Self-Confidence, Interest, Beliefs, and Metacognition: Key Influences on Problem-Solving Behavior

Abstract

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

Undergraduate Students’ Understanding of Proof: Relationships Between Proof Conceptions, Beliefs, and Classroom Experiences with Learning Proof

“…McLeod (1992), in a review of the literature on affective issues, used a variation of the classification of beliefs proposed by Lester et al (1989), who described beliefs in terms of the subjective knowledge of students regarding mathematics, beliefs about self as learner of mathematics, and beliefs about mathematics teaching. He concluded that beliefs in the first two of these dimensions play a central role in mathematics learning.…”

“…To identify students' beliefs about proof, following the theoretical framework proposed by Lester et al (1989) the survey focused on three contexts of proof: (a) beliefs about proof, (b) beliefs about themselves as learners of proof, and (c) students' previous experiences with proof in instructional settings. Table 10 shows examples of items that were designed to address each of these three contexts of proof.…”

“…He concluded that beliefs in the first two of these dimensions play a central role in mathematics learning. Indeed, a brief review of this literature suggests that there is a significant correlation between achievement in mathematics and confidence in doing mathematics, as well as between achievement and perceived motivation and personal control (Fennema and Sherman 1977;Lester et al 1989;Reyes 1984;Schoenfeld 1989). However, McLeod noted a void in the literature on students' beliefs about mathematics instruction (and, as a result, how beliefs on mathematics instruction may relate to students' achievement in mathematics)-a void that remains two decades later.…”

“…Even less is known regarding the possible relationship between these beliefs and students' achievement in either constructing or evaluating proof. One of our goals in this study was to gain further insights into students' beliefs with respect to proof in the three dimensions suggested by Lester et al (1989)-verification, explanation, communication-and the relationship of these beliefs to students' proof conceptions. …”

Meeting the Needs of Students With Learning Disabilities in Inclusive Mathematics Classrooms: The Role of Schema-Based Instruction on Mathematical Problem-Solving

“…Most textbooks do not provide opportunities to discriminate among problems that require different solutions, because all problems presented on a page can be solved using the same procedure (e.g., multiplication). A second concern is the use of superficial cues such as key words [(e.g., in all suggests addition, left suggests subtraction, share suggest division (Lester, Garofalo, & Kroll, 1989)] that "send a terribly wrong message about doing math" ( Van de Walle, 2004, p. 152). The reliance on keywords to select an operation or a solution procedure (e.g., "cross multiply") can SCHEMA-BASED INSTRUCTION 5 lead to systematic errors.…”