Psychological Assessment 2014, Vol. 26, No. 4, 1225-1234
© 2014 American Psychological Association 1040-3590/14/$ 12.00 http://dx.doi.org/10.1037/a0037688
Measurement of Math Beliefs and Their Associations With Math Behaviors in College Students Helen M. Hendy, Nancy Schorschinsky, and Barbara Wade Penn State University, Schuylkill Campus
Our purpose in the present study was to expand understanding of math beliefs in college students by developing 3 new psychometrically tested scales as guided by expectancy-value theory, self-efficacy theory, and health belief model. Additionally, we identified which math beliefs (and which theory) best explained variance in math behaviors and performance by college students and which students were most likely to have problematic math beliefs. Study participants included 368 college math students who completed questionnaires to report math behaviors (attending class, doing homework, reading textbooks, asking for help) and used a 5-point rating scale to indicate a variety of math beliefs. For a subset of 84 students, math professors provided final math grades. Factor analyses produced a 10-item Math Value Scale with 2 subscales (Class Devaluation, No Future Value), a 7-item single-dimension Math Confi dence Scale, and an 11-item Math Barriers Scale with 2 subscales (Math Anxiety, Discouraging Words). Hierarchical multiple regression revealed that high levels of the newly discovered class devaluation belief (guided by expectancy-value theory) were most consistently associated with poor math behaviors in college students, with high math anxiety (guided by health belief model) and low math confidence (guided by self-efficacy theory) also found to be significant. Analyses of covariance revealed that younger and male students were at increased risk for class devaluation and older students were at increased risk for poor math confidence. Keywords: math beliefs, math value, math confidence, math anxiety, math performance
Many American college students have problems completing the math courses that are required for their professional goals (Ash craft & Faust, 1994; Chipman, Krantz, & Silver, 1992; Scarpello, 2007). Intervention efforts to improve college student math per formance would be enhanced by identification of math beliefs most associated with poor math behaviors so that these problem atic beliefs could be targeted for change. Therefore, our purpose in the present study was to expand understanding of math beliefs in college students with the development of three new psychometri cally tested scales guided by expectancy-value theory, selfefficacy theory, and health belief model. Additionally, the purpose of the present study was to identify which math belief (and which theory) best explained college student investment in recommended math behaviors (such as attending class, doing homework, reading the textbook, and asking for help when confused) as well as their final math course performance. Finally, the present study planned to examine which student demographics (age, gender, and ethnic ity) were associated with high risk for these most problematic math beliefs.
One of the most widely studied math beliefs has been the college student’s perception of “math anxiety” (Hopko, 2003; Plake & Parker, 1982; Richardson & Suinn, 1972), which is often described as a feeling of tension, confusion, and frustration when attempting to do math problems. Meta-analysis results have doc umented that such math anxiety is negatively correlated (r = -.27) with math grades in school-age children (Hembree, 1990; Ma, 1999), explaining a relatively small 7.3% of the variance in student math performance (-.27 X -.27 = R2 = percentage of variance explained), with similar patterns across student gender and ethnic ity. Three examples of math anxiety measures include the 98-item Mathematics Anxiety Rating Scale (MARS; Richardson & Suinn, 1972), the 24-item Revised MARS (Plake & Parker, 1982), and the 12-item revised MARS (MARS-R; Hopko, 2003). Besides the affective component of math anxiety in student math beliefs, other measures have been developed to capture the more cognitive components of math value and math confidence. Two examples of these measures include the 20-item Perceived Usefulness of Math ematics Scale (Pajares & Miller, 1994) and the five-item Math Confidence Subscale identified within the 180-item SelfDescription Questionnaire (Marsh & O’Neill, 1984). Most of the existing math belief measures have not included complete psychometric examination such as factor analysis, tests for goodness of fit, internal reliability, test-retest reliability and support for convergent or criterion validity (see Table 1.) How ever, the most important limitation of existing measures is that they have not yet been compared for their ability to explain math behaviors often recommended for college students to improve their math performance such as attending class, doing homework, read-
This article was published Online First August 18, 2014. Helen M. Hendy, Psychology Program, Penn State University, Schuylkill Campus; Nancy Schorschinsky, Mathematics and Chemistry, Penn State University, Schuylkill Campus; Barbara Wade, Psychology Program, Penn State University, Schuylkill Campus. Correspondence concerning this article should be addressed to Helen M. Hendy, Psychology Program, Penn State University, Schuylkill Campus, 200 University Drive, Schuylkill Haven, PA 17972. E-mail: [email protected]
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Table 1 Psychometric Examinations Conducted fo r Previously Available Math Belief Measures Math belief measure
No. of items
Mathematics Anxiety Rating Scale (Richardson & Suinn, 1972) Mathematics Anxiety Rating Scale-Revised (Plake & Parker, 1982) Mathematics Anxiety Rating Scale-Revised (Hopko, 2003) Perceived Usefulness of Mathematics (Pajares & Miller, 1984) Math Confidence (Marsh & O’Neill, 1984)
98 24 12 20 5
ing the textbook, and asking for help when confused. Also, with the most commonly known factor, math anxiety, explaining only 7.3% of the variance in math performance (Hembree, 1990; Ma, 1999), other beliefs may exist that could account for additional variance in college student math behaviors. Understanding a broader range of math beliefs associated with college student neglect of recommended math behaviors would enhance the design of interventions to improve student success in college math courses.
Theories Suggesting Math Beliefs Various theories suggest possible math beliefs held by college students that could affect their investment in recommended math behaviors to improve math performance. For example, the expectancy-value theory (Wigfteld & Eccles, 1992) suggests that some students may not believe that these recommend math behav iors or the mathematical understanding they provide are valuable for their immediate or long-term goals. Students are unlikely to invest much time in day-to-day math behaviors (attending class, doing homework, reading textbooks, asking for help, or monitor ing one’s grade in the class) if they see them as unrelated to their understanding of math concepts, their final course grade, gradua tion from college, future employment, or future financial well being. Alternatively, the self-efficacy theory (Bandura, 1997) suggests that even if students believe that the math behaviors help advance their immediate or long-term goals, they may not have the confi dence (or self-efficacy) that they can do these behaviors. Sources of an individual’s self-efficacy for a specific behavior have been identified by Bandura as including past experience with the be havior, the presence of powerful or peer models for the behavior, verbal commentary from others about the individual’s skill at the behavior, and physiological conditions that help or hinder the behavior (such as difficulty with anxiety or anger). Finally, the health belief model (Janz & Becker, 1984) was originally developed to explain why some individuals fail to per form recommended health-related behaviors (such as exercise or smoking cessation), but it may also help explain why some stu dents fail to perform recommended math behaviors. This theory suggests that besides the perceived value of the suggested behavior (as proposed by expectancy-value theory) and their perceived confidence in doing the behavior (as proposed by self-efficacy theory), perceived barriers to performing the behavior may pre vent individuals from moving forward with the action. For exam ple, individuals may perceive benefits associated with joining a zumba exercise class, and they may perceive themselves as capa ble of performing zumba moves, but they would be unlikely to
Factor analysis —
Yes Yes —
Yes
Goodness of fit — —
Yes —
—
Internal reliability
Test-retest reliability
Support for validity
.97 .98 .94 .93 .89
.85
Yes Yes Yes — Yes
— —
— —
show this behavior if they also perceive barriers such as having a history of failure at learning new dance skills, hearing discourag ing words from others about their dancing abilities, or worrying about the time it takes away from other pursuits; having no parent or peer models for dancing; or experiencing excess anxiety at performing dance moves in front of strangers. Similarly, perceived math barriers could come from having a history of failed math courses, from hearing discouraging words from others about one’s math ability (Aunola, Nurmi, Lerkkanen, & Rasku-Puttonen, 2003; Baker, Gersten, & Lee, 2002; Fumer & Duffy, 2002), from the absence of parent or peer models for effective math behaviors (Hamm & Faircloth, 2005; Mayfield & Vollmer, 2007), or from physiological and emotional reactions to math such as feeling nervous, tense, frustrated, angry, or distracted (Ashcraft & Krause, 2007; Beilock, 2008; Ho et al„ 2000).
Purpose of Present Study Our purpose in the present study was to expand understanding of math beliefs in college students by developing three new psychometrically tested measures guided by expectancy-value theory (Wigfteld & Eccles, 1992), self-efficacy theory (Bandura, 1997), and health belief model (Janz & Becker, 1984). We also sought to identify which math beliefs (and which theory) best explained variance in recommended math behaviors and performance by college students. A third purpose was to identify student demo graphics associated with the most problematic math beliefs. More specifically, the three new math belief measures planned were a Math Value Scale (MVS) derived from concepts of valueexpectancy theory, a Math Confidence Scale (MCS) derived from concepts of self-efficacy theory, and a Math Barriers Scale (MBS) derived from concepts of health belief model. After exploratory factor analysis identified dimensions within each math belief mea sure, the additional psychometric examinations planned were cal culation of goodness-of-fit values, internal reliability, test-retest reliability, and support for convergent validity with one of the most widely used existing measures of math anxiety (Math Anxiety Rating Scale-Revised, or MARS-R, Hopko, 2003). Then, after dimensions and subscales were identified within the three new math belief measures (MVS, MCS, and MBS), they would be compared for their ability to explain unique portions of the vari ance in recommended math behaviors and performance for college students (attending class, doing homework, reading textbooks, asking for help, and final grade earned), after controlling for demographic variables that included age, gender, and ethnicity. Finally, analyses were planned to examine how each math belief was associated with these demographic variables to identify which
MATH BELIEFS
students were at increased risk for the most problematic math beliefs. Interventions to improve college student math performance would be enhanced by the availability of the new psychometrically sound measures of math beliefs provided by the present study and documented to explain variance in math behaviors and perfor mance. Such measures could be used by math professors to iden tify students with math beliefs that put them at high risk for poor math behaviors and performance and then to guide these students to monitor associations among their math beliefs, math behaviors, and math course performance.
Method Participants Study participants included 368 college students from a univer sity campus in south-central Pennsylvania serving students from the local small-town communities, students from the large metro politan areas of Philadelphia and New York, and international students. These 368 college math students were recruited from 13 algebra classes (170 men, 195 women, 3 individuals with no gender reported; mean age = 19.7 years, SD = 3.6; 55.1% White, 37.7% African American, 2.5% Hispanic-Latino, and 1.1% Asian American). Responses from these math students were used in most of the analyses of the present study (such as exploratory factor analysis, descriptive statistics, goodness of fit, internal reliability, and analyses of covariance [ANCOVAs] to examine demographics asso ciated with math beliefs). The math classes were sampled during class time, and there was a response rate of 96.1% of the students completing the questionnaire of the present study. Subsets of the 368 math students were available for specific psychometric evaluations of the new math belief scales. For ex ample, one subset of 50 math students was a convenience sample given the questionnaire on two occasions, at the Week 8 and Week 12 of class, to allow us to calculate test-retest reliability (22 men, 28 women; mean age = 19.1 years, SD = 1.9; 56% White, 40% African American, and 2% Hispanic-Latino). Another subset of 50 math students was a convenience sample that also completed the MARS-R (Hopko, 2003) to examine convergent validity for any subscale of math anxiety that might emerge within the new math belief scales. A third subset of 84 math students was a convenience sample for which teachers provided student final math grades and ratings for student attendance to evaluate the validity of student self-reports and to examine how student math beliefs were asso ciated with final math performance (39 men, 44 women; mean age = 19.2 years, SD = 2.2; 62% White, 32% African American, and 1% Hispanic-Latino). Teachers reported final grades as the percentage of possible points earned by the student (0%-100%) and attendance as the percentage of class meetings the student was present 0%—100%).
Procedures The 368 math students were asked at approximately Week 8 in their 15-week semester to complete a questionnaire that asked for demographic information (age, gender, and ethnicity) and selfratings for a number of math behaviors. For example, they were asked to estimate the percentage of math classes attended (10%,
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20%, 30%, 40%, 50%, 60%; 70%, 80%, 90%, or 100%), the percentage of homework assignments completed (using the same ratings), the number of days per week they used their math textbook, and how often they asked someone for help when con fused about math (1 = almost never, 2 — rarely, 3 = sometimes, 4 = often, 5 = almost always). The subset of 50 math students whose responses were used to evaluate convergent validity for any math anxiety dimension within Perceived Math Barriers was also asked to rate how often (1 = almost never, 2 = rarely, 3 = sometimes, 4 = often, or 5 = almost always) they experienced anxiety in each of the 12 math situations listed in the MARS-R (Hopko, 2003). Students were also asked to use a 5-point rating (1 = strongly disagree, 2 = mildly disagree, 3 = do not know, 4 = mildly agree, and 5 = strongly agree) to report their math beliefs, grouped into three sets, which were used to develop the three new math belief scales: (a) For the 33 items considered for the Math Value Scale (MVS), we were guided by the value-expectancy theory (Wigfield & Eccles, 1992). The 33 possible MVS items were selected to include both short-term and long-term consequences of the stu dents’ math behaviors such as improved math understanding and math grades, improved chances of employment, greater success in their chosen professional future, and improved future financial well-being, (b) For the seven items considered for the Math Con fidence Scale (MCS), we were guided by the self-efficacy concept from self-efficacy theory (Bandura, 1997). The items were se lected to include math behaviors for which students might be expected to vary in confidence such as earning good top grades, earning passing grades, understanding math concepts, learning math concepts on their own if they missed class, and practicing math homework problems, (c) For the 28 items considered for the Math Barriers Scale (MBS), we were guided by the health belief model (Janz & Becker, 1984) proposal that perceived barriers are often the strongest predictor that a person will perform a given behavior, regardless of the “perceived benefit” or “perceived selfefficacy” for that behavior. The 28 possible barriers included their past experiences with math courses, available social models for successful math performance, verbal commentary about their math skills from important others, and physiological factors they expe rience that may distract them from focusing on math concepts (such as sweating, anxiety, tension, frustration, or anger). To keep their questionnaire responses anonymous and confiden tial, we asked students to avoid putting their names anywhere on the forms, to seal their completed questionnaires in a provided envelope, and to deposit their sealed envelopes in a drop box provided to them. However, for the subset of 84 math students for whom teachers provided final grades and ratings of attendance as validity checks of student self-reports, a code number was placed on the questionnaires to identify the students (known only to the third author), then later matched to teacher ratings collected at the end of the semester. For the subset of 50 math students whose responses were used to evaluate test-retest reliability for the three new math belief scales (MVS, MCS, and MBS), a similar code number procedure was used to link their responses given on two occasions 3 weeks apart. The math students received extra credit points worth approximately 1% of their course grade for either completion of the questionnaire or an alterna tive assignment (reading an article about sleep deprivation and answering 10 questions about it).
HENDY, SCHORSCHINSKY, AND WADE
1228 Data Analysis
Exploratory factor analyses to identify subscales for the three math belief scales. The first goal for data analysis was to identify underlying dimensions/subscales for each of the three math beliefs shown by the present sample of 368 college math students: Math Value Scale (MVS), Math Confidence Scale (MCS), and Math Barriers Scale (MBS). For the MVS, exploratory factor analysis was conducted for responses of the 368 math students to the 33 possible items shown in Table 2 that were guided by value-expectancy theory (Wigfield & Eccles, 1992). For the MCS, exploratory factor analysis was conducted for ratings given by the 368 math students to the seven possible items shown in Table 3 that were guided by self-efficacy theory (Bandura, 1997). For the MBS, exploratory factor analysis was conducted for ratings by the 368 math students to the 28 possible items shown in Table 4 that were guided by the health belief model (Janz & Becker, 1984).
Each of the exploratory factor analyses described were con ducted using SPSS Version 19 software but only after we had eliminated items with extreme scores and limited variability, those showing “floor effects” (arbitrarily defined as those for which 75% or more of students gave ratings of 1 [never]), and those showing “ceiling effects” (arbitrarily defined as those for which 75% or more of students gave ratings of 5 [always]). The exploratory factor analysis on the remaining items for each scale was done using varimax rotation to identify dimensions/subscales within each scale that were as conceptually clear and orthogonal to each other as possible. Other strict criteria were used in these explor atory factor analyses to enhance the internal reliability and overall goodness of fit of the subscales to the data: (a) each dimension/ subscale was required to include at least three items, (b) all items within a dimension were required to show factor loadings of .50 + only within that dimension, and (c) items within each dimension
Table 2 Exploratory Factor Analysis Results fo r Two Subscales o f the 10-Item Math Value Scale (MVS) fo r 368 College Math Students No. 22 .
23. 29. 28. 30. 16. 25. 24. 18. 26.
2.
17. 6.
7. 8. 11. 20.
27. 31. 32. 3. 12. 13. 14. 19. 21. 33. 9. 10 .
15. 1.
4. 5.
MVS subscale/item Class Devaluation (six items): I can get a good grade in math even if I skip classes. I can get a good grade in math even if I skip the assigned homework. If I miss math classes, I can always catch up later. If I miss math classes, I can always learn it on my own from the textbook. If I skip homework assignments, I can always catch up later. I can learn the math material without coming to class. No Future Value (four items): Getting a bad grade in math will not seriously affect my future financial well-being. Getting a bad grade in math will not seriously affect my future employment possibilities. Being good at math will help me in my future professional life. Getting a bad grade in math will not seriously affect the completion of my college degree. Items cut because . . . Floor or ceiling effects in ratings: Coming to math class is important so I can see what is most important for the exams, (ceiling effects) I need to pass math to complete my college degree, (ceiling effects) Loaded in dimensions with fewer than 3+ items: Coming to math class just makes me more confused. Reading the math textbook outside class helps me understand the material better. Reading the math textbook just makes me more confused. Doing the assigned homework just makes me more confused. If I fail a math class, I can always repeat it. If I miss math classes, I can always get the notes from my friends. Even if I do poorly on math exams, I can make up for it later and still get a good grade. If I get a bad grade on math exams, I can still get a good grade by doing extra credit. Did not load .50+ on any dimension: Coming to math class is important so I can learn the easiest way to solve math problems. I understand math better when I ask the teacher for help. As long as I come to math class, I do not need to do anything more to learn the material. Understand math better when I ask for help from somebody other than the teacher. Being good at math will help my future financial well-being. If I come to every math class and do all the assigned homework, I will get a good grade. There are simply other things in my life that take priority over math. Loaded .50+ on more than one dimension: Doing the assigned homework helps me understand the material better. Doing the assigned homework helps me get a better grade on exams. Coming to math class helps me get a better grade on exams. In dimension with internal reliability less than .70: Coming to math class is important so I can learn new math concepts. Coming to math class is important so I can practice the math problems. Coming to math class is important so I can ask the teacher questions if I get confused.
Factor loading .80 .80 .78 .72 .72 .69 .87 .86 —.59 .59
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+.
MATH BELIEFS
Table 3 Exploratory Factor Analysis Results fo r the Single Dimension o f the Seven-Item Math Confidence Scale (MCS)Jor 368 College Math Students No.
Math confidence dimension of MCS
2. 1. 5.
I am confident that I can get an A in math. I am confident that I can get a passing grade in math. Math seems easy for me, and I am confident I will get a good grade in this math class. Even if I do not understand a math problem at first, I am confident I will get it eventually. If I miss a math class, I am confident that I can make up the work. If I get a bad grade on a math test, I know I can do better next time with more practice. I am confident I can practice math problems by myself until I understand them.
4. 3. 6. 7.
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grade. In each regression analysis, three demographic variables were entered first into the equation: age in years, gender (effect coded with 1 = male, 0 = female), and ethnicity (effect coded with 1 = White, 0 = other). Then, all math belief subscales
Factor loading .81 .80 .79
Table 4 Exploratory Factor Analysis Results fo r Two Subscales o f the 11-Item Math Barriers Scale (MBS) fo r 368 College Math Students
.77 .69 .68 .60
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+. All seven items considered loaded on one dimension.
No. 21. 22. 20. 24. 23. 19. 8. 25.
were required to show internal reliability of .70 + for the 368 math students included in the present study. Other psychometric characteristics of the three new math belief scales. The second goal of data analysis was to examine the psychometric characteristics for the three new math belief scales (MVS, MCS, and MBS). Descriptive statistics (mean, SD, skewness, and kurtosis) for the 368 college students were calcu lated for each subscale from each of the three new scales, with each subscale score defined as the mean 5-point rating for items within that dimension/subscale (see Table 5). To examine how well ratings from the 368 students fell into the subscales of the three new scales, Amos Version 19 and SPSS Version 19 software were used to calculate two goodness-of-fit values often recom mended for use in combination (Arbuckle, 2007; Hu & Bentler, 1999): Comparative fit index (CFI) with values .95 + suggesting excellent fit and values .90 + suggesting acceptable fit, and root-mean-square error of approximation (RMSEA) with values .06 or less suggesting excellent fit and values .10 or less suggesting acceptable fit. Additionally, internal reliability scores (Cronbach’s alpha) were calculated for each subscale from the MVS, MCS, and MBS as rated by the 368 math students, and test-retest reliability scores (Pearson correlation) were calculated for each subscale as rated by the subset of 50 math students who responded to the scales on two occasions 4 weeks apart. Finally, bivariate Pearson correlations were calculated between possible each pair of sub scale scores from the MVS, MCS, and MBS (see Table 6). Math beliefs most associated with math behaviors and performance. The third goal for data analysis was to determine which math beliefs measured by subscales of the new MVS, MCS, and MBS explained the most variance in math behaviors and performance by college students, controlling for demographic vari ables. Hierarchical multiple regression analyses were conducted (with SPSS Version 19 software) using each measure of math behavior and performance as the outcome variable: attending class, doing homework, reading textbooks, asking for help, and final
16. 18. 17.
2. 3. 10. 12. 13. 14. 27. 28. 7. 9. 11. 15. 26. 1. 4. 5. 6.
MBS subscale/item Math Anxiety (eight items): When I do math problems, I feel frustrated and angry. When I do math problems, I feel stupid. When I do math problems, I feel nervous. When I am taking a math exam, I forget everything that I have practiced. When I get confused about something in math, I feel embarrassed. When I am taking a math exam, I feel tense and have trouble breathing. I have trouble remembering the steps in solving math problems. I cannot concentrate on math for more than short periods of time. Discouraging Words (three items): My parents have told me that I am bad at math. My friends have told me that I am bad at math. My teachers have told me that I am bad at math. Items cut because . . . Loaded in dimensions with fewer than 3+ items: During math exams, the teacher does not give us enough time to complete the problems. The problems included on math exams do not always match the problems we practiced. In the past, I have received F grades on math tests. My parents were good at math. My close friends are good at math. I have had to repeat math courses because of bad grades. If I do well in math, I am afraid that my family will expect me to do well in all my classes. If I do well in math, other people may expect me to help them with their math problems. Did not load .50+ on any dimension: Some math teachers describe the steps to solving problems without explaining why they are done. I have trouble keeping organized. In the past, I have received A grades on math tests. I have had to drop math courses because of bad grades. Loaded .50+ on more than one dimension: I am easily distracted by things around me when I am doing math. In dimension with internal reliability less than .70: I do not have enough time to practice math problems. I do not have a good location to do my math homework. I do not have any friends that are good in math so I can practice with them. I cannot find a math tutor or anyone else who can help me.
Factor loading .82 .81 .81 .80 .75 .73 .71 .68 .89 .86 .84
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+.
HENDY, SCHORSCHINSKY, AND WADE
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Psychometric Characteristics fo r Subscales o f the Three Math Belief Scales Goodness of fit
Math belief scale/subscale Math Value Scale Class Devaluation No Future Value Math Confidence Scale Math Confidence Math Barriers Scale Math Anxiety Discouraging Words
CFI
RMSEA
.90
.10
.92
.12
.95
.09
Mean (SD)
Skewness
Kurtosis
Internal reliability
Test-retest reliability
2.54(1.05) 2.25 (0.95)
.38 .40
-.6 6 -.6 9
.85 .71
.76*** .28*
3.79 (0.90)
-.8 4
.29
.85
.88***
2.69(1.22) 1.63(1.00)
.29 1.55
-1.11 1.48
.91 .87
.90*** .71***
Note. Samples were composed of college math students (N = 50 for test-retest reliability; N = 368 for all other values. CFI = comparative fit index; RMSEA = root-mean-square error of approximation. *p < .05. * * > < .0 0 1 .
identified for the MVS, MCS, and MBS were entered into the equation, with each subscale score again defined as the mean 5-point rating for items within that dimension. Student demographics associated with math beliefs. The fourth goal in data analysis was to examine how student demo graphics were associated with math belief subscale scores (again defined as the m ean 5-point rating for all items in the subscale). For each math belief subscale from the MVS, MCS, and MBS, a 2 X 2 analysis o f covariance (ANCOVA) was conducted to compare the math belief for two genders (male, female) and for two ethnic groups (White, other), with age in years used as a covariate.
Results Exploratory Factor Analyses to Identify Subscales for the Three Math Belief Scales The first exploratory factor analysis conducted with ratings from 368 math students produced a 10-item MVS with two subscales: Class Devaluation (six items) and No Future Value (four items; see Table 2). The second exploratory factor analysis with ratings from these students produced a seven-item, single-dimension MCS (see Table 3). The third exploratory factor analysis conducted with ratings from these students produced an 11-item MBS with two
Table 6
Bivariate Correlations Between Subscales o f the Three Math Belief Scales Subscale 1. 2. 3. 4. 5.
Class Devaluation No Future Value Math Confidence Math Anxiety Discouraging Words
1
2
3
4
5
—
.122*
.406*** -.129*
-.284*** .180** -.601***
-.071 .119* -.327*** .462***
—
Note. Class Devaluation and No Future Value subscales are from the Math Value Scale; Math Confidence subscale is from the Math Confidence Scale, and Math Anxiety and Discouraging Words subscales are from the Math Barriers Scale. N = 368 college math students. *p < .05. **p < .01. ***p < .001.
subscales: M ath Anxiety (eight items) and Discouraging W ords (three items; see Table 4).
Other Psychometric Characteristics of the Three New Math Belief Scales Mean 5-point ratings for the five subscales from the MVS, MCS, and MBS revealed that the most commonly reported math belief by the 368 math students was M ath Confidence (mean rating = 3.79, SD = 0.90), and their least commonly reported math belief was Discouraging W ords (mean rating = 1.63, SD = 1.00; see Table 5). Goodness-of-fit values for the two subscales of the MVS sug gested acceptable fit for the ratings given by the 368 math students according to two goodness-of-fit values (with CFI at .90 or above, RMSEA at .10 or less). Goodness-of-fit values for the one MCS dimension suggested acceptable fit for the math student ratings only according to CFI (with a value of .92), but not according to RMSEA (with a value of .12). Goodness-of-fit values for the two subscales o f the MBS suggested acceptable fit for ratings by the math students according to both goodness-of-fit values (CFI and RMSEA; see Table 5). Internal reliability scores for all five math beliefs from the MVS, MCS, and MBS as rated by the 368 math students were above the recommended .70, as was required of them during their factor analyses to identify their scale dimensions; see Table 5). A ddi tionally, all test-retest reliability correlations for the MVS, MCS, and MBS math beliefs were significant across the two occasions they were rated by the subset o f 50 math students, and all but one o f them showed values above the recommended .70. The one exception was the belief of No Future Value for math, which showed a significant but low test-retest correlation o f .28 (p = .046; see Table 5). The mean rating that these 50 students gave for No Future Value was 2.51 (SD = 0.92) at W eek 8 o f class and 2.34 (SD = 0.87) at W eek 12 o f class. As one check o f the convergent validity for the new math belief measures, the MBS Math Anxiety ratings by subset o f 50 math students showed a strong correlation (r = .82, p = .000) with their ratings for the commonly used M A R S-R (Hopko, 2003). Unfor tunately, no other published measures of math confidence were included in the present study to examine their convergent validity with the MCS M ath Confidence ratings by the math students.
MATH BELIEFS However, support for the validity of the new math belief measures measured by the MVS, MCS, and MBS was also found in results from their bivariate correlations for the full set of 368 math students, which showed patterns that might be expected. For example, moderate positive correlations were found between Class Devaluation and Math Confidence (r = .406) and between Math Anxiety and Discouraging Words (p = .462), and moderate neg ative correlations between Math Confidence and Math Anxiety (P ~ -.601) and between Math Confidence and Discouraging Words {p = -.327; see Table 6).
Math Beliefs Most Associated With Math Behaviors and Performance First, as a simple validity check for the recommended behaviors self-reported by the subset of 84 math students, a strong correlation was found between student and teacher ratings of class attendance (r = .69, p = .000). Unfortunately, no other teacher ratings were available to calculate correlations with the three other self-rated math behaviors considered in the present study (doing homework, reading the textbook, and asking for help). Hierarchical multiple regression analyses revealed that even after demographic variables (age, gender, and ethnicity) were taken into account, math beliefs measured by the new MVS, MCS, and MBS scales significantly explained variance for math behav iors and performance in college students. For example, students with more Class Devaluation belief (from the MVS) tended to show less attending class (p = -.26), less doing homework (p = -.28), less reading textbooks (P = -.33), and less asking for help (P = -.12). Students with less Math Confidence (from the MCS) tended to show less attending class (P = .15), less doing home work (P = .18), less reading textbooks (p = .21), and less asking for help (P = .15). Students with more Math Anxiety (from the MBS) tended to show less attending class (P = -.17) and lower final grades (P = -.28). The math beliefs of No Future Value and Discouraging Words were not significantly associated with any of the five measures of math behavior and performance (see Table 7).
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Student Demographics Associated With Math Beliefs ANCOVA revealed that gender and age were significantly as sociated with the math belief of Class Devaluation, with males (M = 2.80, SD = 1.07; females M = 2.33, SD = 0.98) and younger students reporting more Class Devaluation beliefs (r = -.203, n = 365, p = .000). ANCOVA also revealed that gender and ethnicity were significantly associated with the math belief of No Future Value, with males (M = 2.37, SD = 0.98; females M = 2.16, SD = 0.90) and White students (M = 2.38, SD = 0.93; other ethnicity M = 2.11, SD = 0.95) reporting more No Future Value beliefs. ANCOVA revealed that age was significantly associated with Math Confidence, with older students reporting less Math Confidence (r = -.138, n = 362, p = .009). Finally, ANCOVA showed that none of the demographic variables considered in the present study were significantly associated with the math beliefs of Math Anxiety or Discouraging Words (see Table 8).
Discussion Psychometric Strengths of the Three New Math Belief Scales Strong psychometric characteristics were found for all three new measures of math belief developed in the present study: Math Value Scale (MVS), Math Confidence Scale (MCS), and Math Barriers Scale (MBS). For example, subscales of all three mea sures showed acceptable goodness-of-fit values, strong internal reliability, and strong test-retest reliability values with the excep tion that No Future Value (within the MVS) showed a significant but low value of .28 (p = .046). One interpretation for this finding is suggested by the decrease in the math students’ mean rating for No Future Value from 2.51 (SD = 0.92) at Week 8 to 2.34 (SD = 0.87) at Week 12 of class: Perhaps as students neared the end of their math class, some of them finally came to believe that the math skills they were learning might be valuable for their profes sional futures. Finally, convergent validity of the Math Anxiety
Table 7 Hierarchical Multiple Regression Results fo r Math Beliefs Associated With Recommended Math Behaviors and Performance fo r College Students Class attendance® Variable Step 1: Demographic variables Age Gender (1 = male, 0 = female) Ethnicity (1 = White, 0 = other) Step 2: Math belief variables Class Devaluation (MVS) No Future Value (MVS) Math Confidence (MCS) Math Anxiety (MBS) Discouraging Words (MBS)
3
t
Homeworkb
3
t
Textbook0
3
t
Asking for helpd
Final grade0
3
t
3
t
.03 -.14 12
0 31 1 41 1 20
-.01 .05 .02
0.26 0.93 0.31
.07 -.04 .05
1.41 0.78 1.00
.07 .04 -.10
1.38 0.67 1.84
.12 -.04 -.13
2.22* 0.72 2.35*
-.26 -.01 .15 -.17 -.07
4.44*** 0.23 2.23* 2.47* 1.13
-.28 -.03 .18 -.08 -.11
4.83*’* 0.56 2.60* 1.10 1.92
-.33 .02 .21 .02 .06
5.55*** 0.39 3.03** 0.24 0.97
-.12 -.04 .15 -.02 -.03
1.98* 0.73 2.13* 0.23 0.43
0 76 1 94 1 72 2.03* 0.05 Note. Values were entered into the equation after demographic variables were controlled. MVS = Math Value Scale; MCS = Math Confidence ScaleMBS = Math Barrier Scale. a* 2,„ - lU > 34H - 5-30- hR2 - .131***, F(8 , 343) = 6.45. c R2 = .106***, F(8 , 338) = 4.99. dR2 = .062**, F{8, 343) = 2.82. e R2 = .343 , F(8, 73) = 4.76. *p < .05. * > < .0 1 . ***p
© 2014 American Psychological Association 1040-3590/14/$ 12.00 http://dx.doi.org/10.1037/a0037688
Measurement of Math Beliefs and Their Associations With Math Behaviors in College Students Helen M. Hendy, Nancy Schorschinsky, and Barbara Wade Penn State University, Schuylkill Campus
Our purpose in the present study was to expand understanding of math beliefs in college students by developing 3 new psychometrically tested scales as guided by expectancy-value theory, self-efficacy theory, and health belief model. Additionally, we identified which math beliefs (and which theory) best explained variance in math behaviors and performance by college students and which students were most likely to have problematic math beliefs. Study participants included 368 college math students who completed questionnaires to report math behaviors (attending class, doing homework, reading textbooks, asking for help) and used a 5-point rating scale to indicate a variety of math beliefs. For a subset of 84 students, math professors provided final math grades. Factor analyses produced a 10-item Math Value Scale with 2 subscales (Class Devaluation, No Future Value), a 7-item single-dimension Math Confi dence Scale, and an 11-item Math Barriers Scale with 2 subscales (Math Anxiety, Discouraging Words). Hierarchical multiple regression revealed that high levels of the newly discovered class devaluation belief (guided by expectancy-value theory) were most consistently associated with poor math behaviors in college students, with high math anxiety (guided by health belief model) and low math confidence (guided by self-efficacy theory) also found to be significant. Analyses of covariance revealed that younger and male students were at increased risk for class devaluation and older students were at increased risk for poor math confidence. Keywords: math beliefs, math value, math confidence, math anxiety, math performance
Many American college students have problems completing the math courses that are required for their professional goals (Ash craft & Faust, 1994; Chipman, Krantz, & Silver, 1992; Scarpello, 2007). Intervention efforts to improve college student math per formance would be enhanced by identification of math beliefs most associated with poor math behaviors so that these problem atic beliefs could be targeted for change. Therefore, our purpose in the present study was to expand understanding of math beliefs in college students with the development of three new psychometri cally tested scales guided by expectancy-value theory, selfefficacy theory, and health belief model. Additionally, the purpose of the present study was to identify which math belief (and which theory) best explained college student investment in recommended math behaviors (such as attending class, doing homework, reading the textbook, and asking for help when confused) as well as their final math course performance. Finally, the present study planned to examine which student demographics (age, gender, and ethnic ity) were associated with high risk for these most problematic math beliefs.
One of the most widely studied math beliefs has been the college student’s perception of “math anxiety” (Hopko, 2003; Plake & Parker, 1982; Richardson & Suinn, 1972), which is often described as a feeling of tension, confusion, and frustration when attempting to do math problems. Meta-analysis results have doc umented that such math anxiety is negatively correlated (r = -.27) with math grades in school-age children (Hembree, 1990; Ma, 1999), explaining a relatively small 7.3% of the variance in student math performance (-.27 X -.27 = R2 = percentage of variance explained), with similar patterns across student gender and ethnic ity. Three examples of math anxiety measures include the 98-item Mathematics Anxiety Rating Scale (MARS; Richardson & Suinn, 1972), the 24-item Revised MARS (Plake & Parker, 1982), and the 12-item revised MARS (MARS-R; Hopko, 2003). Besides the affective component of math anxiety in student math beliefs, other measures have been developed to capture the more cognitive components of math value and math confidence. Two examples of these measures include the 20-item Perceived Usefulness of Math ematics Scale (Pajares & Miller, 1994) and the five-item Math Confidence Subscale identified within the 180-item SelfDescription Questionnaire (Marsh & O’Neill, 1984). Most of the existing math belief measures have not included complete psychometric examination such as factor analysis, tests for goodness of fit, internal reliability, test-retest reliability and support for convergent or criterion validity (see Table 1.) How ever, the most important limitation of existing measures is that they have not yet been compared for their ability to explain math behaviors often recommended for college students to improve their math performance such as attending class, doing homework, read-
This article was published Online First August 18, 2014. Helen M. Hendy, Psychology Program, Penn State University, Schuylkill Campus; Nancy Schorschinsky, Mathematics and Chemistry, Penn State University, Schuylkill Campus; Barbara Wade, Psychology Program, Penn State University, Schuylkill Campus. Correspondence concerning this article should be addressed to Helen M. Hendy, Psychology Program, Penn State University, Schuylkill Campus, 200 University Drive, Schuylkill Haven, PA 17972. E-mail: [email protected]
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HENDY, SCHORSCHINSKY, AND WADE
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Table 1 Psychometric Examinations Conducted fo r Previously Available Math Belief Measures Math belief measure
No. of items
Mathematics Anxiety Rating Scale (Richardson & Suinn, 1972) Mathematics Anxiety Rating Scale-Revised (Plake & Parker, 1982) Mathematics Anxiety Rating Scale-Revised (Hopko, 2003) Perceived Usefulness of Mathematics (Pajares & Miller, 1984) Math Confidence (Marsh & O’Neill, 1984)
98 24 12 20 5
ing the textbook, and asking for help when confused. Also, with the most commonly known factor, math anxiety, explaining only 7.3% of the variance in math performance (Hembree, 1990; Ma, 1999), other beliefs may exist that could account for additional variance in college student math behaviors. Understanding a broader range of math beliefs associated with college student neglect of recommended math behaviors would enhance the design of interventions to improve student success in college math courses.
Theories Suggesting Math Beliefs Various theories suggest possible math beliefs held by college students that could affect their investment in recommended math behaviors to improve math performance. For example, the expectancy-value theory (Wigfteld & Eccles, 1992) suggests that some students may not believe that these recommend math behav iors or the mathematical understanding they provide are valuable for their immediate or long-term goals. Students are unlikely to invest much time in day-to-day math behaviors (attending class, doing homework, reading textbooks, asking for help, or monitor ing one’s grade in the class) if they see them as unrelated to their understanding of math concepts, their final course grade, gradua tion from college, future employment, or future financial well being. Alternatively, the self-efficacy theory (Bandura, 1997) suggests that even if students believe that the math behaviors help advance their immediate or long-term goals, they may not have the confi dence (or self-efficacy) that they can do these behaviors. Sources of an individual’s self-efficacy for a specific behavior have been identified by Bandura as including past experience with the be havior, the presence of powerful or peer models for the behavior, verbal commentary from others about the individual’s skill at the behavior, and physiological conditions that help or hinder the behavior (such as difficulty with anxiety or anger). Finally, the health belief model (Janz & Becker, 1984) was originally developed to explain why some individuals fail to per form recommended health-related behaviors (such as exercise or smoking cessation), but it may also help explain why some stu dents fail to perform recommended math behaviors. This theory suggests that besides the perceived value of the suggested behavior (as proposed by expectancy-value theory) and their perceived confidence in doing the behavior (as proposed by self-efficacy theory), perceived barriers to performing the behavior may pre vent individuals from moving forward with the action. For exam ple, individuals may perceive benefits associated with joining a zumba exercise class, and they may perceive themselves as capa ble of performing zumba moves, but they would be unlikely to
Factor analysis —
Yes Yes —
Yes
Goodness of fit — —
Yes —
—
Internal reliability
Test-retest reliability
Support for validity
.97 .98 .94 .93 .89
.85
Yes Yes Yes — Yes
— —
— —
show this behavior if they also perceive barriers such as having a history of failure at learning new dance skills, hearing discourag ing words from others about their dancing abilities, or worrying about the time it takes away from other pursuits; having no parent or peer models for dancing; or experiencing excess anxiety at performing dance moves in front of strangers. Similarly, perceived math barriers could come from having a history of failed math courses, from hearing discouraging words from others about one’s math ability (Aunola, Nurmi, Lerkkanen, & Rasku-Puttonen, 2003; Baker, Gersten, & Lee, 2002; Fumer & Duffy, 2002), from the absence of parent or peer models for effective math behaviors (Hamm & Faircloth, 2005; Mayfield & Vollmer, 2007), or from physiological and emotional reactions to math such as feeling nervous, tense, frustrated, angry, or distracted (Ashcraft & Krause, 2007; Beilock, 2008; Ho et al„ 2000).
Purpose of Present Study Our purpose in the present study was to expand understanding of math beliefs in college students by developing three new psychometrically tested measures guided by expectancy-value theory (Wigfteld & Eccles, 1992), self-efficacy theory (Bandura, 1997), and health belief model (Janz & Becker, 1984). We also sought to identify which math beliefs (and which theory) best explained variance in recommended math behaviors and performance by college students. A third purpose was to identify student demo graphics associated with the most problematic math beliefs. More specifically, the three new math belief measures planned were a Math Value Scale (MVS) derived from concepts of valueexpectancy theory, a Math Confidence Scale (MCS) derived from concepts of self-efficacy theory, and a Math Barriers Scale (MBS) derived from concepts of health belief model. After exploratory factor analysis identified dimensions within each math belief mea sure, the additional psychometric examinations planned were cal culation of goodness-of-fit values, internal reliability, test-retest reliability, and support for convergent validity with one of the most widely used existing measures of math anxiety (Math Anxiety Rating Scale-Revised, or MARS-R, Hopko, 2003). Then, after dimensions and subscales were identified within the three new math belief measures (MVS, MCS, and MBS), they would be compared for their ability to explain unique portions of the vari ance in recommended math behaviors and performance for college students (attending class, doing homework, reading textbooks, asking for help, and final grade earned), after controlling for demographic variables that included age, gender, and ethnicity. Finally, analyses were planned to examine how each math belief was associated with these demographic variables to identify which
MATH BELIEFS
students were at increased risk for the most problematic math beliefs. Interventions to improve college student math performance would be enhanced by the availability of the new psychometrically sound measures of math beliefs provided by the present study and documented to explain variance in math behaviors and perfor mance. Such measures could be used by math professors to iden tify students with math beliefs that put them at high risk for poor math behaviors and performance and then to guide these students to monitor associations among their math beliefs, math behaviors, and math course performance.
Method Participants Study participants included 368 college students from a univer sity campus in south-central Pennsylvania serving students from the local small-town communities, students from the large metro politan areas of Philadelphia and New York, and international students. These 368 college math students were recruited from 13 algebra classes (170 men, 195 women, 3 individuals with no gender reported; mean age = 19.7 years, SD = 3.6; 55.1% White, 37.7% African American, 2.5% Hispanic-Latino, and 1.1% Asian American). Responses from these math students were used in most of the analyses of the present study (such as exploratory factor analysis, descriptive statistics, goodness of fit, internal reliability, and analyses of covariance [ANCOVAs] to examine demographics asso ciated with math beliefs). The math classes were sampled during class time, and there was a response rate of 96.1% of the students completing the questionnaire of the present study. Subsets of the 368 math students were available for specific psychometric evaluations of the new math belief scales. For ex ample, one subset of 50 math students was a convenience sample given the questionnaire on two occasions, at the Week 8 and Week 12 of class, to allow us to calculate test-retest reliability (22 men, 28 women; mean age = 19.1 years, SD = 1.9; 56% White, 40% African American, and 2% Hispanic-Latino). Another subset of 50 math students was a convenience sample that also completed the MARS-R (Hopko, 2003) to examine convergent validity for any subscale of math anxiety that might emerge within the new math belief scales. A third subset of 84 math students was a convenience sample for which teachers provided student final math grades and ratings for student attendance to evaluate the validity of student self-reports and to examine how student math beliefs were asso ciated with final math performance (39 men, 44 women; mean age = 19.2 years, SD = 2.2; 62% White, 32% African American, and 1% Hispanic-Latino). Teachers reported final grades as the percentage of possible points earned by the student (0%-100%) and attendance as the percentage of class meetings the student was present 0%—100%).
Procedures The 368 math students were asked at approximately Week 8 in their 15-week semester to complete a questionnaire that asked for demographic information (age, gender, and ethnicity) and selfratings for a number of math behaviors. For example, they were asked to estimate the percentage of math classes attended (10%,
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20%, 30%, 40%, 50%, 60%; 70%, 80%, 90%, or 100%), the percentage of homework assignments completed (using the same ratings), the number of days per week they used their math textbook, and how often they asked someone for help when con fused about math (1 = almost never, 2 — rarely, 3 = sometimes, 4 = often, 5 = almost always). The subset of 50 math students whose responses were used to evaluate convergent validity for any math anxiety dimension within Perceived Math Barriers was also asked to rate how often (1 = almost never, 2 = rarely, 3 = sometimes, 4 = often, or 5 = almost always) they experienced anxiety in each of the 12 math situations listed in the MARS-R (Hopko, 2003). Students were also asked to use a 5-point rating (1 = strongly disagree, 2 = mildly disagree, 3 = do not know, 4 = mildly agree, and 5 = strongly agree) to report their math beliefs, grouped into three sets, which were used to develop the three new math belief scales: (a) For the 33 items considered for the Math Value Scale (MVS), we were guided by the value-expectancy theory (Wigfield & Eccles, 1992). The 33 possible MVS items were selected to include both short-term and long-term consequences of the stu dents’ math behaviors such as improved math understanding and math grades, improved chances of employment, greater success in their chosen professional future, and improved future financial well-being, (b) For the seven items considered for the Math Con fidence Scale (MCS), we were guided by the self-efficacy concept from self-efficacy theory (Bandura, 1997). The items were se lected to include math behaviors for which students might be expected to vary in confidence such as earning good top grades, earning passing grades, understanding math concepts, learning math concepts on their own if they missed class, and practicing math homework problems, (c) For the 28 items considered for the Math Barriers Scale (MBS), we were guided by the health belief model (Janz & Becker, 1984) proposal that perceived barriers are often the strongest predictor that a person will perform a given behavior, regardless of the “perceived benefit” or “perceived selfefficacy” for that behavior. The 28 possible barriers included their past experiences with math courses, available social models for successful math performance, verbal commentary about their math skills from important others, and physiological factors they expe rience that may distract them from focusing on math concepts (such as sweating, anxiety, tension, frustration, or anger). To keep their questionnaire responses anonymous and confiden tial, we asked students to avoid putting their names anywhere on the forms, to seal their completed questionnaires in a provided envelope, and to deposit their sealed envelopes in a drop box provided to them. However, for the subset of 84 math students for whom teachers provided final grades and ratings of attendance as validity checks of student self-reports, a code number was placed on the questionnaires to identify the students (known only to the third author), then later matched to teacher ratings collected at the end of the semester. For the subset of 50 math students whose responses were used to evaluate test-retest reliability for the three new math belief scales (MVS, MCS, and MBS), a similar code number procedure was used to link their responses given on two occasions 3 weeks apart. The math students received extra credit points worth approximately 1% of their course grade for either completion of the questionnaire or an alterna tive assignment (reading an article about sleep deprivation and answering 10 questions about it).
HENDY, SCHORSCHINSKY, AND WADE
1228 Data Analysis
Exploratory factor analyses to identify subscales for the three math belief scales. The first goal for data analysis was to identify underlying dimensions/subscales for each of the three math beliefs shown by the present sample of 368 college math students: Math Value Scale (MVS), Math Confidence Scale (MCS), and Math Barriers Scale (MBS). For the MVS, exploratory factor analysis was conducted for responses of the 368 math students to the 33 possible items shown in Table 2 that were guided by value-expectancy theory (Wigfield & Eccles, 1992). For the MCS, exploratory factor analysis was conducted for ratings given by the 368 math students to the seven possible items shown in Table 3 that were guided by self-efficacy theory (Bandura, 1997). For the MBS, exploratory factor analysis was conducted for ratings by the 368 math students to the 28 possible items shown in Table 4 that were guided by the health belief model (Janz & Becker, 1984).
Each of the exploratory factor analyses described were con ducted using SPSS Version 19 software but only after we had eliminated items with extreme scores and limited variability, those showing “floor effects” (arbitrarily defined as those for which 75% or more of students gave ratings of 1 [never]), and those showing “ceiling effects” (arbitrarily defined as those for which 75% or more of students gave ratings of 5 [always]). The exploratory factor analysis on the remaining items for each scale was done using varimax rotation to identify dimensions/subscales within each scale that were as conceptually clear and orthogonal to each other as possible. Other strict criteria were used in these explor atory factor analyses to enhance the internal reliability and overall goodness of fit of the subscales to the data: (a) each dimension/ subscale was required to include at least three items, (b) all items within a dimension were required to show factor loadings of .50 + only within that dimension, and (c) items within each dimension
Table 2 Exploratory Factor Analysis Results fo r Two Subscales o f the 10-Item Math Value Scale (MVS) fo r 368 College Math Students No. 22 .
23. 29. 28. 30. 16. 25. 24. 18. 26.
2.
17. 6.
7. 8. 11. 20.
27. 31. 32. 3. 12. 13. 14. 19. 21. 33. 9. 10 .
15. 1.
4. 5.
MVS subscale/item Class Devaluation (six items): I can get a good grade in math even if I skip classes. I can get a good grade in math even if I skip the assigned homework. If I miss math classes, I can always catch up later. If I miss math classes, I can always learn it on my own from the textbook. If I skip homework assignments, I can always catch up later. I can learn the math material without coming to class. No Future Value (four items): Getting a bad grade in math will not seriously affect my future financial well-being. Getting a bad grade in math will not seriously affect my future employment possibilities. Being good at math will help me in my future professional life. Getting a bad grade in math will not seriously affect the completion of my college degree. Items cut because . . . Floor or ceiling effects in ratings: Coming to math class is important so I can see what is most important for the exams, (ceiling effects) I need to pass math to complete my college degree, (ceiling effects) Loaded in dimensions with fewer than 3+ items: Coming to math class just makes me more confused. Reading the math textbook outside class helps me understand the material better. Reading the math textbook just makes me more confused. Doing the assigned homework just makes me more confused. If I fail a math class, I can always repeat it. If I miss math classes, I can always get the notes from my friends. Even if I do poorly on math exams, I can make up for it later and still get a good grade. If I get a bad grade on math exams, I can still get a good grade by doing extra credit. Did not load .50+ on any dimension: Coming to math class is important so I can learn the easiest way to solve math problems. I understand math better when I ask the teacher for help. As long as I come to math class, I do not need to do anything more to learn the material. Understand math better when I ask for help from somebody other than the teacher. Being good at math will help my future financial well-being. If I come to every math class and do all the assigned homework, I will get a good grade. There are simply other things in my life that take priority over math. Loaded .50+ on more than one dimension: Doing the assigned homework helps me understand the material better. Doing the assigned homework helps me get a better grade on exams. Coming to math class helps me get a better grade on exams. In dimension with internal reliability less than .70: Coming to math class is important so I can learn new math concepts. Coming to math class is important so I can practice the math problems. Coming to math class is important so I can ask the teacher questions if I get confused.
Factor loading .80 .80 .78 .72 .72 .69 .87 .86 —.59 .59
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+.
MATH BELIEFS
Table 3 Exploratory Factor Analysis Results fo r the Single Dimension o f the Seven-Item Math Confidence Scale (MCS)Jor 368 College Math Students No.
Math confidence dimension of MCS
2. 1. 5.
I am confident that I can get an A in math. I am confident that I can get a passing grade in math. Math seems easy for me, and I am confident I will get a good grade in this math class. Even if I do not understand a math problem at first, I am confident I will get it eventually. If I miss a math class, I am confident that I can make up the work. If I get a bad grade on a math test, I know I can do better next time with more practice. I am confident I can practice math problems by myself until I understand them.
4. 3. 6. 7.
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grade. In each regression analysis, three demographic variables were entered first into the equation: age in years, gender (effect coded with 1 = male, 0 = female), and ethnicity (effect coded with 1 = White, 0 = other). Then, all math belief subscales
Factor loading .81 .80 .79
Table 4 Exploratory Factor Analysis Results fo r Two Subscales o f the 11-Item Math Barriers Scale (MBS) fo r 368 College Math Students
.77 .69 .68 .60
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+. All seven items considered loaded on one dimension.
No. 21. 22. 20. 24. 23. 19. 8. 25.
were required to show internal reliability of .70 + for the 368 math students included in the present study. Other psychometric characteristics of the three new math belief scales. The second goal of data analysis was to examine the psychometric characteristics for the three new math belief scales (MVS, MCS, and MBS). Descriptive statistics (mean, SD, skewness, and kurtosis) for the 368 college students were calcu lated for each subscale from each of the three new scales, with each subscale score defined as the mean 5-point rating for items within that dimension/subscale (see Table 5). To examine how well ratings from the 368 students fell into the subscales of the three new scales, Amos Version 19 and SPSS Version 19 software were used to calculate two goodness-of-fit values often recom mended for use in combination (Arbuckle, 2007; Hu & Bentler, 1999): Comparative fit index (CFI) with values .95 + suggesting excellent fit and values .90 + suggesting acceptable fit, and root-mean-square error of approximation (RMSEA) with values .06 or less suggesting excellent fit and values .10 or less suggesting acceptable fit. Additionally, internal reliability scores (Cronbach’s alpha) were calculated for each subscale from the MVS, MCS, and MBS as rated by the 368 math students, and test-retest reliability scores (Pearson correlation) were calculated for each subscale as rated by the subset of 50 math students who responded to the scales on two occasions 4 weeks apart. Finally, bivariate Pearson correlations were calculated between possible each pair of sub scale scores from the MVS, MCS, and MBS (see Table 6). Math beliefs most associated with math behaviors and performance. The third goal for data analysis was to determine which math beliefs measured by subscales of the new MVS, MCS, and MBS explained the most variance in math behaviors and performance by college students, controlling for demographic vari ables. Hierarchical multiple regression analyses were conducted (with SPSS Version 19 software) using each measure of math behavior and performance as the outcome variable: attending class, doing homework, reading textbooks, asking for help, and final
16. 18. 17.
2. 3. 10. 12. 13. 14. 27. 28. 7. 9. 11. 15. 26. 1. 4. 5. 6.
MBS subscale/item Math Anxiety (eight items): When I do math problems, I feel frustrated and angry. When I do math problems, I feel stupid. When I do math problems, I feel nervous. When I am taking a math exam, I forget everything that I have practiced. When I get confused about something in math, I feel embarrassed. When I am taking a math exam, I feel tense and have trouble breathing. I have trouble remembering the steps in solving math problems. I cannot concentrate on math for more than short periods of time. Discouraging Words (three items): My parents have told me that I am bad at math. My friends have told me that I am bad at math. My teachers have told me that I am bad at math. Items cut because . . . Loaded in dimensions with fewer than 3+ items: During math exams, the teacher does not give us enough time to complete the problems. The problems included on math exams do not always match the problems we practiced. In the past, I have received F grades on math tests. My parents were good at math. My close friends are good at math. I have had to repeat math courses because of bad grades. If I do well in math, I am afraid that my family will expect me to do well in all my classes. If I do well in math, other people may expect me to help them with their math problems. Did not load .50+ on any dimension: Some math teachers describe the steps to solving problems without explaining why they are done. I have trouble keeping organized. In the past, I have received A grades on math tests. I have had to drop math courses because of bad grades. Loaded .50+ on more than one dimension: I am easily distracted by things around me when I am doing math. In dimension with internal reliability less than .70: I do not have enough time to practice math problems. I do not have a good location to do my math homework. I do not have any friends that are good in math so I can practice with them. I cannot find a math tutor or anyone else who can help me.
Factor loading .82 .81 .81 .80 .75 .73 .71 .68 .89 .86 .84
Note. Varimax rotation was used with requirements that items show no floor or ceiling effects in ratings (75+% students gave rating of 1 or 5), that subscales have 3+ items, that all items show factor loadings .50+ on one subscale only, and that subscale internal reliability is .70+.
HENDY, SCHORSCHINSKY, AND WADE
1230 Table 5
Psychometric Characteristics fo r Subscales o f the Three Math Belief Scales Goodness of fit
Math belief scale/subscale Math Value Scale Class Devaluation No Future Value Math Confidence Scale Math Confidence Math Barriers Scale Math Anxiety Discouraging Words
CFI
RMSEA
.90
.10
.92
.12
.95
.09
Mean (SD)
Skewness
Kurtosis
Internal reliability
Test-retest reliability
2.54(1.05) 2.25 (0.95)
.38 .40
-.6 6 -.6 9
.85 .71
.76*** .28*
3.79 (0.90)
-.8 4
.29
.85
.88***
2.69(1.22) 1.63(1.00)
.29 1.55
-1.11 1.48
.91 .87
.90*** .71***
Note. Samples were composed of college math students (N = 50 for test-retest reliability; N = 368 for all other values. CFI = comparative fit index; RMSEA = root-mean-square error of approximation. *p < .05. * * > < .0 0 1 .
identified for the MVS, MCS, and MBS were entered into the equation, with each subscale score again defined as the mean 5-point rating for items within that dimension. Student demographics associated with math beliefs. The fourth goal in data analysis was to examine how student demo graphics were associated with math belief subscale scores (again defined as the m ean 5-point rating for all items in the subscale). For each math belief subscale from the MVS, MCS, and MBS, a 2 X 2 analysis o f covariance (ANCOVA) was conducted to compare the math belief for two genders (male, female) and for two ethnic groups (White, other), with age in years used as a covariate.
Results Exploratory Factor Analyses to Identify Subscales for the Three Math Belief Scales The first exploratory factor analysis conducted with ratings from 368 math students produced a 10-item MVS with two subscales: Class Devaluation (six items) and No Future Value (four items; see Table 2). The second exploratory factor analysis with ratings from these students produced a seven-item, single-dimension MCS (see Table 3). The third exploratory factor analysis conducted with ratings from these students produced an 11-item MBS with two
Table 6
Bivariate Correlations Between Subscales o f the Three Math Belief Scales Subscale 1. 2. 3. 4. 5.
Class Devaluation No Future Value Math Confidence Math Anxiety Discouraging Words
1
2
3
4
5
—
.122*
.406*** -.129*
-.284*** .180** -.601***
-.071 .119* -.327*** .462***
—
Note. Class Devaluation and No Future Value subscales are from the Math Value Scale; Math Confidence subscale is from the Math Confidence Scale, and Math Anxiety and Discouraging Words subscales are from the Math Barriers Scale. N = 368 college math students. *p < .05. **p < .01. ***p < .001.
subscales: M ath Anxiety (eight items) and Discouraging W ords (three items; see Table 4).
Other Psychometric Characteristics of the Three New Math Belief Scales Mean 5-point ratings for the five subscales from the MVS, MCS, and MBS revealed that the most commonly reported math belief by the 368 math students was M ath Confidence (mean rating = 3.79, SD = 0.90), and their least commonly reported math belief was Discouraging W ords (mean rating = 1.63, SD = 1.00; see Table 5). Goodness-of-fit values for the two subscales of the MVS sug gested acceptable fit for the ratings given by the 368 math students according to two goodness-of-fit values (with CFI at .90 or above, RMSEA at .10 or less). Goodness-of-fit values for the one MCS dimension suggested acceptable fit for the math student ratings only according to CFI (with a value of .92), but not according to RMSEA (with a value of .12). Goodness-of-fit values for the two subscales o f the MBS suggested acceptable fit for ratings by the math students according to both goodness-of-fit values (CFI and RMSEA; see Table 5). Internal reliability scores for all five math beliefs from the MVS, MCS, and MBS as rated by the 368 math students were above the recommended .70, as was required of them during their factor analyses to identify their scale dimensions; see Table 5). A ddi tionally, all test-retest reliability correlations for the MVS, MCS, and MBS math beliefs were significant across the two occasions they were rated by the subset o f 50 math students, and all but one o f them showed values above the recommended .70. The one exception was the belief of No Future Value for math, which showed a significant but low test-retest correlation o f .28 (p = .046; see Table 5). The mean rating that these 50 students gave for No Future Value was 2.51 (SD = 0.92) at W eek 8 o f class and 2.34 (SD = 0.87) at W eek 12 o f class. As one check o f the convergent validity for the new math belief measures, the MBS Math Anxiety ratings by subset o f 50 math students showed a strong correlation (r = .82, p = .000) with their ratings for the commonly used M A R S-R (Hopko, 2003). Unfor tunately, no other published measures of math confidence were included in the present study to examine their convergent validity with the MCS M ath Confidence ratings by the math students.
MATH BELIEFS However, support for the validity of the new math belief measures measured by the MVS, MCS, and MBS was also found in results from their bivariate correlations for the full set of 368 math students, which showed patterns that might be expected. For example, moderate positive correlations were found between Class Devaluation and Math Confidence (r = .406) and between Math Anxiety and Discouraging Words (p = .462), and moderate neg ative correlations between Math Confidence and Math Anxiety (P ~ -.601) and between Math Confidence and Discouraging Words {p = -.327; see Table 6).
Math Beliefs Most Associated With Math Behaviors and Performance First, as a simple validity check for the recommended behaviors self-reported by the subset of 84 math students, a strong correlation was found between student and teacher ratings of class attendance (r = .69, p = .000). Unfortunately, no other teacher ratings were available to calculate correlations with the three other self-rated math behaviors considered in the present study (doing homework, reading the textbook, and asking for help). Hierarchical multiple regression analyses revealed that even after demographic variables (age, gender, and ethnicity) were taken into account, math beliefs measured by the new MVS, MCS, and MBS scales significantly explained variance for math behav iors and performance in college students. For example, students with more Class Devaluation belief (from the MVS) tended to show less attending class (p = -.26), less doing homework (p = -.28), less reading textbooks (P = -.33), and less asking for help (P = -.12). Students with less Math Confidence (from the MCS) tended to show less attending class (P = .15), less doing home work (P = .18), less reading textbooks (p = .21), and less asking for help (P = .15). Students with more Math Anxiety (from the MBS) tended to show less attending class (P = -.17) and lower final grades (P = -.28). The math beliefs of No Future Value and Discouraging Words were not significantly associated with any of the five measures of math behavior and performance (see Table 7).
1231
Student Demographics Associated With Math Beliefs ANCOVA revealed that gender and age were significantly as sociated with the math belief of Class Devaluation, with males (M = 2.80, SD = 1.07; females M = 2.33, SD = 0.98) and younger students reporting more Class Devaluation beliefs (r = -.203, n = 365, p = .000). ANCOVA also revealed that gender and ethnicity were significantly associated with the math belief of No Future Value, with males (M = 2.37, SD = 0.98; females M = 2.16, SD = 0.90) and White students (M = 2.38, SD = 0.93; other ethnicity M = 2.11, SD = 0.95) reporting more No Future Value beliefs. ANCOVA revealed that age was significantly associated with Math Confidence, with older students reporting less Math Confidence (r = -.138, n = 362, p = .009). Finally, ANCOVA showed that none of the demographic variables considered in the present study were significantly associated with the math beliefs of Math Anxiety or Discouraging Words (see Table 8).
Discussion Psychometric Strengths of the Three New Math Belief Scales Strong psychometric characteristics were found for all three new measures of math belief developed in the present study: Math Value Scale (MVS), Math Confidence Scale (MCS), and Math Barriers Scale (MBS). For example, subscales of all three mea sures showed acceptable goodness-of-fit values, strong internal reliability, and strong test-retest reliability values with the excep tion that No Future Value (within the MVS) showed a significant but low value of .28 (p = .046). One interpretation for this finding is suggested by the decrease in the math students’ mean rating for No Future Value from 2.51 (SD = 0.92) at Week 8 to 2.34 (SD = 0.87) at Week 12 of class: Perhaps as students neared the end of their math class, some of them finally came to believe that the math skills they were learning might be valuable for their profes sional futures. Finally, convergent validity of the Math Anxiety
Table 7 Hierarchical Multiple Regression Results fo r Math Beliefs Associated With Recommended Math Behaviors and Performance fo r College Students Class attendance® Variable Step 1: Demographic variables Age Gender (1 = male, 0 = female) Ethnicity (1 = White, 0 = other) Step 2: Math belief variables Class Devaluation (MVS) No Future Value (MVS) Math Confidence (MCS) Math Anxiety (MBS) Discouraging Words (MBS)
3
t
Homeworkb
3
t
Textbook0
3
t
Asking for helpd
Final grade0
3
t
3
t
.03 -.14 12
0 31 1 41 1 20
-.01 .05 .02
0.26 0.93 0.31
.07 -.04 .05
1.41 0.78 1.00
.07 .04 -.10
1.38 0.67 1.84
.12 -.04 -.13
2.22* 0.72 2.35*
-.26 -.01 .15 -.17 -.07
4.44*** 0.23 2.23* 2.47* 1.13
-.28 -.03 .18 -.08 -.11
4.83*’* 0.56 2.60* 1.10 1.92
-.33 .02 .21 .02 .06
5.55*** 0.39 3.03** 0.24 0.97
-.12 -.04 .15 -.02 -.03
1.98* 0.73 2.13* 0.23 0.43
0 76 1 94 1 72 2.03* 0.05 Note. Values were entered into the equation after demographic variables were controlled. MVS = Math Value Scale; MCS = Math Confidence ScaleMBS = Math Barrier Scale. a* 2,„ - lU > 34H - 5-30- hR2 - .131***, F(8 , 343) = 6.45. c R2 = .106***, F(8 , 338) = 4.99. dR2 = .062**, F{8, 343) = 2.82. e R2 = .343 , F(8, 73) = 4.76. *p < .05. * > < .0 1 . ***p
