Efficacy Beliefs, Problem Posing, and Mathematics Achievement
Nicolaou, Aristoklis A., Philippou, George N., Focus on Learning Problems in Mathematics
Perceived self-efficacy beliefs have been found to be a strong predictor of mathematical performance while problem posing is considered fundamental in mathematical learning. In this study we examined the relation among efficacy in problem posing, problem-posing ability, and mathematics achievement. Quantitative data were collected from 176 fifth and sixth grade students, and interview data from six students selected on the basis of hierarchical cluster analysis. Students' perceived efficacy to construct problems was found to be a strong predictor of the respective performance as well as of the general mathematics achievement. A strong correlation was also found between ability in problem posing and general mathematics performance. The students constructed problems of greater variety and complexity on the basis of informal tasks rather than on the basis of formal tasks. Significant differences were found in problem posing ability, between fifth and sixth grade students. The findings provide support to earlier studies indicating the predictive power of context-specific efficacy beliefs. Implications are drawn about strategies for enhancing students' efficacy beliefs and problem-posing ability.
Theoretical Background and Aims
Research on mathematics teaching and learning has recently focused on affective variables, which were found to play an essential role that influences behavior and learning (Bandura, 1997). The affective domain is a complex structural system consisting of four main components: emotions, attitudes, beliefs, and values (Goldin, 2002). Beliefs can be defined as one's subjective knowledge, theories, and conceptions and include whatever one considers as true knowledge, although he or she cannot provide convincing evidence to support it (Pehkonen, 2001). Self-beliefs can be described as one's beliefs regarding personal characteristics and abilities and include dimensions such as self-concept, self-efficacy, and self-esteem. Self-efficacy can be defined as "one's belief that he/she is able to organize and apply plans in order to achieve a certain task" (Bandura, 1997, p. 3). This study focuses on self-efficacy of primary students with respect to problem posing.
Self-efficacy is a task-specific construct and there is a correspondence between self-efficacy beliefs and the criterial task being assessed; in contrast, self-concept is the sense of ability with respect to more global goals (Pajares, 2000; Bandura, 1986), while self-esteem is a measure of feeling proud about a certain trait, in comparison with others (Klassen, 2004; Bong & Skaalvik, 2003). The task-specificity of efficacy beliefs implies that related studies are more illuminating when they refer to certain tasks, such as problem posing; the predictive power of self-efficacy is in this case maximized (Pajares & Schunk, 2002). On the other hand, the level of specificity could not be unlimited; as Lent and Hackett (1987) have rightly observed, specificity and precision are often purchased at the expense of practical relevance and validity.
The construct self-efficacy is tightly connected to motivation and plays a prominent role in human development since it directly influences behavior. According to Bandura's social cognitive theory, every individual possess a system that exerts control on his/her thoughts, emotions and actions. Among the various mechanisms of human agency, none is more central or pervasive than self-efficacy beliefs (Bandura & Locke, 2003; Pajares, 2000).
Research on self-efficacy has recently been accumulated providing among other things notable theoretical advances that reinforce the role attributed to this construct in Bandura's social cognitive theory. Several studies have indicated a strong correlation between mathematics self-efficacy and mathematics achievement (Klassen, 2004). It was further found that mathematics self-efficacy is a good predictor of mathematics performance irrespective of the indicators of performance (Pajares, 1996; Bandura, 1986) and regardless of any other variables (Bandura & Locke, 2003; Pajares & Graham, 1999). …