A Continuous Assessment Scheme for Statistics Courses

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A continuous assessment scheme for statistics courses for social scientists F. R. Jolliffe

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Department of Statistics and Operational Research , Brunei University , Uxbridge, Middlesex Published online: 09 Jul 2006.

To cite this article: F. R. Jolliffe (1976) A continuous assessment scheme for statistics courses for social scientists, International Journal of Mathematical Education in Science and Technology, 7:1, 97-103, DOI: 10.1080/0020739760070115 To link to this article: http://dx.doi.org/10.1080/0020739760070115

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INT. J. MATH. EDUC. SCI. TECHNOL., 1976, VOL. 7, NO. 1, 9 7 - 1 0 3

A continuous assessment scheme for statistics courses for social scientists by F. R. JOLLIFFE Department of Statistics and Operational Research, Brunei University, Uxbridge, Middlesex

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(Received 27 June 1975)

This paper discusses the pros and cons of assessment schemes used in statistics courses for social scientists. It then goes on to describe the use of essays and of work-books in continuous assessment. Essays would be on social science topics, but with a statistical or quantitative content. Examples for the work-books would use real data of relevance to the students' main studies, be fairly open-ended, and discussed in classes, so that students would write up attempts at them in a work-book in a similar manner to writing up practical work. The scheme described is thought to have advantages over other assessment schemes, to provide motivation to students, and to have an important part to play in the teaching process.

1. Introduction The teaching of statistics to social science students is a topic which generates many long and frequently heated discussions. Two of the main problems from the teacher's point of view are that in general the students lack mathematical expertise, and to some extent linked with this, they lack motivation to learn statistics [1]. Another related problem is that of examinations. It is sometimes thought that the stress of learning for a written examination in statistics would destroy any slight interest in the subject that students might have had. In addition, if the course is of a traditional introductory type [2-4], and the examination is mainly numerate in character, results obtained by the less mathematically able students will tend to be poor, and passes in the examination as much due to luck as understanding. A study by the author of marks obtained in the statistics examination, by first year social science undergraduates at Southampton University in two consecutive years, showed that mostly, the marks of students who had not taken a mathematics subject at GCE Advanced Level fell around the pass mark, some below and some above, and were on the whole lower than the marks of the other students. One idea is to make statistics courses for social science students non-examinable, and let the students enjoy the subject without the worry of whether or not they have any ability at it. The objection to this is that passing the examination is often the only motivation students have for attending the course. If there were no examination the majority of students would opt out completely, since they find the subject difficult and do not, at an early stage in their studies, see its relevance to their disciplines. An alternative is to examine students by

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means of continuous assessment, that is, grade the students on work set to do in their own time at intervals throughout the course. This ensures that the student makes some attempt to follow the course, and provides a measure of his attainment on it. In addition, continuous assessment can be used as a means of illustrating the use of statistics in the students' main area of study, and can be made an integral part of the teaching process. 2. Continuous assessment Continuous assessment is in itself open to abuse, however, and one of the main objections to its use in elementary statistics courses, is that it is difficult to tell to what extent the contribution handed in by the student for marking is his own work. When the same wrong notation, or the same slip in arithmetic, occurs in several students' work we suspect that students have been blindly copying one another, rather than working together honestly, which should be encouraged. If the mistakes are sufficiently bad, copiers and copied will all fail, which is probably in accordance with their ability. The worry is that some weak students will be lucky enough or clever enough to copy work which is correct, and this will usually be undetected, so that students who have little or no statistical knowledge will be passed as having some degree of competence. Several procedures are available for removing doubts as to the validity of a pass list in the event of students copying one another. Some involve an element of written examinations, such as combining assignments done in the student's own time with quick tests in class, or requiring some or all assignments to be done under examination conditions, or having continuous assessment as part of an assessment procedure which includes also a written examination of the conventional type. But all of these ideas bring back the problems of examinations which the continuous assessment procedure was intended to solve. Another alternative is to set students the same task, but with different data, so that they might discuss methods together, but are more likely to do their own thinking and arithmetic. This means ensuring that the assignments set to different students are comparable as regards interest, level of difficulty, and amount of computation—no small task. Note too that although there will then be many different examples to discuss, there will be no one example which all students have studied. The best procedure perhaps, is to remove the environment which makes students find it necessary to copy. To achieve this, give students sufficient help in tutorials so that each is able to produce a reasonably satisfactory attempt at work set, establish through small group teaching a good relationship with students, and make the computational aspects of assignments minimal. Last, but not least, make the assignments sufficiently interesting that each student would feel motivated to attempt them and even follow up initial problems along lines of particular appeal to him. Continuous assessment is used, and fairly successfully, in social science and arts subjects where assignments consist mainly of essays, and in experimental science where assignments involve practical work and its written presentation. Statistics assignments can usefully borrow some of the best ideas of these methods. Possibly the main advantage of the essay as a means of assessing students, is that it is open-ended in its requirement. All students are able to attempt

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it, and very good students may, if they wish, write something in the nature of a monograph. As a teaching instrument, setting essays is a means of sending students to books and papers and ensuring that they read round a subject, and even delve deeper into it. From the students' point of view, if the essay topics are interesting, and seem to be relevant to their studies, they will need no other motivation to make honest and willing attempts at the assignments. The carrying out of practical work involves close contact between teachers and students, and experiments illustrate points of theory and provide motivation. Written up records of the practicals are useful to the student as illustrations of methods, and enable the teacher to assess what the student has gained from the practical. Practical work and its written presentation are essentially continuous in nature. 3. Quantitative essays The idea of setting students essays can be taken over directly into statistics assessment, with the proviso that the essays submitted should have a quantitative or statistical content. For given topics the student would be expected to find relevant data, present the data as tables and/or diagrams, discuss the facts using summary or other measures as far as possible, and suggest further analysis where appropriate. The emphasis in the essay would be on the ' numbers ', whereas in essays written for the students' main subjects the emphasis would be on the ' words '. From suggestions of broad subject areas of possible interest the students (and possibly teachers of the students' main subjects) can be asked to guide the statistics lecturer to specific topics. It will then be necessary to check that appropriate data and publications are available in the library, before deciding on a final short-list of topics. Student participation in this way should ensure initial motivation to do the assignment. Essay titles set to Brunei University Government and Sociology students in 1974/5 as a result of discussion with the students were (a) What do local authorities spend on different parts of the education service? (b) Is defence a major item of government expenditure? (c) A study of trade union membership. (d) The measurement of poverty. Perhaps the main difficulty in setting an essay of this type is in guiding students as to exactly what is expected of them. At the one extreme a student might hand in 20 pages of prose and argue that his essay is quantitative because it includes a couple of numerical facts. At the other extreme a student might hand in one table of data, represented also in diagram form, and argue that this is an essay because he has written one explanatory sentence. It might be advisable to give students examples of papers or books where the mix between words and statistical aspects is about right for a quantitative essay, possibly taking these from the reading list issued to students with the essay titles. On the whole, the nearer to the end of the statistics course that the essay is written, the better will the student be able to make use of his knowledge of statistics in preparing it. On the other hand, the author's experience suggests that students tend to enjoy an assignment of this nature and find it relatively

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easy, and an essay might be timed near the start of the course to generate an interest in statistics, or even in the middle of the course when students are beginning to feel the strain of exposure to numeracy. 4. Work-books Statistics is essentially a practical subject, and experimental work can be made an important part of the course, and used in a continuous assessment procedure. However, the extent to which it is feasible to do this is open to question [5]. Experiments can be very time consuming. The course given by Jowett and Davies [5] at Sheffield University had 3 hours of lectures, and 5 hours of practical, every week throughout the year. Compare this with the introductory courses given to social science students, which are typically 10 to 20 hours of lectures in total, with possibly the same number of tutorials or classes. Devoting more than 4 or 5 hours of this to practical work would mean a serious reduction in the content of the syllabus. Increasing the amount of time available for statistics, which is usually thought to be of secondary importance in the student's studies, is not often a realistic possibility. There is also the question of the suitability of experiments. Social science students are typically very concerned about the relevance of statistics courses to their main subjects. One feels that elaborate scientific experiments would not appeal, and coin or die tossing exercises would be thought a waste of time once the novelty had worn off. There is a good case for judicious use of films [6] as a substitute for experiments. The obvious exception to the non-relevance of experiments is the social survey, but a survey would usually be a group exercise over a lengthy period of time, and would not easily fit into an assessment procedure. The next best thing, perhaps, is to use the results, and study the methods, of someone else's survey. Many social science investigations are published, and many official statistics, some collected from surveys, are of direct relevance to social scientists. Instead of spending time performing an experiment, the student might spend time finding published data, or he can be presented with the data if the teacher feels this would be more efficient. The time which would be spent in practicals discussing analysis, and presentation of results, is now spent on discussing the interpretation of the published figures. Students would write up class discussions in a work-book, and carry out suggested computations and any additional ones they thought necessary. Ideally, the examples sheets used as a basis for the work-book should be open-ended in their requirements, so that the student has a chance to show his individual ability and understanding of methods. Thus there would be many correct solutions, and the risk of students copying one another's work is avoided to some extent. The student is also encouraged to think about the suitability of different measures and techniques to the data under consideration. For example, instead of giving the student a small frequency distribution with an instruction of the type ' Calculate the arithmetic mean and standard deviation ', the student might be given a table involving several frequency distributions, such as, say, income ranges by number of members in the household, and an instruction 'Discuss fully, using diagrams and summary measures as appropriate'. It would be stressed that comments on the data explaining the meaning of the

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figures were as important as elaborate calculations. The students would also be encouraged to find other information relating to data on the sheets. A disadvantage of leaving students with very open-ended instructions is that a student who writes good comments, perhaps calculates a few percentages, and draws a diagram, has produced a piece of work which is worthy of a pass, and yet has not shown that he has any real knowledge of statistics. This may be unsatisfactory from the point of view of the lecturers in the student's main subject areas, who might be particularly anxious that their students are able to perform various routine calculations at the end of the course. Thus it may be necessary to indicate to the student, for each example sheet, what specific things he is expected to do. This could be done in classes by leading students in the right direction, say towards finding an arithmetic mean. Alternatively, or possibly as well, the open-ended instructions with each example could include a comment such as ' Your write-up of this sheet must demonstrate that you are able to calculate an arithmetic mean and interpret it'. Any student who did not attempt the task required—here calculation of an arithmetic mean— would fail that assignment. Admittedly assessing students' work will take longer, and be more difficult, if they hand in individual attempts at the exercise sheets, than if they all perform an identical specified calculation, but since the kinds of things they might do are limited in number and relatively simple (bearing in mind that we are discussing an introductory course for social science students) it is not too great a task for the lecturer to prepare a fairly complete set. In any case there is quite a good argument for having results of possible computations available to the students on request, to enable them to spend more time on interpretation than arithmetic [7]. In general, comparability of different students is tricky, and it is hard to say whether a student who finds a small number of relevant statistics should be commended for his parsimonious summary of the data, or condemned for his laziness or inability to produce more. It might be best to mark each student's contribution as a piece of work in its own right, and grade it fairly subjectively, instead of trying to quantify just what the student had done. Standards can be maintained through a set of work since one quickly develops a feel for what constitutes a particular grading. In cases of doubt cross-checks can be made with work of about the same standard. With assignments of this nature, which are discussed in class and attempted over a period of time, it would be nice to free the students from constraints of having to hand in assignments by given dates, and, say, require to see the workbooks for grading only at the end of each term. However, since in statistics the learning comes through doing, and students who do not keep up to date with assignments will soon find the lectures incomprehensible, it might be better to collect assignments more frequently, say every 2 or 3 weeks. Often the social science lecturers will be able to suggest areas of interest, or publications, which they would like their students to look at in their statistics course, and these could possibly be used in the work-book assignments. Sometimes a student might suggest a topic, and, if so, he is more likely to be motivated to attempt some statistics in connection with it than if it is chosen for him. However, not all suggestions will be suitable, in the sense that they might be of no use for illustrating a particular statistical technique. Qualitative tables

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may be interesting, but they cannot be used for teaching anything about grouped frequency distributions. So at the end of the day it may still be left to the statistics teacher to find examples. 5. Conclusions Both the quantitative essay and the work-book, as described above, have a considerable number of points in their favour as a means of continuous assessment of social science undergraduates, and as an aid in teaching elementary statistics to these students. Both can be used to demonstrate the usefulness of statistics to areas of social science interest, and to some extent follow suggestions that statistics courses for social scientists should be directed towards problems arising in their disciplines [8]. They are akin to problem-oriented methods of teaching statistics [9], but since they are part of a more formal course, avoid some of the difficulties caused by the unstructured nature of the former. They avoid too much stress on numeracy, and are very much in accordance with a commonly occurring aim, which is to teach the students enough for them to be able to read and understand the statistics relating to their own fields. A separate, but related aim, is that the students should learn about relevant sources of statistical data. The essay encourages students to read about sources of data and to find and use data for themselves. The work-book exercises are more useful in teaching students about the existence of sources, by giving them information taken from different publications. Similar ideas are used in one of the Open University courses [10] which uses sources of data as the motivation for learning statistical skills. The method of continuous assessment described in this paper would seem to be more satisfactory than the commonly used methods, in that less numerate students are not handicapped by it, and less interested students are more likely to make honest attempts at it. Apart from its use in measuring students' ability and knowledge, it has a positive contribution to make as a means of motivating students, and of teaching them facts and methods of direct use to them as social scientists. In addition, the kind of programme described is less demanding for the teacher, in time of preparation and instigation, than some of the more ambitious schemes sometimes advocated. The author tried a scheme of this type with Social Science students at Brunei University in 1974/5, and some of the recommendations made in this paper are a direct result of experience in implementing it. It was sufficiently successful that it will be used again next session. The students will, in fact, for administrative reasons, be assessed by an examination, but this provides an opportunity for comparing the two methods of assessment, and of developing an examination along the lines of the assignments. References [1] [2] [3]

KALTON, G., 1973, Int. J. Math. Educ. Sci. Technol., 4, 7. BLALOCK, H. M., 1972 Social Statistics, second edition (New York: McGraw Hill). JOLLIFFE, F. R., 1974, Commonsense Statistics for Economists and Others (London,

[4]

YEOMANS,

Boston: Routledge & Kegan Paul). K. A., 1968, Statistics for the Social Scientist, Vols. 1 and 2 (Harmondsworth: Penguin Books).

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[5] [6] [7]

JOWETT G. H., and DAVIES, H. M. I960, Jl. R. statist. Soc. A, 123, 10. AUSTWICK, K., HINE, J., and WETHERILL, G. B., 1971, Rev. Int. statist. Inst., 39, 287. FOSTER, F. G., and SMITH, T. M. F., 1969, Appl. Statist., 18, 264. [8] ROSENBAUM, S., 1971, Jl. R. statist. Soc. A, 134, 534. [9] COMMITTEE ON THE UNDERGRADUATE PROGRAM IN MATHEMATICS, 1972, Introductory

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[10]

Statistics without Calculus (California: Mathematical Association of America). 1975, Statistical Sources D291 (Milton Keynes: Open University).

OPEN UNIVERSITY,

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