The purposes of this study were as follows:
(1) to investigate how high school students integrate and accept sources of mathematics self-efficacy when they judge and feel self-efficacy in mathematics. (2) to figure out the differences in mathematics self-efficacy, mathematics attitude, and mathematics achievement depending on how to combine and accept sources of self-efficacy. (3) to analyze whether achievement goal orientation led to different recognition of information of self-efficacy.
Research questions for the purpose of this study were as follows:
First, how many heterogeneous relations of high school students can be classified between mathematics self-efficacy and sources of mathematics self-efficacy?
Second, what differences do the latent classes that present heterogeneous relations of high school students between self-efficacy and sources of self-efficacy have in self-efficacy, mathematics attitude, and mathematics achievement?
Third, can achievement goal orientation predict the latent classes that present heterogeneous relations of high school students between self-efficacy and sources of self-efficacy?
The subjects were 360 third-grade hight school students in Daegu(206 boys and 154 girls). And they were surveyed on mathematics self-efficacy, achievement goal orientation, learning attitude in mathematics, and academic achievement in mathematics. To verify that there were heterogeneous relations between the four sources of mathematics self-efficacy and mathematics self-efficacy and to derive the latent classes of sources of mathematics self-efficacy, I used a regression mixture model analysis. To analyze differences among the latent classes in academic achievement in mathematics, one-way ANOVA was used. In the cases of self-efficacy and mathematics attitude, homogeneity of variance was not assumed. So, robust ANOVA(Welch test) was used. Because self-efficacy was a factor that could affect mathematics attitude and mathematics achievement, I used ANCOVA to control the influence of self-efficacy and compared values before and after control. Follow-up tests were performed for all analyses. Also, multinomial logistic regression analysis was used to find out how achievement goal orientation influenced the classification of the latent classes.
The major findings of this study and conclusions were as follows:
First, regression mixture analysis identified four latent classes with differential associations between four sources of mathematics self-efficacy on the one hand and mathematics self-efficacy on the other; Individual sources class, Multi sources class, Social sources class, and Non-source class. Individual sources class referred to cases in which self-efficacy was influenced only by the internal sources such as the experience of success and the physiological state among the sources of self-efficacy. Multi sources class was the class in which self-efficacy was affected by all sources of self-efficacy. Social sources class referred to when self-efficacy was affected only by social sources such as vicarious experience and verbal persuasion. Finally, Non-source class had no sources that affected self-efficacy. These results were confirmation of Bandura(1986a)’s argument that people accepted the self-efficacy information through cognitive processing rather than accepting it as it was.
Second, self-efficacy, mathematics attitude, and mathematics achievement were different depending on the four latent classes. There were significant differences in each class. Individual sources class was highest in all three variables, multi sources class was the second highest, social sources class was the next, and non-source class was the lowest. In the end, it was suggested that students should select and combine various high level sources in order to increase their self-efficacy and improve their mathematics attitude and mathematics achievement.
Third, this study identified significant predictive variables that affected the relations between sources of self-efficacy and self-efficacy of high school students. The task approach goal was the most important factor for predicting the four latent classes, and the other achievement goal orientations excluding the task avoidance goal significantly predicted the four latent classes. In this way, it was found that it was necessary to increase the students'' self-efficacy by setting a classroom goal structure that increased the task approach goal and self approach goal and minimized the motivation to avoid.

Key words : mathematics self-efficacy, sources of self-efficacy, achievement goal theory, regression mixture model, mathematics attitude, mathematics achievement