u n t , w h a t i s t h e p r e s e n t v a l u e o f t h i s p u r c h a s e ? A s s u m e a 1 0 % i n t e r e s t rate. Problem # 7 — H a y e s ' Disease Y o u a r e a p u b l i c h e a l t h o f f i c e r i n y o u r last y e a r a t y o u r p r e s e n t p o s t . A s y o u r final act, y o u m u s t d e c i d e w h e t h e r o r n o t t o l a u n c h a o n e - s h o t (i.e., o n e t i m e ) i m m u n i z a t i o n c a m p a i g n against a n e w l y d i s c o v e r e d respiratory affliction c a l l e d " H a y e s ' D i s e a s e . " H a y e s ' D i s e a s e i s u n c o m f o r t a b l e b u t g e n e r a l l y n o t s e r i o u s . Its most usual s y m p t o m s are fever and c o u g h i n g and temporary h e a r i n g loss, although

Cost Benefit Analysis

795

sometimes more serious complications occur. H o w e v e r , it is extremely contagious. I n a n e p i d e m i c y e a r , e s s e n t i a l l y e v e r y s u s c e p t i b l e p e r s o n w i l l c o n t r a c t it. E p i d e m ics a r e b e l i e v e d t o o c c u r e v e r y 5 y e a r s . S i n c e t h e d i s e a s e h a s o n l y r e c e n t l y b e e n r e c o g n i z e d , y o u d o n ' t k n o w w h e n t h e last e p i d e m i c i n y o u r r e g i o n o c c u r r e d . T o m a k e y o u r d e c i s i o n , y o u a s s e m b l e t h e f o l l o w i n g facts c o n c e r n i n g c o s t s a n d benefits of the program. Costs v a c c i n e $3/patient ( a s s u m e t h a t i t i s 1 0 0 % e f f e c t i v e ) office costs $4/patient a d v e r t i s i n g for c a m p a i g n $ l / p a t i e n t Benefits T h e b e n e f i t s o f t h e p r o g r a m are c a l c u l a t e d f r o m t h e c o s t s w h i c h p a t i e n t s w o u l d h a v e incurred if t h e y h a d contracted the disease. T h e s e costs are as follows: 65% of patients—no expense 2 5 % — o n e $ 1 5 office visit 5 % c o n t r a c t p n e u m o n i a ; o f t h e s e , 1/20 r e q u i r e $500 h o s p i t a l i z a t i o n , 19/20 r e q u i r e 2 o f f i c e v i s i t s @ $ 1 5 a n d $5 m e d i c a t i o n 5% contract otitis (ear trouble) r e q u i r i n g 2 office visits @ $ 1 5 a n d $5 m e d i c a t i o n Since y o u don't k n o w w h e n the next e p i d e m i c will occur, these benefits m a y be r e a l i z e d this y e a r or in a n y of the four f o l l o w i n g years. F o r p u r p o s e s of computation, y o u a s s u m e that 2 0 % of the benefits w i l l be realized this year and 2 0 % in e a c h of t h e four f o l l o w i n g y e a r s . Y o u a s s u m e a n interest rate o f 1 0 % t o find the p r e s e n t v a l u e of the future benefits. Do a cost-benefit analysis on the basis of these figures to d e c i d e w h e t h e r or not y o u s h o u l d l a u n c h this c a m p a i g n . REFERENCES C o h e n , B . L . " R e l a t i v e R i s k s o f S a c c h a r i n a n d C a l o r i e I n g e s t i o n . " Science, 1 9 9 ( 3 ) , 983, 1978. C o m e y , D . D . " H o w W e A l m o s t L o s t A l a b a m a . " Chicago Tribune, A u g u s t 3 1 , 1 9 7 5 . Federal Register, 4 2 ( 7 3 ) , 1 9 9 9 6 - 2 0 0 0 6 , 1 9 7 7 . Kirkland, Toxicity

J. Journal and

of the

American

Carcinogenicity

Study

Medical Association, of

238(5),

399,

Orthotoluenesulfonamide

and

1977. Saccharin,

Project E405/405D, D e p a r t m e n t of Health and Public Welfare of the Canadian Government, February, 1974. Z e c k h a u s e r , R . " P r o c e d u r e s for V a l u i n g L i v e s . " Public Policy, 2 3 , 4 1 9 - 4 6 4 , 1 9 7 5 .

TEN.

C O G N I T I V E PROCESSES IN CREATIVE A C T S

In the next t w o chapters, we w i l l approach creativity from t w o very different points of v i e w . In this chapter, we w i l l d e f i n e creativity and discuss four of the c o g n i t i v e p r o c e s s e s u n d e r l y i n g creative acts: p r o b l e m finding, i d e a generation, planning, and preparation. In addition, we will describe some procedures which can increase your p r o b l e m finding and idea generation skills. In the next chapter, w e w i l l discuss t h e social conditions o f c r e a t i v i t y — t h a t is, w e w i l l d e s c r i b e h o w society treats p e o p l e differently a c c o r d i n g to their sex, r e l i g i o n , a n d race, a n d h o w this d i f f e r e n c e i n t r e a t m e n t e n c o u r a g e s o r d i s c o u r a g e s c r e a t i v e a c t i v i t y . T h e m a i n point that we w a n t to m a k e in both of t h e s e chapters is that y o u can exercise some control over the cognitive processes and the social conditions w h i c h i n f l u e n c e y o u r c r e a t i v e abilities. T o this extent, y o u c a n i n c r e a s e y o u r c h a n c e s o f becoming a creative person. W H A T IS A CREATIVE A C T ? C r e a t i v e acts c o m e in a great variety of forms. A creative act m a y be quite ordinary and i n c o n s e q u e n t i a l — f o r example, it m i g h t be s o m e t h i n g as simple as m a k i n g up a b e d t i m e story to tell our c h i l d r e n — o r it m a y be w o r l d s h a k i n g — a s w a s Galileo's invention of the science of physics. A creative act m a y involve years of concentrated w o r k — c o n s i d e r the decades D a r w i n d e v o t e d to d e v e l o p i n g the evid e n c e for t h e t h e o r y o f e v o l u t i o n — o r i t m a y b e b r i e f — c o n d e n s e d i n t o a s u d d e n flash o f i n s i g h t — t h e sort o f i n s i g h t t h a t d r o v e A r c h i m e d e s n a k e d f r o m h i s b a t h shouting, " E u r e k a ! " W h a t i s t h e r e a b o u t t h e s e v e r y d i f f e r e n t acts t h a t l e a d s u s t o c a l l t h e m all " c r e a t i v e " ? T y p i c a l l y , w e a p p l y fairly stringent criteria i n j u d g i n g creativity. I n most cases, we r e q u i r e an act to pass three tests before we call it creative. First, we m u s t b e l i e v e t h a t t h e a c t i s original. S e c o n d , w e m u s t b e l i e v e t h a t i t i s valuable. A n d third, it m u s t s u g g e s t to us that the p e r s o n w h o p e r f o r m e d the act has special m e n t a l abilities. F o r e x a m p l e , w h e n w e s e e w h a t t h e p e r s o n h a s d o n e , w e a s k o u r s e l v e s , " H o w d i d s h e e v e r t h i n k o f t h a t ? " or, " H o w d i d h e h a v e t h e p a t i e n c e t o w o r k all t h a t o u t ? " Let's e x a m i n e t h e s e conditions in order.

T h e C o m p l e t e Problem Solver

798 Originality

We certainly w o u l d n ' t j u d g e a painter creative w h o simply c o p i e d the pictures of other painters. To be j u d g e d creative, painters must use their o w n resources to s h a p e t h e p a i n t i n g . T h e y m u s t p a i n t t h e i r own p i c t u r e s . We d o n ' t m e a n t h o u g h that e v e r y t h i n g in a c r e a t i v e w o r k m u s t be original. Painters, writers, and inventors routinely use ideas b o r r o w e d from others. H o w e v e r , the creative person c o m b i n e s or interprets these b o r r o w e d ideas in w a y s t h a t a r e o r i g i n a l . R e n a i s s a n c e artists v e r y f r e q u e n t l y p a i n t e d t h e M a d o n n a a n d C h i l d , b u t e a c h g r e a t artist p r e s e n t e d t h e t h e m e i n a n i n d i v i d u a l w a y . S o m e t i m e s a person will do something original w h i c h is not n e w . F o r e x a m p l e , a scientist m a y m a k e a discovery, quite i n d e p e n d e n t of other p e o p l e , only to find later that the d i s c o v e r y has b e e n m a d e several times b e f o r e . In our society n e w n e s s i s i m p o r t a n t . W e a r e v e r y c a r e f u l t o g i v e s p e c i a l c r e d i t t o t h e p e r s o n w h o i s first t o invent something. Still, we attribute creativity to the scientist a b o v e on the grounds of originality, e v e n though the discovery is not new. Value E v e n if an act is original, we w o n ' t c o n s i d e r it creative unless we also j u d g e it t o b e v a l u a b l e . S u p p o s e , for e x a m p l e , y o u w e r e t o t u r n all t h e f u r n i t u r e i n y o u r house upside d o w n . T h a t w o u l d be original, but it w o u l d hardly be valuable. Your friends w o u l d not ask admiringly, " H o w did y o u think of t h a t ? " Rather, they w o u l d ask, " W h y d i d y o u think of t h a t ? " and w o r r y a bit a b o u t y o u r m e n t a l health. Judging w h e t h e r something is valuable or not is tricky. Perfectly reasonable p e o p l e m a y d i s a g r e e w i t h e a c h other a b o u t the v a l u e , say, o f c o n t e m p o r a r y m u s i c . Further, opinions c h a n g e o v e r time. In the 1860's, both the critics a n d the p u b l i c m u c h preferred the painters of the F r e n c h A c a d e m y to the Impressionists. N o w b o t h critical a n d p o p u l a r j u d g m e n t is reversed. T o d a y the w o r k of Impressionists such as R e n o i r , D e g a s , and M o n e t is m u c h better k n o w n than that of F r e n c h A c a d e m y painters such as G r e u z e or Gerome. If j u d g m e n t s of value can change with time, then j u d g m e n t s of creativity can c h a n g e , too. An act w h i c h is j u d g e d creative by one generation m a y not s e e m so to the next. Abilities O u r final c o n d i t i o n for j u d g i n g a n a c t c r e a t i v e i s t h a t i t m u s t s u g g e s t t h a t t h e p e r s o n w h o p e r f o r m e d the act has special m e n t a l abilities. I m a g i n e the f o l l o w i n g s c e n e : A h o u s e p a i n t e r is r e t o u c h i n g the c e i l i n g in a m u s e u m gallery. Just as he is a b o u t to finish, his foot slips. He k n o c k s o v e r the ladder, splattering paint everyw h e r e a n d d o e s a d o u b l e flip into a potted palm. He regains c o n s c i o u s n e s s several hours later, j u s t i n t i m e t o h e a r m e m b e r s o f the m u s e u m s e l e c t i o n c o m m i t t e e saying, "Brilliant!!" "A work of genius!" " W h a t freedom of m o v e m e n t ! " Peering t h r o u g h the p a l m fronds, he sees that t h e y are referring to the aftermath of his accident on the wall. N o w , w h i l e the housepainter produced something original and valuable, we c a n ' t c a l l h i s a c t c r e a t i v e . F a l l i n g off t h e l a d d e r d o e s n ' t i n a n y w a y c a u s e u s t o admire the housepainter's mental abilities. It only suggests that he may have a tendency toward clumsiness. Y o u m i g h t b e i n c l i n e d t o say t h a t t h e h o u s e p a i n t e r ' s a c t w a s n ' t c r e a t i v e b e c a u s e i t w a s u n i n t e n t i o n a l . Y o u s h o u l d n o t i c e , t h o u g h , t h a t s o m e v e r y c r e a t i v e acts a r e unintentional. For example, Becquerel's discovery of radioactivity was unintentional. He had no i d e a that u r a n i u m ore w o u l d fog a p h o t o g r a p h i c film. We a d m i r e

Cognitive Processes in Creative Acts

799

B e c q u e r e l for m a k i n g t h e d i s c o v e r y b e c a u s e , w h i l e i t w a s u n i n t e n t i o n a l , h e h a d t h e w i t t o r e c o g n i z e its s i g n i f i c a n c e . CREATIVITY A N D

IQ

I f c r e a t i v e acts r e q u i r e s p e c i a l m e n t a l a b i l i t i e s , w e m i g h t e x p e c t t h a t c r e a t i v e p e o p l e h a v e e s p e c i a l l y h i g h I Q ' s . W h i l e creative p e o p l e d o h a v e h i g h e r than average IQ's, the relation b e t w e e n creativity and IQ is complex. T h e simplest w a y to s u m m a r i z e it is to say that p e o p l e w i t h b e l o w a v e r a g e I Q ' s t e n d not to be creative. H o w e v e r , if we look just at p e o p l e a b o v e a certain IQ l e v e l , s u c h as 120, then there is v e r y little relation b e t w e e n creativity a n d I Q . It is as if there is a certain m i n i m u m I Q r e q u i r e d for c r e a t i v i t y , after w h i c h I Q d o e s n ' t m a t t e r . An alternative v i e w (see H a y e s , 1978) is that IQ has n o t h i n g to do w i t h creative a b i l i t y , a n d t h a t t h e r e a s o n p e o p l e w i t h l o w I Q ' s are n o t c r e a t i v e l y p r o d u c t i v e i s that they aren't g i v e n the opportunity. IQ does predict h o w w e l l a person will do in school. P e o p l e w h o d o p o o r l y i n school m a y h a v e difficulty getting into good schools and getting g o o d jobs in w h i c h there is a c h a n c e to be creative. T h u s , our s o c i e t y m a y a c t u a l l y prevent p e o p l e w i t h l o w I Q ' s f r o m b e i n g c r e a t i v e . T h e important p o i n t a b o u t creativity and I Q that y o u s h o u l d u n d e r s t a n d i s this: Y o u shouldn't g i v e up on y o u r s e l f as a creative person just b e c a u s e y o u r IQ is not outstanding. For example, the IQ's of such famous people as Copernicus, Rembrandt, and Faraday have b e e n estimated at 110 or less! CREATIVITY A N D ILL-DEFINED PROBLEMS In d i s c u s s i n g the nature of a creative act, we w i l l f o l l o w the l e a d of N e w e l l , S h a w , a n d S i m o n (1964). T h e s e authors p r o p o s e d that a creative act is a special kind of p r o b l e m solving, that it is the act of s o l v i n g an ill-defined p r o b l e m . In Chapter 1, we defined ill-defined problems as ones w h i c h require problem s o l v e r s t o c o n t r i b u t e t o t h e d e f i n i t i o n o f t h e p r o b l e m f r o m t h e i r own r e s o u r c e s . T o solve an ill-defined problem, y o u may be required either to make decisions based o n y o u r o w n k n o w l e d g e a n d v a l u e s ( g a p - f i l l i n g d e c i s i o n s ) o r t o d i s c o v e r n e w information t h r o u g h y o u r o w n active exploration o f t h e p r o b l e m (jumping in), o r both. Y o u r s o l u t i o n s t o i l l - d e f i n e d p r o b l e m s a r e v e r y m u c h your s o l u t i o n s . T h e y d e p e n d on your k n o w l e d g e and your values. Other p e o p l e w o u l d almost certainly h a v e a r r i v e d a t d i f f e r e n t s o l u t i o n s b e c a u s e t h e i r k n o w l e d g e a n d v a l u e s d i f f e r from yours. It is just b e c a u s e we put our o w n k n o w l e d g e and v a l u e s into the solution of ill-defined p r o b l e m s that it is possible to solve t h e m creatively. It is by d r a w i n g on private r e s o u r c e s — d i f f e r e n t in e a c h person and largely h i d d e n from the outside v i e w e r — t h a t w e are a b l e t o p r o d u c e solutions w h i c h d a z z l e a n d astound. C O G N I T I V E PROCESSES U N D E R L Y I N G CREATIVE A C T S C r e a t i v e acts d e p e n d o n a g r e a t m a n y o f o u r c o g n i t i v e p r o c e s s e s w o r k i n g together in h a r m o n y — p r o c e s s e s of representation, search, memory, and decision making. W e w i l l focus o n just four o f t h e s e p r o c e s s e s w h i c h are e s p e c i a l l y important for c r e a t i v i t y : p r o b l e m f i n d i n g , i d e a g e n e r a t i o n , p l a n n i n g , a n d p r e p a r a t i o n . Problem Finding M a n y p r o b l e m s c o m e t o u s w i t h n e a t l a b e l s w h i c h say i n effect, " S o l v e m e . I'm a p r o b l e m ! " P r o b l e m s i n e x a m s , I Q t e s t s , a n d p u z z l e b o o k s a r e l i k e that. W h e n w e o p e n a n e x a m b o o k l e t , w e are l i k e l y t o f i n d l a b e l s s u c h a s " Q u e s t i o n # 1 " o r " P r o b l e m 3 . " W h e t h e r w e can solve the p r o b l e m s o r not, w e have n o trouble finding them.

T h e C o m p l e t e Problem Solver

200

S o m e p r o b l e m s aren't as neatly l a b e l e d , though. T h e r e is a special class of ill-defined problems in w h i c h what we have to contribute to the definition of the p r o b l e m is t h e d i s c o v e r y that t h e r e is a p r o b l e m . H e r e is an u n c o m p l i c a t e d e x a m p l e in w h i c h I d i s c o v e r e d a problem: O v e r the years, I have searched through the theater pages of the n e w s p a p e r hundreds of times to find w h a t w a s p l a y i n g at my local theater. On the average, the search i n v o l v e d e x a m i n i n g about half of the ads before finding the one I wanted. W h i l e this w a s a m i n o r a n n o y a n c e , until r e c e n t l y I h a d n ' t s e e n it as a p r o b l e m — t h a t is, a s a d i f f i c u l t y for w h i c h o n e c o u l d f i n d a s e n s i b l e s o l u t i o n . W h e n I d i d s e e i t a s a p r o b l e m — w h e n I finally said to myself, " S o m e t h i n g c o u l d be d o n e about t h a t " — i t w a s e a s y e n o u g h to think of solutions, e.g., standardize the positions of the ads or a l p h a b e t i z e t h e m . F i n d i n g this p r o b l e m w a s a great deal m o r e difficult than finding m e t h o d s for its s o l u t i o n . PROBLEM F I N D I N G A N D CREATIVITY T h e p r o c e s s o f p r o b l e m f i n d i n g p l a y s a v e r y i m p o r t a n t r o l e b o t h i n artistic a n d i n s c i e n t i f i c c r e a t i v i t y . F o r e x a m p l e , i n t h e e a r l y 1 9 0 0 ' s , a g r o u p o f A m e r i c a n artists called the "ashcan school" discovered esthetic values in the everyday appearance of the c i t y — p e o p l e s h o p p i n g or crossing at an intersection. T h e y took it as their p r o b l e m to capture t h e s e v a l u e s in their paintings. No o n e told t h e m to do this. In fact, a h o r r i f i e d a r t i s t i c e s t a b l i s h m e n t t o l d t h e m n o t to. T h e y h a d t o d i s c o v e r t h e i r artistic p r o b l e m for t h e m s e l v e s . P r o b l e m finding is important in scientific discovery as w e l l . Einstein and Infeld (1938) c o m m e n t : G a l i l e o formulated the problem of determining the velocity of light, but did not solve it. T h e formulation of a problem is often more essential than its solution, w h i c h may be m e r e l y a matter of mathematical or experimental skill. To raise n e w questions, n e w possibilities, to regard old problems from a n e w angle, requires creative imagination and marks real advance in science. (p. 95) H e r e are three cases in point. Case

1

In the h e y d a y of the telegraph, h u n d r e d s of operators listened to the dots and dashes of the M o r s e ticker and transcribed t h e m into w o r d s on paper. W h e n the c o n n e c t i o n w a s bad, the m e s s a g e s w e r e hard to hear. M o s t s i m p l y s h r u g g e d their shoulders a n d d i d the b e s t t h e y c o u l d in a difficult situation. E d i s o n d i s t i n g u i s h e d h i m s e l f from the others by s e e i n g the difficulty as a p r o b l e m to be solved. He s o l v e d it by constructing a d e v i c e w h i c h w o u l d record the dots and dashes as visible marks on a rotating disk. W h e n the d e v i c e was c o m p l e t e , h o w e v e r , others didn't n e c e s s a r i l y s e e i t a s t h e s o l u t i o n t o a p r o b l e m . E d i s o n ' s e m p l o y e r s , for e x a m p l e , thought that r e a d i n g from a disk rather than listening to the ticker w a s a c o m p l e t e waste of time. T h e pattern of e v e n t s that l e d to E d i s o n ' s invention has b e e n repeated in a n u m b e r o f s c i e n t i f i c d i s c o v e r i e s . W h a t m a n y h a d o b s e r v e d a n d d i s m i s s e d a s a trivial annoyance, one person recognized as an important problem to be solved. Case

2

As many bacteriologists do, Alexander Fleming was growing colonies of bacteria o n sterile agar plates. A n a c c i d e n t , s u c h a s m u s t h a v e h a p p e n e d t h o u s a n d s o f times in other laboratories, contaminated some of the plates w i t h dust. T h e bacteria F l e m i n g w a s trying to grow died in the neighborhood of the dust specks. Rather

207

Cognitive Processes in Creative Acts

t h a n t h r o w i n g t h e p l a t e s o u t a s " s p o i l e d , " F l e m i n g s a w t h a t t h e y p o s e d a n interest^ i n g p r o b l e m — " W h y d i d the bacteria d i e ? " T h e a n s w e r w a s that the dust contained the mold w h i c h produces P e n i c i l l i n — a substance w h o s e existence was unknown at that t i m e . T h u s a major m e d i c a l d i s c o v e r y d e p e n d e d on s o m e o n e s e e i n g a p r o b l e m in s o m e s p o i l e d agar plates. Case

3

As a part of a study of digestion, the physiologist, Ivan Pavlov, investigated the salivary reflex in dogs. D o g s salivate automatically w h e n food is put into their m o u t h s . T h e e x p e r i m e n t s w e n t w e l l a t first, b u t t h e n t o P a v l o v ' s s u r p r i s e , t h e d o g s b e g a n to salivate b e f o r e t h e y h a d any food in their m o u t h s at all. He found that they w o u l d salivate at the sight or s o u n d of the food d i s h e s or to a n y other signal that had frequently b e e n associated with feeding. T h e s e developments seriously complicated Pavlov's e x p e r i m e n t s . But, rather than s e e i n g t h e m s i m p l y as an annoyance, a s most w o u l d h a v e d o n e , h e s a w that t h e y r e v e a l e d a n important p r o b l e m — " W h a t was the nature of these anticipatory r e s p o n s e s ? " Pavlov r e c e i v e d s t e m warning from his c o l l e a g u e s that h e w o u l d b e risking his v e r y p r o m i s i n g c a r e e r i n p h y s i o l ogy if he p u r s u e d these "unscientific" psychological interests. Fortunately, Pavlov had the courage to ignore these warnings and to continue to work on the problem he had discovered. T h i s work on conditioned reflexes earned him the N o b e l prize.

I M P R O V I N G Y O U R P R O B L E M - F I N D I N G SKILLS T h e three p r o c e d u r e s we are about to d e s c r i b e are not l i k e l y to earn y o u a N o b e l p r i z e , b u t t h e y c a n h e l p t o m a k e y o u a b e t t e r p r o b l e m f i n d e r . T h e first i s i n t e n d e d a s a n a i d t o a s p i r i n g i n v e n t o r s . T h e s e c o n d a n d t h i r d are critical thinking t e c h n i q u e s , t h a t i s , t e c h n i q u e s for f i n d i n g f l a w s i n a r g u m e n t s . C r i t i c a l t h i n k i n g s k i l l s a r e i m p o r t a n t for c r e a t i v i t y b e c a u s e d e t e c t i n g a f l a w i n a n a c c e p t e d t h e o r y c a n b e a v e r y p o w e r f u l s o u r c e o f i n s p i r a t i o n for c r e a t i n g a n e w t h e o r y . C r i t i c a l t h i n k i n g skills are also v e r y useful in s u c h m u n d a n e activities as d e f e n d i n g o u r s e l v e s against politicians and e n c y c l o p e d i a salesmen. Bug Listing B u g listing is a t e c h n i q u e that A d a m s (1974) r e c o m m e n d s to h e l p inventors find promising p r o b l e m s to w o r k on. T h e basic idea is that things w h i c h bother y o u , such as ice cream c o n e s that drip or typewriters that w o n ' t spell, p r o b a b l y bother other p e o p l e as w e l l . An invention created to solve one of y o u r b u g s c o u l d have a w i d e market. A b u g l i s t i s s i m p l y a l i s t o f t h i n g s t h a t b o t h e r y o u . T o m a k e a b u g list, y o u s h o u l d carry a n o t e b o o k w i t h y o u s o t h a t y o u c a n r e c o r d t h e b u g s w h e n y o u n o t i c e t h e m . ( B u g s are o f t e n s o c o m m o n p l a c e t h a t t h e y a r e h a r d t o r e m e m b e r . ) H e r e i s a representative b u g list: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

r e m e m b e r i n g to mail letters taking out the trash fastening seatbelts putting tops b a c k on toothpaste, k e t c h u p , etc. changing the cat b o x washing dishes making my b e d k e e p i n g clean and dirty laundry separate t u r n i n g off l i g h t s s l i d i n g doors that stick

The C o m p l e t e Problem Solver

202 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

cupboard doors that don't close crumbs on the table, counter, and floor c a t fur p e o p l e w i t h dirty ears hanging up my coat restaurants w i t h d i m lights that m a k e y o u r e y e s go b u g g y p l a c e s that are c r o w d e d stupid teachers p e o p l e i n p a n t s t h a t are t o o t i g h t shorts that are too short m e n w h o w e a r t h e i r shirts o p e n t o t h e n a v e l music in supermarkets fight songs stupid radio and television personalities sour milk sunburn tangled hair razor stubble dirty glasses dirt u n d e r contact l e n s e s food that drips on y o u r clothes (like tacos)

32. 33. 34. 35. 36. 37. 38. 39. 40.

getting teeth c l e a n e d at the dentist dark nail polish dirty fingernails r e g i s t e r i n g for c l a s s e s drying my hair humid days crying babies dripping faucets passport photos

E a c h of these b u g s can be v i e w e d as a problem to be solved, and as the potential source of a useful invention. SEARCHING

FOR COUNTEREXAMPLES

The Employment Argument T h i s a r g u m e n t is not v e r y popular today, b u t it w a s w i d e l y a c c e p t e d in the '50's. " W h e n a c o m p a n y is filling a job, it should hire m e n in preference to w o m e n , because m e n have to support families." A c o m m o n w a y for p e o p l e t o t e s t a n a r g u m e n t i s t o s e a r c h for p o s i t i v e i n s t a n c e s (Wason, 1 9 6 8 ) — t h a t is, to think of a case in w h i c h the a r g u m e n t s e e m s right. F o r e x a m p l e , a p e r s o n m i g h t say, " W e l l , if a m a n is supporting a w i f e a n d three kids, it i s m o r e i m p o r t a n t for h i m t o h a v e a j o b t h a n for a s i n g l e w o m a n t o h a v e o n e . " H a v i n g found a positive instance, m a n y stop their search and accept the argument. W h i l e s e a r c h i n g for p o s i t i v e i n s t a n c e s i s a v e r y c o m m o n t e c h n i q u e , i t i s , unfortunately, a v e r y uncritical o n e . It ignores cases in w h i c h t h e a r g u m e n t is false. A m u c h m o r e critical approach to testing arguments is o n e w i d e l y u s e d by philoso p h e r s a n d m a t h e m a t i c i a n s — s e a r c h i n g for c o u n t e r e x a m p l e s . A c o u n t e r e x a m p l e i s a c a s e i n w h i c h t h e a r g u m e n t w e are t e s t i n g i s f a l s e . A p e r s o n s e a r c h i n g for a c o u n t e r e x a m p l e to t h e e m p l o y m e n t a r g u m e n t m i g h t say, " W e l l , a w o m a n w h o is supporting h e r c r i p p l e d h u s b a n d certainly d e s e r v e s to h a v e a j o b as m u c h as a b a c h e l o r d o e s . " C l e a r l y , we are m u c h less likely to a c c e p t a faulty a r g u m e n t if we t e s t i t b y s e a r c h i n g for c o u n t e r e x a m p l e s . H e r e i s a n o t h e r c a s e .

203

Cognitive Processes in Creative Acts The Pre-cognition A r g u m e n t

S o m e claim that certain p e o p l e can have accurate k n o w l e d g e of events before they happen. Positive Instance:

" Y e a h ! L a s t w e e k m y aunt d r e a m t that s o m e t h i n g good w a s going to h a p p e n to me and today I w o n the lottery!"

Counterexample:

" S u r e , b u t t w o w e e k s ago, she d r e a m t that y o u r turtle was going to have puppies."

S e a r c h i n g for c o u n t e r e x a m p l e s i s o n l y o n e o f m a n y t e c h n i q u e s w h i c h p h i l o s o p h e r s h a v e d e v e l o p e d for d e t e c t i n g p r o b l e m s i n a r g u m e n t s . A n i n t e r e s t i n g a n d highly readable introduction to some of these techniques is provided by Thomas S c h w a r t z in h i s b o o k , The Art of Logical Reasoning ( 1 9 8 0 ) . SEARCHING

FOR ALTERNATIVE INTERPRETATIONS

O u r s e c o n d t e c h n i q u e for f i n d i n g p r o b l e m s i n a r g u m e n t s b o r r o w s f r o m t h e c r i t i c a l s p i r i t o f s c i e n t i f i c r e s e a r c h . I t i s t h e t e c h n i q u e o f s e a r c h i n g for a l t e r n a t i v e interpretations. Case

1:

Singing

the

Baby

to

Flab

S u p p o s e t h a t s o m e o n e t r i e d t o c o n v i n c e u s that s i n g i n g t o c h i l d r e n m a k e s t h e m g r o w . T h e y tell u s that t h e y h a v e systematically o b s e r v e d 2 0 k i n d e r g a r t e n c h i l d r e n w h o s e mothers sing to t h e m e v e r y day, and that o v e r a p e r i o d of e i g h t m o n t h s , these c h i l d r e n g a i n e d a n a v e r a g e o f four p o u n d s . F o r t u n a t e l y , t h e m a i n p r o b l e m i n t h i s a r g u m e n t i s e a s y t o f i n d . W e k n o w that kindergarten c h i l d r e n are g r o w i n g rapidly, so we can readily formulate an alternate interpretation o f the data. T h o s e c h i l d r e n m i g h t h a v e g a i n e d four p o u n d s e v e n i f t h e y hadn't b e e n s u n g to. T h e p r o b l e m w i t h t h e a r g u m e n t i s t h a t t h e o b s e r v a t i o n s d o n o t i n c l u d e control m e a s u r e m e n t s , for e x a m p l e , w e i g h t g a i n s o f 2 0 k i n d e r g a r t e n c h i l d r e n w h o s e m o t h e r s d o not s i n g t o t h e m . W e w o u l d b e m u c h m o r e i n c l i n e d t o a c c e p t t h e a r g u m e n t i f t h e c h i l d r e n w h o w e r e s u n g t o g a i n e d more t h a n t h e c h i l d r e n w h o w e r e not. Case

2:

Hypnotizing

Chickens

O n c e , in my w a s t e d y o u t h , I ran across an e n t e r p r i s i n g s a l e s m a n w h o w a s selling a d e v i c e for h y p n o t i z i n g c h i c k e n s . I n d e e d , t h e d e v i c e w a s i m p r e s s i v e . T h e s a l e s m a n h e l d a c h i c k e n d o w n o n a t a b l e a n d p l a c e d t h e d e v i c e , w h i c h e m i t t e d a soft b u z z i n g , next to the c h i c k e n ' s h e a d . W h e n he r e l e a s e d the c h i c k e n , it just lay there a n d s t a r e d a s i f all a c t i v e c h i c k e n t h o u g h t s h a d b e e n c h a s e d f r o m its h e a d . I t w a s c o m p l e t e l y g o r k e d o u t . T h e n , after 3 0 s e c o n d s o f c o m p l e t e i m m o b i l i t y , i t s u d d e n l y s c r a m b l e d u p a n d r a n off. O n e can easily u n d e r s t a n d that m a n y p e o p l e , p e r h a p s e v e n the salesman himself, w o u l d b e c o n v i n c e d o f t h e e f f i c a c y o f t h e d e v i c e — s o m e t o t h e e x t e n t o f b u y i n g one. I wasn't c o n v i n c e d , h o w e v e r , b e c a u s e I had an alternative interpretation of the demonstration. F o r no g o o d reason, at that t i m e , I w a s the local e x p e r t on " a n i m a l h y p n o s i s . " I k n e w that various a n i m a l s — g u i n e a p i g s , snakes, alligators, and y e s , chickens, c o u l d be " h y p n o t i z e d " simply by putting t h e m into certain postures. In p a r t i c u l a r , a c h i c k e n c a n b e h y p n o t i z e d b y h o l d i n g its h e a d d o w n o n a t a b l e for a few s e c o n d s . M y alternative interpretation w a s that w h a t w e w e r e s e e i n g w a s just a n o t h e r c a s e o f a n i m a l h y p n o s i s a n d that a c o n t r o l o b s e r v a t i o n i n w h i c h t h e c h i c k e n ' s h e a d w a s h e l d o n t h e t a b l e without t h e b u z z e r w o u l d h a v e p r o d u c e d t h e s a m e r e s u l t , i.e., a g o r k e d - o u t c h i c k e n .

204

The C o m p l e t e Problem Solver Case

I

Revisited:

Singing

to

Children

S u p p o s e that the person s t u d y i n g singing and w e i g h t gain had m a d e the control o b s e r v a t i o n w e s u g g e s t e d a b o v e . S u p p o s e that t h e y h a d f o u n d t h a t c h i l d r e n of mothers w h o sing to them gain t w o pounds more on the average than children of mothers w h o d o not. W o u l d w e t h e n b e forced t o a c c e p t the a r g u m e n t that singing t o c h i l d r e n m a k e s t h e m g r o w ? T h e a n s w e r i s " N o . " T h e a r g u m e n t still h a s p r o b lems as we can see by considering some alternative interpretations. Perhaps there w e r e o t h e r d i f f e r e n c e s b e t w e e n t h e t w o g r o u p s o f m o t h e r s . P e r h a p s i f w e interv i e w e d the m o t h e r s , the non-singers w o u l d say, " S i n g to my k i d ? H e l l , I d o n ' t e v e n f e e d ' m ! " , w h i l e the singers insist that a healthy child n e e d s at least n i n e meals a day. C l e a r l y , s e a r c h i n g for a l t e r n a t e i n t e r p r e t a t i o n s c a n h e l p u s t o f i n d p r o b l e m s i n persuasive arguments. Summary I n s o m e c a s e s f i n d i n g a p r o b l e m i s t h e m o s t d i f f i c u l t p a r t o f s o l v i n g it. M a n y d i s c o v e r i e s d e p e n d on finding a p r o b l e m that others h a v e i g n o r e d . We f r e q u e n t l y h e a r p e r s u a s i v e a r g u m e n t s t h a t h a v e p r o b l e m s h i d d e n i n t h e m . I f w e fail t o f i n d t h e p r o b l e m s , we are l i k e l y to be p e r s u a d e d of s o m e t h i n g that is false. By asking the f o l l o w i n g three questions, we can increase our c h a n c e s of finding such problems: 1. C a n I t h i n k of a c o u n t e r e x a m p l e ? 2. H a v e appropriate control measurements b e e n m a d e ? 3. A r e there alternate interpretations of the result? Exercises 1. D e s i g n a study of s i n g i n g and w e i g h t gain that w o u l d a v o i d the p r o b l e m s we found above. 2. W h a t p r o b l e m s can y o u find in the following arguments? a. N i n e out of 10 doctors i n t e r v i e w e d said that t h e y p r e s c r i b e our brand in p r e f e r e n c e t o all o t h e r s . B u y o u r b r a n d . b . I n c l i n i c a l t e s t s , e i g h t o u t o f 1 0 f e l t r e l i e f w i t h i n 1 0 m i n u t e s after u s i n g o u r b r a n d . B u y it. c. F o u r l e a d i n g pain relievers contain 650 mg of aspirin p e r tablet, but our b r a n d c o n t a i n s 800 m g . B u y o u r b r a n d . GENERATING

IDEAS

M a n y of the difficulties in creative p r o b l e m solving arise not in finding the p r o b l e m b u t i n g e n e r a t i n g i d e a s for a s o l u t i o n . W h e n f a c e d w i t h a p r o b l e m l i k e T h e L o s e r ( R e m e m b e r the m a n w h o a l w a y s lost a t g a m b l i n g b e c a u s e the fortune teller c a s t a s p e l l o n h i m ? ) , s o m e p e o p l e are t e r r i f i c a t t h i n k i n g u p i d e a s a n d s o m e are t e r r i b l e . T h i s s e c t i o n i s d e s i g n e d for p e o p l e w h o h a v e t r o u b l e g e n e r a t i n g i d e a s . M a n y methods h a v e b e e n proposed to help m a k e ideas flow. We will discuss just two: 1. b r a i n s t o r m i n g , a n d 2. finding analogies. Brainstorming Brainstorming is a t e c h n i q u e d e v e l o p e d by Alex Osborn, an executive of a m a j o r N e w Y o r k a d v e r t i s i n g firm, a n d first d e s c r i b e d i n h i s b o o k , Your Creative Power ( 1 9 4 8 ) . I t i s d e s i g n e d t o i n c r e a s e t h e f l o w o f i d e a s i n s m a l l g r o u p m e e t i n g s .

Cognitive Processes in Creative Acts

205

T h e m o s t important principle u n d e r l y i n g brainstorming is that the process of generating i d e a s is c o m p l e t e l y s e p a r a t e d from the p r o c e s s of e v a l u a t i n g t h e m . B r a i n s t o r m i n g s e s s i o n s t a k e p l a c e i n t w o p h a s e s — a n idea generation p h a s e a n d a n idea evaluation p h a s e . D u r i n g t h e i d e a g e n e r a t i o n p h a s e , all j u d g i n g a n d c r i t i c i s m of the p r o d u c e d ideas is e l i m i n a t e d — o r rather d e f e r r e d — u n t i l the evaluation phase. In most c o n f e r e n c e s , such as t o w n m e e t i n g s and informal p l a n n i n g sessions, t h e s t a n d a r d f o r m a t i s d e b a t e , t h a t is, p r o p o s a l a n d c r i t i c i s m . T h i s f o r m a t i s n o t notably successful in producing n e w ideas. Typically in such sessions, each n e w i d e a is m e t w i t h a w e l t e r of criticism. B r a i n s t o r m i n g is q u i t e different from most m e e t i n g situations. During the idea generation phase: 1. 2. 3. 4.

All criticism of ideas is w i t h h e l d until the evaluation phase. W i l d or e v e n silly ideas are w e l c o m e d . Quantity of ideas is encouraged. Participants are e n c o u r a g e d to c o m b i n e or i m p r o v e on ideas already suggested. 5. T h e g r o u p acts as a w h o l e , not b r e a k i n g up into several small groups. 6 . O n e p e r s o n acts a s a s e c r e t a r y t o r e c o r d t h e l i s t o f i d e a s .

A b r a i n s t o r m i n g g r o u p n e e d s a l e a d e r w h o w i l l e n f o r c e t h e r u l e s . T h e first a n d most important task of the l e a d e r is to be sure that criticism is w i t h h e l d . T h e r e seems to be a strong t e n d e n c y in m a n y p e o p l e to respond to an idea by saying, " O h , that w o n ' t w o r k b e c a u s e " An idea w h i c h is proposed as a j o k e m a y not be useful in itself as a solution of the problem, and yet it may aid the solution by suggesting a n e w dimension of the problem or by opening a n e w line of inquiry. I n a t y p i c a l b r a i n s t o r m i n g s e s s i o n , t h e m e m b e r s o f t h e g r o u p are a l l o w e d t o p r o p o s e ideas w h e n e v e r t h e y p l e a s e . T h e possibility exists in this situation that one or a f e w of the g r o u p m e m b e r s w i l l d o m i n a t e the session, w i t h the result that others m a y b e p r e v e n t e d f r o m c o n t r i b u t i n g all t h a t t h e y c o u l d . T o e l i m i n a t e t h i s p o s s i b i l ity, B o u c h a r d ( 1 9 7 2 ) h a s m o d i f i e d t h e t y p i c a l b r a i n s t o r m i n g p r o c e d u r e b y a d d i n g a s e q u e n c i n g rule in w h i c h the m e m b e r s of the group take turns in offering ideas. He reported that groups u s i n g the s e q u e n c i n g rule p r o d u c e d more ideas than groups t h a t d i d n o t u s e it. If the group b e g i n s to run out of ideas during the idea-generating phase, it s o m e t i m e s h e l p s t o r e v i e w t h e list o f i d e a s a l r e a d y s u g g e s t e d . W h e n t h e g r o u p ' s ideas have b e e n exhausted, it is time to m o v e to the evaluation phase. In the evaluation phase, each idea is r e v i e w e d critically to d e t e r m i n e if it is in fact a p r a c t i c a l s o l u t i o n . A l i s t o f t h e i d e a s t h a t t h e g r o u p c o n s i d e r s m o s t p r a c t i c a l is then submitted as the group's problem-solving recommendations. O s b o r n f e e l s t h a t b r a i n s t o r m i n g i s h e l p f u l i n p r o d u c i n g n e w i d e a s for t w o reasons. First, the reduction in criticism during the generation phase allows ideas to be born and d e v e l o p e d that otherwise m i g h t n e v e r h a v e b e e n suggested, or might have b e e n rejected before they had r e c e i v e d sufficient positive consideration. S e c o n d , O s b o r n f e e l s t h a t b r a i n s t o r m i n g s e s s i o n s p r o m o t e a k i n d o f s o c i a l c o n t a g i o n i n w h i c h o n e p e r s o n ' s i d e a i n s p i r e s a b e t t e r i d e a i n a n o t h e r — a n i d e a that the second person w o u l d n ' t have thought of otherwise. Studies r e v i e w e d b y Stein (1975) indicate that g r o u p s u s i n g the brainstorming t e c h n i q u e do p r o d u c e more ideas than groups that generate and evaluate ideas simultaneously. Further, m o s t of t h e s e studies also s h o w that the brainstorming groups p r o d u c e more high-quality ideas. W e s h o u l d n o t e t h a t O s b o r n r e c o m m e n d s b r a i n s t o r m i n g o n l y for c e r t a i n t y p e s of "simple and talkable" p r o b l e m s — p r o b l e m s like, " H o w can we prevent stealing

The C o m p l e t e Problem Solver

206

from the l i b r a r y ? " and, " H o w can w e g e t m o r e foreign visitors t o Come t o the U n i t e d S t a t e s ? " I n d e e d , research on group p r o b l e m solving suggests that groups do better t h a n i n d i v i d u a l s o n s o m e k i n d s o f p r o b l e m s b u t n o t o n o t h e r s . I n d i v i d u a l s are a s g o o d or b e t t e r than g r o u p s in s o l v i n g arithmetic p r o b l e m s ( H u d g i n s , 1960), and in s o l v i n g p r o b l e m s w h e r e e a c h i n d i v i d u a l h a s all t h e n e c e s s a r y i n f o r m a t i o n . G r o u p s are s u p e r i o r to i n d i v i d u a l s in tasks w h e r e the p o o l i n g of skills a n d information is i m p o r t a n t . F o r e x a m p l e , g r o u p s are s u p e r i o r t o i n d i v i d u a l s for r e m e m b e r i n g a c o m p l e x s t o r y ( P e r l m u t t e r , 1 9 5 3 ) , a n d for s o l v i n g p r e j u d i c e - p r o v o k i n g s y l l o g i s m s (Barnlund, 1959). B a r n l u n d suggests that the g r o u p is m o r e o b j e c t i v e than the individual b e c a u s e prejudices are not c o m p l e t e l y shared a m o n g the m e m b e r s of the group. Exercise F o r m a g r o u p and c o n d u c t a brainstorming session. S u g g e s t e d topics are: • w a y s to save time • w a y s to k e e p the city clean • u s e s for d i s c a r d e d s t y r o f o a m c u p s Individual Brainstorming W h i l e the brainstorming t e c h n i q u e w a s d e s i g n e d to be u s e d w i t h groups, it can also be u s e d by a single person in private idea-generating sessions. T h e principles t o b e a p p l i e d a r e t h e s a m e . S e p a r a t e i d e a g e n e r a t i o n f r o m e v a l u a t i o n . Start w i t h t h e idea generation phase, writing d o w n ideas as they occur, w i t h o u t criticism. Y o u s h o u l d w e l c o m e w i l d o r s i l l y i d e a s , a n d y o u s h o u l d try t o c o m b i n e o r i m p r o v e i d e a s that w e r e g e n e r a t e d earlier. T h e hard part in this p h a s e is to control y o u r internal e d i t o r — t h e i n t e r n a l v o i c e o f c r i t i c i s m w h i c h m a y l e a d y o u t o i g n o r e a n i d e a that s e e m s too d u m b or trivial. Just a s w i t h g r o u p b r a i n s t o r m i n g , w h e n y o u b e g i n t o r u n o u t o f i d e a s , y o u c a n r e v i e w t h e list a s a s o u r c e t o s t i m u l a t e f u r t h e r p r o d u c t i o n . W h e n t h e i d e a s r e a l l y have stopped coming, it is time to m o v e on to the evaluation phase. Here you r e v i e w e a c h i d e a t o s e l e c t t h o s e t h a t s e e m b e s t for s o l v i n g t h e p r o b l e m . Exercise C o n d u c t a brainstorming session by yourself. S u g g e s t e d topics: • h o w to increase your o w n efficiency • h o w to m a k e your favorite annoyance less a n n o y i n g • h o w to p e r s u a d e s o m e o n e to g i v e y o u a j o b I n d i v i d u a l brainstorming sessions can b e v e r y helpful w h e n y o u are w r i t i n g (see F l o w e r , 1980). S u p p o s e that y o u are p l a n n i n g a m a g a z i n e article on architect u r e for a t e e n a g e a u d i e n c e . T o b r a i n s t o r m , first g e n e r a t e all o f t h e i d e a s y o u c a n think of that a t e e n a g e r m i g h t find interesting or important a b o u t architecture. As y o u d o t h i s , s c r a t c h d o w n r a p i d n o t e s i n t h e f o r m o f s c a t t e r e d w o r d s a n d p h r a s e s that will r e m i n d y o u of the ideas. W h e n idea generation is c o m p l e t e , evaluate the i d e a s — t h a t is, d e c i d e w h i c h ideas y o u w a n t t o i n c l u d e i n y o u r a r t i c l e — a n d then o r g a n i z e t h e m into an outline. At this point, y o u are w e l l started in p r o d u c i n g your article. DISCOVERING

ANALOGIES

A n a l o g i e s a r e a n i m p o r t a n t s o u r c e o f i d e a s w h e n w e a r e s e a r c h i n g for p r o b l e m s o l u t i o n s . S e v e r a l s y s t e m s h a v e b e e n p r o p o s e d for s t i m u l a t i n g t h e f o r m a t i o n o f u s e f u l a n a l o g i e s . I n e s s e n c e , all o f t h e s e s y s t e m s e m p l o y s o m e c h e c k l i s t o f a n a l o g y

Cognitive Processes in Creative Acts

207

types. T h e user of the system works through the checklist a n d tries to find analogies of each type. I n G o r d o n ' s s y n e c t i c s s y s t e m ( G o r d o n , 1 9 6 1 ) , t h e c h e c k l i s t c o n s i s t s o f four a n a l o g y t y p e s : p e r s o n a l , direct, s y m b o l i c , a n d fantasy. S u p p o s e that w e w e r e m e m b e r s o f a s y n e c t i c s g r o u p l o o k i n g for i d e a s t o i m p r o v e a u t o m o b i l e b r a k e s . F i r s t , w e w o u l d try t o f o r m p e r s o n a l a n a l o g i e s i n w h i c h w e p u t o u r s e l v e s d i r e c t l y i n t o t h e p r o b l e m s i t u a t i o n . F o r e x a m p l e , w e c o u l d i m a g i n e o u r s e l v e s a s t h e b r a k e s o f a car. N e x t , w e w o u l d s e a r c h for a d i r e c t a n a l o g y i f t h e s a m e f u n c t i o n i s a c c o m p l i s h e d i n s o m e o t h e r s e t t i n g , s u c h a s a c a t t r y i n g t o s t o p o n a s l i p p e r y floor. T h i r d , w e w o u l d try t o f o r m s y m b o l i c a n a l o g i e s . I w o u l d t e l l y o u a b o u t t h i s k i n d o f a n a l o g y i f I c o u l d , b u t u n f o r t u n a t e l y , I h a v e b e e n u n a b l e t o f i n d a n i n t e l l i g i b l e d e s c r i p t i o n o f it. Finally, we think of fantasy analogies in w h i c h anything i n c l u d i n g magic and s c i e n c e f i c t i o n are a l l o w e d , for i n s t a n c e , c l a w s r e a c h i n g o u t o f t h e r o a d t o g r a b o u r wheels. I n K o b e r g a n d B a g n a l l ' s ( 1 9 7 4 ) a t t r i b u t e a n a l o g y s y s t e m , t h e c h e c k l i s t i s a list o f a t t r i b u t e s o f a n o b j e c t — i t s n a m e , f o r m , f u n c t i o n , c o l o r , a n d m a t e r i a l . I f w e are t r y i n g t o i m p r o v e s o m e o b j e c t , s a y , a f i r e p l a c e , w e first l i s t its a t t r i b u t e s a n d t h e n attach a n a l o g i e s to e a c h one. T a b l e 1 illustrates the process. T a b l e 1. T h e Attribute A n a l o g y System (from K o b e r g and Bagnall, 1 9 7 4 ) A s s u m i n g t h e p r o b l e m i s t o i m p r o v e a f i r e p l a c e , its a t t r i b u t e s a r e : Name: Form: Function: Color: Material:

Fireplace G e o m e t r i c , angular, conical, etc. H e a t room, p s y c h o l o g i c a l l y soothing, etc. Black, brick red, etc. Steel, masonry, etc.

Analogy Chains (similarities) Name: C o m b u s t i o n c h a m b e r , tea pot, auto e n g i n e , cigarette lighter, etc. Form: Architectural constructions, crystals, prisms, etc. Function: C a t o n l a p , r o b e , i n t i m a t e f r i e n d , e t c . Ideas Produced (for i m p r o v i n g f i r e p l a c e ) C h a n g e n a m e to e n e r g y transformer. T r y forms w h i c h are d e r i v e d from crystal structures. Use robe insulation principle to c o n s e r v e radiant heat, etc.

As y e t there appears to be no solid experimental e v i d e n c e that either synectics or attribute a n a l o g i e s actually work. H o w e v e r , the s y n e c t i c s s y s t e m has r e c e i v e d a g o o d deal of favorable c o m m e n t from users in industry (see Stein, 1975). P L A N N I N G A N D CREATIVITY A plan is a set of directions we use to g u i d e us in solving a p r o b l e m . T h e more effort w e p u t i n t o p l a n n i n g , t h e m o r e l i k e l y w e a r e t o c o n s t r u c t a g o o d m a p w h i c h w i l l g u i d e u s e f f i c i e n t l y t o t h e b e s t s o l u t i o n . S i n c e c r e a t i v e acts a r e p r o b l e m - s o l v i n g a c t s , i t s h o u l d n ' t s u r p r i s e u s t h a t p l a n n i n g i s a l s o i m p o r t a n t for c r e a t i v i t y . F l o w e r a n d H a y e s (1980) h a v e s h o w n t h a t g o o d e x p o s i t o r y w r i t e r s p l a n m u c h m o r e e f f e c tively than do poor writers. A p a t h - b r e a k i n g s t u d y b y G e t z e l s a n d C s i k s z e n t m i h a l y i ( 1 9 7 6 ) i n d i c a t e s that p l a n n i n g i s c r i t i c a l l y i m p o r t a n t i n art. T h e y s h o w e d t h a t t h e a m o u n t o f p l a n n i n g * * T h e s e authors claim to measure problem finding rather than planning. However, as these terms are used in this text, and, I b e l i e v e , in cognitive science generally, it is more appropriate to say that they have measured planning.

The C o m p l e t e Problem Solver

208

t h a t a n art s t u d e n t d i d i n p r e p a r i n g t o d r a w a p i c t u r e p r e d i c t e d n o t o n l y t h e q u a l i t y of the resulting picture, but also w h e t h e r or not the student w o u l d b e c o m e a p r o d u c t i v e artist y e a r s l a t e r . T h e investigators tested 31 m a l e second- and third-year students at a prestig i o u s art s c h o o l . E a c h s t u d e n t w a s b r o u g h t i n t o a n e x p e r i m e n t a l r o o m s u p p l i e d with d r a w i n g materials, an e m p t y table, and a variety of objects. T h e students w e r e asked to select any of the objects they wanted, arrange t h e m on the table as they chose, and t h e n to d r a w a picture. T h e y w e r e told, " T h e important thing is that the drawing should be pleasing to you." T h e experimenter then noted three things about the students' behavior before they started d r a w i n g a n d three things about their b e h a v i o r w h i l e they w e r e drawing. T h e s e six b e h a v i o r s (listed in T a b l e 2) w e r e c h o s e n to reflect the a m o u n t of planning the students did in executing the drawing. For example, students w e r e s c o r e d h i g h o n p l a n n i n g i f t h e y e x a m i n e d m a n y o b j e c t s o r i f t h e y m a d e m a n y drafts a n d t o o k c o n s i d e r a b l e t i m e i n a r r i v i n g a t t h e final s t r u c t u r e o f t h e i r d r a w i n g s .

T a b l e 2. Six Behaviors Used to M e a s u r e Planning BEFORE Bl. B2. B3.

DRAWING:

H o w many objects w e r e manipulated? W e r e u n u s u a l o b j e c t s c h o s e n — t h a t is, d i d a s t u d e n t c h o o s e o b j e c t s w h i c h f e w other students chose? H o w carefully did the student examine the objects?

WHILE

DRAWING:

Wl.

H o w m u c h time e l a p s e d from the b e g i n n i n g o f d r a w i n g t o the time a t w h i c h t h e final s t r u c t u r e a p p e a r e d ? ( j u d g e d l a t e r b y l o o k i n g a t a s e q u e n c e o f t i m e d photographs) W 2 . D i d t h e s u b j e c t start t h e d r a w i n g o v e r o r c h a n g e t h e a r r a n g e m e n t o f t h e objects? W 3 . W a s the d r a w i n g simply a c o p y of the arrangement of objects, or w e r e the objects in the d r a w i n g modified in size, position, or n u m b e r ?

T h e d r a w i n g s that the students p r o d u c e d w e r e then e v a l u a t e d i n d e p e n d e n t l y b y f i v e art c r i t i c s for o r i g i n a l i t y , c r a f t s m a n s h i p , a n d o v e r a l l v a l u e . T a b l e 3 s h o w s t h e correlations b e t w e e n the average of the five critics' j u d g m e n t s and the p l a n n i n g b e h a v i o r s . F i v e o f t h e six p l a n n i n g b e h a v i o r s s h o w s t r o n g c o r r e l a t i o n s w i t h t h e c r i t i c s ' j u d g m e n t s . ( I n t h i s s t u d y , a c o r r e l a t i o n o f 0.3 o r l a r g e r s h o u l d b e c o n s i d e r e d significant.) Planning, then, appears to be very helpful in creating a g o o d d r a w i n g . I n a f o l l o w - u p s t u d y s e v e n y e a r s later, G e t z e l s a n d C s i k s z e n t m i h a l y i t r i e d t o d e t e r m i n e h o w s u c c e s s f u l t h e 3 1 s t u d e n t s w e r e i n p u r s u i n g artistic c a r e e r s . B y c o n t a c t i n g art c r i t i c s , d i r e c t o r s o f art g a l l e r i e s , a n d t h e s t u d e n t s t h e m s e l v e s , t h e y f o u n d t h a t a b o u t h a l f o f t h e s t u d e n t s h a d d r o p p e d o u t o f art c o m p l e t e l y . T h e r e s t w e r e p u r s u i n g c a r e e r s i n t h e f i n e arts w i t h v a r y i n g d e g r e e s o f s u c c e s s . Seven* w e r e u s i n g t h e i r s k i l l s i n r e l a t e d p r o f e s s i o n s , s u c h a s t e a c h i n g art, b u t h a d n o t y e t e x h i b i t e d t h e i r w o r k p u b l i c l y . T h e r e m a i n i n g n i n e h a d all e x h i b i t e d . S o m e w e r e r e p r e s e n t e d by major galleries and o n e had a c h i e v e d a v e r y notable d e g r e e of success. H i s w o r k is h u n g in the b e s t galleries, and articles a b o u t his w o r k a p p e a r in the most r e s p e c t e d art j o u r n a l s . T h e l a s t c o l u m n o f T a b l e 1 s h o w s t h e c o r r e l a t i o n b e t w e e n s u c c e s s a s a n artist a n d t h e p l a n n i n g b e h a v i o r s m e a s u r e d s e v e n y e a r s e a r l i e r . F o u r o f t h e six b e h a v i o r s s h o w strong correlations w i t h s u c c e s s . T h e s i m p l e s t w a y t o interpret this remarka b l e r e s u l t i s t o a s s u m e t h a t t h e s u c c e s s f u l artists h a b i t u a l l y m a k e p l a n n i n g p a r t o f

Cognitive Processes in Creative Acts

209

t h e i r a p p r o a c h t o a r t i s t i c p r o b l e m s . P l a n n i n g l e a d s t o h i g h q u a l i t y i n all t h e i r artistic work just as it led to high quality in the experimental drawings. CREATIVITY

AND

PREPARATION

I n C h a p t e r 2 , w e r e v i e w e d r e s e a r c h i n d i c a t i n g that skillful c h e s s players emp l o y a n e n o r m o u s a m o u n t o f chess-pattern k n o w l e d g e . T o a c q u i r e this k n o w l e d g e , the chess p l a y e r must s p e n d thousands of hours of p r e p a r a t i o n — p l a y i n g chess, reading chess magazines, and studying chess positions. Simon and C h a s e (1973) n o t e t h a t i t i s v e r y r a r e for a p e r s o n t o r e a c h t h e g r a n d m a s t e r l e v e l o f s k i l l w i t h l e s s than 10 years of i n t e n s i v e study. If chess masters require a long and intensive period of preparation to acquire their skill, w h a t about painters a n d c o m p o s e r s ? C e r t a i n l y painting a n d composition require large amounts of pattern k n o w l e d g e . T h e painter must k n o w h o w a face will look from a m u l t i t u d e of a n g l e s a n d h o w colors clash or h a r m o n i z e . T h e c o m p o s e r must k n o w the timbres of the various instruments and the sound, look, and feel of c h o r d s a n d k e y s t r u c t u r e s . D o c r e a t i v e artists t h e n a l s o r e q u i r e i n t e n s e p r e p a r a t i o n before they can p r o d u c e w o r k s of real merit? D o e s creative productivity d e p e n d on preparation, or can g e n i u s e s p r o d u c e masterworks right from the b e g i n n i n g of their careers? To a n s w e r this q u e s t i o n , I d e c i d e d to e x a m i n e the l i v e s of famous c o m posers. I started my i n v e s t i g a t i o n w i t h t h e i n c r e d i b l y p r e c o c i o u s Mozart, b e c a u s e he is the c o m p o s e r w h o s e e m s least likely to have required a l o n g period of preparat i o n . H e b e g a n t o s t u d y m u s i c a t f o u r a n d w r o t e h i s first s y m p h o n y a t t h e a g e o f eight. I h a v e g r a p h e d the n u m b e r of w o r k s that M o z a r t p r o d u c e d in e a c h year of his career in F i g u r e 1. T h e figure s h o w s that Mozart's productivity increased steadily for t h e first 1 0 o r 1 2 y e a r s o f h i s c a r e e r . I t a l s o s h o w s t h a t M o z a r t d i d p r o d u c e w o r k s in the very early part of his career w h e n he had had o n l y a y e a r or t w o of preparation. I f t h e s e are w o r k s o f v e r y h i g h q u a l i t y , t h e n w e c o u l d c o n c l u d e , for M o z a r t a t least, that h e d i d n ' t r e q u i r e l o n g preparation t o b e o u t s t a n d i n g l y c r e a t i v e . H o w e v e r ,

270

The C o m p l e t e Problem Solver

2T i i i I l i i i II i i i t I i i i i I 0

5

10

15

i

i i i

20

I 25

i i i i | i 30

N u m b e r of y e a r s into c a r e e r Figure 1. T h e y e a r m a r k e d 0 on t h e g r a p h is 1760, the y e a r w h e n Mozart w a s four and b e g a n intens i v e m u s i c a l training. T h e solid line in t h e figure is b a s e d on information from Grove's Dictionary of Music (1955). T h e d a s h e d line is b a s e d on K o e c h e l ' s listings (1965) as revised b y m o d e r n m u s i c o l o g i s t s . T h e s e two s o u r c e s a r e i n r e a s o n a b l e a g r e e m e n t a b o u t what w o r k s w e r e p r o d u c e d w h e n . these early works may not be of outstanding quality. Perhaps they have b e e n p r e s e r v e d for t h e i r h i s t o r i c a l r a t h e r t h a n t h e i r m u s i c a l v a l u e . To obtain some measure of the quality of Mozart's work, I turned, oddly e n o u g h , to S c h w a n n ' s Record and Tape Guide. I r e a s o n e d t h a t an e x c e l l e n t w o r k is l i k e l y to be r e c o r d e d m o r e often than a poor one. T h e d e c i s i o n to record a w o r k p r e s u m a b l y reflects b o t h musical j u d g m e n t and popular t a s t e — t h a t is, it reflects the musical j u d g m e n t by a c o n d u c t o r that the w o r k is w o r t h w h i l e and the b e l i e f of the record c o m p a n i e s that the record will sell. Figure 2 shows the n u m b e r of recordings listed in Schwann's guide (August, 1979) of works written in each year of Mozart's career. W h i l e about 12 percent of M o z a r t ' s w o r k s w e r e w r i t t e n i n t h e first 1 0 y e a r s o f h i s c a r e e r , o n l y 4.8 p e r c e n t o f the recordings c a m e from this early period. Further, m a n y of the recordings of early w o r k s are i n c l u d e d i n c o l l e c t i o n s w i t h l a b e l s s u c h a s , " T h e C o m p l e t e S y m p h o n i e s o f M o z a r t . " P e r h a p s t h e e a r l y w o r k s w e r e i n c l u d e d for r e a s o n s o f c o m p l e t e n e s s r a t h e r t h a n e x c e l l e n c e . W h e n r e c o r d i n g s i n c l u d e d i n c o m p l e t e c o l l e c t i o n s are o m i t t e d from t h e c a l c u l a t i o n s , the p e r c e n t a g e of r e c o r d i n g s in this early p e r i o d

Cognitive Processes in Creative Acts

211

N u m b e r o f y e a r s into c a r e e r Figure 2. d r o p s t o 2.4. T h e s e o b s e r v a t i o n s s u g g e s t t h a t M o z a r t ' s e a r l y w o r k s are n o t o f t h e same h i g h quality as his later o n e s . T h e m u s i c critic, H a r o l d S c h o n b e r g , is of the same opinion. He says (1970) I t i s s t r a n g e t o s a y o f a c o m p o s e r w h o s t a r t e d w r i t i n g a t six, a n d l i v e d o n l y 3 6 y e a r s , t h a t h e d e v e l o p e d l a t e , b u t t h a t i s t h e truth. F e w o f M o z a r t ' s early w o r k s , e l e g a n t as t h e y are, h a v e the personality, concentration, and r i c h n e s s t h a t e n t e r e d h i s m u s i c after 1 7 8 1 " In 1 7 8 1 , Mozart w a s in the 2 1 s t y e a r of his career.

272

The C o m p l e t e Problem Solver

S o m e w o r k s are r e c o r d e d t w o or three times in different c o m p l e t e collections. T h e r e f o r e , t o w e e d o u t w o r k s w h i c h m a y b e r e c o r d e d for r e a s o n s o t h e r t h a n m u s i c a l q u a l i t y , I d e f i n e d a m a s t e r w o r k (for t h e p u r p o s e s o f t h i s s t u d y ) a s o n e for w h i c h f i v e different r e c o r d i n g s are currently listed in S c h w a n n ' s g u i d e . By this definition, M o z a r t ' s first m a s t e r w o r k w a s w r i t t e n i n t h e 1 2 t h y e a r o f h i s c a r e e r . To explore the question about creativity and preparation more generally, I s e a r c h e d for b i o g r a p h i c a l m a t e r i a l a b o u t all o f t h e c o m p o s e r s d i s c u s s e d i n S c h o n b e r g ' s The Lives of the Great Composers ( 1 9 7 0 ) . F o r 76 of t h e s e c o m p o s e r s , I w a s a b l e t o d e t e r m i n e w h e n t h e y s t a r t e d i n t e n s i v e s t u d y o f m u s i c . I n c i d e n t a l l y , all o f these composers had at least one work listed in Schwann's guide, and 64 had one or more w o r k s w h i c h w e r e available on five different records. I n F i g u r e 3 all o f t h e c a r e e r s o f t h e c o m p o s e r s a r e s h o w n o n t h e s a m e s c a l e . T h a t is, the 10th y e a r of H a n d e l ' s c a r e e r is g r a p h e d in the s a m e p l a c e as the 10th y e a r o f B r a h m s ' career. T h e figure s h o w s that v e r y f e w c o m p o s e r s p r o d u c e d masterw o r k s w i t h l e s s t h a n 1 0 y e a r s o f p r e p a r a t i o n . T h e r e are j u s t t h r e e e x c e p t i o n s : S a t i e ' s "Trois G y m n o p e d i e s , " written in year 8; Shostakovich's S y m p h o n y # 1 , and Paganini's Caprices, both written in year 9. B e t w e e n year 10 and year 25, there is an enormous increase in productivity.

0 2

10

18

26

34

42

N u m b e r o f y e a r s into c a r e e r Figure 3. F i g u r e 4 s h o w s that c o m p o s e r s m a i n t a i n their p r o d u c t i v i t y t h r o u g h the 40th year of their careers. F i g u r e 5 indicates that a d e c l i n e in productivity b e g i n s at a b o u t t h e 5 0 t h y e a r o f t h e i r c a r e e r s . P r o d u c t i v i t y c a n c o n t i n u e far b e y o n d t h e 5 0 t h y e a r , h o w e v e r . F o r e x a m p l e , A l b e n i z ' s first m a s t e r w o r k w a s w r i t t e n i n t h e 7 2 n d year of his career!

Cognitive Processes in Creative Acts

213

It is r e a s o n a b l e to ask w h e t h e r the important factor in the c o m p o s e r s ' productivity is r e a l l y p r e p a r a t i o n or if p e r h a p s t h e important factor is s i m p l y a g e . It is c o n c e i v a b l e , for e x a m p l e , t h a t c o m p o s e r s h a v e t o b e , s a y , 1 6 o r 2 2 , b e f o r e t h e y c a n write g o o d m u s i c . P e r h a p s it is e x p e r i e n c e in life rather than e x p e r i e n c e in m u s i c that i s c r i t i c a l . T o t e s t t h i s p o s s i b i l i t y , I d i v i d e d t h e c o m p o s e r s i n t o t h r e e g r o u p s . T h e first c o n s i s t e d o f 1 4 c o m p o s e r s w h o h a d b e g u n t h e i r c a r e e r s b e t w e e n t h e a g e s of 3 and 5. T h e second group consisted of 30 composers w h o b e g a n their careers b e t w e e n 6 and 9 years of age. T h e third group consisted of 20 composers w h o b e g a n their careers at 10 or later. I r e a s o n e d t h a t i f a g e w e r e t h e c r i t i c a l factor, t h e n t h o s e w h o s t a r t e d t h e i r careers early w o u l d h a v e to w a i t longer to p r o d u c e g o o d w o r k than those composers w h o s t a r t e d l a t e . I n fact, t h i s w a s n o t t h e c a s e . T h e m e d i a n n u m b e r o f y e a r s t o first n o t a b l e c o m p o s i t i o n w a s 1 6 . 5 for t h e first g r o u p , 2 2 for t h e s e c o n d g r o u p , a n d 2 1 . 5 for t h e t h i r d g r o u p . It appears, t h e n , that w h a t c o m p o s e r s n e e d to write g o o d m u s i c is not maturing but rather m u s i c a l preparation. T h e results m a k e it dramatically c l e a r that no o n e c o m p o s e s o u t s t a n d i n g m u s i c w i t h o u t first h a v i n g a b o u t 1 0 y e a r s o f i n t e n s i v e musical preparation.

274

T h e C o m p l e t e Problem Solver

T h e s e r e s u l t s d o not m e a n t h a t t h e r e i s n o s u c h t h i n g a s g e n i u s . T h e y d o not m e a n that j u s t a n y o n e w i t h 1 0 t o 2 5 years o f e x p e r i e n c e can write great m u s i c . T h e y do m e a n that e v e n a p e r s o n e n d o w e d w i t h the g e n i u s of Mozart or B e e t h o v e n w i l l n e e d 10 years or more of intense preparation to be creative. T h e s e results h a v e the following practical implications: 1 . I f y o u h a v e b e e n w o r k i n g h a r d i n y o u r c h o s e n a r e a for s e v e r a l y e a r s a n d haven't yet r e c e i v e d a N o b e l or e v e n a Pulitzer prize, don't despair. Think in terms of d e c a d e s rather than years. 2 . I f y o u h a v e d e c i d e d t o g o into c r e a t i v e w o r k b e c a u s e y o u are " b a s i c a l l y lazy," y o u h a v e m a d e a ghastly mistake. Creative scientists typically work 70-80 h o u r s a w e e k . Y o u w o u l d d o b e t t e r t o b e a p l u m b e r o r m e c h a n i c . T h e s e are h o n o r a b l e professions and t h e y p a y m u c h better p e r hour than s c i e n c e o r art.

SUMMARY A creative act is o n e w h i c h : 1. is o r i g i n a l , 2. valuable, and 3. suggests that the p e r s o n performing the act has u n u s u a l mental abilities. >

Cognitive Processes in Creative Acts

2/5

A c r e a t i v e a c t is a p r o b l e m - s o l v i n g act, a n d , in p a r t i c u l a r it is t h e s o l u t i o n of an i l l - d e f i n e d p r o b l e m . F o u r c o g n i t i v e p r o c e s s e s e s p e c i a l l y i m p o r t a n t for c r e a t i v i t y are: p r o b l e m finding, i d e a generation, planning, a n d preparation. Problem finding—the d i s c o v e r y o f a n e w p r o b l e m n o t s u g g e s t e d b y a n y o n e e l s e — i s i m p o r t a n t i n i n i t i a t i n g n e w d i r e c t i o n s i n s c i e n c e a n d art. T h r e e p r o c e d u r e s t h a t c a n h e l p u s t o f i n d p r o b l e m s a r e : b u g l i s t i n g , s e a r c h i n g for c o u n t e r e x a m p l e s , a n d s e a r c h i n g for a l t e r n a t i v e i n t e r p r e t a t i o n s . S o m e t i m e s , w h e n w e are t r y i n g t o s o l v e a n i l l - d e f i n e d p r o b l e m , \ v e are b l o c k e d b y d i f f i c u l t y i n generating ideas for s o l u t i o n . B r a i n s t o r m i n g a n d d i s c o v e r i n g a n a l ogies c a n h e l p us out of this difficulty. Planning i s i m p o r t a n t i n c r e a t i v e a c t i v i t i e s a s i t i s i n a n y f o r m o f p r o b l e m s o l v i n g . G o o d w r i t i n g a n d g o o d art d e p e n d o n g o o d p l a n n i n g . E x t e n s i v e preparation i s e s s e n t i a l for acts o f o u t s t a n d i n g c r e a t i v i t y . C o m p o s e r s require about 10 years of preparation before they can produce works of outstanding quality. REFERENCES A d a m s , J . L . Conceptual Blockbusting.

San Francisco:

W.H.

Freeman & Co.,

1974.

Barnlund, D . C . " A C o m p a r a t i v e Study o f Individual, Majority, and G r o u p Judgm e n t . " Journal of Abnormal and Social Psychology, 5 8 , 5 5 - 6 0 , 1 9 5 9 . B o u c h a r d T . J . , Jr. Journal of Applied

"A Comparison of T w o Group Brainstorming Procedures." Psychology, 5 6 , 4 1 8 - 4 2 1 , 1972.

Einstein A., and Infeld, Schuster, Inc., 1938. F l o w e r , L. Problem Solving Jovanovich, Inc., 1980.

L.

The

Evolution

Strategies for

of Physics.

Writing.

New

New

York:

York:

Simon

Haxcourt,

and

Brace,

F l o w e r , L., a n d H a y e s , J.R. " T h e C o g n i t i o n o f D i s c o v e r y : D e f i n i n g a R h e t o r i c a l P r o b l e m . " College Composition and Communication, 2 ( 3 1 ) , 2 1 - 3 2 , 1980. G e t z e l s , J.W., a n d C s i k s z e n t m i h a l y i , M. The Creative Vision: A Longitudinal Study of Problem Finding in Art. N e w Y o r k : J o h n W i l e y & S o n s , I n c . , 1 9 7 6 . G o r d o n W.J.J. Synectics. N e w Y o r k :

Collier,

Grove's Dictionary of Music and Musicians, Press, Inc., 1955. H a y e s , J.R. Cognitive Dorsey Press, 1978.

Psychology:

1961.

Fifth

Thinking

Edition.

and

Creating.

N e w York:

St.

Homewood,

Martin's

IL:

The

Hudgins, B.B. "Effects of Group Experience on Individual Problem Solving." Journal of Educational Psychology, 5 1 , 3 7 - 4 2 , 1960. K o b e r g , D . , a n d B a g n a l l , J . The Universal Traveler, T h i r d E d i t i o n . L o s A l t o s , C A : William Kaufmann, Inc., 1974. KoechelABC,

Fifth Edition.

Wiesbaden:

Breitkopf and Hartel,

1965.

N e w e l l , A., S h a w , J.C., a n d S i m o n , H.A. " T h e P r o c e s s e s o f C r e a t i v e T h i n k i n g . " I n Contemporary Approaches to Creative Thinking, T h i r d E d i t i o n , e d i t e d by H . E . Gruber, G. Terrell, and M. Wertheimer. N e w York: Atherton Press, 1964.

ELEVEN.

H O W SOCIAL C O N D I T I O N S AFFECT CREATIVITY*

Creativity can o c c u r on a variety of l e v e l s , as p o i n t e d out in C h a p t e r 10. W h e n w e speak o f creativity i n this chapter, w e m e a n the v e r y h i g h e s t l e v e l o f c r e a t i v i t y — t h e l e v e l at w h i c h a s y m p h o n y is p r o d u c e d , a scientific discovery made, or a P u l i t z e r - p r i z e - w i n n i n g p l a y is w r i t t e n . W e are i n n o w a y s u g g e s t i n g t h a t p e o p l e w h o a s p i r e t o a n d a c h i e v e t h i s h i g h l e v e l of creativity are better, h a p p i e r , or m o r e p e r s o n a l l y fulfilled than the rest of us. A n d we certainly h a v e no m a g i c formula to insure that p e o p l e can b e c o m e a R e m b r a n d t or an E i n s t e i n or a Mozart. S t i l l , w e d o k n o w s o m e factors t h a t are i m p o r t a n t for a c h i e v i n g h i g h - l e v e l c r e a t i v i t y . I n t h e last c h a p t e r w e d i s c u s s e d t h e i m p o r t a n c e o f k n o w l e d g e , p l a n n i n g , a n d h a r d w o r k ; i n t h i s c h a p t e r w e w i l l d i s c u s s h o w s o c i a l c o n d i t i o n s affect creativity. INTRODUCTION W h e n w e c o m p a r e groups o f p e o p l e i n the U n i t e d States, w e find enormous differences a m o n g t h e m in creative productivity. M e n a c h i e v e more than w o m e n ; Jews, more than Christians; Orientals, more than w h i t e s ; and w h i t e s , more than blacks and native Americans. S o m e c l a i m that t h e s e differences are innate. F o r e x a m p l e , J e n s e n (1971) has argued that racial differences in intellectual p e r f o r m a n c e , particularly on IQ tests, are d u e i n p a r t t o h e r e d i t y . H o w e v e r , B r o d y a n d B r o d y ( 1 9 7 6 ) h a v e c a r e f u l l y r e v i e w e d the research on racial differences in i n t e l l i g e n c e . T h e y found no e v i d e n c e t o s u g g e s t t h a t t h e d i f f e r e n c e s are h e r e d i t a r y . W e b e l i e v e t h a t d i f f e r e n c e s i n h i g h - l e v e l c r e a t i v i t y b e t w e e n g r o u p s are d u e largely to differences in e n v i r o n m e n t . S o m e e n v i r o n m e n t s foster creativity and s o m e d o n o t . A s w e s e e it, s o c i e t y h a s b e e n c o n d u c t i n g s o m e v e r y l a r g e - s c a l e " n a t u r a l " e x p e r i m e n t s o n c r e a t i v i t y . M e n a n d w o m e n i n o u r s o c i e t y are r a i s e d differently, e v e n t h o u g h t h e y m a y b e raised i n the same h o u s e h o l d . Jews and C h r i s t i a n s are t a u g h t d i f f e r e n t a t t i t u d e s t o w a r d l e a r n i n g . B l a c k s a n d w h i t e s are given different e c o n o m i c opportunities. A l l of t h e s e differences can influence crea* T h i s chapter was written w i t h Sandra J. B o n d , of C a r n e g i e - M e l l o n University.

278

T h e C o m p l e t e Problem Solver

tive productivity. By examining the experiences of p e o p l e in these groups, we can gain s o m e insight into h o w this h a p p e n s . MEN A N D W O M E N Historically, m e n have b e e n more creative than w o m e n , as T a b l e 1 shows. This i s t r u e n o t o n l y for t h e s c i e n c e s a n d p o l i t i c s , b u t for t h e arts a s w e l l . A l t h o u g h t h e r e a r e a b o u t e q u a l n u m b e r s o f m e n a n d w o m e n i n o u r p o p u l a t i o n (49 p e r c e n t , m a l e ; 51 percent, female), m e n have clearly overshadowed w o m e n in creative accom-

T a b l e 1 . A c h i e v e m e n t s b y M e n and W o m e n Nobel

Prizes

(1901-1979)

Men

physics chemistry physiology or medicine economics

a i» ***

to

National Academy of Science Awards Carty M e d a l in Science (1932-1978) D r a p e r M e d a l in P h y s i c s (1886-1980) Elliot Medal in Zoology or Paleontology (1917-1979) Kovalenko M e d a l in Medical Science (1952-1979) Thompson Medal in Geology or Paleontology (1921-1976) National AMA

Medal

of Science

Distinguished

Service

(1962-1979) Award

(1938-1979)

Total

to

Pulitzer Prizes fiction drama poetry music biography (music)

Total

"S

of

the

House

of Representatives

Governors Total

Listings

0

15 41 46 12 27

0 1 1 0 0

153

2

42

0 11 (2%)

75

5

53 66 56 34 64

18 5 11 2

131

8

6

55 (10%)

(1901-1980)

Senators

Members

15

515 (90%)

Nobel Prize peace U.S.

2 3 2

(1901-1979)

Prix de Rome

CO

112 90 122

635 (98%)

Nobel Prize literature

Women

65

6

99

1

419

16

48

2

631 (96%) in Who's Who

in America

(1980-1981)

94%

25 (4%) 6%

Percent of population in the United States: m e n , 49 percent; w o m e n , 51 percent. Population figures for 1 9 7 8 : males, 1 0 6 . 5 million; females, 1 1 2 . 0 million.

279

H o w Social Conditions Affect Creativity

p l i s h m e n t . A r e w o m e n less c r e a t i v e than m e n b y nature? W e feel that there i s really n o r e a s o n t o b e l i e v e s o . O u r c u l t u r e treats m e n a n d w o m e n v e r y d i f f e r e n t l y . W o m e n are g i v e n far l e s s o p p o r t u n i t y a n d e n c o u r a g e m e n t t o b e c r e a t i v e t h a n a r e m e n . I n the f o l l o w i n g sections, w e w i l l argue that t h e s e differences i n treatment can m a k e an enormous difference in creative output. S o m e Prerequisites for Creativity: Interest, S e l f - c o n f i d e n c e , and T i m e I n C h a p t e r 1 0 w e f o u n d that e v e n M o z a r t a n d B e e t h o v e n r e q u i r e d m o r e than 10 years of intensive preparation before they c o u l d b e g i n to p r o d u c e their masterworks. In any field, p e o p l e w h o w a n t to be creative must e x p e c t to invest enormous a m o u n t s o f t i m e a n d effort i n t h e i r p r o f e s s i o n a n d e v e n t h e n i t m a y n o t b e e n o u g h . T h e effort i n v o l v e d g o e s far b e y o n d t h e h o u r s a n o r m a l j o b r e q u i r e s . F o r e x a m p l e , U n i v e r s i t y o f C a l i f o r n i a p r o f e s s o r s s p e n d a n average o f 6 0 h o u r s w e e k l y o n t e a c h ing and research (Harris, 1972). S o m e s p e n d m u c h more time. Herbert Simon, 1978 N o b e l L a u r e a t e i n E c o n o m i c s , s p e n t a b o u t 1 0 0 h o u r s a w e e k for y e a r s d o i n g t h e w o r k for w h i c h h e e v e n t u a l l y w o n t h e N o b e l p r i z e ( p e r s o n a l c o m m u n i c a t i o n ) . S i n c e there are o n l y 168 hours in a w e e k , it is o b v i o u s that i n t e n s i v e c r e a t i v e w o r k of this type necessarily takes p r e c e d e n c e over everything e l s e — i n c l u d i n g sleep! P e o p l e w h o c a n i n v e s t t h i s sort o f i n t e n s e effort i n a f i e l d m u s t h a v e : 1 . S u f f i c i e n t i n t e r e s t i n t h e f i e l d t h a t t h e y want t o i n v e s t t h e t i m e ; 2. sufficient self-confidence to b e l i e v e that t h e y are not s i m p l y w a s t i n g their time; and 3. sufficient f r e e d o m from other r e s p o n s i b i l i t i e s so that n e c e s s a r y t i m e is available to them. O u r c u l t u r e m a k e s i t far m o r e l i k e l y t h a t m e n w i l l m e e t t h e s e c r i t e r i a t h a n women. Education

of Females:

The

Early

Years

By tradition, males and females have v e r y different roles in our c u l t u r e — r o l e s they are e x p e c t e d to p l a y from v e r y early c h i l d h o o d on. W h e n we hear of a child b e i n g b o r n , w e a r e a l m o s t a l w a y s t o l d its s e x . S e x u a l i d e n t i t y i s t a k e n v e r y s e r i o u s l y , a n d t h e c h i l d i s r a r e l y a l l o w e d t o f o r g e t it. A s i n f a n t s , b o y s a r e d r e s s e d i n b l u e a n d h a v e b o y s ' n a m e s , w h i l e g i r l s a r e d r e s s e d i n p i n k a n d h a v e g i r l s ' n a m e s . A s fouryear-olds, b o y s are e x p e c t e d to be interested in trucks; and girls, in dolls. In the primary grades, b o y s are e x p e c t e d to s h o w more interest in s c i e n c e than girls do. T h e s e e x p e c t a t i o n s m o l d c h i l d r e n ' s i n t e r e s t s . T o r r a n c e ( 1 9 6 0 ) f o u n d t h a t g i r l s are r e l u c t a n t t o w o r k w i t h s c i e n c e t o y s a n d o f t e n s a y t h a t t h e s e t o y s a r e n ' t s u i t a b l e for t h e m , a n d that b o y s s u g g e s t t w i c e a s m a n y i d e a s a s g i r l s i n e x p e r i m e n t s w i t h science materials. By c h a n g i n g the teaching conditions, though, Torrance was able to r e d u c e this difference significantly. He c o n c l u d e d that social forces w e r e at w o r k e v e n at early a g e s to turn girls a w a y from certain fields. Not only d o e s society g u i d e girls' interests a w a y from s c i e n c e — i t also directs them very p o w e r f u l l y t o w a r d other occupations in w h i c h it is difficult to pursue high-level creative activities. C h i e f among these " w o m e n ' s j o b s " is "wife-andmother." A c c o r d i n g to the cultural ideal, the wife-and-mother is a h o m e m a k e r — she cleans, sews, cooks, and w a s h e s at h o m e , is " f u l f i l l e d " by raising babies, and admires and supports h e r h u s b a n d . It is e a s y to g u e s s that t h e r e is little about this o c c u p a t i o n t h a t fosters t h e c r e a t i v e e f f o r t n e c e s s a r y t o p r o d u c e a m a s t e r w o r k . I f a wife-and-mother does work outside the home, society encourages her to enter women's occupations such as domestic helper, nurse, secretary, and teacher. T h e s e o c c u p a t i o n s h a v e r e l a t i v e l y l o w status a n d l o w p a y , a n d offer l i t t l e o p p o r t u n i t y for creative a c h i e v e m e n t .

T h e C o m p l e t e Problem Solver

220 Self-confidence

E x p e r i e n c e s during the early years not only shape children's interests, they also h e l p to d e t e r m i n e w h e t h e r children will d e v e l o p sufficient confidence in t h e m s e l v e s to c o m p e t e in creative fields. I t i s t r a d i t i o n a l i n o u r c u l t u r e t o b e l i e v e t h a t a s a g r o u p w o m e n suffer f r o m certain intellectual deficits. It is said that w o m e n can't h a n d l e abstract ideas and that t h e y can't think logically. A corollary of these b e l i e f s is that w o m e n can't do w e l l in math- or science-related fields such as engineering, m e d i c i n e , or finance. If a y o u n g g i r l ' s p a r e n t s b e l i e v e t h e s e t h i n g s , i t m a y b e d i f f i c u l t for h e r t o d e v e l o p c o n f i d e n c e that she c o u l d s u c c e e d in any of these fields. In a study of successful and unsuccessful w o m e n , B o n d and H a y e s (1978) found that parents of successful w o m e n don't share this cultural s t e r e o t y p e . O n e of the successful w o m e n , a psychiatrist, says: M y father s h a r e d the v i e w that w o m e n s h o u l d d o w h a t e v e r w o r k t h e y w e r e e q u i p p e d t o d o a n d g a v e m e faith i n m y s e l f , d e t e r m i n a t i o n , a n d m o r a l support. (Rossi and C a l d e r w o o d , 1973) A n o t h e r , a s u c c e s s f u l c h e m i s t , states: It w a s a l w a y s a s s u m e d that I w o u l d p u r s u e a scientific career, since I s h o w e d s o m e talent in that direction. I cannot say that I w a s e n c o u r a g e d to h a v e professional aspirations, as m u c h as that it w a s a s s u m e d that I w o u l d fulfill t h e m . (Kundsin, 1974) A pediatrician recalls: To my father and mother, it w a s important that p e o p l e are w h a t they a r e — a n d that b e i n g w h o and w h a t I w a s w a s good. As a w o m a n , I w a s told, I w o u l d be able to do w h a t e v e r I wanted. (Kundsin, 1974) A noted physician says: D u r i n g a d o l e s c e n c e a n d c o l l e g e , i t w a s t a k e n for g r a n t e d b y e v e r y o n e , i n c l u d i n g myself, that I w o u l d go into m e d i c a l school. No o n e s e e m e d to h a v e a n y d o u b t s a t all t h a t I w o u l d g o , a n d r e t r o s p e c t i v e l y , t h e m o s t e x traordinary factor I c a n n o w r e c o g n i z e is that n e v e r o n c e w h i l e I w a s g r o w i n g up do I r e m e m b e r a n y o n e at any time e v e r s u g g e s t i n g that there was s o m e t h i n g I c o u l d not do simply b e c a u s e I w a s female. E v e r y o n e I k n e w o r e v e r c a m e i n c o n t a c t w i t h s i m p l y t o o k i t for g r a n t e d t h a t w h a t e v e r I w a n t e d to d o , I c o u l d do. (Kundsin, 1974) Role M o d e l s I t i s h e l p f u l for p e o p l e p l a n n i n g t o e n t e r a d i f f i c u l t f i e l d t o h a v e r o l e m o d e l s — t h a t is, p e o p l e t h e y a d m i r e a n d w a n t t o i m i t a t e , w h o c a n s h o w t h e m w h a t i t i s t o l e a d t h e sort o f l i f e t o w h i c h t h e y a s p i r e . I f y o u w a n t t o b e a p h y s i c i a n , y o u w i l l h a v e a r e a l a d v a n t a g e if a p a r e n t , c l o s e r e l a t i v e , or f r i e n d is a p h y s i c i a n . Y o u w i l l b e a b l e t o l e a r n f i r s t h a n d w h a t sorts o f p r o b l e m s d o c t o r s f a c e a n d h o w t h e y d e a l with them. T h e r e is no q u e s t i o n that the p e o p l e we select as role m o d e l s have a powerful e f f e c t o n t h e d i r e c t i o n o u r life c a n t a k e . T w o o f t h e s u c c e s s f u l w o m e n i n o u r s t u d y — t h e first, a p h y s i c i a n ; t h e s e c o n d , a n e l e c t r i c a l e n g i n e e r — t a l k a b o u t t h e i r role m o d e l s : At the age of 1 1 , 1 was taken in by [a doctor's family] to live with them i n N e w York. T h i s i s w h e r e m y a c a d e m i c intellectual life and m y interest i n m e d i c i n e b e g a n , for [the d o c t o r ] l o v e d t o t a l k o v e r h i s c a s e s w i t h m e . I often w e n t on his rounds w i t h him. (Kundsin, 1974)

H o w Social Conditions Affect Creativity

22 7

M y professional career can b e said t o h a v e started w i t h freshman p h y s i c s . M y p e r f o r m a n c e i n that c l a s s i m p r e s s e d m y t e a c h e r e n o u g h s o that s h e , o n h e r o w n , s t a r t e d t o offer m e p r o f e s s i o n a l c o u n s e l i n g . S h e s u g g e s t e d I p u r s u e a career in science. T h i s professor m a i n t a i n e d an interest in me thoughout my undergraduate days at Hunter College and broadened my horizons by m a n y orders of magnitude. (Kundsin, 1974) W o m e n m a y b e b e t t e r r o l e m o d e l s for o t h e r w o m e n b e c a u s e t h e y c a n s h o w t h e m h o w to deal effectively w i t h the special difficulties that w o m e n e n c o u n t e r — sexism, role conflicts, etc. Unfortunately, b e c a u s e there are relatively f e w w o m e n in creative fields n o w , there are relatively f e w w o m e n w h o can serve as role m o d e l s for y o u n g f e m a l e s w h o w a n t t o e n t e r t h e s e f i e l d s . F o r a w o m a n , her m o s t significant role m o d e l m a y be her mother. It should c o m e a s n o s u r p r i s e t h a t " m o t h e r s w h o stay h o m e t o r a i s e a f a m i l y u s u a l l y h a v e daughters w h o w a n t to do just that" (Angrist and Almquist, 1975). Of c o l l e g e w o m e n w h o aspire to careers, only 20 percent have mothers w h o had n e v e r w o r k e d outside the home. F o r c o l l e g e w o m e n w h o do not aspire to a career, the comparable figure is 50 percent (Angrist and Almquist, 1975). Mothers w h o work outside the h o m e not only c o n v e y to their d a u g h t e r s that w o r k i n g is v a l u e d , b u t t h e y also s e r v e as examples of how to successfully c o m b i n e career and motherhood. H e r e is h o w two s u c c e s s f u l w o m e n (the first, a p s y c h i a t r i s t ; t h e s e c o n d , a c o l l e g e a d m i n i s t r a t o r ) p u t it: All of the w o m e n in my family w o r k e d . It w a s e x p e c t e d that I w o u l d . B o t h o f m y parents w e r e professionals and this i n f l u e n c e d m y c h o i c e t o take u p a professional occupation. (Rossi and C a l d e r w o o d , 1973) I don't b e l i e v e it has e v e r occurred to a n y o n e on either side of the family that a w o m a n c o u l d be or o u g h t to be 'a m e r e h o u s e w i f e , ' and hardly a m a n in the family w o u l d be l i k e l y to c h o o s e a w i f e w h o s a w h e r s e l f in that single r o l e . I t i s a f a m i l y i n w h i c h e v e r y w o m a n i s s u p p o s e d t o b e somebody. (Rossi and C a l d e r w o o d , 1973) I n contrast, h e r e are t w o statements from n o n - c a r e e r - o r i e n t e d c o l l e g e w o m e n with mothers w h o had n e v e r w o r k e d outside the home: I w o u l d like to get married and h a v e a family and I don't w a n t to be a career w o m a n . T h a t is a v e r y p o t e n t statement. I think my m o t h e r is a g o o d example of what I w o u l d like to be and I have several girlfriends w h o s e m o t h e r s a r e t h e s a m e . T h e y are n o t c a r e e r w o m e n y e t t h e y a r e v e r y a c t i v e in the c o m m u n i t y . I think y o u can k e e p up w i t h w h a t is g o i n g on in the w o r l d a n d y o u d o n ' t h a v e t o w o r k all y o u r l i f e . (Almquist and Angrist, 1971) and: O u r f a m i l y life h a s b e e n k i n d o f s t r a n g e . M y f a t h e r e v e n h a s a l i t t l e b e l l h e rings and m y m o t h e r c o m e s r u n n i n g and brings h i m coffee a n d h e w i l l call her from a n o t h e r r o o m to c h a n g e the t e l e v i s i o n station. A n d it has b e e n so s u c c e s s f u l . H e i s s o h a p p y a n d s h e i s h a p p y d o i n g it. W h y n o t t r e a t h i m l i k e a king b e c a u s e the m a l e e g o is kind of a sensitive thing to go tampering w i t h . M y l i f e i s p r o b a b l y n o t g o i n g t o m a k e t h a t m u c h d i f f e r e n c e o n soc i e t y , b u t m a y b e w h a t m y h u s b a n d a n d c h i l d r e n d o w i l l . I d o n ' t f e e l that I a m that i m p o r t a n t , b u t i f I h a d t h e t i m e left o v e r , I w o u l d l i k e t o d o v o l u n t e e r w o r k o r s p e n d t i m e o n m y o w n h o b b i e s . I f I h a v e t h e t i m e left over. (Angrist and Almquist, 1975) It is hardly surprising that w o m e n from s u c h families m i g h t find it e s p e c i a l l y difficult t o t h i n k o f t h e m s e l v e s i n a c a r e e r o u t s i d e t h e h o m e . H o w e v e r , a s w e w i l l s e e , t h e r e are p e o p l e w h o h a v e o v e r c o m e e v e n m o r e s e r i o u s d i f f i c u l t i e s .

The C o m p l e t e Problem Solver

222 Getting Credentials

S u p p o s e that a w o m a n s u r v i v e s or e s c a p e s early d i s c o u r a g e m e n t and d e c i d e s to obtain professional training. (Some d o — a b o u t one P h . D . in 10 is earned by a female.) N o w she faces another hurdle. In professional school she is likely to feel i s o l a t e d a n d u n w a n t e d . H e r e are s o m e e x p e r i e n c e s r e p o r t e d b y s u c c e s s f u l w o m e n . [They] w e r e i n t e r v i e w i n g graduate students in the political science departm e n t . T h e professor there said to m e : " W e l l I g u e s s we are g o i n g to h a v e t o take y o u . W e don't like older w o m e n , w e don't think y o u are w o r t h our i n v e s t m e n t , b u t t h i s i s a state s c h o o l , y o u h a v e t h e g r a d e s a n d y o u h a v e t h e letters of r e c o m m e n d a t i o n , so we are g o i n g to h a v e to take y o u . " T h a t w a s d i s c o u r a g i n g , so I w a l k e d across the street to the sociology department and asked if they had a graduate advisor. T h e y said: " N o , b u t D r . D a l t o n w i l l talk t o y o u . " W h e n I w e n t in and p r e s e n t e d my credentials, he said: " Y o u k n o w y o u have to take statistics." I said: " Y e s , I u n d e r s t a n d that." T h e n he asked: " Y o u k n o w what getting a Ph.D. is like?" I said: " Y e s , I d o . " H e s a i d , " W e l l , y o u are w i l l i n g t o g o t h r o u g h t h i s ? " I s a i d , " Y e s . " H e s a i d , " W e l l , w e are d e l i g h t e d t o h a v e y o u . " A n d I said, " W e l l , I am o l d . " A n d h e said, " O h , no, no. (He h a p p e n e d t o b e t w e n t y years older t h a n I ) Y o u are n o t o l d . " I w a s afraid t o m e n t i o n t h a t I w a s a w o m a n . So the reason I am a sociologist today is b e c a u s e there w a s sex discrimination in the political science department at U C L A , b u t not in the sociology department. (Cantor, 1974) A physicist reports: D u r i n g m y first y e a r o f g r a d u a t e s c h o o l , w h a t s e e m e d t o m e l i k e a n i n f i n i t e n u m b e r of professors, t e a c h i n g assistants, and c o l l e a g u e s , n o n e of w h o m w e r e w o m e n , told me that w o m e n can't think analytically and I must, therefore, be h u s b a n d hunting. T h e resultant d i s c o u r a g e m e n t w a s as great as or greater than any I've k n o w n since; h e n c e the solid determination w i t h w h i c h I e m e r g e d w i t h my P h . D . . . . It n e e d e d to be solid, b e c a u s e it s e e m s that a w o m a n in p h y s i c s m u s t be at least t w i c e as d e t e r m i n e d as a man with the same competence, in order to achieve as m u c h as he does. (Kundsin, 1974) A mathematician recounts: As a w o m a n mathematician—a mathematical l o g i c i a n — m y existence apparently refutes a c o m m o n l y h e l d conviction. It is generally b e l i e v e d t h a t a w o m a n is n o t l o g i c a l , b u t acts on t h e b a s i s of i n t u i t i o n a l o n e . . . T h i s p o i n t o f v i e w , s o p r e v a l e n t d u r i n g t h e fifties a n d e a r l y s i x t i e s , w a s reinforced w h e n I w e n t from an all-female e n v i r o n m e n t to an all-male e n v i r o n m e n t w h e n I enrolled in Harvard University Graduate School. I r e c a l l v e r y v i v i d l y m y first d a y i n c l a s s : t h r e e s e a t s i n f r o n t o f m e , t h r e e seats in b a c k of m e , and t w o seats on either side w e r e left vacant. I w a s a c o m p l e t e pariah in that social setting. T h e reason w a s quite simple. T h e m e n w e r e positively u n a b l e t o interact w i t h m e . T h e y w e r e accustomed t o dating girls and talking to t h e m about s w e e t things and e v e n speaking to t h e m a b o u t m o r e intellectual subjects such as politics, history, and sociology. B u t to c o n v e r s e about a purely masculine subject such as physics or mathematics as o n e e q u a l to another w a s s o m e t h i n g t h e y had not previously experienced . . . M y first c o l l o q u i u m a t H a r v a r d U n i v e r s i t y w a s a m e m o r a b l e e v e n t . T h e tea, w h i c h p r e c e d e d the actual lecture, w a s h e l d in the library and w a s

H o w Social C o n d i t i o n s Affect Creativity

223

a r a t h e r f o r m a l affair. A s I e n t e r e d , all e y e s s a n k l o w e r i n t o t h e t e a c u p s i n a great effort not to s e e m to notice m e . N e e d l e s s to say, no o n e talked to me at all. At t h e e n d of the tea the chairman flipped the light s w i t c h up and d o w n a s a s i g n a l for t h e c o l l o q u i u m t o b e g i n . A s h e d i d s o h e t u r n e d t o m e a n d said, w i t h a t w i n k l e in his e y e , " Y o u r p r e s e n c e is n o t e d h e r e . " (Kundsin, 1974)

Your presence is noted here." C u r r e n t l y , c o n d i t i o n s a r e far f r o m i d e a l for t h e c r e a t i v e w o m a n . H o w e v e r , w e don't w a n t t o g i v e the i m p r e s s i o n that there has b e e n n o progress. T h i n g s h a v e b e e n worse. M a r y S o m e r v i l l e b e c a m e a scientist in E n g l a n d in the 1800's, b u t she had t o p u t u p w i t h a l o t t o d o it. O n e day Mary saw a magazine in w h i c h she found an algebra problem. She yearned to k n o w w h a t the X's and Y's m e a n t — s h e had n e v e r heard of a l g e b r a — a n d no o n e could tell her anything more. H e r rescue c a m e by a c c i d e n t : s h e o v e r h e a r d a p a i n t e r t e l l i n g p e o p l e t o s t u d y E u c l i d for p e r spective and mathematics. She k n e w , then, she n e e d e d " E u c l i d " — b u t social m o r e s forbade a y o u n g l a d y of 16 to go to a b o o k s e l l e r . S h e found a m a n w h o c o u l d d o for h e r w h a t s h e c o u l d n o t d o h e r s e l f : b u y E u c l i d for h e r a n d h e l p h e r w i t h p r o b l e m s . B u t s h e ran i n t o o p p o s i t i o n a t h o m e . S h e sat u p late r e a d i n g E u c l i d and the servants c o m p l a i n e d a b o u t h o w m a n y candles she w a s using. H e r candles w e r e taken away, so she was forced to memorize Euclid. She worked problems in her mind every night before g o i n g to s l e e p . H e r father w a r n e d that t h e y w o u l d h a v e to stop Mary's s t u d i e s ' o r w e s h a l l h a v e M a r y i n a strait j a c k e t o n e o f t h e s e d a y s . ' F o r w o m e n in the 1800's in E n g l a n d , too m u c h l e a r n i n g w a s c o n s i d e r e d a v e r y dangerous thing. (Toth and T o t h , 1978)

224

The C o m p l e t e Problem Solver

T h i s i s h a r d l y t h e sort o f w a r m , e n t h u s i a s t i c s u p p o r t w h i c h a t a l e n t e d b u t uncertain p e r s o n m a y n e e d to c o n t i n u e on a difficult path. M a r r i a g e and Creativity Since 93 p e r c e n t of m e n and 95 percent of w o m e n marry, most creative p e o p l e c o m b i n e c a r e e r a n d m a r r i a g e . A s t u d y b y B . W . H a y e s (1980) s u g g e s t s t h a t i t i s m u c h e a s i e r for m e n t o c o m b i n e c a r e e r a n d m a r r i a g e t h a n i t i s for w o m e n t o d o so. S h e c o m p a r e d 1 7 4 m e n w i t h 1 7 4 w o m e n l i s t e d i n Who's Who. T a b l e 2 s h o w s t h e percentage of these m e n and w o m e n w h o had ever married. Table 2. P e r c e n t Married in Who's Who Sample [Data compiled by B. W. Hayes (1980).]

single ever married

men

women

9.7%

37.9%

90.3%

62.1%

Successful m e n marry about as often as m e n in the general population, b u t successful w o m e n are m o r e than s e v e n times as l i k e l y to remain single as other women. T a b l e 3 s h o w s that successful married m e n have about the same n u m b e r of children as married m e n in the general population. T a b l e 3. P e r c e n t of M a r r i a g e s in Who's Who Sample with V a r i o u s N u m b e r s of Children [Data compiled by B. W. Hayes (1980).]

No. Children 0 1 2 3 4 5 or m o r e Average no. children per family

Men 9% 12% 29% 18% 15% 7% 2.48

Women 34% 21% 19% 16% 7% 2% 1.46

Successful married w o m e n , h o w e v e r , have f e w e r c h i l d r e n than other married w o m e n . O f a l l t h e w o m e n i n t h e Who's Who s a m p l e , 5 0 p e r c e n t a r e c h i l d l e s s . A similar pattern is found in w o m e n w h o have Ph.D.'s. O n l y 45 percent of w o m e n w i t h doctorates are m a r r i e d ; 3 8 p e r c e n t h a v e n e v e r b e e n (Astin, 1969). O f the female P h . D . ' s w h o are married, 80 p e r c e n t are married to professional m e n , and 30 percent h a v e no children (Simon, Clark, and G a l w a y , 1967). This is twice the c h i l d l e s s r a t e for t h e g e n e r a l p o p u l a t i o n . W o m e n , then, s e e m to have m o r e difficulty than m e n in c o m b i n i n g marriage w i t h a successful career. W h y s h o u l d this be so? I m a g i n e that y o u h a v e just b e g u n your career as a microbiologist at a prest i g i o u s u n i v e r s i t y . Y o u w o r k f u l l - t i m e , are m a r r i e d , a n d h a v e a y o u n g c h i l d . Y o u r b o s s h a s a g r e e d t o p a y y o u r e x p e n s e s for a w e e k - l o n g c o n f e r e n c e that w i l l a l l o w y o u to k e e p abreast of recent d e v e l o p m e n t s in y o u r field. W h a t do y o u d o ?

H o w Social Conditions Affect Creativity

225

If y o u are a m a n , y o u w o u l d p r o b a b l y say to yourself, " W h a t ' s the p r o b l e m ? I'm going!" B u t if you're a w o m a n , the decision w o u l d not be so simple. Most likely it w o u l d i n v o l v e j u g g l i n g a l o t o f factors b o t h i n r e a l i t y a n d i n y o u r m i n d . W h o w o u l d d o t h e c o o k i n g a n d w a s h i n g ? W o u l d y o u b e a b l e t o f i n d a d e q u a t e c h i l d - c a r e arr a n g e m e n t s for a n e n t i r e w e e k ? W o u l d y o u b e a b l e t o l e a v e y o u r c h i l d for a w h o l e w e e k w i t h o u t w o r r y i n g a b o u t the c h i l d or a b o u t the possibility that y o u aren't a g o o d m o t h e r for d o i n g s o ? M a n y m a r r i e d w o m e n w h o f a c e j u s t t h i s t y p e o f s i t u a t i o n f i n d t h a t a s u p p o r t i v e h u s b a n d is a t r e m e n d o u s a s s e t . In t h i s s i t u a t i o n , a s u p p o r t i v e h u s b a n d m a y b e o n e w h o w i l l t e l l y o u , " O f c o u r s e y o u s h o u l d g o . I'll t a k e c a r e o f t h e b a b y , " or, " G r e a t i d e a . I'll g o w i t h y o u . L e t ' s c a l l y o u r f o l k s (or m i n e ) a n d s e e i f t h e y c a n c o m e a n d stay w i t h t h e b a b y for t h a t w e e k . " A n o n - s u p p o r t i v e h u s b a n d m i g h t s a y s o m e t h i n g l i k e , " W e l l , y o u r w o r k i n g i s o n e t h i n g , b u t b e i n g g o n e for a w h o l e w e e k a n d l e a v i n g t h e b a b y h e r e w i t h m e i s a n o t h e r , " or, " I d o n ' t t h i n k it's a g o o d i d e a for y o u t o g o . " Married female Ph.D.'s, e v e n those with no children typically spend about 50 hours a w e e k doing housework. Married male Ph.D.'s, e v e n those w i t h children, s p e n d l e s s t h a n 1 0 h o u r s a w e e k o n h o u s e h o l d c h o r e s (data c o m p i l e d b y G r i b b e n and presented by Sells and Patterson in Rossi and C a l d e r w o o d [1973], pp. 79-91). P e o p l e w h o w o r k 100 hours a w e e k o n r e s e a r c h a n d 5 0 m o r e o n h o u s e w o r k w o n ' t survive very long, let alone w i n a N o b e l prize! A b o u t eight percent of w o m e n Ph.D.'s allow a time lapse of over 27 years b e t w e e n the B.A. a n d the P h . D . O n l y s e v e n p e r c e n t finish i n the n o r m a l four-year period, w h i l e the majority o f m e n do. O n the average, w o m e n P h . D . ' s obtain their d e g r e e s five y e a r s later than m e n (Astin, 1969). T h i s m e a n s that, o n the a v e r a g e , w o m e n Ph.D.'s h a v e five years less than m e n to devote to research. O n l y 81 percent of female Ph.D.'s work full-time, as c o m p a r e d with 98 percent of male Ph.D.'s, and 2 1 p e r c e n t o f w o m e n P h . D . ' s i n t e r r u p t t h e i r l i f e ' s w o r k for a n a v e r a g e o f 1 4 m o n t h s (Astin, 1969). T h e s e differences b e t w e e n m a l e and female P h . D . ' s are not bizarre or unexp l a i n a b l e . T h e y a r e a c c o u n t e d for b y t h e fact t h a t w o m e n t a k e t i m e off t o h a v e a n d c a r e for b a b i e s . O f c o u r s e , m e n s h a r e i n h a v i n g c h i l d r e n , b u t i t r a r e l y i n t e r r u p t s their career. U n t i l n o w , w e h a v e b e e n d i s c u s s i n g factors w i t h i n m a r r i a g e w h i c h c a n i m p e d e the careers of creative w o m e n . S o m e t i m e s , though, marriage can stop a career dead! H e r e are s o m e e x p e r i e n c e s r e p o r t e d b y u n s u c c e s s f u l w o m e n i n t h e s t u d y b y B o n d and Hayes (1978): My h u s b a n d w a s in favor of my returning to work, b u t n o w he is a m b i v a l e n t . H e f e e l s i t i s t o o t i r i n g for m e a n d t h a t i f w e n e e d m o r e m o n e y , he should get a second job. ( G i n z b e r g a n d Y o h a l e m , 1966) I could not complete my doctorate because w h e n my husband comp l e t e d his P h . D . , he found a j o b in another city and I w a s o b l i g e d to take over the care of his four-year-old child by a previous marriage. ( G i n z b e r g a n d Y o h a l e m , 1966) A l l o f m y c r e d i t s for t h e P h . D . w e r e c o m p l e t e d a n d q u a l i f y i n g e x a m s passed by 1952. All that had to be d o n e w a s the thesis. H o w e v e r , my h u s b a n d h a d o b t a i n e d h i s P h . D . a n d w a s l e a v i n g C o l u m b i a for postgraduate w o r k at California Institute of T e c h n o l o g y . Of course, I w e n t w i t h him. ( G i n z b e r g a n d Y o h a l e m , 1966) I n h e r b o o k , Silences, O l s e n ( 1 9 7 8 ) m a k e s a v e r y p o w e r f u l c a s e t h a t m a n y creative w o m e n writers have b e e n silenced by the d e m a n d s placed on them by

226

The C o m p l e t e Problem Solver

their roles as w i v e s and mothers. H e r e she describes her o w n circumstances as a writer and mother: C i r c u m s t a n c e s for s u s t a i n e d c r e a t i o n are a l m o s t i m p o s s i b l e . N o t b e c a u s e t h e c a p a c i t i e s t o c r e a t e n o l o n g e r e x i s t , o r t h e n e e d ( t h o u g h for a w h i l e a s in any fullness of life the n e e d m a y be o b s c u r e d ) , b u t . . . the n e e d cannot b e first. I t c a n h a v e a t b e s t o n l y p a r t self, part t i m e . . . M o t h e r h o o d m e a n s b e i n g instantly interruptible, responsive, responsible. C h i l d r e n n e e d one now ( a n d r e m e m b e r , i n o u r s o c i e t y , t h e f a m i l y m u s t o f t e n try t o b e t h e c e n t e r for l o v e a n d h e a l t h t h e o u t s i d e w o r l d i s n o t ) . T h e v e r y f a c t t h a t t h e s e are n e e d s o f l o v e , n o t d u t y , t h a t o n e f e e l s t h e m a s o n e ' s self; that there i s no one else to be responsible for these needs, g i v e s t h e m p r i m a c y . It is distraction, not meditation, that b e c o m e s habitual; interruption, not continuity; s p a s m o d i c , not constant, toil. W o r k interrupted, d e f e r r e d , postponed makes b l o c k a g e — a t best, lesser accomplishment. U n u s e d capacities atrophy, c e a s e to b e . For the professional w o m a n w h o does marry, marrying s o m e o n e w i t h similar g o a l s , i n t e r e s t s , a n d o c c u p a t i o n a l status i s e x t r e m e l y i m p o r t a n t . D o i n g s o i n c r e a s e s the likelihood that h e r partner w i l l understand, accept, and e n c o u r a g e w h a t she is t r y i n g t o d o . M e n w h o a r e a l s o e n g a g e d i n c r e a t i v e p r o f e s s i o n a l w o r k are i n a g o o d p o s i t i o n t o r e a l i z e its i n h e r e n t v a l u e a n d t h a t i t r e q u i r e s lots o f t i m e a n d h a r d w o r k . E v e n w i t h a s u p p o r t i v e h u s b a n d w h o s h a r e s c h o r e s a n d c h i l d c a r e , i t still i s n ' t e a s y for w o m e n i n o u r s o c i e t y t o c o m b i n e a c a r e e r , m a r r i a g e , a n d a f a m i l y . ( M e n , w e m i g h t n o t e , h a v e b e e n d o i n g s o for y e a r s . ) C h i l d c a r e c a n b e v e r y e x p e n s i v e . S o m e e m p l o y e r s still h a v e n e p o t i s m r u l e s w h i c h p r e v e n t a h u s b a n d a n d w i f e f r o m w o r k i n g a t t h e s a m e i n s t i t u t i o n . W e l l - m e a n i n g p e o p l e m a y still t e l l c a r e e r w o m e n that t h e y a r e r u i n i n g t h e i r c h i l d r e n ' s l i v e s . I t i s n ' t e a s y — b u t i t c a n b e d o n e . O f t h e 17 professional w o m e n in our study, 16 stated that t h e y w e r e glad t h e y chose to c o m b i n e a career w i t h marriage and a family, and that t h e y w e r e h a p p y w i t h their lives. Of the 14 nonprofessional w o m e n in our study, nine reported dissatisfaction with their lives.

Summary Historically, w o m e n h a v e b e e n m u c h less creative than m e n . This does not m e a n t h a t w o m e n a r e l e s s c r e a t i v e t h a n m e n b y n a t u r e . O u r c u l t u r e p r o v i d e s far more opportunity and e n c o u r a g e m e n t to m e n to be creative than to w o m e n . 1. O u r c u l t u r e d i s c o u r a g e s w o m e n from taking an interest in s c i e n c e - r e l a t e d fields and encourages them to be interested in homemaking. 2. O u r culture tends to u n d e r m i n e the confidence of w o m e n in their ability to c o m p e t e in certain creative fields. 3. T h e r e are relatively f e w female role m o d e l s in creative fields. 4. M a l e s m a y resent and discriminate against females in professional education and at work. 5 . I n o u r c u l t u r e i t i s m u c h m o r e d i f f i c u l t for w o m e n t o m i x m a r r i a g e a n d c a r e e r t h a n i t i s for m e n . T h i s i s d u e t o t h e a s s u m p t i o n s t h a t d o m e s t i c d u t i e s s u c h a s c o o k i n g a n d c h i l d c a r e are m o r e t h e r e s p o n s i b i l i t y o f w o m e n t h a n m e n , and that the h u s b a n d ' s career is more important than the wife's and, therefore, any conflict of interest m u s t be r e s o l v e d in favor of the h u s b a n d ' s career. T h e s e f a c t o r s , t a k e n t o g e t h e r , m a y w e l l a c c o u n t for t h e o b s e r v e d d i f f e r e n c e s i n creativity b e t w e e n m e n and w o m e n .

H o w Social Conditions Affect Creativity

227

THE CREATIVE A C C O M P L I S H M E N T OF JEWS In the w e s t e r n w o r l d , J e w s constitute a very small proportion of the general p o p u l a t i o n (less than o n e p e r c e n t in E u r o p e and less than three p e r c e n t in the U n i t e d States). T h e y constitute a m u c h larger proportion of the creative population. T a b l e 4 s h o w s this clearly. W h y a r e J e w s s o c r e a t i v e ? W e b e l i e v e t h a t t h e p l a c e t o l o o k for t h e a n s w e r i s in the J e w i s h family. J e w s h a v e a l o w e r d i v o r c e rate, h i g h e r i n c o m e , and f e w e r c h i l d r e n than other p e o p l e ( G o l d s t e i n and G o l d s c h e i d e r , 1968). M o s t J e w s l i v e in urban areas. All of t h e s e f a c t o r s — s t a b l e h o m e s w i t h both parents present, h i g h i n c o m e , small family s i z e , a n d l i v i n g i n a n u r b a n e n v i r o n m e n t are a s s o c i a t e d w i t h h i g h I Q ( L i p s e t a n d L a d d , 1 9 7 4 ; P a t a i a n d W i n g , 1 9 7 5 ) . H o w e v e r , t h e r e are t w o factors w h i c h w e b e l i e v e are m o r e i m p o r t a n t t h a n all o f t h e s e . T h e first i s J e w i s h r e s p e c t for s c h o l a r ship, and the second is the importance of the child in the J e w i s h family.

Table 4. A c c o m p l i s h m e n t s of Jews [Based on data from Arieti, 1976; Ladd and Lipset, 1975; Lipset and Ladd, 1971, 1974; Patai and Wing, 1975; and Sherman, 1965.]

In the U.S., where less than 3% of the population are Jews: 1. Jews have w o n 2 7 % of Nobel Prizes awarded to Americans. 2. J e w i s h violin virtuosos occur at 12 to 25 times expectancy. 3 . 1 6 % o f J e w s are p r o f e s s i o n a l s , c o m p a r e d t o 1 0 % o f t h e e n t i r e p o p u l a t i o n . 4 . 1 0 % o f all c o l l e g e f a c u l t y m e m b e r s are J e w s ; 1 9 % o f f a c u l t y m e m b e r s a t e l i t e c o l l e g e s are J e w s . 5 . 7 0 % o f J e w i s h f a c u l t y m e m b e r s are a t r e s e a r c h - o r i e n t e d u n i v e r s i t i e s . 6. 3 2 % of J e w s are at the highest quality universities, c o m p a r e d to 9% of nonJ e w i s h faculty. 7. 3 3 % of J e w i s h faculty m e m b e r s have over 10 publications, c o m p a r e d to 1 1 % of C a t h o l i c a n d 1 5 % o f Protestant faculty m e m b e r s . 8 . 2 8 % o f J e w i s h f a c u l t y m e m b e r s are full p r o f e s s o r s a n d a t a y o u n g e r a g e (38 v e r s u s 40) t h a n n o n - J e w i s h f a c u l t y . 9 . 2 5 % o f l a w p r o f e s s o r s are J e w i s h . 10. 2 2 % of m e d i c a l faculty are J e w i s h . 1 1 . 2 1 % o f b i o c h e m i s t r y f a c u l t y are J e w i s h . In Europe, where less than 1% of the population are Jews: 1 . S a r t o n ' s list o f s c h o l a r s i n E u r o p e t o 1 4 0 0 A . D . l i s t e d 1 0 . 6 % J e w s — t h r e e t i m e s their proportion in E u r o p e at the time. 2 . J e w s h a v e w o n 1 6 % o f all N o b e l P r i z e s ( 1 9 0 1 - 1 9 7 2 ) a n d h a v e w o n 6 0 % o f t h e Prizes in Economics, 2 4 % in Physiology and Medicine, and 2 0 % in Physics. T h e ratio o f J e w i s h L a u r e a t e s i s 2 8 t i m e s g r e a t e r t h a n t h e r e s t o f t h e w o r l d population. 3. In pre-Nazi Germany, 2 5 % of mathematicians, medical researchers, physicists w e r e J e w i s h — 3 0 times their population proportion.

and

4. In Italy in the 1930's, 5 0 % of m a t h e m a t i c i a n s w e r e J e w i s h . 5. In 1937, 9% of Soviet scholars w e r e Jewish. 6 . I n 1 9 4 7 , 1 6 % o f S t a l i n s c i e n c e a w a r d s a n d 1 4 % o f S t a l i n art a n d l i t e r a t u r e awards went to Jews. 7 . 3 . 5 % o f B r i t i s h f a c u l t y are J e w i s h , a l t h o u g h J e w s c o n s t i t u t e l e s s t h a n 1 % o f t h e British population (1965). 8. 7% of m e m b e r s of Britain's scientific honor society, the Royal A c a d e m y of S c i e n c e , are J e w i s h ( 1 9 7 1 ) .

H o w Social Conditions Affect Creativity

229

m o t h e r stand b y a n d w a t c h her c h i l d , w i t h s u c h a " g o o d h e a d , " g o straight to h e l l ? (Sarason, 1973) T h e c h a n n e l l i n g of the child's energies toward scholarly pursuits and away f r o m d i s t r a c t i n g a c t i v i t i e s s e e m s t o w o r k . T h e f o l l o w i n g f i g u r e s , b a s e d o n d a t a from L a d d and L i p s e t (1975) and S h e r m a n (1965), s h o w that a larger p e r c e n t a g e of Jews in the U.S. go to c o l l e g e than non-Jews, and w h e n they get there they perform better than non-Jews. 1 . 6 2 o u t o f e v e r y 1 0 0 c o l l e g e - a g e J e w s are i n c o l l e g e , c o m p a r e d t o 2 7 o u t o f e v e r y 100 n o n - J e w s . 2. J e w i s h c o l l e g e students are proportionately in better schools than non-Jewish students. 3. Jews have better grade-point averages than non-Jews. 4 . J e w s are i n P h i B e t a K a p p a a t t w i c e t h e i r u n d e r g r a d u a t e p r o p o r t i o n . In our society, scholarship is an important route to success. A d v a n c e d education often l e a d s to a professional career, a g o o d i n c o m e , and an opportunity to do creative work. Traditionally, scholarship has b e e n highly r e s p e c t e d by J e w s , and the J e w i s h family has b e e n v e r y effective in t e a c h i n g this v a l u e to the J e w i s h child. W e b e l i e v e t h a t i t i s b y t r a n s m i t t i n g r e s p e c t for s c h o l a r s h i p t o t h e y o u n g t h a t J e w i s h culture has s u c c e e d e d in p r o d u c i n g such a large n u m b e r of creative p e o p l e . THE A S I A N S Asians are another h i g h l y creative minority w i t h i n A m e r i c a n culture. W h i l e t h e y c o n s t i t u t e l e s s t h a n o n e p e r c e n t o f o u r p o p u l a t i o n , t h e y are o n e p e r c e n t o f t h e u n d e r g r a d u a t e p o p u l a t i o n and t w o p e r c e n t of the graduate p o p u l a t i o n (Kitano, 1 9 7 6 ) . F o u r A s i a n A m e r i c a n s h a v e w o n N o b e l p r i z e s (all i n P h y s i c s ) . A r e m a r k a b l e number of Asian Americans have b e c o m e outstanding musicians. M a n y Asian A m e r i c a n s are p r o f e s s i o n a l s — 2 1 p e r c e n t of Japanese m a l e s and 30 p e r c e n t of C h i n e s e males, as compared to 15 percent of non-Asian A m e r i c a n males (Almquist, 1979). T h e r e are m a n y s i m i l a r i t i e s b e t w e e n A s i a n a n d J e w i s h s u b c u l t u r e s i n o u r society. B o t h h a v e an ancient heritage w h i c h e s t e e m s scholarship and a stable family e n v i r o n m e n t w h i c h encourages children to pursue scholarly careers. F o r t h e J e w s , s c h o l a r s h i p h a s s t r o n g r e l i g i o u s o v e r t o n e s . F o r t h e A s i a n s , i t has strong moral c o n n e c t i o n s . F o r m a n y c e n t u r i e s [the C h i n e s e ] r e v e r e d w r i t t e n c h a r a c t e r s , w h i c h t h e y b e l i e v e t o h a v e b e e n c r e a t e d b y p a s t s a g e s . T h e r e w e r e m a n y ' s o c i e t i e s for saving papers with written characters.' T h e s e societies e m p l o y e d collectors w h o r o a m e d a r o u n d t o w n , w i t h forks i n h a n d a n d b a s k e t s o n t h e i r backs, gathering s u c h scattered p i e c e s . T h e bits w e r e then b u r n e d at the l o c a l C o n f u c i a n t e m p l e . I t w a s b e l i e v e d that a p e r s o n w h o u s e d i n s c r i b e d p a p e r s for t o i l e t p u r p o s e s w o u l d b e s t r u c k d e a d b y l i g h t n i n g . A n d o n e w h o a c c i d e n t a l l y s t e p p e d on a b o o k m u s t p i c k it up and p l a c e it on his h e a d m o m e n t a r i l y for p r o p i t i a t i o n . ( H s u , 1970) T h e ultimate a c h i e v e m e n t w a s wisdom, and the Confucian w a y to wisdom was by studying. A s i a n students w e r e e x p o s e d to stories of scholars w h o let nothing interfere w i t h their desire to learn: " t h e m a n w h o put a r o p e around his n e c k and tied it to the c e i l i n g to k e e p from falling a s l e e p or the m a n w h o u s e d a cageful of fireflies to study at n i g h t or the m a n w h o p o u r e d ice w a t e r on his b o d y in w i n t e r to stay a w a k e t o s t u d y " ( D o r e , 1 9 6 5 ; H o r i n o u c h i , 1 9 6 7 ) . W h i l e W e s t e r n e r s m a y f i n d these stories e x c e s s i v e or g r u e s o m e , t h e y illustrate the d e g r e e to w h i c h scholarship was esteemed by Asians.

T h e C o m p l e t e Problem Solver

230 T h e Family

A s i a n s h a v e t h e l o w e s t d i v o r c e rate (1.6 p e r c e n t ) o f a n y g r o u p i n the U n i t e d States. T h e y h a v e a h i g h m a r r i a g e rate, h i g h h o m e o w n e r s h i p , h i g h i n c o m e , a n d h i g h o c c u p a t i o n a l s t a t u s . U n e m p l o y m e n t i s rare a n d f a m i l y s i z e i s s m a l l . T h i s pattern r e s e m b l e s that f o u n d a m o n g J e w s . R e s p e c t for l e a r n i n g i s i n t e r w o v e n w i t h r e s p e c t for t h e f a m i l y a n d for t h e C o n f u c i a n v a l u e s o f d i l i g e n c e a n d h a r d w o r k . C o n s i d e r t h e first a d m o n i t i o n i n a Japanese school: To be b o r n h u m a n and not be able to write is to be less than human. Illiteracy is a form of b l i n d n e s s . It brings shame on your teacher, s h a m e on y o u r parents a n d s h a m e on yourself. T h e heart of a c h i l d of three stays w i t h h i m till he is a h u n d r e d as the p r o v e r b says. D e t e r m i n e to s u c c e e d , study w i t h all y o u r m i g h t , n e v e r f o r g e t t i n g t h e s h a m e o f f a i l u r e . (Dore, 1965) T h e p r e c e p t w a s also taught at h o m e : " B y learning w e l l , y o u w i l l honor your family n a m e " (Horinouchi, 1967). O n e y o u n g man recalled: W h e n I w a s s e n t t o s c h o o l I w a s e x p e c t e d t o try m y b e s t a n d t o g e t g o o d g r a d e s — t h e r e w a s no q u e s t i o n a b o u t this in my or my parents' m i n d . I g u e s s it's j u s t l i k e m y m o t h e r a n d f a t h e r — i f M o m i r o n e d a shirt, s h e h a d t o d o b e s t ; i f m y D a d d u g a d i t c h h e h a d t o d o i t j u s t r i g h t . S o d i d all o f t h e other N i s e i kids. I g u e s s that's w h y so m a n y of us w e r e on the H o n o r Roll. (Kitano, 1976) L i k e the Jews, Asians encourage some activities and discourage others. T h e A s i a n c o m m u n i t y r e i n f o r c e s t h e f a m i l y i n e n f o r c i n g its v a l u e s . O n e m a n r e c a l l s h o w the community dealt with an instance of delinquency: I k n e w these t w o brothers w h o w e r e pretty wild. T h e y w o u l d get d r u n k . . . w e r e always fighting, always in trouble and w e r e uncontrollable. Finally, their father c a m e to talk to my father and other J a p a n e s e families in the n e i g h b o r h o o d . . . all a g r e e d t h a t t h e s e b o y s w o u l d h u r t t h e r e p u t a t i o n o f t h e o t h e r J a p a n e s e a n d p r o v i d e p o o r m o d e l s for t h e y o u n g e r b o y s . . . s o e v e n though the brothers w e r e already y o u n g adults and out of high school, t h e y w e r e s e n t b a c k t o J a p a n i n 1 9 3 7 . A s far a s I k n o w , t h e y n e v e r c a m e b a c k to the U n i t e d States. (Kitano, 1976) T h e A s i a n s h a v e t h e l o w e s t c r i m e rate i n t h e U n i t e d S t a t e s . I t i s p r o b a b l y n o a c c i d e n t t h a t t h e y h a v e t h e l o w e s t s c h o o l d r o p - o u t rate o f a n y g r o u p i n t h e c o u n t r y . D r o p p i n g out of high school "is considered a disgrace in an Asian c o m m u n i t y " (Horinouchi, 1967). Conclusion Asians, like J e w s , form a remarkably creative subculture in A m e r i c a . L i k e the Jews, Asians have an ancient heritage w h i c h places a high value on scholarship a n d , l i k e t h e J e w s , t h e y p r o v i d e a s t a b l e f a m i l y e n v i r o n m e n t i n w h i c h c h i l d r e n are e n c o u r a g e d to w o r k hard and to pursue scholarly careers. Further, in both groups d e v i a t i o n s from this i d e a l pattern, e.g., d r o p p i n g out of s c h o o l , are a c t i v e l y discouraged. BLACKS T h e r e are 1. lack of 2. lack of 3. lack of

s e v e r a l factors w h i c h c a n s e r i o u s l y t h w a r t c r e a t i v e effort: education opportunity support

H o w Social Conditions Affect Creativity

23 7

Unfortunately, blacks h a v e b e e n the victim of all three. P h y s i c a l l y r e m o v e d against their w i l l from their native land, t h e y w e r e m a d e to serve a society w h i c h w o u l d n e i t h e r t e a c h t h e m h o w t o b e c o m e its m e m b e r s , n o r a l l o w t h e m t o r e t a i n their heritage. As a c o n s e q u e n c e , blacks have b e e n doubly d e n i e d a culture and cultural values. I n a d e q u a t e Education W h e n t h e s l a v e s w e r e f r e e d , t h e y w e r e b a r r e d b y l a w f r o m m a n y j o b s a n d from most schools. E a g e r to learn, t h e y set up their o w n schools, b u t these w e r e genera l l y staffed b y b l a c k s w i t h l i t t l e m o r e e d u c a t i o n t h a n t h e i r p u p i l s . S e g r e g a t e d s c h o o l s w e r e t h e n o r m u n t i l t h e S u p r e m e C o u r t r u l e d i n 1 9 5 4 that s e p a r a t e i s n o t e q u a l . I t s t i l l i s n ' t . M o s t b l a c k s still a t t e n d s e g r e g a t e d s c h o o l s . I n Alabama, Louisiana, and Mississippi, less than one p e r c e n t of black students attend s c h o o l w i t h w h i t e s ( P i n k n e y , 1 9 6 9 ) . I n t h e N o r t h i n 1 9 6 6 , 6 5 p e r c e n t o f b l a c k first g r a d e r s a t t e n d e d s c h o o l s w h i c h w e r e 9 0 t o 100 p e r c e n t b l a c k ( U . S . D e p a r t m e n t o f Health, E d u c a t i o n , a n d Welfare, 1966). A b o u t 60 p e r c e n t of b l a c k c o l l e g e students a t t e n d b l a c k c o l l e g e s , o f w h i c h t h e r e are 1 2 3 . H o w e v e r , o n l y 6 3 p e r c e n t o f t h e m are a c c r e d i t e d ( U . S . D e p a r t m e n t o f C o m m e r c e , 1966). I n five s o u t h e r n states i n 1954, t h e r e w e r e 1.8 b o o k s p e r p u p i l i n b l a c k s c h o o l l i b r a r i e s , b u t 4.8 p e r p u p i l i n w h i t e school libraries (Ashmore, 1954). In black high schools, in 1966, 20 percent w e r e w i t h o u t p h y s i c s e q u i p m e n t c o m p a r e d t o six p e r c e n t o f w h i t e s c h o o l s ( U . S . D e p a r t m e n t o f H e a l t h , E d u c a t i o n , a n d W e l f a r e , 1966). A l t h o u g h o n l y a b o u t 25 p e r c e n t of b l a c k s l i v e in rural areas today, in 1900, 90 p e r c e n t did (U.S. B u r e a u of the C e n s u s , 1966). M a n y adult blacks a t t e n d e d a school l i k e t h e o n e d e s c r i b e d b e l o w in an e x c e r p t f r o m Growing Up in the Black Belt. A n d it is not as t h o u g h urban b l a c k s h a v e it any better. F u l l y two-thirds are m e m b e r s of the l o w e r class a n d attend p r e d o m i n a t e l y black schools ( P i n k n e y , 1969). It is in a d i l a p i d a t e d b u i l d i n g , o n c e w h i t e w a s h e d , standing in a rocky field u n f i t for c u l t i v a t i o n . D u s t - c o v e r e d w e e d s s p r e a d a c a r p e t all a r o u n d , e x c e p t for a n u n e v e n , b a r e a r e a o n o n e s i d e w h i c h l o o k s l i k e a b a l l f i e l d . B e h i n d the school is a small b u i l d i n g w i t h a broken, sagging door. As we approach, a nervous m i d d l e - a g e d w o m a n c o m e s to the door of the school. She greets us in a discouraged voice marred by a s p e e c h i m p e d i m e n t . Escorted inside, w e o b s e r v e that the b r o k e n b e n c h e s are c r o w d e d t o three t i m e s their normal c a p a c i t y . O n l y a f e w b a t t e r e d b o o k s are i n s i g h t , a n d w e l o o k i n v a i n for m a p s o r c h a r t s . W e l e a r n t h a t f o u r g r a d e s are a s s e m b l e d h e r e . (Johnson, 1941) In the face of all this, it is a tribute to the p e r s e v e r a n c e of b l a c k p e o p l e that they have reduced their illiteracy 70 percent in about 70 years. Unfortunately, to be creative in a t e c h n o l o g i c a l l y a d v a n c e d society like ours, that is not g o o d e n o u g h . To c o m p e t e for t h e c r e a t i v e o c c u p a t i o n s , b l a c k s n e e d e d u c a t i o n a n d t r a i n i n g e q u a l t o w h i t e s . S o far, s o c i e t y h a s n o t p r o v i d e d that. Lack of O p p o r t u n i t y M a l c o l m X r e c o u n t e d i n h i s a u t o b i o g r a p h y a n all t o o t y p i c a l e x a m p l e : I k n o w t h a t h e [the t e a c h e r ] p r o b a b l y m e a n t w e l l i n w h a t h e h a p p e n e d t o a d v i s e me that day. I d o u b t that he m e a n t any harm. It w a s j u s t in his nature as an A m e r i c a n w h i t e m a n . I w a s o n e of his top students, o n e of the school's t o p s t u d e n t s — b u t a l l h e c o u l d s e e for m e w a s t h e k i n d o f f u t u r e ' i n y o u r p l a c e ' t h a t a l l w h i t e p e o p l e s e e for b l a c k p e o p l e . He told m e , " M a l c o l m , y o u ought to be thinking about a career. Have you been giving it thought?"

232

The C o m p l e t e Problem Solver T h e t r u t h is, I h a d n ' t . I n e v e r h a v e f i g u r e d o u t w h y I t o l d h i m , " W e l l , y e s , sir, I ' v e b e e n t h i n k i n g I ' d l i k e t o b e a l a w y e r . " L a n s i n g c e r t a i n l y h a d no N e g r o l a w y e r s — o r doctors e i t h e r — i n those days, to hold up an image I m i g h t h a v e a s p i r e d to. A l l I r e a l l y k n e w for c e r t a i n w a s t h a t a l a w y e r didn't w a s h dishes, as I was doing. Mr. O s t r o w s k i l o o k e d surprised, I r e m e m b e r , a n d l e a n e d b a c k in his chair a n d c l a s p e d his hands b e h i n d his h e a d . H e k i n d o f half-smiled and s a i d , " M a l c o l m , o n e o f l i f e ' s first n e e d s i s for u s t o b e r e a l i s t i c . D o n ' t m i s u n d e r s t a n d m e , n o w . W e a l l h e r e l i k e y o u , y o u k n o w that. B u t y o u ' v e got to be realistic about b e i n g a nigger. A l a w y e r — t h a t ' s no realistic goal for a n i g g e r . Y o u n e e d t o t h i n k a b o u t s o m e t h i n g y o u can b e . Y o u ' r e g o o d with your h a n d s — m a k i n g things. E v e r y b o d y admires your carpentry shop work. W h y don't y o u plan on carpentry? People like y o u as a p e r s o n — y o u ' d g e t all k i n d s o f w o r k . " T h e m o r e I t h o u g h t afterwards about w h a t he said, the m o r e u n e a s y it made me. It just kept treading around in my mind. What m a d e it really begin to disturb me was Mr. Ostrowski's advice to others in my c l a s s — a l l of t h e m white. Most of t h e m had told him they w e r e p l a n n i n g to b e c o m e farmers, like their p a r e n t s — t o one day take over their f a m i l y f a r m s . B u t t h o s e w h o w a n t e d t o s t r i k e o u t o n t h e i r o w n , t o try something n e w , he had encouraged. S o m e , mostly girls, w a n t e d to be teachers. A few w a n t e d other professions, such as one boy w h o wanted to b e c o m e a c o u n t y agent; another, a veterinarian; a n d o n e girl w a n t e d to be a n u r s e . T h e y all r e p o r t e d t h a t M r . O s t r o w s k i h a d e n c o u r a g e d w h a t e v e r they had wanted. Yet nearly none of them had earned marks equal to mine. It w a s a surprising t h i n g that I h a d n e v e r t h o u g h t of it that w a y before, b u t I r e a l i z e d t h a t w h a t e v e r I w a s n ' t , I was s m a r t e r t h a n n e a r l y all of t h o s e w h i t e k i d s . B u t a p p a r e n t l y I w a s still n o t i n t e l l i g e n t e n o u g h , i n t h e i r e y e s , to b e c o m e w h a t e v e r I w a n t e d to be. It w a s t h e n that I b e g a n to c h a n g e — i n s i d e . ( M a l c o l m X , 1964) No matter h o w bright or h a r d w o r k i n g an i n d i v i d u a l is, if opportunity to be

creative is d e n i e d , there can be no creativity. Lack of Support Blacks h a v e h i g h e r u n e m p l o y m e n t , l o w e r i n c o m e , less education, and more c h i l d r e n than w h i t e p e o p l e . O n l y one-third o w n their o w n h o m e s . T h e d i v o r c e rate a m o n g b l a c k s i s 2 3 p e r c e n t , t h e i l l e g i t i m a c y rate i s 2 5 p e r c e n t , a n d 2 5 p e r c e n t o f b l a c k h o u s e h o l d s are h e a d e d b y females w i t h n o m a l e p r e s e n t (Pinkney, 1969; U.S. D e p a r t m e n t o f L a b o r , 1965). H o w d o t h e s e grim statistics affect the b l a c k c h i l d ? F i r s t , s u c h a p a t t e r n a l l o w s l i t t l e c h a n c e for e s c a p e . C h i l d r e n w h o g r o w u p i n an i m p o v e r i s h e d e n v i r o n m e n t h a v e little e x p o s u r e to the range of options society c a n offer. R o l e m o d e l s a r e l i k e l y t o b e t h e t o u g h e s t k i d s o n t h e b l o c k , r a t h e r t h a n scholars. C h i l d r e n in an i m p o v e r i s h e d e n v i r o n m e n t are p r o b a b l y not s u r r o u n d e d b y b o o k s o r s t i m u l a t i n g toys. T h e y m a y n e i t h e r b e e n c o u r a g e d t o study nor rew a r d e d for d o i n g s o . C h a n c e s a r e t h a t t h e p e o p l e t h e y i n t e r a c t w i t h o n a d a i l y b a s i s — t h e i r friends a n d n e i g h b o r s — a r e i n the same p r e d i c a m e n t t h e y are. T h e y p r o b a b l y a t t e n d a n i l l - e q u i p p e d s c h o o l w h i c h offers l i t t l e s t i m u l a t i o n . A s a r e s u l t o f b e i n g raised in a b r o k e n h o m e , they may not learn the values and behaviors necessary to a c h i e v e s u c c e s s in our s o c i e t y ( C o m e r , 1967). F u r t h e r , the a b s e n c e of a s t r o n g m a l e r o l e m o d e l m a y m a k e i t d i f f i c u l t for m a l e c h i l d r e n t o d e v e l o p g o o d w o r k habits (Ginzberg, 1956). C h i l d r e n f r o m s u c h a n e n v i r o n m e n t m a y still a s p i r e t o a c h i e v e — i n d e e d , m a n y do (Pinkney, 1969), and m a n y have. B u t to do so requires o v e r c o m i n g tremendous o d d s . W i t h o u t m o n e y , a g o o d e d u c a t i o n , a s t a b l e h o m e , a s t r o n g v a l u e for s c h o l a r ship, a n d w i t h o u t e n c o u r a g i n g friends a n d family, the task s e e m s e n o r m o u s .

H o w Social Conditions Affect Creativity

233

T h e r e i s h o p e o f b r e a k i n g this pattern, b u t m o s t o f t h e c h a n g e s m u s t c o m e from sources external to blacks. M o r e and better jobs, especially jobs which pay and promote nondiscrimin a t e l y , w o u l d h e l p . I n 1 9 6 6 , a b l a c k w i t h s o m e c o l l e g e e d u c a t i o n w a s still e a r n i n g less than a w h i t e w i t h an eighth-grade education (Fein, 1966). Better housing and more a c c e s s i b l e h o m e o w n e r s h i p could add a sense of stability and security. An i m p r o v e d e n v i r o n m e n t can increase a child's I Q ; a d e p r e s s e d environment can m a k e i t g o d o w n (Patai a n d W i n g , 1 9 7 5 ) . B e t t e r e q u i p p e d a n d s t a f f e d s c h o o l s c o u l d h e l p e q u a l i z e the educational situation. Studies s h o w that as schools improve, p e r f o r m a n c e i m p r o v e s (Patai a n d W i n g , 1975). T h e s e c h a n g e s , t h o u g h , m u s t c o m e from the larger society. F r o m w i t h i n the b l a c k c o m m u n i t y , h e l p can c o m e from blacks w h o h a v e succ e e d e d and w h o are w i l l i n g t o w o r k w i t h those w h o h a v e not, a n d t h e r e b y serve a s role models. H e l p can also c o m e from the c h u r c h , as it has in the past, rights. T h e c h u r c h has b e e n the outstanding social institution nity, the p l a c e w h e r e b l a c k s h a v e f o u n d refuge and e m o t i o n a l future m u s t b e e v e n m o r e d e m a n d i n g i f blacks are t o b r e a k discussed.

i n t h e d r i v e for c i v i l in the black commur e l i e f . Its r o l e i n t h e the pattern we have

B l e a k as t h e p i c t u r e m a y b e , it has s o m e p o s i t i v e notes. B l a c k s are p r o g r e s s i n g a n d i n d e e d h a v e m a d e t r e m e n d o u s contributions in literature, m u s i c , sports, entertainment, and dance. F o u r blacks (two Americans) h a v e w o n the N o b e l prize. About 10 p e r c e n t of b l a c k s are professionally e m p l o y e d (Almquist, 1979). Further, fully 80 p e r c e n t of mothers of black college students work outside the h o m e (Willie and M c C o r d , 1972). While many probably do so by necessity, nonetheless, as we h a v e seen in our discussion of w o m e n , a w o r k i n g mother is the best assurance that h e r c h i l d r e n , e s p e c i a l l y h e r f e m a l e c h i l d r e n , w i l l also work. In those black families w h i c h are stable and m i d d l e class, education is emphas i z e d a n d e n c o u r a g e d . M o r e blacks are attending c o l l e g e , setting h i g h e r goals, and getting better jobs than ever before (Willie and M c C o r d , 1972). T h e major difference b e t w e e n b l a c k a n d w h i t e families a p p e a r s t o b e o n e o f class, not color. " A s soon as blacks enter the m i d d l e class, family patterns associated with the l o w e r class t e n d to disappear. T h e differences b e t w e e n black and white f a m i l y p a t t e r n s , t h e n , a r e l a r g e l y t h e r e s u l t o f status i n t h e s o c i e t y " ( P i n k n e y , 1 9 6 9 ) . B e c a u s e t h e b l a c k l o w e r c l a s s i s s o l a r g e , p r o b l e m s l o o m l a r g e . P e r h a p s a s status improves, the creative potential of blacks will likewise improve. C O M P A R I N G A S I A N S A N D BLACKS Asians are s o m e t i m e s h e l d up to blacks as " m o d e l minorities" (Almquist, 1979), the i m p l i c a t i o n b e i n g that if Asians c o u l d b e c o m e successful, blacks can, too. H o w e v e r , as A l m q u i s t points out, there are a n u m b e r of w a y s in w h i c h this implication is unfair. First, m a n y A s i a n s c a m e to this country as h i g h l y s k i l l e d w o r k e r s . O v e r 60 percent of A s i a n immigrants in the 1950's w e r e classified as w h i t e collar workers ( A l m q u i s t , 1 9 7 9 ) . S e c o n d , A s i a n s a r e l e s s t h a n o n e p e r c e n t o f o u r p o p u l a t i o n , o r 1.3 million p e o p l e . B l a c k s , n u m b e r i n g 20 million, are not as easily a b s o r b e d by the work force. T h i r d , w h i l e A s i a n s h a v e b e e n s u b j e c t e d t o d i s c r i m i n a t i o n (for e x a m p l e , t h e internment of Japanese A m e r i c a n s during W o r l d W a r II), it can in no w a y be c o m p a r e d to 200 years of slavery w h i c h d e p r i v e d an entire p e o p l e of their dignity, heritage, civil rights, and property. Fourth, Asians chose to c o m e here and v i e w e d the United States as a place of opportunity. Fifth, the strong Asian communities can accommodate newcomers and provide them with housing and jobs.

T h e C o m p l e t e Problem Solver

234

Asian creative a c h i e v e m e n t is certainly to be e m u l a t e d , b u t it is unfair to suggest that if Asians h a v e " m a d e it," blacks s h o u l d h a v e " m a d e it," too. T h e p r o b l e m s that b l a c k s h a v e faced are e x t r e m e l y s e v e r e ones a n d t h e y are b y n o means solved. D e s p i t e t h e s e difficulties, blacks h a v e m a d e v e r y respectable progress. AMERICAN

INDIANS

Statistics on A m e r i c a n Indians are not encouraging. T h e a v e r a g e educational l e v e l a m o n g I n d i a n s i s 9.8. O n l y o n e - t h i r d f i n i s h h i g h s c h o o l . M a n y a r e f u n c t i o n a l l y illiterate (Wax, 1971). One-third live in poverty, m a k i n g t h e m the ranking poverty group in this country (Almquist, 1979). U n e m p l o y m e n t is chronic, and has at times r e a c h e d 80 p e r c e n t a m o n g those l i v i n g on reservations, as o v e r 60 p e r c e n t do (Wax, 1 9 7 1 ) . I n d i a n s h a v e t h e h i g h e s t birth rate a n d l o w e s t life e x p e c t a n c y o f a n y g r o u p in our country. W h e n E u r o p e a n s o c c u p i e d the North American continent, the Indian populat i o n w a s g r e a t l y r e d u c e d — f r o m o v e r o n e m i l l i o n i n 1 4 9 2 t o 220,000 i n 1 9 1 0 (Almquist, 1 9 7 9 ) — a n d their usual means of livelihood was removed. W h e n the buffaloes w e r e destroyed . . . the Sioux w e r e d e p r i v e d not only of food, b u t also of culturally significant activities. T h e tribal societies c o n c e r n e d w i t h w a r a n d h u n t i n g lost their functions a n d atrophied. T h e arts a n d t e c h n i q u e s s u r r o u n d i n g t h e b u f f a l o h u n t . . . w h i c h h a d o n c e b e e n s o u r c e s o f s o c i a l status a n d o f p r i d e i n w o r k m a n s h i p , w e r e n o w r e n d e r e d useless. (Barber, 1941) Indian cultures are quite diverse. T h e r e are h u n d r e d s of distinct tribes w h i c h m a y differ greatly from o n e another in l a n g u a g e and c u s t o m s . U n l i k e J e w i s h culture, though, Indian cultures typically do not m e s h w e l l w i t h the dominant American culture. C o n s i d e r this letter sent to the Virginia C o m m i s s i o n in 1744 regarding a n i n v i t a t i o n for six I n d i a n c h i l d r e n t o a t t e n d W i l l i a m a n d M a r y C o l l e g e : Several of our y o u n g p e o p l e w e r e formerly brought up at C o l l e g e s of the N o r t h e r n P r o v i n c e s ; t h e y w e r e i n s t r u c t e d i n all y o u r s c i e n c e s ; b u t w h e n they came back to us, they w e r e bad runners, ignorant of every means of living in the woods, unable to bear either cold or hunger, k n e w neither h o w to b u i l d a cabin, take a deer, or kill an e n e m y , spoke our language i m p e r f e c t l y , w e r e t h e r e f o r e n e i t h e r fit for h u n t e r s , w a r r i o r s , o r c o u n s e l l o r s , t h e y w e r e t o t a l l y g o o d for n o t h i n g . W e a r e , h o w e v e r , n o t t h e l e s s o b l i g e d b y y o u r k i n d offer, t h o u g h w e d e c l i n e a c c e p t i n g it; a n d t o s h o w o u r g r a t e f u l S e n s e o f it, i f t h e G e n t l e m e n o f V i r g i n i a w i l l s e n d u s a D o z e n o f t h e i r S o n s w e w i l l t a k e c a r e o f t h e i r e d u c a t i o n , i n s t r u c t t h e m i n all w e k n o w , a n d make M e n of them. (Noel, 1968) A l t h o u g h t h i s l e t t e r w a s w r i t t e n o v e r 200 y e a r s a g o , i t v o i c e s t h e a t t i t u d e s o f m a n y I n d i a n s t o d a y . T h e " W h i t e M a n ' s " e d u c a t i o n i s still n o t v i e w e d a s w o r t h w h i l e . F o r I n d i a n s , the school d r o p o u t rate i s v e r y h i g h . I n m a n y c a s e s , d r o p p i n g out can be attributed directly to conflicts b e t w e e n the local Indian culture and the dominant culture. For example, many southwest Indians do not b e l i e v e in making d e c i s i o n s for o t h e r p e o p l e o r i n a d v i s i n g t h e m . I n d i a n s do not b e l i e v e that o n e p e r s o n s h o u l d tell a n o t h e r w h a t to do o r t h a t e n d l e s s h o u r s s h o u l d b e w a s t e d i n p e r s u a d i n g a n o t h e r . F r o m inf a n c y , I n d i a n s a r e t a u g h t t o r e s p e c t t h e r i g h t s o f o t h e r s a n d t o a v o i d interfering with other people. E v e n if another person is placing himself or herself in great physical danger, the Indian will not tell the person what to do. Passengers in an automobile will remain silent rather than w a r n the driver of a rock slide or a steer in the road. To w a r n the driver w o u l d constitute interference. (Almquist, 1979)

H o w Social Conditions Affect Creativity

235

T h i s a s p e c t o f t h e i r c u l t u r e l e a d s I n d i a n c h i l d r e n t o d r o p o u t o f s c h o o l a n d , further, to make the decision to drop out of school on their own, without discussion or parental guidance. B e i n g b a r e l y l i t e r a t e a n d n o t e s t e e m i n g t h e v a l u e s o f t h e l a r g e r s o c i e t y has m a d e f i n d i n g e m p l o y m e n t d i f f i c u l t for I n d i a n s . W a x ( 1 9 7 1 ) s a y s He m a y be a s h r e w d j u d g e of h u m a n character, be strong, loyal, reliable, and w i l l i n g to work, b u t he will be lucky if he gains e v e n the most menial and poorly paying employment. T h e i r o w n c u l t u r e m a k e s i t d i f f i c u l t for n a t i v e A m e r i c a n s t o o b t a i n e d u c a t i o n and e m p l o y m e n t in the U n i t e d States. Further, it m a k e s it u n l i k e l y that t h e y w o u l d be able or i n c l i n e d to be creative w i t h i n the framework of our culture.

CONCLUSION If your society 1. values intellectual activities, and 2. encourages you to be interested in intellectual activities, and 3. e n c o u r a g e s y o u to b e l i e v e that y o u can s u c c e e d in intellectual activities, and 4. helps y o u to obtain necessary education, and 5. does not impose other occupations on y o u w h i c h p r e e m p t your time, then your chances of b e c o m i n g a creative person will be m u c h better than average. H o w e v e r , i f a n y o f t h e s e c o n d i t i o n s i s v i o l a t e d b e c a u s e y o u are a b l a c k , a w o m a n , a N a t i v e A m e r i c a n , o r for w h a t e v e r r e a s o n , t h e n y o u r c h a n c e s o f b e c o m i n g a c r e a tive person will be greatly reduced. T h e s e observations a b o u t the social conditions of creativity h a v e implications b o t h for t h e i n d i v i d u a l a n d for s o c i e t y a s a w h o l e . First, i n d i v i d u a l s should be aware of social pressures w h i c h m a y t e n d to b l o c k their creative d e v e l o p m e n t . Ask yourself these questions about your upbringing: • In h i g h s c h o o l , w a s it more important to be a g o o d s t u d e n t or a g o o d athlete? • D i d your friends consider it unfeminine to be intellectual? • W e r e y o u w o r r i e d that y o u m i g h t lose y o u r religion if y o u t h o u g h t too d e e p l y about science or philosophy? • Have you m o d e l e d your o w n career on someone you k n e w w e l l ? • Was there pressure on y o u at h o m e to get good grades? • D i d your parents e x p e c t y o u to b e c o m e a professional? • W e r e your parents interested in intellectual things? In a n s w e r i n g t h e s e questions, most p e o p l e can identify m a n y social pressures w h i c h h a v e p u s h e d t h e m either toward o r a w a y from intellectual pursuits. B e i n g aware of the pressures w h i c h have influenced you and w h i c h may influence you in the future can h e l p y o u to take a m o r e active role in d e t e r m i n i n g your o w n creativity. I f s o c i e t y s u g g e s t s t h a t y o u c a n ' t b e c r e a t i v e b e c a u s e o f y o u r r a c e o r sex, y o u n e e d n ' t be d e t e r r e d if y o u u n d e r s t a n d that the fault is in society a n d not in you. Y o u c a n s e t g o a l s for y o u r s e l f — a s M a l c o l m X a n d M a r y S o m e r v i l l e d i d — f a r b e y o n d t h e g o a l s o t h e r s m i g h t try t o s e t for y o u . Y o u c a n c o n s c i o u s l y s e e k a r o l e m o d e l for y o u r s e l f a m o n g t h e p e o p l e y o u k n o w w h o are o u t s t a n d i n g i n y o u r f i e l d o r b y r e a d i n g b i o g r a p h i e s . Y o u can s e e k out p e o p l e i n y o u r field and ask t h e m h o w t h e y

236

The C o m p l e t e Problem Solver

" m a d e it." Y o u can resist p r e s s u r e s from y o u r friends t o b e " r e g u l a r " a n d c h o o s e instead to be special. T h e observations in this chapter should lead us to ask s o m e v e r y serious q u e s tions about our culture. C l e a r l y , our culture d e p e n d s on the w o r k of creative p e o p l e for its w e l l - b e i n g . I t d e p e n d s o n t h e m t o s o l v e p r o b l e m s o f p o l l u t i o n , o f p o p u l a t i o n , of food and e n e r g y production, and of disease. It d e p e n d s on t h e m to p r o d u c e n e w m u s i c , art, d a n c e , a n d l i t e r a t u r e . W h i l e o u r s o c i e t y d e p e n d s o n c r e a t i v e p o e p l e , i t does not appear to be very interested in fostering their d e v e l o p m e n t . For example, w h y do basketball superstars earn more than N o b e l prize w i n n e r s ? Is it b e c a u s e basketball players do more to reduce disease or increase food production? W h y do advertising e x e c u t i v e s earn more than c o l l e g e professors? Do t h e y contribute more t o art a n d k n o w l e d g e ? W h y d o e s o u r s o c i e t y i n s i s t t h a t w o m e n , r e g a r d l e s s o f t h e i r potential creativity, raise their c h i l d r e n personally, rather than be p r o v i d e d w i t h a d e q u a t e c h i l d - c a r e f a c i l i t i e s ? W h y are t h e r e a t h l e t i c s c h o l a r s h i p s ? W o u l d n ' t t h e m o n e y b e b e t t e r s p e n t o n p e o p l e w h o are g o o d a t i n t e l l e c t u a l r a t h e r t h a n a t h l e t i c s k i l l s ? W h y d o e s o u r s c h o o l s y s t e m p u t s u c h e m p h a s i s o n its a t h l e t i c p r o g r a m s ? W h y is it that w h e n we look at our h i g h schools, we are v e r y l i k e l y to s e e a sign saying, " H o m e of the Fighting W o m b a t s , " but nothing about the school's academic achievements? T h e r e a r e m a n y s u c h q u e s t i o n s t o ask. A s k i n g t h e m a n d p r o v i d i n g s e n s i b l e a n s w e r s c o u l d b e c r u c i a l for t h e s u r v i v a l o f o u r s o c i e t y . REFERENCES Almquist,

E . M . Minorities, Gender, and Work.

Lexington,

MA:

Lexington Books,

1979. A l m q u i s t , E . M . , and Angrist, S.S. " R o l e M o d e l I n f l u e n c e s o n C o l l e g e W o m e n ' s C a r e e r A s p i r a t i o n s . " I n The Professional Game, e d i t e d b y A . T h e o d o r e . C a m b r i d g e , M A : S c h e n k m a n P u b l i s h i n g C o . , Inc., 1 9 7 1 . A n g r i s t , S . S . , a n d A l m q u i s t , E . M . Careers and Contingencies. 1975. A r i e t i , S. Creativity: The Magic Synthesis. 1976.

N e w York:

Dunellen,

N e w York: Basic B o o k s , Inc., Publishers,

A s h m o r e , H . S . The Negro and the Schools. North Carolina Press, 1954.

Chapel

A s t i n , H . S . The Woman Doctorate in America. 1969.

N e w York: Russell Sage Foundation,

B a r b e r , B. " A c c u l t u r a t i o n a n d view, 6 ( 1 0 ) , 6 5 3 - 6 7 3 , 1 9 4 1 .

Messianic

Hill, N C :

M o v e m e n t s . " American

B o n d , S.J., a n d H a y e s , J.R. A Comparison of Successful sional Women. U n p u b l i s h e d m a n u s c r i p t , 1 9 7 8 . B r o d y , E . B . , a n d B r o d y , N. Intelligence: N e w York: A c a d e m i c Press, Inc., 1976.

The

Nature,

and

Determinants,

U n i v e r s i t y of

Sociological Re-

Unsuccessful

and

Profes-

Consequences.

C a n t o r , M. " W h y I am a S o c i o l o g i s t . " In Graduate and Professional Education of Women. P r o c e e d i n g s o f A A U W C o n f e r e n c e , W a s h i n g t o n , D . C . , M a y , 1 9 7 4 . C o m e r , J.P. 1967.

"The

S o c i a l P o w e r o f t h e N e g r o . " Scientific American, 2 1 6 ( 4 ) , 2 1 - 2 7 ,

A P P E N D I X I.

ON

THE USE OF TIME

T i m e m a n a g e m e n t is important because it can influence your problem-solving efficiency in a n u m b e r of w a y s : 1. T h e s i m p l e s t and m o s t o b v i o u s w a y is that t h r o u g h p o o r s c h e d u l i n g , y o u allow i n s u f f i c i e n t t i m e t o w o r k a p r o b l e m t h r o u g h t o s o l u t i o n or, p e r h a p s , y o u n e v e r get to it at all. No matter h o w bright y o u are, if y o u don't w o r k on a p r o b l e m , y o u w o n ' t s o l v e it. 2. E v e n if y o u do s c h e d u l e e n o u g h time to solve the p r o b l e m , y o u may not schedu l e t h e b e s t t i m e t o s o l v e it. F o r e x a m p l e , y o u m a y s c h e d u l e w o r k o n a p r o b l e m a t t h e v e r y l a s t m i n u t e b e f o r e t h e p r o b l e m i s d u e . T h e p r e s s u r e t h a t r e s u l t s from such s c h e d u l i n g can cause inefficient problem solving by limiting the amount of p l a n n i n g that y o u feel y o u can do at the b e g i n n i n g of the solution process or by discouraging y o u from e x a m i n i n g alternative solution paths in sufficient depth. C e r t a i n l y , s u c h s c h e d u l i n g w o u l d e l i m i n a t e t h e v e r y v a l u a b l e p r a c t i c e o f rev i e w i n g a n d c r i t i c i z i n g t h e s o l u t i o n p r o c e s s i m m e d i a t e l y after t h e s o l u t i o n i s a c h i e v e d . A far b e t t e r t i m e t o s c h e d u l e p r o b l e m s o l u t i o n i s r i g h t after c l a s s . A t t h i s t i m e , facts a n d i d e a s w h i c h m a y b e f o r g o t t e n i n a f e w h o u r s c a n b e u s e d a n d consolidated. 3. If your time m a n a g e m e n t is generally poor, your ability to solve problems may suffer b e c a u s e y o u h a v e f a i l e d t o a c q u i r e a n e s s e n t i a l p i e c e o f k n o w l e d g e o r b e c a u s e y o u h a v e not attained sufficient mastery of a basic skill. Students somet i m e s try t o s a v e t i m e b y a t t e m p t i n g t o s o l v e p r o b l e m s after o n l y a q u i c k g l a n c e at the relevant chapter or without working through the practice problems. While this corner-cutting t e c h n i q u e s o m e t i m e s a l l o w s y o u to g e t b y , in the l o n g run it is likely to reduce your problem-solving efficiency. W h i l e there is v e r y little g l a m o u r in t i m e m a n a g e m e n t , the t e c h n i q u e s are simple and they work. For some, they can produce considerable gains in problemsolving efficiency. A c c o r d i n g to C r a w l e y (1936), time m a n a g e m e n t is the most useful skill that students acquire in courses on study-skills. T E C H N I Q U E S O F TIME M A N A G E M E N T T i m e m a n a g e m e n t e m p l o y s t w o major t e c h n i q u e s : Time planning: a l l o t t i n g s p e c i f i c b l o c k s o f t i m e d u r i n g t h e d a y o r w e e k t o specific tasks a c c o r d i n g to their importance a n d the time t h e y require. Efficiency: g e t t i n g a g r e a t d e a l a c c o m p l i s h e d i n t h e a v a i l a b l e t i m e . 239

The C o m p l e t e Problem Solver

240

L e t ' s start o u r d i s c u s s i o n w i t h t i m e p l a n n i n g . T h e first s e r i o u s s t e p i n t i m e p l a n n i n g is to find out h o w y o u s p e n d your time n o w . Perhaps the simplest w a y to d o this i s t o k e e p a t i m e c h a r t s u c h a s t h i s : TIME

SCHEDULE

FOR

WEEK Date:

Hour

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

On the Use of Time

241

Y o u s h o u l d r e c o r d y o u r a c t i v i t i e s i n t h e c h a r t for a t l e a s t o n e c o m p l e t e w e e k , s i n c e for m o s t p e o p l e t i m e u s a g e v a r i e s a g o o d d e a l f r o m d a y t o d a y . W e e k e n d s , for e x a m p l e , are u s u a l l y s c h e d u l e d q u i t e d i f f e r e n t l y from w e e k d a y s . S i n c e t h e r e c o r d i s e n t i r e l y for y o u r b e n e f i t , i t i s w o r t h m a k i n g i t a s c o m p l e t e a n d a c c u r a t e a s y o u c a n . T a k e t h e d a t a s h e e t w i t h y o u w h e r e v e r y o u g o s o that y o u c a n m a k e t h e e n t r i e s a s s o o n after t h e a c t i v i t y a s p o s s i b l e . B e u n s p a r i n g l y h o n e s t a b o u t s u c h q u e s t i o n s as, " W a s I studying or really just reading m a g a z i n e s ? " O n e w o u l d e x p e c t most c o l l e g e students to s p e n d 12 to 15 hours in class e a c h w e e k and 25 to 30 hours in study. O n c e y o u have an idea of h o w you use your time, you may be delighted and i m p r e s s e d , o r y o u m a y d e c i d e that y o u w a n t t o d e s i g n a b e t t e r s c h e d u l e . Y o u m a y f i n d , for e x a m p l e , that y o u s p e n d m o r e t i m e i n c a s u a l r e c r e a t i o n t h a n y o u t h o u g h t a n d w a n t t o c o n v e r t s o m e o f t h o s e h o u r s t o s t u d y u s e . Y o u m a y find that y o u are p u t t i n g u n d u e e m p h a s i s o n o n e t o p i c a t t h e e x p e n s e o f o t h e r s o r that s o m e t o p i c s are s u f f e r i n g b e c a u s e t h e y are b e i n g left for t h e last m i n u t e . I f a n y o f t h e s e t h i n g s i s t r u e , y o u w i l l l i k e l y profit b y m a k i n g a t i m e p l a n a n d s t i c k i n g t o it. To m a k e a time plan, y o u should establish a fixed w e e k l y s c h e d u l e in w h i c h s p e c i f i c t o p i c s are a s s i g n e d t o s p e c i f i c t i m e slots. T h e a m o u n t o f t i m e y o u a s s i g n t o e a c h t o p i c s h o u l d d e p e n d o n its i m p o r t a n c e t o y o u a n d o n t h e d i f f i c u l t y y o u h a v e w i t h it. D e c i d i n g o n t h e r e l a t i v e i m p o r t a n c e o f t h e t o p i c s i s v e r y i m p o r t a n t a n d therefore worth considerable thought. M a k i n g these decisions sensibly may be the most valuable aspect of time planning. T r y t o p l a c e t h e t i m e slot f o r s t u d y i n g a t o p i c a s c l o s e a s p o s s i b l e after t h e l e c t u r e o n that t o p i c . T r y t o s c h e d u l e t o p i c s that r e q u i r e a g r e a t d e a l o f p r o b l e m s o l v i n g o r c o n c e n t r a t i o n a t t i m e s w h e n y o u are m o s t a l e r t . L e a v e r o u t i n e a c t i v i t i e s a n d r e c r e a t i o n for t i m e s w h e n y o u are l i k e l y t o n e e d r e l a x a t i o n . Your schedule should include a w e e k l y review and planning session to see w h e r e y o u h a v e g o t t e n d u r i n g t h e p a s t w e e k a n d w h e r e y o u are g o i n g d u r i n g t h e next. Y o u r s c h e d u l i n g of recreation should be realistic rather than idealistic. Y o u m a y f e e l t h a t i t i s n o b l e t o s p e n d n o t i m e a t all o n f o o l i s h t h i n g s l i k e T V w a t c h i n g , b u t i f m i s s i n g y o u r f a v o r i t e p r o g r a m s turns y o u into a s e l f - p i t y i n g b l o b , t h e sacrifice p r o b a b l y i s n ' t w o r t h i t a c a d e m i c a l l y o r o t h e r w i s e . O n t h e o t h e r h a n d , i f y o u find that y o u r e w a r d y o u r s e l f w i t h a n h o u r o f T V for e v e r y h o u r y o u s t u d y , y o u are c l e a r l y b e i n g w i l d l y s e l f - i n d u l g e n t . B e r e a s o n a b l e ! G i v e y o u r s e l f e n o u g h r e c r e a t i o n t o stay efficient but not too m u c h more. O n c e y o u h a v e e s t a b l i s h e d y o u r t i m e p l a n , y o u s h o u l d try t o f o l l o w i t c l o s e l y . O f c o u r s e , i t m a y n e e d a d j u s t m e n t s . Y o u m a y find that y o u h a v e t o o m u c h o f s o m e t h i n g o r t o o l i t t l e o f s o m e t h i n g e l s e . I f a n e m e r g e n c y c o m e s u p , say i n t h e form o f a n e x a m , y o u h a v e t o b o r r o w t i m e from o n e t o p i c t o w o r k o n a n o t h e r . T h e borrowed time should always be paid back, however. Y o u m a y find i t h e l p f u l t o c o n t i n u e y o u r t i m e d i a r y s o t h a t y o u c a n s e e t h e extent to w h i c h your actual time usage approximates y o u r plan. K e e p i n g the diary for a t l e a s t o n e w e e k o n c e y o u r p l a n i s i n o p e r a t i o n c e r t a i n l y s h o u l d b e i n f o r m a t i v e . G i v e n a satisfactory s c h e d u l e o f w o r k a n d r e c r e a t i o n , t h e n e x t p r o b l e m i s t o i n s u r e that y o u g e t t h e m a x i m u m b e n e f i t o u t o f t h e h o u r s y o u w o r k . E f f i c i e n c y requires both organization and concentration. ORGANIZING YOUR WORK Most of us have had the experience of settling d o w n to work only to discover that n e c e s s a r y p a p e r s are m i s s i n g o r that w e h a v e l o s t t h e a s s i g n m e n t . O r w e m a y b e w o r k i n g a l o n g a t h i g h s p e e d o n a p a p e r a n d h a v e t o s t o p for 1 0 m i n u t e s t o s e a r c h for a d i c t i o n a r y u n d e r h e a p s o f p a p e r s a n d r e c o r d j a c k e t s . T h e p u r p o s e o f o r g a n i z i n g your work is to reduce the amount of time you spend doing things like searching

242

The C o m p l e t e Problem Solver

t h r o u g h v a r i o u s h e a p s for t h i n g s y o u n e e d r i g h t n o w . S o m e o f t h e f o l l o w i n g r u l e s may help: 1 . K e e p a s h e l f (or f i x e d p l a c e ) for f r e q u e n t l y u s e d r e f e r e n c e b o o k s a n d m a k e a h a b i t o f r e t u r n i n g t h e m t o t h e i r p l a c e s after u s e . 2 . K e e p a s e p a r a t e , c l e a r l y m a r k e d n o t e b o o k o r f o l d e r for e a c h t o p i c . 3. Date your papers, and n u m b e r the pages. 4. K e e p y o u r p e r s o n a l library in s o m e reasonable order, e.g., arranged by topic or alphabetically by author. 5. If y o u can't c o m p l e t e a j o b in o n e session, d i v i d e it into subparts w h i c h can be completed in one session. 6. W h e n y o u finish a w o r k session on a g i v e n topic, y o u should plan w h a t to do next w h e n y o u return to that topic and l e a v e yourself a note as a reminder. It m a y save y o u s o m e time in reorienting yourself to the topic. CONCENTRATION T h e other a s p e c t o f efficiency that w e w i l l c o n s i d e r i s concentration. T r y i n g t o m a k e y o u r s e l f c o n c e n t r a t e i s a m u c h m o r e d i f f i c u l t task t o g e t h o l d o f t h a n a t t e m p t ing t o o r g a n i z e y o u r w o r k . S o m e authors s i m p l y t h r o w u p their hands and say, " T h e only w a y to concentrate is to c o n c e n t r a t e . " T h e r e are s o m e items of a d v i c e in the " H o w to S t u d y " literature, h o w e v e r , w h i c h y o u m a y find useful. 1 . O n c e y o u e s t a b l i s h y o u r w o r k s c h e d u l e , s t i c k t o it. O n e a u t h o r d r a w s a n a n a l o g y with eating. If y o u regularly eat at certain times of the day, y o u will be hungry at those times. If y o u regularly work at certain times of the day, y o u will be ready for w o r k a t t h o s e t i m e s . 2. W h i l e it is not absolutely essential, a quiet work place does help. K e e p TV and friends out of your study area. U s e a radio only if the m u s i c d r o w n s out sounds t h a t are m o r e d i s t r a c t i n g . 3 . U s e y o u r w o r k s p a c e o n l y for w o r k . I f y o u h a v e a m o m e n t for r e l a x a t i o n , d o i t somewhere else. 4. S c h e d u l e short rests a n d m a k e g o o d u s e of t h e m : stretch, w a l k around, do s o m e t h i n g different from w h a t y o u h a v e b e e n doing. 5 . D o n ' t w o r k m o r e t h a n t w o h o u r s a t a t i m e o n t h e s a m e t o p i c i f y o u c a n h e l p it. 6 . S c h e d u l e y o u r w o r k h o u r s for t h o s e t i m e s o f t h e d a y w h e n y o u are m o s t e f f i c i e n t . I t m a y b e w o r t h t r y i n g d i f f e r e n t t i m e s t o d i s c o v e r if, e v e n t h o u g h y o u p a s s y o u r s e l f off as a n i g h t o w l , y o u are really a latent m o r n i n g p e r s o n , a n d v i c e versa. 7. W h e n y o u h a v e w o r k e d t h r o u g h a p r o b l e m , take t i m e to r e v i e w w h a t it w a s that g a v e y o u d i f f i c u l t y a n d h o w y o u o v e r c a m e it. R e f l e c t i o n j u s t after y o u h a v e s o l v e d a p r o b l e m is a v e r y v a l u a b l e aid in r e m e m b e r i n g n e w l y d i s c o v e r e d solution t e c h n i q u e s , so that y o u don't have to discover t h e m a n e w e a c h time the s a m e sort o f p r o b l e m c o m e s u p . 8. A v o i d daydreaming. If daydreaming b e c o m e s a problem in r e a d i n g — i f you notice that your e y e s have m o v e d over the page b u t y o u haven't understood a t h i n g — t r y s u m m a r i z i n g e a c h p a r a g r a p h t o y o u r s e l f a s y o u c o m e t o t h e e n d o f it. If d a y d r e a m i n g is a p r o b l e m at other times, e.g., w h i l e w o r k i n g p h y s i c s probl e m s , try u s i n g a k i t c h e n t i m e r w h i c h y o u k e e p r e s e t t i n g b e f o r e i t r i n g s — w h e n i t d o e s ring, i t w i l l b r i n g y o u b a c k from y o u r d a y d r e a m . 9. If d a y d r e a m i n g b e c o m e s a s e v e r e p r o b l e m d u r i n g a study session, w o r k on a m o r e i n t e r e s t i n g t o p i c , o r s t o p s t u d y i n g a l t o g e t h e r for a w h i l e . REFERENCES Crawley, 1936.

S . L . , Studying

Efficiently.

Englewood

Cliffs,

NJ:

Prentice-Hall,

Inc.,

A P P E N D I X II

PROBABILITIES

T h i s a p p e n d i x i s d e s i g n e d for t h e r e a d e r w h o f e e l s t h a t h i s k n o w l e d g e o f e l e m e n t a r y p r o b a b i l i t y t h e o r y is shaky or absent. After s t u d y i n g this a p p e n d i x , the r e a d e r s h o u l d b e a b l e t o d o t h e f o l l o w i n g four t h i n g s : 1. d e f i n e p r o b a b i l i t y , 2 . state t h r e e e l e m e n t a r y p r o p e r t i e s o f p r o b a b i l i t i e s , 3 . e s t i m a t e p r o b a b i l i t i e s for s o m e s i m p l e e v e n t s , a n d 4 . c o m p u t e p r o b a b i l i t i e s for s o m e c o m p l e x e v e n t s . PROBABILITY

DEFINED

M a n y , perhaps most, of the events in our e v e r y d a y e x p e r i e n c e h a v e outcomes that w e can't p r e d i c t w i t h certainty. F o r e x a m p l e , w e can't b e certain that our favorite football t e a m will w i n today's g a m e or that a r e c e n t l y p u r c h a s e d plant will survive our care. We can think of plant raising as an activity w h i c h can have either of two outcomes: T h e plant will live or it will die. If we have had poor success in the past, w e m a y h a v e little c o n f i d e n c e that the n e w plant w i l l l i v e . I f w e h a v e b e e n g e n e r a l l y successful i n the past, w e m a y b e m o d e r a t e l y o r e v e n strongly confident o f s u c c e s s . I n a n y c a s e w e w i l l r e a l i z e t h a t w e c a n ' t p r e d i c t with certainty w h e t h e r or not the n e w plant will live. In s o m e cases, it is possible to d e s c r i b e the uncertainty of an e v e n t by specifyi n g p r o b a b i l i t i e s for t h e v a r i o u s p o s s i b l e o u t c o m e s o f t h e e v e n t . T h e p r o b a b i l i t y o f a n o u t c o m e i s d e f i n e d a s t h e p r o p o r t i o n o f t i m e s t h a t o u t c o m e w i l l o c c u r g i v e n that w e o b s e r v e a n i n d e f i n i t e l y large n u m b e r o f e v e n t s . T h u s , i f the probability that our t e a m w i l l w i n i s a c t u a l l y 0.80, t h e n i f w e o b s e r v e m a n y g a m e s , o u r t e a m w i l l w i n about 80 percent of them. W h e n w e k n o w t h e p r o b a b i l i t y , w e still d o n ' t k n o w w i t h c e r t a i n t y w h e t h e r o r n o t o u r t e a m w i l l w i n its n e x t g a m e . T h e u s e f u l t h i n g t h a t w e d o k n o w i s that i f w e b e t o n our team m a n y t i m e s , w e will w i n e i g h t bets out o f t e n . I n general, k n o w i n g t h e p r o b a b i l i t y o f a n o u t c o m e i s u s e f u l for p r e d i c t i n g t h e n u m b e r o f t i m e s that outcome will occur w h e n the e v e n t is repeated m a n y times. Probabilities are related to odds, but the t w o shouldn't be confused. T h e odds o f a n o u t c o m e i s t h e ratio o f t h e n u m b e r o f t i m e s t h e o u t c o m e o c c u r s , N , t o t h e n u m b e r of times it d o e s n ' t occur, N. T h a t is, O d d s (outcome) = — N

244

T h e C o m p l e t e Problem Solver

T h e p r o b a b i l i t y of an o u t c o m e is the ratio of t h e n u m b e r of t i m e s the o u t c o m e o c c u r s , N , t o t h e total n u m b e r o f e v e n t s , T . S i n c e t h e total n u m b e r o f e v e n t s i s t h e sum of N and N, then i i -i. / , N N Probability (outcome) = — = — — — T N +N t>

U s u a l l y probabilities are stated as d e c i m a l fractions, w h e r e a s odds are stated as the ratio o f w h o l e n u m b e r s . T h u s , t h e p r o b a b i l i t y t h a t o u r t e a m w i l l w i n i s 0.80, b u t t h e o d d s t h a t i t w i l l w i n a r e e i g h t t o t w o , or, r e d u c e d t o l o w e s t t e r m s , f o u r t o o n e . THREE PROPERTIES OF PROBABILITIES 1. A probability is a n u m b e r b e t w e e n zero and one. An outcome w h i c h can n e v e r occur has a probability of zero, and an o u t c o m e w h i c h always occurs has a probabili t y o f o n e . M o s t o u t c o m e s h a v e p r o b a b i l i t i e s t h a t fall b e t w e e n t h e s e t w o . A p r o b a bility can n e v e r be a negative number. 2 . T h e p r o b a b i l i t i e s o f a l l t h e o u t c o m e s o f a n e v e n t sum t o one. S u p p o s e t h a t a n e v e n t has a n u m b e r o f p o s s i b l e o u t c o m e s — s a y five. S u p p o s e that w e h a v e obs e r v e d the e v e n t a n indefinite n u m b e r o f times, T , s o that w e can calculate the probabilities of the outcomes. If outcome 1 occurred N times, outcome 2, N times, t

and so on, then P = Nj/T, P

2

2

= N / T , etc. Since e v e r y e v e n t must have one of the 2

five outcomes, N j + N j + N g + ^ + N s - T . N o w , the sum of the probabilities of the outcomes is given by: T,

T~>

Tl *•

x

T,

°

^

N

T,

"

rj-i

_N

t

t

N rj-i

+ N

2

2

N

N

3

+ N

4

rj-i

rji

3

+ N

N

5

rji

4

+ N

5

T

3.

The law of large numbers. T a b l e 1 s h o w s t h e r e s u l t s of t o s s i n g a c o i n r e p e a t e d l y .

A s t h e n u m b e r o f t o s s e s i n c r e a s e s , t h e p r o p o r t i o n o f h e a d s o b t a i n e d a p p r o a c h e s 0.5. T h u s , i n a m i l l i o n t o s s e s o f a fair c o i n , w e s h o u l d o b t a i n a p r o p o r t i o n o f h e a d s v e r y c l o s e t o 0.5. I f t h e c o i n h a d b e e n b i a s e d , t h e p r o p o r t i o n o f h e a d s w o u l d h a v e s e t t l e d d o w n to some other value. T a b l e 2 shows the result of tossing a biased coin. F r o m these results, we can e s t i m a t e t h a t for t h i s c o i n , t h e p r o b a b i l i t y o f a h e a d i s a b o u t 0 . 5 3 . T h e v a l u e , 0 . 5 3 , is o n l y an estimate of t h e probability, since we can n e v e r toss the c o i n t h e infinite n u m b e r of times r e q u i r e d to find the " t r u e " probability. H o w e v e r , the estimate will approach the true probability more and more closely as we increase the number of tosses. T h e fact that the e s t i m a t e w i l l a p p r o a c h t h e true p r o b a b i l i t y in this c a s e is a n i n s t a n c e o f t h e law o f large numbers: T h e p r o p o r t i o n o f t i m e s t h a t a n e v e n t h a s

Table 1 Number of tosses

10

100

300

500

1,000

10,000

4

54

138

235

484

4,983

0.40

0.54

0.46

0.47

0.48

0.498

Number of heads Proportion of heads

Probabilities

245 Table 2

Number of tosses

10

100

300

500

1,000

10,000

Number of heads

5

57

147

256

519

5,309

0.50

0.57

0.49

0.51

0.52

0.531

Proportion of heads

a p a r t i c u l a r o u t c o m e a p p r o a c h e s t h e t r u e p r o b a b i l i t y for t h a t o u t c o m e a s t h e number of events increases. ESTIMATING PROBABILITIES FOR SIMPLE O U T C O M E S P r o b a b i l i t y e s t i m a t e s c o m e from t w o s o u r c e s : e x p e r i e n c e a n d rational m o d e l s . T h e discussion of the b i a s e d coin a b o v e illustrates the use of e x p e r i e n c e to m a k e probability estimates. An e v e n t is observed a n u m b e r of times; the proportion of times a particular o u t c o m e occurs is c a l c u l a t e d ; and that proportion is u s e d as the estimate of the probability of the o u t c o m e . T h e more important it is that we have an accurate estimate, the larger the n u m b e r of events we observe. If we want to be sure that a d r u g has a l o w probability of c a u s i n g harmful side effects, we give it extensive testing (observe many events) before releasing it to the public. Rational m o d e l s are the other s o u r c e of p r o b a b i l i t y e s t i m a t e s . In a rational m o d e l , w e try t o i d e n t i f y s e t s o f e q u a l l y p r o b a b l e o u t c o m e s for a n e v e n t . S u p p o s e t h a t w e c o m e f r o m a d i s t a n t c o u n t r y w h e r e c o i n f l i p p i n g i s u n h e a r d of. W h e n w e are a p p r o a c h e d for t h e first t i m e t o b e t o n t h e f l i p o f a c o i n , w e w i l l b e a t a d i s t i n c t d i s a d v a n t a g e i f w e c a n ' t e s t i m a t e t h e p r o b a b i l i t y o f o b t a i n i n g a h e a d o r a tail. W h a t can we do with zero experience in coin flipping? By examining the coin, we can see that t h e t w o s i d e s a r e q u i t e s y m m e t r i c a l . T h e r e i s n o r e a s o n t o b e l i e v e i t w i l l l a n d h e a d s m o r e o f t e n t h a n i t l a n d s t a i l s . W e r e a s o n , t h e r e f o r e , t h a t t h e h e a d s a n d tails are e q u a l l y l i k e l y a l t e r n a t i v e s . S i n c e t h e y a r e t h e o n l y a l t e r n a t i v e s , P ( h e a d s ) + P (tails) = 1 and P ( h e a d s ) = P (tails) = 0.5 In the same way, if we w e r e unfamiliar with dice, we could quickly estimate the p r o b a b i l i t y o f r o l l i n g a t h r e e o r a f i v e b y n o t i c i n g t h a t a d i e i s a c u b e w i t h six e q u a l faces. We estimate that e a c h side is e q u a l l y l i k e l y to turn up on a roll and c o n c l u d e t h a t t h e p r o b a b i l i t y for e a c h s i d e i s o n e - s i x t h . I f t h e d i e h a d b e e n i r r e g u l a r , w i t h faces o f various sizes a n d s h a p e s , w e c o u l d not h a v e m a d e this prediction. I n g e n e r a l , r a t i o n a l m o d e l s for e s t i m a t i n g p r o b a b i l i t i e s s u c h a s t h e s e m a k e extensive use of arguments about symmetry or e q u i v a l e n c e of alternatives. COMPUTING

PROBABILITIES FOR C O M P L E X O U T C O M E S

A s i m p l e o u t c o m e i s o n e for w h i c h w e c a n e s t i m a t e t h e p r o b a b i l i t i e s e i t h e r from e x p e r i e n c e o r f r o m a r a t i o n a l m o d e l . I f A a n d B a r e t w o s i m p l e o u t c o m e s , w e m a y d e f i n e a c o m p l e x e v e n t , J , a s t h e o c c u r r e n c e o f e i t h e r A o r B . T h a t is, J = A or B E v e n t s c o m b i n e d in this w a y are said to be O R e d together. G e t t i n g a n u m b e r greater than three in rolling a die is a c o m p l e x o u t c o m e w h i c h consists of three s i m p l e o u t c o m e s : r o l l i n g a f o u r or a f i v e or a six. T h e s e s i m p l e o u t c o m e s are O R e d

The C o m p l e t e Problem Solver

246

together to form the c o m p l e x outcome. We can define a different c o m p l e x event, K, as t h e o c c u r r e n c e of b o t h A and B. K = A and B E v e n t s c o m b i n e d i n t h i s w a y are s a i d t o b e A N D e d t o g e t h e r . I n r o l l i n g a p a i r of dice, getting a 12 is a c o m p l e x e v e n t w h i c h consists of t w o simple events: rolling a six o n t h e first d i e and r o l l i n g a six o n t h e s e c o n d d i e . T h e s e s i m p l e o u t c o m e s are A N D e d together to form the c o m p l e x outcome. M a n y c o m p l e x o u t c o m e s i n v o l v e both m e t h o d s o f c o m b i n i n g simpler outc o m e s . R o l l i n g a 1 0 o n a p a i r o f d i c e c o n s i s t s o f r o l l i n g a f o u r o n t h e first d i e and a six on t h e s e c o n d , or a f i v e on t h e first d i e and a f i v e on t h e s e c o n d , or a six on t h e first d i e and a f o u r o n t h e s e c o n d . H e r e t h e s i m p l e o u t c o m e s a r e A N D e d t o g e t h e r to form c o m p l e x o u t c o m e s w h i c h are, in turn, O R e d t o g e t h e r to form m o r e c o m p l e x outcomes. In this section, we will show h o w to c o m p u t e the probabilities o f c o m p l e x e v e n t s from the probabilities o f the c o m p o n e n t s i m p l e r e v e n t s . T o d o this a d e q u a t e l y , we w i l l n e e d to c o n s i d e r a fairly c o m p l e x e x a m p l e s u c h as the following. S u p p o s e that y o u h a v e a b o x c o n t a i n i n g 100 identical balls l a b e l e d from 1 to 100. I m a g i n e that y o u shake the b o x t h o r o u g h l y and t h e n d r a w out one ball w i t h o u t l o o k i n g . I t i s e a s y t o c o n s t r u c t a r a t i o n a l m o d e l for t h i s e v e n t , w h i c h i n d i c a t e s t h a t t h e 1 0 0 a l t e r n a t i v e o u t c o m e s are e q u a l l y p r o b a b l e a n d t h a t e a c h h a s a p r o b a b i l i t y o f 1/100. S u p p o s e that balls 1 t h r o u g h 30 are red, balls 31 t h r o u g h 60 are w h i t e , and balls 6 1 t h r o u g h 1 0 0 are b l u e . F u r t h e r , s u p p o s e t h a t b a l l s 1 t h r o u g h 5 0 are m a d e o f w o o d a n d t h a t b a l l s 5 1 t h r o u g h 1 0 0 are m a d e o f p l a s t i c ( s e e F i g u r e 1 ) . T o c o m p u t e p r o b a b i l i t i e s for o u t c o m e s w h i c h a r e O R e d t o g e t h e r : 1 . I d e n t i f y c l e a r l y t h e s e t o f o u t c o m e a n d t h e r e l a t i o n s for w h i c h y o u w a n t t o c o m p u t e a probability. F o r e x a m p l e , if a s k e d to find the p r o b a b i l i t y of rolling a 3 or a 5 y o u can w r i t e " F i n d P r o b (3 or 5 ) . " 2 . D e t e r m i n e w h e t h e r o r n o t t h e o u t c o m e s a r e mutually exclusive. T w o o u t c o m e s are m u t u a l l y e x c l u s i v e if t h e y can't occur together. It is i m p o s s i b l e to get both a h e a d a n d a tail w i t h a s i n g l e toss o f a c o i n . T h e r e f o r e , " h e a d " a n d " t a i l " a r e m u t u a l l y e x c l u s i v e o u t c o m e s of this e v e n t . In the s a m e w a y , g e t t i n g a 3 a n d a 5 on t h e roll o f a s i n g l e d i e a r e m u t u a l l y e x c l u s i v e o u t c o m e s . I n p r o b a b i l i t y t e r m s , i f A a n d B are m u t u a l l y e x c l u s i v e o u t c o m e s , t h e n P r o b ( A and B ) = 0 ; t h a t i s , t h e p r o b a b i l i t y t h a t the two occur together is zero. 3A. If the o u t c o m e s are m u t u a l l y e x c l u s i v e , then the probability of the O R e d o u t c o m e s is the s u m of t h e probabilities of the separate o u t c o m e s . T h a t is, P r o b (A or B or . . . K) = P r o b (A) + P r o b (B) + . . . + P r o b (K).

247

Probabilities Example:

W h a t is t h e p r o b a b i l i t y of r o l l i n g a t h r e e or a f i v e on a s i n g l e r o l l of a d i e ? Solution S t e p 1. F i n d P r o b (3 or 5). S t e p 2 . T h e s e o u t c o m e s are m u t u a l l y e x c l u s i v e a s w e n o t e d a b o v e . S t e p 3. P r o b (3 or 5) = P r o b (3) + P r o b (5) = 1/6 + 1/6 = 1/3 Example: W h a t is the probability of d r a w i n g a w h i t e ball in the situation d e s c r i b e d above? Solution S t e p 1. F i n d P r o b (1 or 2 or 3 . . . 30). S t e p 2 . T h e s e o u t c o m e s are m u t u a l l y e x c l u s i v e s i n c e w e h a v e a s s u m e d that o n l y o n e ball is to be d r a w n at a time. S t e p 3. P r o b (1 or 2 or 3 or .. . 30) = P r o b (1) + P r o b (2) + P r o b (3) . . . + P r o b (30) = 1/100 + 1/100 + 1/100 . . . + 1/100 = 30/100 = 0.30 3B. I f t w o o u t c o m e s a r e not m u t u a l l y e x c l u s i v e , t h e n t h e p r o b a b i l i t y o f t h e O R e d o u t c o m e s i s t h e s u m o f t h e p r o b a b i l i t i e s o f t h e s e p a r a t e o u t c o m e s minus t h e p r o b a bility that the t w o o c c u r together. T h a t is, P r o b (A or B) = P r o b (A) + P r o b (B) - P r o b (A and B) Example: F i n d the probability of d r a w i n g a w h i t e ball or a w o o d e n ball. Solution Step 1. F i n d Prob (white or w o o d e n ) . S t e p 2 . T h e s e o u t c o m e s a r e not m u t u a l l y e x c l u s i v e s i n c e w e c a n d r a w a ball w h i c h is both white and wooden. Step 3. Prob (white or wooden) = P r o b ( w h i t e ) + P r o b ( w o o d e n ) — P r o b ( w h i t e and w o o d e n ) = 0.30 + 0.50 - P r o b ( 3 1 or 32 or . . . 50) = 0.80 - 0.20 = 0.60 T h e r e a s o n t h a t w e m u s t s u b t r a c t P r o b ( A and B ) for o u t c o m e s t h a t are n o t mutually e x c l u s i v e is that o t h e r w i s e s o m e o u t c o m e s w o u l d be c o u n t e d t w i c e . In the e x a m p l e a b o v e , to find the probability that a ball is w o o d e n , we a d d up probabilities for o u t c o m e s 1 t h r o u g h 5 0 . T o f i n d t h e p r o b a b i l i t y t h a t a b a l l i s w h i t e , w e a d d u p t h e p r o b a b i l i t i e s o f o u t c o m e s 3 1 t h r o u g h 60. N o w , t o find the p r o b a b i l i t y o f d r a w i n g a ball w h i c h is w h i t e or w o o d e n , what we really w a n t to do is to add the probabilities of o u t c o m e s 1 through 60. By a d d i n g the probability of d r a w i n g a w h i t e ball and d r a w i n g a w o o d e n ball, we are in effect c o u n t i n g o u t c o m e s 31 t h r o u g h 50 t w i c e . T h e s e a r e j u s t t h e o u t c o m e s w h i c h are b o t h w h i t e and w o o d e n . T h i s e r r o r i s c o r r e c t e d by s u b t r a c t i n g P r o b ( w h i t e and w o o d e n ) = P r o b ( 3 1 ) + P r o b (32) + . . . + P r o b (50). F o r s o m e , the d i a g r a m s in F i g u r e 2 h e l p to clarify the difference b e t w e e n o u t c o m e s w h i c h are m u t u a l l y e x c l u s i v e a n d t h o s e w h i c h are n o t . T o c o m p u t e p r o b a b i l i t i e s for o u t c o m e s w h i c h are A N D e d t o g e t h e r : 1 . I d e n t i f y c l e a r l y t h e s e t o f o u t c o m e s a n d t h e i r r e l a t i o n s for w h i c h y o u w a n t t o c o m p u t e a p r o b a b i l i t y , e . g . , P r o b (A and B and C ) .

248

The C o m p l e t e Problem Solver

Probabilities

249

2 . D e t e r m i n e w h e t h e r o r n o t t h e o u t c o m e s are independent o f e a c h o t h e r . A n o u t c o m e i s i n d e p e n d e n t o f a n o t h e r o u t c o m e i f t h e p r o b a b i l i t y o f t h e first o u t c o m e is not influenced by the occurrence or non-occurrence of the second. For example, the probability that there w i l l b e a n e c l i p s e o f the m o o n t o m o r r o w d o e s not d e p e n d o n w h e t h e r o r n o t I a m k i n d t o m y cat. T h e t w o o u t c o m e s a r e u n r e l a t e d a n d i n d e p e n d e n t . H o w e v e r , t h e p r o b a b i l i t y t h a t I w e a r a r a i n c o a t t o m o r r o w is dependent o n t h e w e a t h e r . I f t h e w e a t h e r i s r a i n y o r o v e r c a s t , t h e p r o b a b i l i t y t h a t I w i l l w e a r a raincoat is m u c h h i g h e r than if it is sunny. 3 A . I f t h e o u t c o m e s are i n d e p e n d e n t , t h e n t h e p r o b a b i l i t y o f t h e A N D e d o u t c o m e s is the product of the probabilities of the separate o u t c o m e s . T h a t is, P r o b (A and B and . . . K) = P r o b (A) x P r o b (B) x . . . x P r o b (K) Solution S t e p 1 . F i n d P r o b ( h e a d and h e a d and h e a d ) . S t e p 2 . S u c c e s s i v e c o i n flips a r e i n d e p e n d e n t o f e a c h o t h e r . C o i n s h a v e n o m e m o r y for p r e v i o u s e v e n t s . S t e p 3. P r o b ( h e a d and h e a d and h e a d ) = Prob (head) x Prob (head) x Prob (head) = 1/2 x 1/2 x 1/2 = 1/8 3B. I f t h e o u t c o m e s a r e not i n d e p e n d e n t t h e n w e h a v e t o t a k e t h e d e p e n d e n c i e s into account. We can e x p r e s s the d e p e n d e n c e of o u t c o m e B on o u t c o m e A as follows: Prob ( B | A ) * P r o b , ( B | 5 ) T h i s should be read as the probability of B g i v e n that A has occurred is not e q u a l t o t h e p r o b a b i l i t y o f B g i v e n t h a t A h a s n o t o c c u r r e d ( K m a y b e r e a d a s not A ) . P r o b (B | A) is c a l l e d t h e conditional p r o b a b i l i t y of B g i v e n A. S u p p o s e w e k n o w that a w o o d e n ball has b e e n c h o s e n b u t w e don't k n o w w h a t c o l o r i t i s . T h e p r o b a b i l i t y t h a t i t i s w h i t e i s t h e c o n d i t i o n a l p r o b a b i l i t y o f white g i v e n wooden; t h a t is, P r o b ( w h i t e | w o o d e n ) . W h e n w e l e a r n t h a t a w o o d e n b a l l h a s b e e n c h o s e n we k n o w that o n e of the o u t c o m e s of 1 through 50 has occurred b u t we d o n ' t k n o w w h i c h o n e . O f t h e s e 5 0 o u t c o m e s , 2 0 are w h i t e . W e c a n c o n c l u d e t h e n t h a t P r o b ( w h i t e | w o o d e n ) = 20/50 = 0.40. I f t w o o u t c o m e s A a n d B are n o t i n d e p e n d e n t , t h e p r o b a b i l i t y o f t h e A N D e d o u t c o m e is the probability of A times the conditional probability of B g i v e n A, that i s , P r o b (A and B) = P r o b (A) x P r o b (B | A ) . Example: W h a t is the p r o b a b i l i t y of d r a w i n g a ball that is b o t h w o o d e n a n d w h i t e ? Solution S t e p 1 . F i n d P r o b ( w o o d e n and w h i t e ) . S t e p 2. T h e s e o u t c o m e s are not i n d e p e n d e n t since P r o b ( w o o d e n | white) T- Prob ( w o o d e n | not white) S t e p 3 . P r o b ( w o o d e n and w h i t e ) = Prob (wooden) x Prob (white | wooden) = 0.50 x 0.40 = 0.20 N o t i c e that the order i n w h i c h w e c o n s i d e r the o u t c o m e d o e s n ' t matter. T h u s , Prob (white and wooden) = Prob (white) x Prob (wooden | white) = 0.30 x 2/3 = 0.20 M a n y problems involve both A N D i n g and O R i n g of Outcomes. S u c h problems can be solved by successive applications of the rules we have b e e n using. 4

250

T h e C o m p l e t e Problem Solver

Example: W h a t is the probability of getting a 10 w h e n rolling a pair of d i c e ? Solution S t e p 1. F i n d P r o b (10) = P r o b [(4 and 6) or (5 and 5) or S t e p 2 . T h e A N D e d o u t c o m e s are i n d e p e n d e n t a n d t h e are m u t u a l l y e x c l u s i v e . S t e p 3 . P r o b (10) = [ P r o b (4 and 6) or (5 and 5) or (6 and 4)] = P r o b [(4 and 6) + P r o b (5 and 5) + P r o b (6 and = P r o b (4) x P r o b (6) + P r o b (5) x P r o b (5) + P r o b = 1/6 x 1/6 + 1/6 x 1/6 + 1/6 x 1/6 = 1/36 + 1/36 + 1/36 = 3/36 = 1/12

(6 and 4 ) ] . O R e d outcomes

4)] (6) x P r o b (4)