Academic Exchange Quarterly Summer 2004: Volume 8, Issue 2
An Instrument to Measure Mathematics Attitudes Martha Tapia, Berry College, GA
[email protected] George E. Marsh II, The University of Alabama
[email protected]
KEYWORD: MATH
Martha Tapia is assistant professor of mathematics education at Berry College where she teaches mathematics and mathematics education courses. Her research agenda includes attitudes toward mathematics, technology in education, and emotional intelligence.
George E. Marsh II is a professor of instructional technology in the Institute of Interactive Technology at The University of Alabama. He teaches research and technology courses. His research agenda includes attitudes toward mathematics, technology in education, and distance education.
ABSTRACT This article is a report of the development of a new instrument to measure students’ attitudes toward mathematics, and to determine the underlying dimensions of the instrument by examining the responses of 545 students. The data represent all grade levels and subjects of the secondary mathematics curriculum. The reliability coefficient alpha was .97. A maximum likelihood factor analysis with a varimax rotation yielded four factors: self-confidence; value of mathematics; enjoyment of mathematics; and motivation. Psychometric properties were sound and the instrument, Attitudes Toward Mathematics Inventory (ATMI), can be recommended for use in the investigation of students' attitudes toward mathematics.
Introduction Conventional wisdom and some research suggest that students with negative attitudes toward mathematics have performance problems simply because of anxiety. Attitudinal research in the field of mathematics has dealt almost exclusively with anxiety or enjoyment of subject matter, excluding other factors. One of the first instruments developed was the Dutton Scale (Dutton, 1954; Dutton & Blum, 1968), which measured “feelings” toward arithmetic. Unidimensional scales were developed by Gladstone, Deal, and Drevdahl (1960) and Aiken and Dreger (1961). Later Aiken (1974) constructed scales designed to measure enjoyment of mathematics and the value of mathematics. Multidimensional attitude scales were developed by Michaels and Forsyth (1977) and by Sandman (1980). Some researchers developed scales dealing exclusively with math anxiety. Examples of such scales are the Mathematics Anxiety Rating Scale (Richardson & Suinn, 1972), the Mathematics Anxiety Rating Scale–Revised (Plake & Parker, 1982) and the Mathematics Anxiety Questionanaire (Wigfield & Meece, 1988). The Fennema-Sherman Mathematics Attitude Scales (1976) were developed in 1976, and it has become one of the most popular instruments used in research over the last three decades. The Fennema-Sherman Mathematics Attitude Scales consist of a group of nine instruments: (1) Attitude Toward Success in Mathematics Scale, (2) Mathematics as a Male Domain Scale, (3) and (4) Mother/Father Scale, (5) Teacher Scale, (6) Confidence in Learning Mathematics Scale, (7) Mathematics Anxiety Scale, (8) Effectance Motivation Scale in Mathematics, and (9) Mathematics Usefulness Scale. Ashcraft and Kirk (2001) describe the common belief that because of “long-term avoidance of math, and their lesser mastery of the math that couldn't be avoided, high-math-anxiety individuals are simply less competent at doing math” (p. 224). The “competence explanation” is central to Fennema’s model (Fennema, 1989), which explains math performance as merely an interaction of affect (attitudes and math anxiety) and behavior during learning tasks. Ashcraft and Kirk regard this explanation as simplistic.
Fennema’s theory is based on research with the Fennema-Sherman Mathematics Attitudes Scales, which has clearly been the most popular instrument in research about attitudes toward math (Fennema & Sherman, 1976). The instrument is nearly thirty years old, has 108 items, and takes 45 minutes to complete. It purports to have nine scales, but subsequent research has questioned the validity, reliability (Suinn and Edwards, 1982), and integrity of its scores (O’Neal, Ernest, McLean, & Templeton, 1988). Melancon, Thompson, and Becnel (1994) isolated eight factors rather than nine, and they were unable to find a perfect fit with the model proposed by Fennema and Sherman. Mulhern and Rae (1998) identified only six factors, and suggested that the scales might not gauge what they were intended to measure.
Other researchers suggest that students may find math to be simply unappealing or socially unacceptable, although they may actually have high aptitude. In any case, it is crucial that any investigation of attitudes be assessed with an instrument that has good technical
characteristics if research conclusions are to be meaningful. The relationship of affect to course selection, performance, achievement, and cognitive processes must be based solidly on a valid, reliable measure of attitudes. Attitude scales must withstand factor analysis, tap important dimensions of attitudes, and require a minimum amount of time for administration. Finding a need for a shorter instrument with a straightforward factor structure, the Attitudes Toward Mathematics Inventory (ATMI) was developed.
Item Development The Attitudes Toward Mathematics Inventory was designed to investigate the underlying dimensions of attitudes toward mathematics. The 49-items of the ATMI were constructed in the domain of attitudes toward mathematics to address factors reported to be important in research. Items were constructed to assess confidence, anxiety, value, enjoyment, motivation, and parent/teacher expectations. Consideration was given to previous research as follows: 1. Confidence (Goolsby, 1988; Linn & Hyde, 1989; Randhawa, Beamer, & Lundberg, 1993). The confidence category was designed to measure students’ confidence and self-concept of their performance in mathematics. 2. Anxiety (Hauge, 1991; Terwilliger & Titus, 1995). The anxiety category was designed to measure feelings of anxiety and consequences of these feelings. 3. Value (Longitudinal Study of American Youth (1990). The value of mathematics category was designed to measure students’ beliefs on the usefulness, relevance and worth of mathematics in their life now and in the future. 4. Enjoyment (Ma, 1997; Thorndike-Christ, 1991). The enjoyment of mathematics category was designed to measure the degree to which students enjoy working mathematics and mathematics classes. 5. Motivation (Singh, Granville, & Dika, 2002; Thorndike-Christ, 1991). The motivation category was designed to measure interest in mathematics and desire to pursue studies in mathematics. 6. Parent/teacher expectations (Kenschaft, 1991; Dossey, 1992). The parent/teacher expectations category was designed to measure the beliefs and expectations parents and teachers have of the students’ ability and performance in mathematics
Method Subjects The subjects were 545 high school students, 302 boys and 243 girls, enrolled in mathematics high school classes, including 135 freshmen, 153 sophomores, 168 juniors, 84 seniors, and
five 8th-grade students. Only students taking mathematics were included in the sample. In situ classes were used in the sample.
Materials The ATMI was originally a 49-item scale. The items were constructed using a Likert-scale format with the following anchors: 1 strongly disagree, 2 disagree, 3 neutral, 4 agree, and 5 strongly agree. Twelve items were reversed, which were given the appropriate value for data analysis. The score was the sum of the ratings.
Procedure Teachers administered a 49-item inventory to the subjects during their classes. Four months later, the inventory was re-administered to 64 subjects who had previously taken the survey.
Results
To estimate internal consistency of the scores, Cronbach alpha coefficient was calculated. For scores on the 49 items alpha was .96, indicating a high degree of internal consistency for group analyses. Of the 49 items, 40 had item-to-total correlations above .50, the highest being .82. This suggested that most of the items contributed to the total inventory. The mean and standard deviation of the total score were 169.74 and 32.06 respectively. The standard error of measurement was 6.07.
The value of alpha was .96 for the 49 items, showing a high degree of internal consistency. An item deletion process was performed in order to increase the value of alpha. Items were deleted based on their item-to-total correlation. Nine items had correlations lower than .50. Items were deleted one at a time starting with the one with the lowest item-to-total correlation. After deleting these nine items, alpha reached a value of .97.
The revised inventory had a mean of 137.36, a standard deviation of 28.93 and a standard error of measurement of 5.28. All 40 items had item-to-total correlation above .50, with the highest being .82. This suggested that all items contributed significantly. The test items are homogeneous, which is interpreted to mean that they tend to measure a common trait
Nunally (1973) and Gorsuch (1983) maintain that factor analysis is essential to the evaluation of data and construct elaboration. Responses were subjected to a factor analysis using the maximum likelihood method of extraction and a varimax, orthogonal, rotation. Based on
Gorsuch’s recommendation (1983) to consider both the Kaiser-Guttman (Kaiser, 1970) criterion of retaining factors with eigenvalues greater than 1.0 and Cattell’s (1966) scree test. Four factors were retained, which accounted for 55% of the variance. The convergence criterion was satisfied after nine iterations. Table 1 shows factor loadings, eigenvalues, and percentage of variance for the four-factor solution.
Table 1 Exploratory Factor Analysis of the Attitudes Toward Mathematics Inventory: A Four-Factor Solution
Item number Estimates
Factor I
Factor II
Factor III
Factor IV Final Communality
___________________________________________________________________________ 12 11 14 16 .35 10 22 18 17 21 19 9 15 23 49 20 1 7 5 6 2 48 38 4 37 8 30 25 31 32
.77 .76 .75 .74
.30 .19 .22 .21
.13 .06 .13 .09
.08 .04 .09 .18
.71 .61 .63 .64
.74 .74 .67 .67 .65 .65 .64 .63 .63 .59 .53 .13 .18 .14 .14 .16 .17 .20 .18 .23 .15 .43 .38 .33 .26
.29 .19 .10 .13 .20 .02 .24 .35 .14 .20 .15 .76 .68 .62 .61 .59 .58 .55 .54 .54 .51 .37 .36 .33 .51
.10 .21 .37 .29 .18 .34 .35 .40 .39 .40 .36 .14 .11 .15 .10 .18 .14 .19 .21 .11 .23 .64 .56 .51 .50
.12 .09 .19 .14 .15 .23 .07 .16 .28 .28 .19 .14 .14 .05 .14 .24 .33 .13 .01 .36 .04 .21 .19 .14 .17
.66 .64 .64 .57 .53 .60 .59 .70 .64 .63 .48 .64 .53 .43 .42 .46 .49 .40 .37 .49 .34 .77 .62 .50 .61
27 .33 3 .17 26 .32 28 .26 42 .41 41 .36 33 .27 34 .26 35 .35 24 .48 29 .38 Eigenvalues 7.71 Percentage variance 41.14
.35 .45 .41 .24 .21 .15 .35 .44 .37 .29 .39 3.25 28.55
.46 .45 .45 .38 .37 .32 .23 .25 .46 .30 .22 1.42 18.98
.29 .03 .13 .17 .11 .07 .68 .67 .41 .40 .38 1.30 11.33
.53 .44 .49 .30 .36 .26 .72 .78 .64 .57 .49
________________________________________________________________________
The factor structure of the ATMI covers the domain of attitudes toward mathematics, providing evidence of content validity. Content validity was established by relating the items to the variables: confidence, anxiety, value, enjoyment, and motivation. This structure is explained by the four-factor model supporting different interpretations for students’ selfconfidence, value, enjoyment and motivation as underlying dimensions of attitudes toward mathematics. Table 2showssample items from each of the factors. The complete inventory is available from the first author upon request.
Table 2
Sample items by factors ___________________________________________________________________________
Item Number and Content by Factor ___________________________________________________________________________ Self-confidence 11. Studying mathematics makes me feel nervous. 14. I am always under a terrible strain in a math class. 19. I am able to solve mathematics problems without too much difficulty. Value
5.
Mathematics is important in everyday life.
6.
Mathematics is one of the most important subjects for people to study.
7.
High school math courses would be very helpful no matter what I decide to study,
Enjoyment 25. I have ussually enjoyed sutdying mathematics in school. 26. Mathematics is dull and boring. 31. I am happier in a math class than in any other class. Motivation 29. I would like to avoid using mathematics in college. 33. I am willing to take more than the required amount of mathematics. 34. I plan to take as much mathematics as I can during my education.
In factor analysis, the four-factor solution provided the best simple structure, so four factors were retained. Two of the original six variables were combined to form a single factor, anxiety and confidence, a result also reported by O’Neal, Ernest, McLean & Templeton (1988), Melancon, Thompson, & Becnel (1994) and Mulhern and Rae (1998). One variable was irrelevant due to low correlations, parent/teacher expectations.
Having retained four factors, Cronbach alpha was calculated to estimate internal consistency and reliability of the scores on the subscales. Factor I contains 15 items with a mean of 51.10(SD = 13.13). Factor I is characterized by students’ self-confidence (Self-confidence factor). Items in this factor were derived from those generated for the anxiety and confidence categories. The scores for these items had a Cronbach alpha of .95. Factor II contains 10 items with a mean of 38.37 (SD = 6.74), the value of mathematics factor. These items produced a Cronbach alpha of .89. Factor III contains 10 items with a mean of 31.91 (SD = 8.06). Factor III is characterized by enjoyment of mathematics. The scores on these items produced a Cronbach alpha of .89. Factor IV contains 5 items with a mean of 15.99 (SD = 4.95), the motivation factor. These items, when scored and summed, produced a Cronbach alpha of .88. These data indicate high level of reliability of the scores on the subscales.
The Pearson correlation coefficient was used for test-retest reliability in a four-month followup of the 40-item inventory, administered to 64 students who had previously taken the survey.
The coefficient for test-retest for the total scale was .89, and coefficients for the subscales were as follows: Self-confidence .88; Value .70; Enjoyment .84; and Motivation .78.These data indicate that the scores on the inventory and the subscales are stable over time.
Discussion Four subscales were identified as self-confidence, value, enjoyment, and motivation. Scores on the 40-item scale developed through factor analysis showed good internal reliability, and test-retest reliability showed stability over time. With only 40 items, the estimated time to complete the instrument ranges from 10 to 20 minutes.
Deletion of the parent/teacher items was surprising. In previous research, attitudes of parents and teachers about math have been regarded as extremely important, even to the extent that some studies suggest that a teacher’s or parent’s attitude can motivate or discourage students from pursuing math or may encourage them to do so (Dwyer, 1993; Kenschaft, 1991; Shashaani, 1995). Nonetheless, these items were dropped because of extremely low item-tototal correlations, which requires some consideration.
Kenschaft (1991) reported that parents’ support or lack of support is an important in students’ attitudes and participation in mathematics instruction. Similarly, Dossey (1992) considered teachers to play an important role in shaping attitudes toward mathematics. More recent theories about the influence of adults on children have focused attention on peer group effects. For example, Harris (1995) concluded that peer affiliations become increasingly more influential on shaping attitudes than parents and teachers. Effects vary widely from one sibling to another within the same family (Maccoby & Martin, 1983, p. 82). Wilder (1986) reported that peer group members themselves are responsible for group contrast effects, forming attitudes, behaviors, dress codes, manners, and other social behaviors. Thus, within a particular peer group, attitudes toward educational aspirations are likely to be similar; the attitudes of their parents (if they belong to the same social network) will also be similar; but correlations between individual parents and their children should be insignificant. Day and colleagues (1992) and Kindermann (1993) reported precisely this result. If children are greatly influenced by their peers, they may avoid the pursuit of mathematics if the peer group regards it negatively for any reason.
While it would be absurd to contend that parents have no influence on their children’s attitudes toward math or any other subject matter, it is clear from this sample that parental influence did not hold up. Perhaps this sample was atypical with regard to parent/teacher expectations, and perhaps not. Therefore, the instrument should be tested with a more representative sample. It is also possible that parents and teachers have varying degrees of influence at different developmental ages, a factor that should be seriously considered in future research. Manner of speech, the clothes children and teens wear, and even the schools they attend can be socially beneficial or stigmata in the youth culture. The extent to which a
child’s peer culture values or denigrates mathematics and careers associated with it may greatly determine a child’s choices.
The study was conducted with adolescents, so the ages and characteristics of the subjects in this study cannot be accepted as “normative” because controls were not applied for demographic data such as gender, ethnicity, achievement, and so forth. However, useful information can be obtained corresponding to a variety of demographic classifications because the four scales measure distinct aspects of attitudes toward mathematics.
Attitudinal research should concern more than anxiety and competence, because it is clear that other factors are also important. Although there is a substantial body of research about attitudes toward mathematics, much of it is based on results with tools developed prior to current statistical standards for instrument development. In the meanwhile, factor analysis has matured as a method to examine interrelationships among a number of variables with minimal loss of information. The ATMI was constructed using these standards and may be an efficient and effective research tool to assess factors that influence expectations and performance in math because of its content validity, reliable factor scores, test-retest reliability, and brevity.
Efforts to improve math instruction over the last decade has degenerated into a debate about traditional or constructivist teaching methodologies, the kind of instructional materials to use, including or banishing calculators, ways to improve teacher training, and the best sequencing of math courses in the curriculum. Far less attention has been directed to the investigation of student attitudes. Although there is a body of research about attitudes toward mathematics, most of it is concerned only with anxiety. Most of this research is also based on results derived from instruments that predated modern statistical standards for factor analysis that currently guide the examination of interrelationships among variables. Use of the ATMI may be important for teachers and researchers, because success or failure in math performance is greatly determined by personal beliefs. Regardless of the teaching method used, students are likely to exert effort according to the effects they anticipate, which is regulated by personal beliefs about their abilities, the importance they attach to mathematics, enjoyment of the subject matter, and the motivation to succeed.
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