relationship between students' self-assessment of

Psychological Reports, 2006,98,765-778. O Psychological Reports 2006

RELATIONSHIP BETWEEN STUDENTS' SELF-ASSESSMENT O F THEIR CAPABILITIES AND THEIR TEACHERS' JUDGMENTS OF STUDENTS' CAPABILITIES I N MATHEMATICS PROBLEM-SOLVING ' PEGGY P. CHEN

Hunter College The City University of New York Summa y.-The study examined the judgments made by four seventh-grade mathematics teachers of their 107 students' competence in solving mathematics problems. Simultaneously, the 107 students made self-efficacy judgments about their capability in solving mathematics problems. The two sets of judgments were tested for predicting students' mathematics performance. Also, students' prior mathematics achievement was studied for its influence on both teachers' and students' judgments and students' mathematics performance. Teachers were asked to make judgments of each student for every mathematics problem solved. Results were consistent with prior research indicating that students' mathematics self-efficacy beliefs were highly predictive of their performance. Path analysis indicated that the mathematics teachers' judgments were also highly predictive of students' performance and self-efficacy. In turn, these variables predicted students' postperformance judgments. Combining students' self-efficacy judgments and teachers' judgments of students increased predictiveness for students' mathematics performance. Educational implications were also discussed.

Although studies have been conducted on teachers' perceived knowledge of their students' capabilities and mathematics thinking processes (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Fennema & Franke, 1992; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996) and students' mathematics self-efficacy and self-evaluation (Schunk, 1998; Zimmerman, 2OOO), no study has yet examined the consistency between students' self-assessment of their capabilities and their teachers' assessment of students' capabilities in solving mathematics problems. Studies have consistently shown that adolescents generally grossly overestimated their abilities to solve mathematics problems (Ewers & Wood, 1993; Pajares & Kranzler, 1995; Pajares & Graham, 1999; Chen, 2003)) and college students were generally overconfident in predicting and postdicting their test performance (Hacker, Bol, Horgan, & Rakow, 2000; Bol & Hacker, 2001; Bol, Hacker, O'Shea, & Allen, 2005). Even though, to some extent, overconfidence may enhance students' motivation, a reasonable match between self-confidence and actual performance is desired. As Bandura (1986) posited, reasonable agreement be'Address correspondence to Peggy P. Chen, Hunter College, 695 Park Avenue, New York, NY 10021 or e-mail ([email protected]).

DO1 10.2466rnR0.98.3.765-778

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tween self-efficacy judgments and actions is most desirable, even though higher self-efficacy judgments can enhance motivation to improve later performance. The present study was done to test agreement between students' perceived capabilities and their actual performance and the agreement between teachers' perceptions of students' capabilities and students' actual performance. Specifically, there were two objectives, to assess the relationship between mathematics teachers' judgments of their students' capabilities and students' own perceived capabilities and to use ~ a t hanalysis modeling to posit direct and indirect effects of both teachers' and students's judgments of their own mathematics capabilities on their actual performance.

Self-efficacy Bandura (1986) defined self-efficacy as "people's judgments of their capabilities to organize and execute courses of action required to attain designated types of performances" (p. 391). As noted by Zimmerman (2000)) self-efficacy specifically refers to later functioning and is assessed prior to performing the target activity. In addition, self-efficacy judgments are taskand domain-specific; thus, measuring self-efficacy should be tailored to the specific activity in question and be context-specific (Bandura, 1986; Pajares & Miller, 1995; Zimmerman, 2000). "Ill-defined global measures of perceived self-efficacy or defective assessments of performance will yield discordances" (Bandura, 1986, p. 397). Pajares and Miller (1995) showed that global measures of self-efficacy were less predictive than task-specific self-efficacy measures. Mathematics self-efficacy beliefs are typically assessed using self-ratings of skill for specific mathematics problems (Schunk, 1981). Research in self-efficacy has shown that students who have higher selfefficacy beliefs or who perceive themselves as more capable achieve higher than those with lower self-efficacy beliefs (Schunk, 1991, 1998). In mathematics, models consistently showed that students' mathematics self-efficacy has a positive association with their performance (Pajares & Miller, 1995; Chen, 2003) and plays a mediating role in this mathematics performance (Pajares & Miller, 1994). Recent research on mathematics self-efficacy has focused on the issue of accuracy between students' self-efficacy, i.e., perceived capability, and their actual performance in mathematics (Ewers & Wood, 1993; Pajares & Graham, 1999; Chen, 2003). Ewers and Wood (1993) studied fifth-grade gifted and regular students' mathematics self-efficacy and their prediction accuracy. They found that gifted students made fewer overestimations than regular students; higher-achieving students were more accurate in judgments of their capabilities than lower-achieving students. Also, Pajares and Graham (1999) found that middle-school students who were more accurate in assessing their own self-efficacy performed better on their mathematics tests than those who were less accurate. Students (regular and

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gifted) were overconfident in their self-efficacy judgments, but gifted students were less overconfident than regular students. Working with seventh graders in mathematics classes, Chen (2003) also observed that students overestimated their mathematics self-efficacy when solving mathematics problems. These studies examining the accuracy of students' self-efficacy consistently showed that students overestimate and are poor judges of their own capabilities for solving mathematics problems.

Teachers' Judgments and Perceptions As for teachers' beliefs, research has focused on teachers' judgments and perceptions of students' thinking about mathematics and of their students' beliefs about and competence in mathematics (Fennema & Franke, 1992; Fennema, et al., 1996; Giwin, Stipek, Salmon, & MacGyvers, 2001; Ball, 2002). The importance of teachers' perceptions of their students' knowledge of mathematics was well documented in the Cognitively Guided Instruction studies (Fennema & Franke, 1992; Fennema, et al., 1996). Fennema and colleagues developed this model, which focused on training mathematics teachers to use children's thinking processes to guide class instruction and tested the model extensively over past decades. Carpenter, et al. (1989) showed that teachers who were training under this model used children's mathematical thinking processes to encourage their students to engage in problem-solving by using a variety of ways to solve problems and listened to students' problem-solving strategies. Fennema, et al. (1996) showed that teachers did change their beliefs over time, and teachers' beliefs had an important effect on their decision-making and classroom instruction. Research using the Cognitively Guided Instruction methodology (Carpenter, et al., 1989; Peterson, Carpenter, & Fennema, 1989; Fennema, et al., 1996) has consistently shown that teachers' knowledge of mathematics as well as their perceptions of students' knowledge of mathematics affect how they plan appropriate classroom activities and assist students in reaching a deeper understanding of mathematics concepts and higher achievement. Teachers' judgments of their students' academic performance may be influenced by students' motivational behaviors in the classroom, current academic performance, and prior achievement records (Giwin, et al., 2001). Equally crucial to students' learning mathematics and achievement is examination of agreement between teachers' and students' judgments of students' motivation in mathematics. Giwin, et al. (2001) examined both teachers' and students' judgments of students' motivation in learning mathematics over time. Comparisons were made between teachers' and students' judgments of student motivation in variables such as self-efficacy, learning goals, and positive and negative emotions. They found teachers were consistent in their judgments of students across the motivational variables throughout an aca-

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demic year and for different topics in mathematics, e.g., fractions and measurement. By contrast, students' own self-judgments of their motivation fluctuated and varied across these motivational measures. Giwin, et a l . ) ~study (2001) showed that comparing agreement between teachers' and students' judgments on students' motivation can provide a more accurate diagnosis of the causes of students' underlying motivational problems in learning mathematics. Postperformance Judgments Using path analysis, Chen (2003) showed that middle-school children who performed better on their mathematics problems as well as those with higher self-efficacy beliefs had higher postpeLformance self-evaluations. Students who reported higher mathematics self-efficacy did solve more mathematics problems correctly and, in turn, had favorable postperformance selfevaluations. Similarly, Bol and Hacker (2001) showed that higher-achieving college students were more accurate in both their preperformance judgments, i.e., prediction, and postperformance judgments, i.e., postdiction. Bol, et al. (2005) showed that higher-achieving students were more accurate in their prediction of number of questions on a test they would likely answer correctly than lower-achieving students but showed no difference on the postdiction measure. Interestingly, lower-achieving students were more accurate in their postdictions, yet still overly confident, whereas higher-achieving students were less accurate and under-confident. Even though Bol and Hacker (2001) and Bol, et al. (2005) examined the calibration accuracy of college students' performance on tests and quizzes, their measures of prediction and postdiction were global measures. In other words, students only needed to estimate the number or percent of questions they were likely to answer correctly before and after a test, which is a gross judgment. Item-specific predictions and postdiction judgments would provide a more precise measure of calibration accuracy, as the present study was designed to test. Prior Achievement According to Bandura (1997), people's beliefs in their present capabilities to perform a task are also influenced by their prior experiences with attempting and completing similar tasks. Previous studies on self-efficacy have shown that students' prior mathematics performance is associated with their self-efficacy and their present mathematics performance (Pajares & Miller, 1994; Chen, 2003). Thus, students' prior achievement is considered an antecedent variable, which contributes to students' self-efficacy beliefs and possible influences on teachers' judgments. In testing a causal model, path analyses are performed to estimate the magnitude of direct and indirect effects of the variables and to test the hypothesized links among them (Stage, Carter, & Nora, 2004). In the present

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study, it was first hypothesized that the exogenous variable, students' prior achievement in mathematics, would have a positive influence on teachers' judgments and students' self-efficacy beliefs. Second, research has shown that students' self-efficacy would influence their performance in mathematics (Schunk, 1991; Pajares & Graham, 1999) as well as their postperformance judgments (Bol & Hacker, 2001; Chen, 2003). Thus, the present researcher hypothesized that students with higher self-efficacy would have higher performance and, in turn, higher postperformance judgments. As indicated earlier, studies on teachers' knowledge of their students' knowledge (Peterson, et al., 1989; Fennema & Franke, 1992; Fennema, et al., 1996) have indicated that teachers who knew their students' cognitions were better at instructional planning, spent more time on problem-solving activities, and could explain the mental processes students used to solve mathematics problems more clearly. However, these studies did not test how teachers' judgments or knowledge of their students actually influenced students' mathematics performance. Therefore, the third hypothesis of the current study was that teachers' judgments would have a positive influence on students' performance; in turn, students' higher performance would produce higher postperformance judgments.

This study was conducted with four seventh-grade mathematics teachers and their 107 (65 female, 42 male) students from four parochial schools in Nashville, Tennessee. The schools were predominately middle-class in socioeconomic status, and about 98% of the students were Euro-American. The schools' administrators indicated that the four participating mathematics teachers were considered highly experienced, with a range of 5 to 15 years of teaching experience. All four teachers were Euro-American women. To assess whether the sample size provided sufficient statistical power, Cohen's table (1992) was consulted, which suggested that for conducting multiple and partial correlation analyses, this sample size would be sufficient to detect medium and large effect sizes at the .05 level with 80% power. Stage, et al. (2004) suggested that for path analysis, a ratio of 20 cases per variable would be adequate. In the present study, with five variables examined, 100 cases were sufficient. Measures Mathematics performance.-In this study, seven mathematics problems, containing items of varying difficulty and designed for the seventh and eight grades, were adapted from the Third International Mathematics and Science Study (IEA TIMSS, 1995). The seven items were chosen based on the math-

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ematics topics they intended to measure, which were also most often taught in U.S. seventh grades: (a) fractions and number sense, (b) algebra, (c) measurement, (d) geometry, (e) data representation, analysis and probability, and ( f ) proportionality. An example item for the topic of geometry was "In a quadrilateral, two of the angles each have a measure of 110°, and the measure of a third angle is 90". What is the measure of the remaining angle?" To obtain more accurate information about students' performance and judgments of their capability and to avoid the possibility of guessing, the researcher modified the items into an open-ended format (Pajares & Miller, 1997). However, for consistency with the mathematics self-efficacy rating scale, which required students to judge whether they could solve each mathematics problem correctly, the items were scored either as correct or incorrect. No partial credit was given. The coefficient alpha estimate of internal consistency for this measure was .65. Mathematics self-efficacy.-As indicated in the self-efficacy literature (Bandura, 1986), self-efficacy measures are to be obtained prior to the actual performance of the target tasks. Before solving each mathematics item, students were asked to rate their self-efficacy for each question: "How confident are you about solving this mathematics problem correctly?" The rating scale had anchors of 1: Not at all and 8: Completely. Cronbach alpha for internal consistency reliability of responding to this scale was .83. Postperformance judgment.-After students solved each mathematics problem, they were asked to render their postperformance judgments. The scale asked students "How confident are you that you solved this mathematics problem correctly?" using a rating scale with anchors of 1: Not at all and 8: Completely. Cronbach alpha was .83. Previous achievement in mathematics.-The participating schools used the Iowa Test of Basic Skills to measure students' achievement. The schools provided the researcher participating students' sixth-grade mathematics scores in the form of grade equivalence and national percentile. According to Impara and Plake (1998))the internal consistency reliability of these subscales ranges from .80 to .90. Mathematics teachers' judgments.-Here, teachers' judgments were related to knowing which particular mathematics skills and concepts were needed to solve specific questions and knowing which skills and concepts their students have mastered to solve those questions. Following the line of research on judgments of mathematics self-efficacy, which are typically assessed with specific target problems or are context-specific, evidence shows that predictability of self-efficacy measures depends on their specificity and correspondence to actual performance tasks (Schunk, 1991; Zimmerman, 1995). This scale was item- and student-specific. For each mathematics problem and each participating student, the mathematics teacher was to answer


77 1

"How likely w d the following student solve this question correctly?" by rating using anchors 1: Not likely and 4: Most likely. Cronbach alpha for this scale of seven items was .92.

Procedure Data were collected in the middle of second (i.e., Spring) semester of the students' seventh grade. Participating students completed the study during their mathematics classes. They were given the following materials: (1) an example of the self-efficacy scale, (2) an answer sheet to record their mathematics self-efficacy judgments, and (3) the seven mathematics problems described above, each followed by a postperformance self-evaluation scale. Using an overhead projector, the researcher presented a sample mathematics problem on the screen and read it aloud. The participating students then made self-efficacy judgments for the sample mathematics item without solving it. Once they understood the procedures, the researcher showed the first mathematics problem and read it aloud. Then, the researcher covered the target problem and instructed students to render their self-efficacy judgments without solving the problem. The brief exposure to each problem allowed students to assess its difficulty without having time to solve it. The same procedure was carried out for all problems. After self-efficacy judgments were made on the seven problems, students were given the actual problems, one per answer sheet. Accompanying each mathematics problem was a postperformance judgment scale presented at the bottom of the answer sheet. Students were instructed to solve one item at a time and fill out the corresponding postperformance scale before going to the next item. Ample time was given to solve the problems. While students were participating in the study, the teachers were asked to make their own judgments on how likely each of their participating students could solve each target question correctly. Teachers were first asked to identify for each mathematics question which concepts, skills, and procedures were necessary to solve the problem successfully; then, they were asked to reflect on each student's acquired mathematics knowledge and skills to assess whether the student could successfully answer each target question. Such a context-specific assessment of teachers' perceptions and judgments of their students' mathematics capabilities may contribute greatly to predicting students' performance better than students' own assessments of their capability. Such data can also offer implications for teachers in more effective ways to make instructional decisions. Each mathematics teacher completed seven forms consisting of the specific mathematics questions (one form per question). O n these forms, which included rating scales, teachers made their judgments of their students' competence in mathematics. The specific mathematics question was printed on

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each form, followed by a list of participating students' names and case numbers. The directions, which were printed on each form above each target problem, stated that the teacher was to read the problem and reflect on the specific mathematics knowledge and skills needed to solve it. Then, the teacher was instructed to rate how likely each participant on the form would be able to solve the target problem correctly. O n average, each teacher made 25 to 28 judgments, i.e., for 26 to 28 participating students, per mathematics question; this process was repeated seven times for each of the problems. Even though the teachers recorded their judgments on the forms, they had no knowledge of how well their students had actually performed on the mathematics problems. The teachers turned in their completed forms to the researcher on either the same or the next day. Before the teachers returned the completed forms, they removed all students' names, leaving only case numbers and their corresponding judgments. Participating teachers and students learned about the students' actual performance only after the study had been completed. Data Analyses The participating students completed seven mathematics problems and made self-efficacy and postperformance judgments on these items. The same seven targeted questions for which mathematics teachers rendered their judgments of their students' capabilities were subjected to analysis. Path analysis was conducted using LISREL 8.5 (Joreskog & Sorbom, 2002) to test the hypothesized model. Model fit was assessed with the chi-squared statistic normed fit index (NFI), and comparative fit index (CFI). These indices are used to evaluate whether the estimated covariance matrix is a good presentation of the sample covariance matrix. An acceptable fit is indicated by a small chi-square value and a nonsignificant chisquare ( p > .05), and by CFI and NFI fit index values greater than .90 (Tabachnik & Fidell, 2001).

(xi),

Model Testing Descriptive statistics of the measures are shown in Table 1 with Pearson intercorrelations among the variables. Chi-square for the model tested was not significant (N = 107) = 1.61, nsl, suggesting a good fit. Other indices CFI and NFI were 1.00 and .99, respectively, also suggesting a good fit of the model tested (Stage, et al., 2004). Fig. 1 shows the model with the standardized path coefficients. Table 2 shows the squared multiple correlations (R2) for the endogenous variables in the model. In this model, R2 values ranged from small to large; variables predicting self-efficacy accounted for 19% of the variance,

[x2

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TABLE 1 MEANS,STANDARD DEVIATIONS, AND PEARSON CORRELATIONS FORMEASURES (N= 107)

M

Measure Mathematics Performance Students' Mathematics Self-efficacy Teachers' Judgment Self-evaluation Prior Achievement

2.42 6.12 2.64 5.54 66.9

SD 1.64 1.30 0.78 1.56 23.09

Y

1

2

3

4

.42 .57 .53 .61

.40 .71 .39

.36 .67

.34

5

-

Note.-Mathematics performance was based on seven items. Prior Achievement was expressed as national percentile ranks for students in sixth-grade mathematics scores on the Iowa Test of Basic Skills.

variables predicting teachers' judgments accounted for 45% of the variance, variables predicting mathematics performance accounted for 44 % of the variance, and variables predicting postperformance judgments accounted for 57% of the variance. In addition, the effect sizes f were reported indicating medium and large effect sizes of endogenous variables in the present model. As Cohen (1992) suggested, effect size ( f 2 ) greater than .35 is considered large, greater than .15 is medium, and greater than .O2 is small. Self-efficacy .17

1-

-

-f i FIkype Performance

rformance D~II-evaluation

FIG. I. A final path model showing the influence of students' self-efficacy and teachers' judgments on mathematics performance (+ significant path, p < .05)

In the model, the exogenous variable, prior achievement in mathematics, had a direct effect on teachers' judgments, suggesting that the higher the students' sixth-grade Iowa Test of Basic SkiUs scores in mathematics were, the higher the teachers' judgments of students' capabilities in solving such problems. Prior mathematics achievement had a nonsignificant direct effect (p = .22, p > .05) and a significant indirect effect on mathematics self-efficacy, as mediated by teachers' judgments. This suggests that students' prior achievement may have influenced teachers' judgments, which in turn influenced students' mathematics self-efficacy. Prior mathematics achievement, mathematics self-efficacy, and teachers' judgments had statistically significant

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TABLE 2 DECOMPOSITION OF EFFECTS FROMPATHANALYSIS Effect

Standardized Coefficient (P)

SE

t

R2

f2

Prior Mathematics Achievement Teachers' Judgments on Self-efficacy Prior Mathematics Achievement on Teachers' Judgments Prior Mathematics Achievement Teachers' Judgments Self-efficacy on Mathematics Performance Self-efficacy Performance on Postperformance Self-evaluation " p < .05.

direct effects on students' mathematics performance. Prior mathematics performances also had a statistically significant indirect effect on mathematics performances mediated by teachers' judgments and students' mathematics self-efficacy, suggesting that prior achievement may affect mathematics performance in part through its effect on teachers' judgments. Prior mathematics achievement and teachers' judgments also had statistically significant indirect effects on students' postperformance judgments. Students' self-efficacy and mathematics performances had direct effects on postperformance judgments, suggesting that self-efficacy and mathematics performance affect students' postperformance judgments. Overall, results of the path analyses showed that teachers' judgments and students' mathematics self-efficacy beliefs were important variables affecting students' mathematics performance, which in turn affected postperformance judgments.

DISCUSSION Guided by Bandura's framework of self-efficacy (1986, 1997)' the research on teachers' knowledge of students' cognition (Fennema, et al., 1996)' and studies on the calibration accuracy of test performances (Bol, et al., 2005)' in this study a model was proposed to test relationships between students' self-efficacy beliefs and their teachers' judgments of students' capabilities in solving mathematics problems. The measures were item-specific to capture a more accurate assessment of students' self-efficacy beliefs, postperformance judgments, and teachers' judgments of students' capabilities in solving each mathematics item. Overall, the findings supported the path model tested. The three fit indices showed that the model fit the data very well, suggesting that both students' self-efficacy and teachers' judgments were strongly associated with students' performance in mathematics.


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Three specific hypotheses were tested in the path model. The first hypothesis was that students' prior achievement would affect mathematics selfefficacy and teachers' judgments; this was partially confirmed. Path analysis showed that students' prior mathematics achievement, i.e., sixth-grade Iowa Test of Basic Skills mathematics scores, has a positive effect on teachers' judgments, but no statistically significant direct effect on students' self-efficacy. However, prior achievement has a statistically significant indirect effect on students' self-efficacy through teachers' judgments. This is a very important finding since teachers' judgments can influence students' beliefs in their capabilities to solve mathematics problems. One possible explanation for this could be that teachers' judgments, although covertly expressed during this study, may have already been externalized and expressed to students throughout the school year, as suggested by the direct effect of teachers' judgments on students' self-efficacy (cf. Fig. 1). This finding is almost certain because teachers regularly provide informal and formal feedback to students regularly throughout the school year. Especially since this study was conducted in the second semester of the academic year, teachers had knowledge of their students' overall mathematics skills and achievement. Although prior academic achievement did not have a statistically significant direct association with students' self-efficacy as hypothesized, the path model indices were strengthened when this path was included in the model. As for predicting current mathematics performance, students' prior achievement was a strong predictor in the present study, as has been consistently shown in research on mathematics self-efficacy (Pajares & Miller, 1994; Bandura, 1997; Chen, 2003). The second hypothesis tested in the model was based on the relationships between students' mathematics self-efficacy and mathematics performance and between their self-efficacy and postperformance judgments. This hypothesis was confirmed. Students' self-efficacy beliefs positively affected their mathematics performance, as has been consistently found (Schunk, 1998; Pajares & Graham, 1999; Zimmerman, 2000). As Bandura (1997) indicated, students with higher self-efficacy performed better than those with lower self-efficacy. The findings also showed that students' self-efficacy beliefs are directly associated with their postperformance judgments and indirectly with their postperformance judgments through performance in mathematics. In other words, the findings suggested that students' self-efficacy beliefs rendered prior to solving mathematics problems were more closely related than actually doing and solving the problems. Chen (2003) also reported that students' self-efficacy beliefs affected their postperformance judgments more than their mathematics performance. The third hypothesis tested concerned the relationship between teachers' judgments of students' capabilities and their students' performance. The

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hypothesis was confirmed. The path model showed that teachers' judgments of their students can indeed have an effect not only on students' performance, but on their beliefs of their own capabilities and on their postperformance judgments. The results suggested that the teachers were fairly accurate in predicting their students' performance on mathematics items. As the findings suggested, compared to students' own assessment of their capabilities, i.e., self-efficacy beliefs, teachers' judgments of students' capabilities had a stronger effect on students' mathematics performance. Since these teachers were considered experienced, their judgments may be more perceptive than their students' own judgment. As prior studies indicated, experienced teachers' knowledge of their students' cognition and belief systems could be invaluable for instructional decisions and planning (Carpenter, et al., 1989; Peterson, et al., 1989; Fennema, et al., 1996; Giwen, et al., 2001). In sum, the data fit the path model tested in this study very well. Students' self-efficacy beliefs and the four teachers' judgments influenced students' mathematics performance. This study used item-specific and contextspecific measures consistent with research on self-efficacy (Bandura, 1986; Pajares & Miller, 1995; Zimmerman, 2000). Such item-specific measures are likely to assess more accurately students' belief systems and teachers' perceptions of their students' mathematics skills and knowledge. However, the data gathered in the present study should be interpreted cautiously. The study used only seven mathematics items and the reliability of that performance was low. More items should be included in research to increase the measure's reliability (Linn & Miller, 2005). Findings showed that teachers' judgments of their students' understanding of mathematics concepts and skds could be invaluable for how students learn mathematics since they were fairly accurate in their assessments (Carpenter, et al., 1989; Peterson, et al., 1989; Fennema, et al., 1996; Giwin, et al., 2001). However, since the study was conducted with only 107 students and 4 teachers from parochial schools -which is statistically sound for testing a path model, replicating the study with more heterogeneous samples is suggested. In addition, it is very possible that the teachers' judgments are static or general. Even though the present researcher attempted to gather item- or context-specific judgments, i.e., with a unique mathematics concept and skills embedded in each item, further research should test whether teachers' judgments are general and static for each student or context-specific. In conclusion, this study provided additional understanding of teachers' judgments and their possible association with students' performance in mathematics, as well as students' own self-assessment of their capabilities in mathematical problem-solving. REFERENCES BANDURA, A. (1986) Social foundations of thought and action: a social cognitive theory. Englewood Cliffs, NJ: Prentice-Hall.


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Accepted May 10, 2006.