Latent Structure of the Sources of Mathematics Self-Efficacy

JOURNAL OF VOCATIONAL BEHAVIOR ARTICLE NO.

49, 292–308 (1996)

0045

Latent Structure of the Sources of Mathematics Self-Efficacy ROBERT W. LENT University of Maryland

FREDERICK G. LOPEZ Michigan State University AND

STEVEN D. BROWN AND PAUL A. GORE, JR. Loyola University Chicago General social cognitive theory and its career-specific elaborations posit four primary sources through which self-efficacy beliefs are acquired and modified: personal performance accomplishments, vicarious learning, social persuasion, and physiological states and reactions. We present two studies exploring the dimensionality of these sources within the context of career-relevant mathematics activities. In Study 1, 295 college students completed measures of the source variables. Testing two- through five-factor models, we found strongest support for a four-factor latent structure of the efficacy sources. In Study 2, involving 481 high school students, a five-factor model fit the data well. We also found evidence of a higher order factor structure in both samples. Several directions for further research on the sources of efficacy information are considered, along with implications for career and academic interventions. q 1996 Academic Press, Inc.

Bandura’s (1986) social cognitive theory has inspired a good deal of research on academic and career development phenomena over the past decade (Borgen, 1991; Hackett & Lent, 1992; Russell & Petrie, 1992). From a career development perspective, one of the theory’s most important aspects is its focus on the means by which people exercise personal agency, or self-direction, in their pursuit of particular activities and life paths (Lent, Brown, & Correspondence concerning this article should be addressed to Robert W. Lent, Department of Counseling and Personnel Services, College of Education, University of Maryland, College Park, Maryland 20742. 292 0001-8791/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Hackett, 1994). Bandura’s theory highlights several agentic variables—in particular, self-efficacy beliefs, outcome expectations, and goal concepts— that are seen as promoting self-regulation, change, and plasticity in behavior. Among these variables, self-efficacy (personal beliefs regarding one’s performance capabilities) is posited as playing an especially central and pervasive role, influencing one’s choice of activities, as well as effort expenditure, persistence, thought patterns, and emotional reactions in the face of obstacles (Bandura, 1986, 1989). Self-efficacy beliefs have been found to predict a variety of career entry indices, such as perceived career options, academic achievement and persistence, and career indecision (Hackett & Lent, 1992; Multon, Brown, & Lent, 1991). There is also growing evidence relating self-efficacy to important postentry outcomes, such as work performance, attendance, and reemployment following job loss (Sadri & Robertson, 1993). While this emergent research base is impressive, the primary focus thus far has been on charting the general predictive terrain of self-efficacy by exploring its basic relations with a variety of career-relevant outcomes. Hackett and Lent (1992) have noted that a number of vital, finer-grained issues have received limited study to this point. One such issue involves the processes whereby people develop and revise their career self-efficacy beliefs. Assuming that self-efficacy helps form a cognitive pathway through which career choices are made and pursued (Lent et al., 1994) and that diverse career interventions achieve their effects partly through their impact on self-efficacy (Eden & Kinnar, 1991; Frayne & Latham, 1987; Wood & Bandura, 1989), it is important to learn more about the types of experience that may inform or transform self-efficacy estimates. Research on the hypothesized antecedents of self-efficacy would test an important aspect of social cognitive theory. Further, on a practical level, research aimed at identifying and confirming the mechanisms underlying self-efficacy development and maintenance may aid in the design of maximally efficient career and educational interventions. According to Bandura (1986), efficacy percepts are acquired and modified largely through four major informational sources: (a) personal performance accomplishments, which include one’s pattern of successes and failures at particular tasks or activities; (b) vicarious learning, or observation of other people’s performance attainments; (c) social persuasion, involving the encouragement or discouragement that one receives from significant others for engaging in particular activities; and (d) physiological states and reactions, including the pleasant or unpleasant emotional and physical sensations (e.g., anxiety, fatigue, composure) that one experiences while performing particular tasks. These four sources are seen as interacting dynamically to affect selfefficacy judgments. The present study explored the antecedents of self-efficacy within the context of mathematics achievement. In particular, we sought to assess the latent structure of, and the interrelations among, the primary sources of experiential

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information that, according to social cognitive theory, should underlie students’ self-efficacy percepts. We elected to focus on mathematics as a skill domain primarily because adequate preparation in mathematics is required for entry into a wide variety of career fields and, thus, serves as a critical determinant of students’ range of career options (Betz, 1992). Several prior studies have explored the theorized precursors of mathematics self-efficacy percepts. Examining the relation of prior performance experience to self-efficacy with path analysis procedures, Hackett (1985) and Lapan, Boggs, and Morrill (1989) found support for a model in which mathematics performance influences mathematics self-efficacy, as predicted by social cognitive theory. Likewise, experimental manipulation of mathematics success or failure experiences has also been shown to produce corresponding gains or reductions in task-specific mathematics self-efficacy (Campbell & Hackett, 1986; Hackett, Betz, O’Halloran, & Romac, 1990). Three studies have examined all four hypothesized sources of mathematics self-efficacy using structured questionnaires. Matsui, Matsui, and Ohnishi (1990) found that performance accomplishments, vicarious learning, and emotional arousal (a form of physiological reaction), but not social persuasion, each explained significant increments in mathematics self-efficacy variation. Lent, Lopez, and Bieschke (1991) found that the four source variables each correlated significantly with mathematics self-efficacy though, after controlling for personal performance, the remaining three sources did not explain significant additional variance in self-efficacy. Lopez and Lent (1992) reported that emotional arousal complemented personal performance in explaining variation in high school students’ mathematics self-efficacy percepts. In each of these studies, the four source variables were found to be significantly and, with the exception of vicarious learning, substantially interrelated. Although the four hypothesized sources of self-efficacy have been found to relate to mathematics self-efficacy, there has, to date, been little effort to confirm the purported structure of the efficacy source variables. Given the observed interrelations among the source variables, it would be especially useful to explore whether they represent four or fewer constructs. Drawing upon theory and research, we developed and tested several plausible alternative models. First, because Bandura’s (1986) theory suggests that each of the four efficacy sources provides relatively distinct, though additive, data for appraising one’s efficacy, we examined a four-factor model of the efficacy source variables. Alternatively, findings that three of the four efficacy source variables intercorrelate substantially (Lent et al., 1991) suggest a more parsimonious two-factor model, wherein one factor derives from direct, personal experience (combining performance accomplishments, emotional arousal, and social persuasion) and the second factor reflects vicarious learning. Because personal attainments are often accompanied by social persuasion (e.g., verbal reinforcement from teachers and parents), a three-factor model was also developed in which personal performance and social persuasion sources constitute

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a single direct experience factor, and vicarious learning and emotional arousal represent two separate factors (Matsui et al., 1990). Finally, earlier work using the Sources of Mathematics Self-Efficacy Scales (Lent et al., 1991) revealed that the vicarious learning scale may not be homogeneous. In fact, this scale contains items reflecting two broad types of models that students may use in estimating their personal efficacy: peers/ friends and adults (e.g., teachers, parents). Therefore, we also examined a five-factor model, identical to the four-factor model except that the vicarious source was subdivided into distinct peer and adult modeling components. In sum, we tested four different latent structure possibilities to determine which representation of the efficacy source variables would provide the best fit to the data. Because it is possible that level of education and mathematics experience might influence the source variables’ latent structure, we tested the four models in two separate studies, one involving college students and the other high school students. Our assumption was that, by clarifying the structure underlying efficacy source information, our findings might shed additional light on those experiences that give rise to self-efficacy and offer suggestions for the composition of interventions aimed at strengthening students’ mathematics self-efficacy percepts. STUDY 1

Method Subjects. Research participants were 295 students (110 men and 185 women) enrolled in introductory psychology courses at a large midwestern university. They were predominantly White (81%; Blacks, Hispanics, and Asians/Pacific Islanders made up 14, 4, and 1% of the sample, respectively) and either freshmen or sophomores (77%). Their mean age was 19.37 years, SD Å 1.60, and their average high school rank was at the 78th percentile, SD Å 15.84. Procedure. Subjects completed demographic and self-report measures in group testing sessions, receiving experimental credit for their participation. Part of the data for this study included the efficacy source ratings of Lent et al.’s (1991) subjects (N Å 138). Our secondary analyses of these data were directed at substantively distinct research questions; the focus herein was on the underlying structure of the sources of mathematics self-efficacy, whereas the prior study largely explored the relation of the source variables to selfefficacy. Combining previously obtained and newly gathered source variable data enabled us to create a larger sample in support of our covariance structure analyses. We have minimized any overlap between the two studies in analyses and reported findings. Instruments. All subjects completed the mathematics self-efficacy and perceived sources of mathematics self-efficacy measures described by Lent et al. (1991). (Participants in the Lent et al., 1991, study also provided data on

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outcome expectations, interests, and career aspirations, which were not used in the present study.) The sources measure consisted of four rationally developed 10-item scales corresponding to the four sources of efficacy described by Bandura (1986). Sample items included ‘‘I received good grades in my high school math classes’’ (Personal Performance Accomplishments Scale), ‘‘My favorite teachers were usually math teachers’’ (Vicarious Learning Scale), ‘‘My friends have discouraged me from taking math classes’’ (Verbal Persuasion Scale), and ‘‘I get really uptight while taking math tests’’ (Emotional Arousal Scale). Items for the performance, vicarious, and persuasion scales were developed by Lent et al. (1991) based on theoretical accounts (Bandura, 1986) and prior experimental operationalizations (e.g., Campbell & Hackett, 1986) of the efficacy sources. The arousal scale consisted of the Fennema–Sherman Math Anxiety Scale, as revised by Betz (1978), which appeared consistent with descriptions of emotional arousal as a source of self-efficacy (e.g., Hackett & Betz, 1981). Subjects responded by indicating their level of agreement with each statement on a 5-point scale, with higher scores reflecting greater agreement. Half of the items were positively worded and half were negatively worded; the latter were reverse-scored so that higher scores on each scale connote more favorable mathematics experiences (e.g., more support, less anxiety). Scale scores, which are formed by summing item responses, can range from 10 to 50. Lent et al. (1991) reported coefficient alphas for the personal performance, vicarious learning, social persuasion, and emotional arousal scales of .86, .56, .74, and .90, respectively, and 2-week test–retest correlations that ranged from .85 to .96. In order to form the two vicarious factors for the five-factor model test in this study, we divided the vicarious items into those referring to adult models (5 items, a Å .62) and peer models (4 items, a Å .66), omitting one item that did not specify the model’s identity. Lent et al.’s (1991) findings also provided initial support for the validity of the sources instrument. For example, Mathematics ACT scores were found to correlate significantly with three of the four source variables, yielding the highest correlation with perceived personal performance, and each of the four sources related to students’ mathematics self-efficacy estimates. Previous research on the emotional arousal (mathematics anxiety) scale has found that it possesses adequate internal consistency and correlates with measures of mathematics achievement, test anxiety, and trait anxiety (Betz, 1978). Analysis. We tested the fit of each of the four models of efficacy source information with the confirmatory factor analysis (CFA) procedures of EQS 4.0 (Bentler, 1989). These analyses used the covariance matrix and maximum likelihood solutions. CFA offers several advantages over exploratory factor analysis when one wishes to test theoretically derived hypotheses about factor structure (Long, 1983; Tinsley & Tinsley, 1987). For example, CFA permits the a priori specification of relations between observed variables and latent

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factors. Individual variables can be fixed to load or not to load on specific factors, thereby clarifying the interpretation of factors. CFA also provides statistical tests to determine how well obtained relations among variables in sample data correspond to hypothesized models (Fassinger, 1987). In testing the two-factor model, we set (i.e., freed) all items from the personal performance, social persuasion, and emotional arousal scales to load on a single common factor, while items from the vicarious learning scale were freed to load on a second factor. The three-factor model was specified by setting the personal performance and social persuasion items to load on one factor and the vicarious learning and emotional arousal items to load on two separate factors. The four-factor model of efficacy source information was formed by freeing the items from the four source scales to load on four separate factors. The five-factor model was identical to the four-factor model, except that the vicarious source items were freed to load on two distinct factors. Factor covariances were freed to be estimated in the multiple factor models because theory and prior research suggest that the source variables are distinct but related constructs. Reported factor intercorrelations were drawn from the Pearson correlation matrix. Listwise deletion methods were used, and errors were specified to be uncorrelated, in all analyses. Prior to testing the goodness of fit of the alternative models, we derived three indicators for each latent factor (i.e., source of self-efficacy). CFA requires that each latent dimension be represented by multiple observed variables or indicators. The following strategy was employed both to create strong multiple indicators for each construct (for use in testing the measurement models or relations between the observed variables and latent constructs) and to maximize sample size in relation to number of freed parameters (an important consideration in the replicability of sample-based CFA model tests; Bentler, 1980). First, single-factor solutions were fit to each separate source scale with exploratory factor analyses. (Previous research with the sources measure suggested that single-factor solutions were appropriate for three of the four scales; see internal consistency estimates, above.) Next, item factor loadings were used to create composite items representing each source construct. Items with the highest and lowest loadings were averaged together to form the first indicator, items with the next highest and lowest loadings were averaged together to form the second indicator, and so on until all items with significant loadings were assigned to one of the three indicators. (Details regarding the assignment of specific items to indicators, together with item factor loadings, may be obtained from the first author.) Each indicator thus served as a composite estimate of one of the efficacy source constructs. This sort of strategy (i.e., calculation of a small set of multiple indicators from a larger pool of items) has been used in prior confirmatory factor analytic tests of hypothesized measurement models (e.g., Brook, Russell, & Price, 1988; Mathieu & Farr, 1991) and is similar to methods often used to construct

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parallel test forms. l coefficients, indexing the relationship between measured indicators and latent constructs, were significant (p õ .01) in each of the model tests, suggesting that the indicators were adequate to support our theory testing objectives (coefficients ranged from .32 to .94, with most over .70). Fit indices. While there is currently no consensus concerning the best index of fit for covariance structural models, the use of multiple indices of overall fit has generally been encouraged (e.g., Bollen, 1989; Hu & Bentler, 1995; Tanaka, 1993). Accordingly, the fit of each model in the current study was assessed with the Nonnormed Fit Index (NNFI; Bentler & Bonett, 1980) and the Comparative Fit Index (CFI; Bentler, 1990). The NNFI, which is mathematically equivalent to Tucker and Lewis’ (1973) index, was developed to correct for the strong association between previous indexes (e.g., the Normed Fit Index) and sample size. The NNFI provides an estimate of fit of a hypothesized model relative to a baseline independence model, and appears to be relatively independent of sample size when maximum likelihood estimations are employed. The CFI indexes the relative change in model fit as estimated by the noncentral x2 of a target model versus the independence model. It has the added advantage of being corrected to fall between 0 (reflecting poor fit) and 1 (indicating perfect fit). Bentler (1990) reported that this fit index performed well under conditions of small sample size using maximum likelihood estimation methods. The CFI also has the advantage over the NNFI in estimating fit in relatively smaller samples, such as that used in this study, where the NNFI may slightly underestimate the fit of well-specified models. In addition to the above two indices, we directly compared the fit of the alternative models using procedures outlined by Bentler (1980). Bentler indicated that, where one can specify an alternate model that is nested within a hypothesized model, the difference in the x2 values of the two models is distributed as a x2 with degrees of freedom equal to the difference in degrees of freedom of the two models. Hoyle and Panter (1995) have recommended the direct test of x2 differences between nested models over the common practice of deciding between models by simply comparing incremental fit indexes, such as the CFI. Since the two- and three-factor models are subsets of (i.e., nested within) the theoretically derived four-factor model, we were able to perform tests comparing the latter with each of its nested alternative models. However, the fit of the four- and five-factor models could not be compared directly because they were not nested models. Results Prior to the factor analyses, we examined potential relations between gender and the source variables and also explored whether gender influenced the relation of the source variables to self-efficacy. We found small but significant correlations of gender to personal performance and emotional arousal (in both cases r Å 0.12, p õ .05), with men tending to report somewhat more

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mathematics-related accomplishments and favorable arousal levels than did women. Correlations involving the vicarious learning and social persuasion sources were, respectively, 0.03 and 0.04 (p ú .05). To assess the effect of gender on source/self-efficacy relations, we performed a regression analysis predicting self-efficacy from gender, the sources, and the interaction of gender with each source variable. Gender and the source variables each explained significant variance in self-efficacy, but the interaction terms did not account for additional significant variance. Thus, although men and women differed to a small degree on two of the efficacy sources, there was no evidence that the sources related to self-efficacy differentially by gender. We therefore combined the source data over gender in our subsequent factor analyses (details of the regression may be obtained from the first author). Table 1 presents the fit indices for the alternative models of the efficacy sources in the college sample. These indices each reflect a progressively better fit to the data as one moves from the two- through the four-factor models. Among the first three models, the four-factor model yielded the highest NNFI and CFI values. Our direct comparison of these models, using the x2 difference test, indicated that the four-factor model provided a significantly better fit to the data than did the two- or three-factor model. The five-factor model did not improve over the four-factor model on either of the fit indices, yielding somewhat smaller NNFI and CFI values. x2 values (and degrees of freedom) for the two- through five-factor models were, respectively, 276.32 (53), 104.70 (51), 63.70 (48), and 120.51 (80) (with the exception of the four-factor model, all were significant at p õ .01). TABLE 1 Fit Indices for Four Models of Efficacy Source Information in College and High School Students Model

NNFI

CFI

x2 diff

.889 .973 .992 .982

212.62* 41.00* — —

College sample (N Å 295) Two-factor Three-factor Four-factor Five-factor

.864 .965 .989 .976

High school sample (N Å 481) Two-factor Three-factor Four-factor Five-factor

.838 .892 .941 .953

.870 .916 .957 .964

262.67* 123.72* — —

Note. NNFI is Nonnormed Fit Index; CFI is Comparative Fit Index; x2 diff is the difference in x2 compared to the four-factor model. * p õ .001.

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Because of the inherent tendency of confirmatory factor analysis to favor more complex models, we also calculated parsimony-adjusted fit indices, which control for the number of free parameters in competing models (Hoyle & Panter, 1995). Akaike’s (1987) information criterion (AIC) values for the two- through four-factor models, in order, were 168.32, 2.70, and 032.30 (lower AIC values reflect better fit). These findings indicated that the four-factor model provides improved fit over the two- and three-factor models, taking model complexity into account. Table 2 presents the factor intercorrelations for the three- through fivefactor models in the college sample (see the upper right triangle in each matrix). Examining the four-factor model specifically, it may be seen that the personal performance, social persuasion, and emotional arousal factors were highly interrelated. With increasing levels of performance success, subjects tended to perceive greater interpersonal support and less aversive arousal in relation to mathematics tasks. The vicarious learning factor was moderately

TABLE 2 Factor Intercorrelations for the Three-, Four-, and Five-Factor Models in College and High School Students Three-factor model Factor

1

2

3

1. Performance/persuasion 2. Vicarious learning 3. Emotional arousal

— .608 .865

.331 — .517

.836 .192 —

Four-factor model

1. 2. 3. 4.

Personal performance Vicarious learning Social persuasion Emotional arousal

1

2

3

4

— .502 .854 .918

.235 — .706 .471

.905 .452 — .655

.845 .193 .768 —

Five-factor model

1. 2. 3. 4. 5.

Personal performance Adult modeling Peer modeling Social persuasion Emotional arousal

1

2

3

4

5

— .551 .194 .854 .918

.435 — .365 .643 .533

.031 .208 — .411 .182

.905 .542 .123 — .656

.844 .335 0.017 .767 —

Note. Values above the diagonals are for the college sample (N Å 295); values below the diagonals are for the high school sample (N Å 481).

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related to social persuasion, yielding smaller correlations with personal performance and arousal. In examining correlations involving the two modeling component factors (see the five-factor model matrix), we observed that the adult modeling factor related moderately to the personal performance, social persuasion, and emotional arousal sources (r Å .34–.54), while corresponding correlations for the peer modeling factor were quite small (r Å 0.02–.12). The correlation between the two modeling factors was modest, r Å .21. The fact that the four-factor model achieved a better fit than the twofactor model—despite the high correlations among three of the four primary factors—is somewhat puzzling. We therefore examined this issue further by exploring the possibility that the performance accomplishments, social persuasion, and emotional arousal factors, though apparently reflecting distinct primary latent dimensions, might each load on a higher order latent dimension representing direct, personal experiences with mathematics. The fit indices for this higher order analysis (CFI Å .982, NNFI Å .977) suggest that a hierarchical factor structure is plausible. The secondary, performancebased factor correlated .29 with the vicarious factor. This hierarchical model is illustrated in Fig. 1.

FIG. 1. Model representation and standardized parameter estimates of the second order confirmatory factor analyses. Parameter estimates for the college and high school samples are presented in order, with the latter in parentheses. The parameter value of 1.0 obtained for the path between the second order factor and the first order personal performance factor resulted from the disturbance of this factor being fixed at the lower bound of 0.0. The nonstandardized parameter estimate for this path Å .90, p õ .01.

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STUDY 2

Method Subjects. Research participants were 481 (231 male, 250 female) students enrolled in mathematics courses in a high school located in a middle-class, suburban community in the Midwest. They were primarily White (92%; Blacks, Hispanics, Asians/Pacific Islanders, and Native Americans each accounted for roughly 2% of the sample), with an average age of 15.98 years (SD Å .99). Students were enrolled in a variety of mathematics courses (e.g., basic algebra, geometry, pre-calculus). Fifty percent of the students were sophomores, 41% were juniors, and 9% were seniors. Procedure. All subjects completed demographic, mathematics self-efficacy, and perceived sources of mathematics self-efficacy measures in their mathematics classes. Part of the sample (n Å 296) had participated in a separate study of the social cognitive models of academic/career interest and performance (Lopez, Lent, Brown, & Gore, 1994); subjects in that investigation also completed measures of outcome expectations and interests. As in Study 1, our secondary analysis of previously obtained data addressed a substantively unique set of research questions (i.e., the latent structure of the efficacy source variables versus the explanatory adequacy of the social cognitive models of interest and performance) and allowed us to achieve larger subject to parameter estimate ratios in testing the factor structures. Instruments. The efficacy source variables were assessed with a version of Lent et al.’s (1991) sources instrument (see Study 1), as revised slightly by Lopez and Lent (1992) for use with high school students. Lopez and Lent found that the four scales correlated in expected directions with one another and with grades in an algebra course. They were also strongly predictive of algebra self-efficacy beliefs. Cronbach a coefficients for the performance accomplishments (.82; 9 items), vicarious learning (.59; 10 items), social persuasion (.74; 9 items), and emotional arousal (.90; 10 items) scales were similar to the internal consistency estimates obtained with college students (Lent et al., 1991). Self-efficacy was assessed with measures linked to the skill content of particular mathematics courses (e.g., geometry) (see Lopez et al., 1994). Analysis. Exploratory factor analytic procedures identical to those used in Study 1 were first employed to generate three indicators for each latent efficacy source construct. All l coefficients were significant in each model test (p õ .01; the coefficients ranged from .41 to .95, with most above .70), showing that the measured indicators were reliably related to their respective latent constructs. We then assessed the fit of each of the alternative models of efficacy source information in the high school students, using the same confirmatory factor analysis procedures that were employed with the college sample.

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Results Gender differences in the source variables followed a different pattern in the high school than in the college sample. Gender correlated with vicarious learning (r Å .13, p õ .01) and social persuasion (r Å .10, p õ .05) among high school students, with females reporting somewhat more modeling and persuasive experiences than did their male peers. Correlations of gender to performance accomplishments (r Å .03) and emotional arousal (r Å 0.03) were not significant. In a regression equation predicting self-efficacy, only the source variables explained significant variation; neither gender nor the gender 1 source interaction accounted for significant predictive variance. Because the gender–source correlations were small, and the source variables did not relate to self-efficacy differently by gender, we combined source data over gender in computing the factor analyses. (Regression findings are available from the first author.) The fit indices for the alternative models of the efficacy sources in the high school sample are shown in Table 1 (lower panel). As in Study 1, fit improves incrementally in moving from the two through the four-factor models. Similarly, direct comparison of these models with the x2 difference test (see the last column of Table 1) indicates that the four-factor model provided a significantly better fit to the data than did the two- or three-factor model. Parsimony-adjusted indices also favored the four-factor representation over the less complex models (AIC values, in order, were 331.82, 196.88, and 79.17). In contrast to the first study, however, the five-factor model achieved somewhat larger NNFI and CFI values than did the four-factor model; it also yielded a smaller AIC value (35.61). x2 statistics (and degrees of freedom) for the two- through five-factor models were, respectively, 437.83 (53), 298.88 (51), 175.16 (48), and 195.61 (80) (all significant, p õ .01). Table 2 shows the factor intercorrelations for the three-, four-, and fivefactor models in the high school sample (see lower left triangles in each matrix). Examining the four-factor model specifically, it may be noted that the personal performance, social persuasion, and emotional arousal factors were highly intercorrelated, as in Study 1. Though the pattern of relations between vicarious learning and the other factors was similar to that in the first study (with the largest correlation occurring between vicarious learning and social persuasion), the magnitude of these relations was uniformly higher than that in the first study. Also, as before, when the peer and adult modeling factors were examined separately (see the five-factor correlation matrix), the adult factor tended to produce larger correlations with the other factors (r Å .53–.64) than did peer modeling (r Å .18–.41). The two modeling factors were moderately interrelated, r Å .37. As in Study 1, we explored the possibility that the intercorrelations among the performance accomplishments, social persuasion, and emotional arousal factors might signal the presence of a second order latent dimension reflecting

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direct, personal experience. Each of these primary factors did achieve substantial loadings on the second order dimension, with fit indices also indicating support for a hierarchical factor structure (CFI Å .936, NNFI Å .915). The secondary, performance-based factor correlated .59 with the vicarious factor (this model is illustrated in Fig. 1). GENERAL DISCUSSION

The goal of this study was to clarify the structure of the sources of mathematics self-efficacy in college and high school students using hypothesis testing procedures. Our results largely support theoretical assumptions (Bandura, 1986) regarding the dimensionality of the primary sources of mathematics self-efficacy. In particular, the four-factor model offered a relatively good fit to the data in both samples, producing significantly better fit than the two- or three-factor models, based on x2 difference tests. The four-factor model also appeared to provide a somewhat better fit than the five-factor model in the college sample, though the reverse held in the high school sample. It may be that, given their developmental level, high school students are somewhat more sensitive to the different information provided by peer and adult models. In interpreting these findings, it seems important to distinguish between their theoretical and practical significance (and between model plausibility and parsimony). Theoretically, the confirmatory factor analysis fit indices support the conclusion that the efficacy sources represent relatively discrete types of information. On a practical level, however, the strong intercorrelations among the personal performance, social persuasion, and emotional arousal sources must be considered. Contrary to the factor analytic findings, this factor interdependence—and earlier findings of large correlations among two or three of the efficacy source variables (Lent et al., 1991; Lopez & Lent, 1992; Matsui et al., 1990)—would seem to argue for a more parsimonious model wherein there are fewer than four basic efficacy source dimensions. In essence, though our fit indices suggest that there are valid conceptual distinctions among the source factors, the high factor intercorrelations imply that these distinctions may have been extremely subtle from our respondents’ perspective. The correlations among the personal performance, social persuasion, and emotional arousal factors may be understandable, theoretically, within the context of skill development. It may be that these three primary efficacy sources tend naturally to occur together as people develop skills in relatively structured performance domains, such as mathematics. For example, over time, one individual may tend to do well on mathematics exams and, concomitantly, receive praise for such accomplishments and face future exams relatively free of worry; conversely, another individual, with a checkered performance history in mathematics, will likely experience both less social recognition, and more debilitating arousal, in relation to such performances. Thus, while our fit indices suggest that they reflect distinct primary latent source dimensions, it seems likely that social persuasion and emotional arousal

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become more or less reliably linked to performance-based indicators, such as grades, over the course of one’s education. In developing particular academic skills, children and adolescents rely, in part, on social feedback and perceived anxiety (or calm) to infer how well they are performing. Eventually, performance indicators, social encouragement, and physiological state are likely to provide convergent information about one’s efficacy. This theoretical analysis is supported by our findings of a hierarchical factor structure wherein performance accomplishments, social persuasion, and emotional arousal each load on a secondary, or higher order, dimension reflecting direct, personal experience. In contrast to the other efficacy sources, the vicarious factor(s) yielded a somewhat more differentiated pattern of relationships. In both samples, the general vicarious factor tended to relate most highly to social persuasion, yielding lesser correlations with the performance accomplishment and arousal factors. When the vicarious factor was divided into peer and adult modeling components, the latter achieved consistently higher relations with the other efficacy sources. The peer and adult factors were, themselves, intercorrelated in both samples, albeit not strongly. These findings involving vicarious learning deserve some comment. Observing others perform well at mathematics tasks does not necessarily translate into personal performances that are successful or worry-free. Model influences may support personal mastery experiences, but do not ensure them. Ultimately, students may rely more on indicators of personal efficacy that are based on their own attainments rather than on those of their significant others. The generally stronger relation of the adult (versus peer) modeling factor to the other efficacy sources may suggest that adult models are particularly instrumental in encouraging students’ efforts at mathematics (e.g., by offering verbal support or serving as occupational exemplars). Prior studies have also found perceived adult model influences to relate more strongly than peer model influences to career-relevant variables(e.g., Hackett, Esposito, & O’Halloran, 1989; Orput, O’Brien, & Brown, 1990). Several caveats and directions for further research might be noted. First, while the latent structure of the source factors was reasonably consistent across our high school and college studies, replication of these findings using other samples and efficacy source indicators would increase confidence in their accuracy and stability and address the possibility that the findings were an artifact of our measurement device or sampling. Caution should also be exercised in view of the tendency of the fit indices to favor more complex models. Second, it would be useful to examine whether the theoretical distinctions among the primary efficacy sources translate into practical differences. For instance, should interventions aimed at performance enhancement incorporate separate components aimed at the three direct experience-related factors, or are these factors so intertwined in students’ perceptions that it would make better sense to treat them jointly (e.g., by providing skill-building, support, and anxiety management ingredients as part of a multicomponent treatment)?

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Third, the vicarious and physiological factors might be assessed with more refined measurement devices. Our sources instrument reflected only a single aspect of physiological state (i.e., arousal); other aspects, such as mood, may also be relevant to the self-appraisal of intellectual capabilities. Also, in assessing vicarious influences, the current instrument aggregated diverse models by referring, for example, to parents in a collective sense. Because individual parents (or other role models) may have a unique influence on students, it may be preferable to isolate particular models, such as mothers and fathers, rather than assume that they represent a monolithic source of influence (cf. Hackett et al., 1989). Fourth, although the current methodology is useful in suggesting the types of experiences that influence self-efficacy beliefs, alternative methods, such as cognitive assessment, may help to illuminate how students process efficacy-relevant information or arrive at efficacy estimates under more natural conditions (Lent, Brown, Gover, & Nijjer, 1996). When taken together with previous findings (e.g., Campbell & Hackett, 1986; Hackett et al., 1990), our results offer some tentative implications for practice—in particular, for boosting unrealistically low mathematics selfefficacy. Efficient routes may include challenging clients’ interpretation of past performance data (Brown & Lent, 1996) or structuring new mastery experiences, for example, by assisting students to identify appropriate remedial or refresher classes. Social support and anxiety management may offer useful additional intervention ingredients for many students, though they seem unlikely to effect durable changes in mathematics self-efficacy in the absence of compelling performance-based feedback. Adult models may also be used to facilitate intervention, particularly for students who lack exposure to nontraditional exemplars. In sum, the present findings support a four-factor model of the primary sources of mathematics self-efficacy in college students, and a five-factor model in high school students. However, three of the source factors were substantially interrelated, and there was evidence of a hierarchical factor structure in both samples, with a higher order latent dimension reflecting direct, personal experiences (as distinct from indirect, vicarious experiences) with mathematics. Future research is needed to clarify how the efficacy sources are structured in other populations and performance domains and whether the theoretical distinctions among the sources can inform interventions aimed at modifying self-efficacy percepts. REFERENCES Akaike, H. (1987). Factor analysis and AIC. Psychometrika, 52, 317–332. Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ: Prentice–Hall. Bandura, A. (1989). Human agency in social cognitive theory. American Psychologist, 44, 1175– 1184.

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