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Mathematics Self-efficacy Performance Discrepancies of Underprepared (Developmental) and Regular Admission College Students.


Forty-eight regular admission (RA) and 42 underprepared (DEV) college students estimated their own ability to solve 10 mathematics problems and then attempted to solve the same 10 problems. The average difference between each participant's self-efficacy for solving each problem and the successful solution yielded a discrepancy score. As expected, the DEV group had significantly lower mathematics performance and efficacy scores than did the RA group. A test of the two groups' discrepancy scores indicated their comparable discrepancies between estimations and performances contrary to expectations. Female RA students had a higher mean discrepancy score than male RA students and female DEV students had a lower mean discrepancy score than male DEV; the male groups had comparable means.

We asked the question would underprepared (developmental) students mathematical self-efficacy and performance be similar to that of normal achieving students or would it be different as in the case of higher achieving students? The developmental student presence on campuses and the importance of self-efficacy in mathematics achievement make a compelling case for answering this question. The purpose of the current study was to identify the discrepancy between self-efficacy and performance in very low achieving, that is, developmental, college students and to compare the discrepancy to the discrepancy in regular admission college students.

Educators often lament student weak mathematical performance; the problem is so widespread that 72% of the post secondary institutions with freshmen offered remedial (also known as developmental) mathematics courses during fall 1995 (U. S. Department of Education, 1996). The relationship between mathematical self-efficacy, the ability to estimate personal performance, and mathematical achievement or actual performance, has been established for elementary school children (Marsh, Walker, & Debus, 1991; Relich, Debus, & Walker, 1986; Schunk, 1984), high school students (Pajares & Kranzler, 1995), and regular admission college students (Hackett & Betz, 1989; Lent, Brown, & Larkin, 1996; Lent, Lopez, & Bieschke, 1991; Matsui, Matsui, & Ohnishi, 1990; Pajares & Miller, 1994; 1995) but not for developmental college students. The role of developmental college student self-efficacy in mathematics achievement remains unclear although it may be an important one. Mathematics self-efficacy has been a stronger predictor of math performance than math anxiety or gender (Kranzler & Pajares, 1997). The purpose of this study was to determine if developmental students would overestimate their academic performances as they have overstated their ability to self regulate (Ley & Young, 1998).

Theory has described and research results have underscored self-efficacy as an important academic performance determinant (Pajares, 1996; Schunk, 1989; Zimmerman, Bandura & Martinez-Pons, 1992). Self-efficacy refers to personal beliefs about one's capabilities to learn or perform skills at designated levels (Bandura, 1986).

Mathematics self-efficacy is a more specific estimate of confidence within one's ability to perform well with regard to particular mathematics tasks (Matsui, Matsui & Ohnishi, 1990). It may well be defined as "a situational or problem-specific assessment of an individual's confidence in her or his ability to successfully perform or accomplish a particular [mathematical] task or problem" (Hackett & Betz, 1989, p 262).

Self-efficacy and academic achievement, persistence, and choice

Mathematics self-efficacy is related to mathematics achievement and other achievement influences such as success expectancy and self-regulation. Mathematical ability perceptions (efficacy) have directly affected student valuing of mathematics as well as their expectancies for success in mathematics (Meece, Wigfield & Eccles, 1990). Gifted elementary, middle, and secondary students reported significantly higher self-efficacy beliefs than have normal achieving students (Pajares & Miller, 1994; Zimmerman & Martinez-Pons, 1990). Self-efficacy has indirectly affected success through apprehension in ninth grade students (Pajares & Johnson, 1996). Mathematics self-efficacy directly influenced fifth, eighth and eleventh grade students' efforts to strategically regulate their learning (Zimmerman & Martinez-Pons, 1990). The students were able to appraise their mathematics competencies accurately and as the students advanced in school their efficacy perceptions and learning strategy use increased.

Self-efficacy actively constructs cognitive representations and an understanding of the objective world (McCombs, 1986, 1989). A learner constructs his or her self-efficacy from information about past performances, observations of others, persuasion by others, and physiological responses (Schunk 1989, 1994). Students should be able to use their own performances as reliable guides for efficacy. Indeed, during interviews about their writing performance, developmental writing students who were less apprehensive about their writing recalled past writing successes more frequently than those who were more apprehensive who conversely recalled failure incidents more often as a source of their self-efficacy beliefs (Wacholz & Etheridge, 1996). Success has increased self-efficacy and failure lowered it although efficacy is usually resilient enough that a single failure will not be detrimental (Schunk, 1989).

Learners also may draw on others' performances to build their own self-efficacy. Observational information may enable a learner to judge his or her own efficacy. A learner who observes peers performing a behavior that the observer has not performed is more likely to believe that he or she too could perform that behavior. Persuasion generally takes the form of feedback from influential others such as teachers, parents or experts. Positive feedback from an instructor has enhanced self-efficacy (c.f., Tuckman & Sexton, 1991) but its effects lessened if the student was not successful. Learner self-efficacy beliefs once constructed from information about others' or personal performances then influence how the learn feels, thinks, behaves and motivates himself.

Self-efficacy beliefs produce effects through four major processes: cognition, motivation, selection, and affective responses (Bandura, 1993). Cognitive processes enable people with a high sense of efficacy to visualize success scenarios that provide support and guidance for performance. Furthermore people who have high self-efficacy for a behavior may not have the ability to perform requisite tasks but nevertheless sense they have the capability to master the task (Bandura, 1993). They tend to set high goals for themselves and the higher self-efficacy is, the greater the goal challenge the person tends to set for him- or herself. Self-efficacy effects motivation in several ways: the goals the person sets for him- or herself, how much effort the person will expend, and how long he or she will persevere and how resilient to failure the person will be (Schunk, 1991).

Self-efficacy is a part of a learner's affective processes that influence achievement. People with low self-efficacy tend to become more anxious and stressed when faced with demands. Academic anxiety, exacerbated by past failures, adversely affects self-efficacy (Meece et al, 1990). Self-efficacy beliefs have been associated with apprehensiveness among developmental writing students (Wacholz & Etheridge, 1996). Unfortunately, straggling students cannot overcome academic anxiety with platitudes about feeling good about themselves; each must build a sense of efficacy, through self-regulative academic skills (Bandura, 1993).

The final process of self-efficacy is selection of the activities and behaviors the learner will attempt. A self-efficacious person perceives that he exercises far greater control over the activities in which he chooses to engage. While people with high self-efficacy avoid situations that they believe overwhelm their capabilities, they still make challenging choices (Bandura, 1993). For example, the stronger the efficacy beliefs among a group of high school equivalency exam students, the more career options the student considered possible (Bores-Rangel, Church, Szendre, & Reeves, 1990). Therefore, those who feel efficacious ought to participate more readily, work harder, and persist longer at the task for which they have self-efficacy than those who doubt their capabilities. Those with low self-efficacy for a behavior may avoid a requisite task altogether (Schunk, 1989).

In his review of early self-efficacy research, Schunk (1989) concluded that "self-efficacy is an important construct for explaining students' learning and performance of cognitive skills in various content areas" (p. 192). Studies involving elementary and secondary school children have consistently confirmed a significant and positive relationship between self-efficacy and skillful performance (Schunk, 1983; 1984; Schunk & Gunn, 1986; Thomas, Iventosch, & Rohwer, 1987; Williams, 1994; Zimmerman, Bandura, & Martinez-Pons, 1992; Zimmerman & Martinez-Pons, 1990). Self-efficacy was the only variable that predicted self-regulation and performance in a path analysis of data from 172 elementary school children (Bouffard & Vezeau, 1996). A meta analysis of 39 studies investigating self-efficacy, performance, and persistence found efficacy beliefs to account for significant variance among participants. The meta-analysis examined students ranging from elementary school to college, and included a variety of achievement levels. It revealed that self-efficacy beliefs accounted for approximately 14% of the variance in academic performance and 12% of the variance in academic persistence across participants, designs, and criterion measures (Multon, Brown & Lent, 1991). They also found stronger relationships among the three factors in lower achievers than in those achieving normal academic progress. However, self-efficacy is not the only influence on achievement behavior. High self-efficacy will not produce competent performance when requisite knowledge and skills are lacking (Schunk 1994).

Pajares and Miller (1994) asked 391 undergraduates to estimate their efficacy on three math confidence scales: solving specific math problems, performing math tasks, and succeeding in math related courses. Students' mathematical, problem solving, self confidence, that is, math self-efficacy, was a more powerful predictor of successful mathematical problem solving than was confidence to perform math related tasks or confidence to succeed in math related courses. In short, the students' judgments about their capability to solve mathematics problems were more predictive of their ability to solve those problems than other variables. In a later study, Pajares and Kranzler (1995) found that mathematics self-efficacy mediated the effect of ability and math experience both on anxiety and performance scores. A study of undergraduate and graduate statistics students found that those who formed perceptions of themselves as inefficacious tended to give up easily, dwell on their perceived deficiencies and suffer from anxiety and stress (Bandalos, Yates, & Thorndike-Christ, 1995). Although there is a significant body of research in the field of mathematics self-efficacy, none to date specifically examines the mathematics self-efficacy of developmental students who are a large proportion of college students.

Low achieving students and self-efficacy

College and universities serve large numbers of low achieving students. More remedial courses in mathematics than in any other basic subject area were offered at public two-year, four-year, and low minority enrollment institutions; 24% of all first-time freshmen enrolled in a developmental mathematics class (U. S. Department of Education, 1996). Institutions most frequently assign students to developmental courses based upon placement test scores alone (63%) or in combination with other criteria, i.e., no or low American College Test/Scholastic Aptitude Test (ACT/SAT) scores or low GPA (22%). The process identifies low achieving students but offers little information about meta-cognitive skills or psychological attributes that the students may lack but which have been correlated to achievement.

College student self-efficacy has been related to achievement. Mathematics ACT scores significantly predicted self-efficacy ([Beta] = .22) in 138 psychology undergraduates even when entered in a regression equation after gender and four other efficacy information sources (Lent, Lopez, & Bieschke, 1991). ACT mathematics scores were significantly correlated to math self-efficacy in 262 psychology undergraduates (Hackett & Betz, 1989).

Although empirical evidence supports the relationship between college student self-efficacy and achievement, these studies may not generalize to developmental students for two reasons. If a college admissions process eliminates low achieving students because of ACT/SAT scores, investigations using college students might control for the prior impact of motivational determinants such as self-efficacy and attenuate the performance and aptitude relationship (Pajares & Kranzler, 1995). Furthermore, even studies with college students may have excluded developmental college students since the researchers frequently draw participants from psychology classes in their own four-year research institutions; such institutions are less likely to include developmental populations (USDOE, 1996). Therefore a study which includes low achieving students who may otherwise not be in college and included in self-efficacy investigations may provide insight about the relationship between self-efficacy and achievement that could be distorted by the restricted achievement range common among many studies using post secondary students.

Despite the dearth of research on developmental student self-efficacy, some evidence supports its role with their poor academic performance. Undergraduate personal performance accomplishments constituted the most influential source of mathematics self-efficacy information (Lent et al., 1991). Since most developmental students' prior mathematical achievement is less than regular admission students, developmental student mathematics self-efficacy should be lower than normal achieving student mathematics self-efficacy.

Developmental students may be less able to judge their own abilities since they may be unable to judge their own study behaviors. Self-efficacy entails a student's ability to accurately judge their own academic abilities or to calibrate their own efficacy. Calibration has usually been biased in favor of overestimation in most studies (Hackett & Betz, 1989; Pajares & Miller, 1994) but the influence of self-efficacy on performance has been stronger in lower achieving students than in normal achieving students (Multon, Brown, & Lent, 1991). Prediction accuracy may correspond to mathematical experience and exposure given that high school mathematics students were more overconfident in their efficacy predictions than college level math students (Pajares & Kranzler, 1995). Developmental students who have poorer mathematics achievement than regular admission students and have often taken fewer mathematics courses in high school than their regular achieving peers may have had less exposure and experience to mathematics so their prediction accuracy may be poorer. On the other hand, gifted students and not low achieving ones have overestimated their performances more than normal achieving students have (Pajares, 1996).

A few studies have raised questions about very low achieving college students' ability to calibrate specific academic abilities (Young & Ley, 1997; Ley & Young, 1998). Interviewed developmental students reported significantly lower self-evaluation, monitoring and record keeping strategies than did regular admission students (Ley & Young, 1998). These three self-regulatory study behaviors enable more self-efficacious learners to reflect upon and judge their own performance capabilities resulting in their correspondingly higher achievement. Since they have been less likely to self evaluate, that is, judge the quality of their work developmental students may calibrate their self-efficacy less accurately than regular admission students do.

Another study provides further evidence that supports the that developmental students may be less able than regular admission students of identify their own study behaviors. Regular admission students reported study behaviors that correlated with their later achievement although low achieving, developmental students' reported study skills not related to later achievement (Nist, Mealey, Simpson, & Kroc, 1990). Study skills measured by self-report Likert items failed to predict achievement among 71 developmental students, although it did predict achievement among 168 regular admission students. Just as developmental students reported study behaviors that did not correspond to their later performance, developmental students may also over estimate their ability to solve mathematical problems. Since developmental students self- judgments about study behaviors were less accurate than regular admission students, the developmental students' overestimation may even be greater than the regular admission students' overestimation when asked to judge, but not to actually solve, whether or not they can solve a mathematical problem.

No research on mathematics performance is complete without addressing gender influences (Fennema, Carpenter, Jacobs, Franke, & Levi, 1998). More women than men are enrolled in college and developmental education male/female ratios are similar. Neither gender disproportionately enrolls in developmental mathematics; about one third are men. Men may be equitably represented in mathematics classes but they command more mathematics self-efficacy than women do. College men have expressed significantly more confidence in ability to perform mathematics tasks than women have (Hackett, G, Betz, N.E., O'Halloran, M.S., & Romac, D.S. 1990; Matsui, Matsui & Ohnishi, 1990). Male college psychology students tended toward overconfidence and women toward under or congruent confidence as it was measured by the discrepancy between mathematics efficacy and performance although the tendency was not significantly different (Hackett & Betz, 1989). Gender differences have existed in regular admission college mathematics students on mathematics self-efficacy although most studies have not investigated self-efficacy in low achieving, underprepared college students.

Self-efficacy studies abound and many of the studies involve college students but very few, if indeed any, studies have investigated the lowest college achiever's relationship between self-efficacy and performance. This study advances self-efficacy research with a lower achieving group often excluded but who comprise a quarter of the college population (USDOE, 1996) and fills a gap in the literature as noted by Pajares (1996). The purpose of our research was to examine the difference between regular admission college students and lower achieving, under prepared (developmental) college students on the discrepancy between their self-efficacy for solving specific mathematical problems and their corresponding performance solving the same problems. We predicted that developmental students would have lower self-efficacy and lower performance than regular admission students but that the developmental students would overestimate their performance more than the regular admission students would.. We predicted that the discrepancy between the problems they through they could solve and the problems they were able to solve would be greater for developmental students than for regular admission students. We expected to confirm gender difference in mathematics self-efficacy substantiated in previous research; men would have higher self-efficacy then women.


We administered the self-efficacy measure to two intact college algebra and two developmental mathematics classes; each of two instructors had an algebra and developmental mathematics class. The instruments were adminstered during the first week of the fall semester. The developmental mathematics class is the entry-level developmental math course for students at this community college. The course covers exponents, solving linear equations and inequalities, functions and graphing linear equations and inequalities. Students are placed into this course with an ACT score of 19 or below and results on an appropriate diagnostic instrument administered prior to enrollment. The college algebra class is the first mathematics course required for degree seeking students. It covers polynomials, factoring, rational expressions, rational exponents, radicals, complex numbers, quadratic and linear equations, inverse functions, exponential and logarithmic functions and linear systems of equations. This course is a prerequisite for all upper level mathematics courses including calculus and trigonometry. Students are placed into this course with an ACT score of 20 or higher, and all students enrolled in the two sections involved in this study had been placed into this course. No students in the college algebra class had been classifed developmental in mathematics. A student may skip this course only by using an Advanced Placement test through their high school or taking a CLEP test. Using these two courses early in the fall semester allowed for comparison between students classified as developmental and requiring remedition, to those who were clearly not developmental. Three hypotheses were tested with three tests for each; efficacy, performance and discrepancy score means. Post hoc data analysis employed additional tests.


Ninety-six participants (see Table 1) from a southeastern urban community in the United States were enrolled in one of 4 mathematics classes taught by one of two math instructors, each of whom taught one of the developmental and one of the regular admission classes at a community college. All classes met weekday mornings and the retention rate for developmental day classes was 96% for the semester in which we collected data. The 48 regular admission and 42 developmental students were classified as freshmen that had completed less than 30 credit hours

The College may place a student in a developmental courses when they are admitted and assigns over 70% of the freshmen students to developmental classes. The College assigns students to developmental education courses based upon two criteria: an examination score below a cut score on the ACT examination and an institutional placement examination. If a student has an unacceptable ACT score, he or she must take an institutional diagnostic test. Students who earn above the cut score on the ACT or the diagnostic test earn exemption from developmental courses.

Ninety-six participants submitted data but six participants were dropped, four of the regular admission and two developmental, because one or more of the efficacy items were blank or completed incorrectly. Of the 90 participants who provided useable data, the 42 developmental included 35 whites, 6 African Americans, and 1 other ethnic group participant; the 48 regular admission included 40 white, 3 African American, and the remaining 5 indicated other ethnicity. The sample average age, 23, was the same as developmental group average age but less than the regular admission participant average age, 24. Fifteen of the developmental and seventeen of the regular admission students were male. Men comprised about one third of the developmental group and regular admission group. Phi coefficients indicated no significant relationship between participant developmental status, and either gender or ethnicity.


One of the researchers reviewed the procedures with the two instructors who collected the data to assure that they followed the specified sequence. During each class during the first week of school, participants were asked to voluntarily participate in the study, signed an informed consent form, and then completed demographic data questions and the mathematics self-efficacy measure (Appendix A). The instructor administered and collected the self-efficacy measure first followed by the mathematics examination (Appendix B). Instructors were reminded not to tell students that the mathematics examination would be the same problems as on the self-efficacy measure.

Self-efficacy Measure and Mathematical Examination

Participants completed a mathematical problem solving self-efficacy scale which had a test-retest coefficients of .78, p [is less than] .02 (Zimmerman & Martinez Pons, 1990) and reliability for the current data, [varies] = .73. The scales included the same 10 items in the same order for each measure; only the student response was different. Mathematical problems were presented in increasing difficulty from simple arithmetic to algebra, probability, and statistics. For the self-efficacy measure participants assigned a percentage to indicate degree of sureness that he or she could solve the mathematical problem. The ratings could be from 0%, completely unsure, to 100%, completely sure. For the mathematical examination, participants had to solve the problem and record the answer.

To determine the difference between self-efficacy and performance, the discrepancy score for each participant was calculated as the mean of the sum of the differences between the ten corresponding self-efficacy scores and performance scores; the score would be negative if a respondent consistently underestimated his performance. The efficacy ratings (0 to 100) on 10 items were summed then a mean calculated. The performance ratings for each case (correct, 100, or incorrect, 0) on 10 items were summed then a mean calculated. The discrepancy was the difference between the two means.

A Levene's test for equality of variances between the developmental and regular admission groups indicated that the group variances for performance scores, efficacy scores and discrepancy scores violated assumptions of normality that rendered t tests inappropriate. Nonparametric Mann-Whitney U tests tested differences between the developmental and regular admission groups on performance scores, efficacy scores, and discrepancy scores. A Mann-Whitney U test detects whether the observed difference between the distribution of scores for each of two groups is statistically significant and is appropriate when the underlying assumptions of a t test are violated (Gall, Borg, & Gall, 1996).


Both the regular admission and the developmental group had higher self-efficacy means than their respective performance means as indicated by negative discrepancy scores. Self-efficacy and performance means were respectively, 66.2 (13.7) and 51.7 (13.6) for the regular admission students and 52.9 (11.7) and 44.5 (11.9) for the developmental group (see Table 1). The regular admission group discrepancy mean, -14.6 (16.9), was larger than the developmental group mean for the -8.36 (14.6) and in opposite the direction hypothesized.

Our first two hypotheses were upheld. Nonparametric Mann-Whitney U tests on efficacy and performance means (see Table 2) indicated a significant difference between developmental and regular admission participants on self-efficacy scores and on performance. The regular admission participants held significantly higher efficacy beliefs and performed significantly better than did the developmental participants. Contrary to our third hypothesis, developmental students did not overestimate their performances more than regular admission students did. A Mann-Whitney U test (see Table 2) on the discrepancy mean scores failed to support our hypotheses that developmental students would report significantly larger over estimation of their abilities than would regular admission participants. Although female mean discrepancy scores were almost fifty percent higher than male scores, both groups overestimated their performances comparably.

We determined the strength and direction of the relationship between developmental status and efficacy, performance, and efficacy-performance discrepancies. A Pearson r correlation indicated the strength of significant relationships between developmental status and efficacy (r = -.46, p [is less than] .001) and between developmental status and performance (r = -.27, p = .01) (Note: Pearson r is mathematically equivalent to a point biserial r when the second variable is dichotomous). Efficacy was more strongly correlated with developmental status than was performance.

Overestimation and underestimation patterns differed between developmental and regular admission groups. Proportionately more regular admission participations (35% or 17) were in the quartile representing greatest overestimation by both groups; 14% or 6 of the developmental participants were in the same quartile. About 26% or 11 of developmental students were in the quartile representing the greatest underestimation of their abilities, about the same proportion as the 25% or 12 for the regular admission students.

Additional Mann Whitney U tests determined differences between developmental and regular admission group efficacy, performance and discrepancy item mean scores (see Table 3). Regular admission students rated their self-efficacy in the predicted direction, higher of the two groups on all but one of the ten efficacy items (item 3) although only five differed significantly (items 4, 6, 7, 8, 10). However, both groups performance was less consistent than their efficacy. Regular admission students performed significantly better on items 1, 4, and 7 but developmental group mean performance scores exceeded regular admission scores on half of the ten items (items 2, 3, 8, 9, 10). The regular admission students had s significantly greater mean discrepancy score on items 8 and 10 and had larger non-significant discrepancy scores on five items (2,3,6,7,9).

We calculated a Mann Whitney U for the gender groups because the groups had unequal variances on each of the three sets of ten item scores; results indicated no gender differences. Mann Whitney U statistics were calculated for gender and performance, (U = 920.0, p = .95; efficacy, U = 922.5, p = .96; and discrepancy, U = 891.0, p = .76) mean scores indicating no significant relationship between gender and the other variables. Developmental men had discrepancy score means (-10.9) very near the discrepancy scores means of regular admission men (-11.3). On the other hand, women developmental students had the lowest mean discrepancy score (-7.0) and women regular admission students had the highest mean discrepancy score (-16.4) among men or women of either status.

Developmental students in this sample had significantly lower mathematical problem solving self-efficacy and lower mathematical problem solving performance than did their regular admission counterparts. The self-efficacy-performance discrepancy score means were statistically comparable, not different, as we had hypothesized. The non-significant difference opposed the direction hypothesized.

For a Discussion and Bibliography see issue's website

Dawn B. Young, Department of Behavioral Sciences, Bossier Parish Community College; Kathryn Ley, Department of Instructional Technology, University of Houston -- Clear Lake.
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Author:Ley, Kathryn
Publication:Academic Exchange Quarterly
Geographic Code:1USA
Date:Jun 22, 2001
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