No abstract

Recent research on cognitive development, memory, and reading indicates that much attention is being given to metacognition. There is also growing support for the view that purely cognitive analyses of mathematical performance are inadequate because they overlook metacognitive actions. This paper is intended as an introduction to metacognition and the role it plays in mathematical performance.

Any good mathematics teacher would be quick to point out that students' success or failure in solving a problem often is as much a matter of self-confidence, motivation, perseverance, and many other noncognitive traits, as the mathematical knowledge they possess. Nevertheless, it is safe to say that the overwhelming majority of problem-solving researchers have been content to restrict their investigations to cognitive aspects of performance. Such a restricted posture may be natural for psychologists and artificial intelligence scientists who are concerned primarily with expert systems or machine intelligence, but it simply will not suffice for the study of problem solving in school contexts.In 1986 we began a study of the role of metacognition in the mathematical problem-solving behavior of seventh-grade students that was preceded by about 5 years of preliminary work in this area. Although we are convinced that metacognition plays a vital role in problem solving, we regard it as but one of a number of driving forces (to use the term coined by Silver [1982] and elaborated on by Schoenfeld [1985, 1987]). At least two other domains seem particularly important: affects and attitudes and beliefs. The purpose of this chapter is to present our ideas about the nature ofthe factors that affect success during problem solving. We begin with a discussion of four types of noncognitive and metacognitive factors. This is followed by several illustrations of the strong influence that certain factors (viz., self-confidence, interest, beliefs, and metacognition) can have on the problem-solving performance of seventh graders. The chapter concludes with a brief discussion of some conjectures related to the nature of the relation of these factors to problem solving. Theoretical ConsiderationsWe begin by postulating that an individual's failure to solve a problem successfully when the individual possesses the necessary knowledge2 stems from the presence of noncognitive and metacognitive factors that inhibit the appropriate utilization of this knowledge. These factors are of at least four types: affects and attitudes, beliefs, control, and contextual factors. In the following sections, we discuss each of these types. D. B. McLeod et al. (eds.), Affect and Mathematical Problem Solving

This study examined arithmetical performance in relation to Luria's theory of the functional organization of the brain and Das, Kirby, and Jarman's model of simultaneous-successive processes. It was found that both computation and problem solving are related to simultaneous synthesis and behavior regulation factors. However, tests of problem solving and quantitative ability loaded higher on the simultaneous synthesis factor than on the behavior regulation factor, while the computation test loaded higher on the regulation factor than on the simultaneous synthesis factor.

Recent research in mathematics education has shown that success or failure in solving mathematics problems often depends on much more than the knowledge of requisite mathematical content. Knowing appropriate facts, algorithms, and procedures is not sufficient to guarantee success. Other factors, such as the decisions one makes and the strategies one uses in connect ion with the control and regulation of one's actions (e.g., deciding to analyze the conditions of a problem, planning a course of action, assessing progress), the emotions one fee ls while working on a mathematical task (e.g., anxiety, frustration, enjoyment), and the beliefs one holds relevant to performance on mathematical tasks, influence the direction and outcome of one's performance (Garofalo and Lester 1985; Schoenfe ld 1985; McLeod 1988). These other factors, although not explicitly addressed in typical mathematics instruction, are nonetheless important aspects of mathematical behavior.