Mathematical ability

Abstract: The goals of this chapter are (1) to outline and substantiate a broad conceptualization of what it means to think mathematically, (2) to summarize the literature relevant to understanding mathematical thinking and problem solving, and (3) to point to new directions in research, development, and assessment consonant with an emerging understanding of mathematical thinking and the goals for instruction outlined here. The use of the phrase “learning to think mathematically” in this chapter’s title is deliberately broad. Although the original charter for this chapter was to review the literature on problem solving and metacognition, the literature itself is somewhat ill defined and poorly grounded. As the literature summary will make clear, problem solving has been used with multiple meanings that range from “working rote exercises” to “doing mathematics as a professional”; metacognition has multiple and almost disjoint meanings (from knowledge about one’s thought processes to self-regulation during problem solving) that make it difficult to use as a concept. This chapter outlines the various meanings that have been ascribed to these terms and discusses their role in mathematical thinking. The discussion will not have the character of a classic literature review, which is typically encyclopedic in its references and telegraphic in its discussions of individual papers or results. It will, instead, be selective and illustrative, with main points illustrated by extended discussions of pertinent examples. Problem solving has, as predicted in the 1980 Yearbook of the National Council of Teachers of Mathematics (Krulik, 1980, p. xiv), been the theme of the 1980s. The decade began with NCTM’s widely heralded statement, in its Agenda for Action, that “problem solving must be the focus of school mathematics” (NCTM, 1980, p. 1). It concluded with the publication of Everybody Counts (National Research Council, 1989) and the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), both of which emphasize problem solving. One might infer, then, that there is general acceptance of the idea that the primary goal of mathematics instruction should be to have students become competent problem solvers. Yet, given the multiple interpretations of the term, the goal is hardly clear. Equally unclear is the role that problem solving, once adequately characterized, should play in the larger context of school mathematics. What are the goals for mathematics instruction, and how does problem solving fit within those goals? Such questions are complex. Goals for mathematics instruction depend on one’s conceptualization of what mathematics is, and what it means to understand mathematics. Such conceptualizations vary widely. At one end of the spectrum, mathematical knowledge is seen as a body of facts and procedures dealing with quantities, magnitudes, and forms, and the relationships among them; knowing mathematics is seen as having mastered these facts and procedures. At the other end of the spectrum, mathematics is conceptualized as the “science of patterns,” an (almost) empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence. The author’s view is that the former perspective trivializes mathematics; that a curriculum based on mastering a corpus of mathematical facts and procedures is severely impoverished—in much the same way that an English curriculum would be considered impoverished if it focused largely, if not exclusively, on issues of grammar. The author characterizes the mathematical enterprise as follows:

Abstract: Path analysis was used to test the predictive and mediational role of self-efficacy beliefs in mathematical problem solving. Results revealed that math self-efficacy was more predictive of problem solving than was math self-concept, perceived usefulness of mathematics, prior experience with mathematics, or gender (N = 350). Self-efficacy also mediated the effect of gender and prior experience on self-concept, perceived usefulness, and problem solving. Gender and prior experience influenced self-concept, perceived usefulness, and problem solving largely through the mediational role of self-efficacy. Men had higher performance, self-efficacy, and self-concept and lower anxiety, but these differences were due largely to the influence of self-efficacy, for gender had a direct effect only on self-efficacy and a prior experience variable. Results support the hypothesized role of self-efficacy in A. Bandura's (1986) social cognitive theory. Social cognitive theory suggests that self-efficacy, "people's judgments of their capabilities to organize and execute courses of action required to attain designated types of performances" (Bandura, 1986, p. 391), strongly influences the choices people make, the effort they expend, and how long they persevere in the face of challenge. According to Bandura (1986), how people behave can often be better predicted by their beliefs about their capabilities than by what they are actually capable of accomplishing , for these beliefs help determine what individuals do with the knowledge and skills they have. Although researchers have established that self-efficacy is a strong predictor of behavior (Maddux, Norton, & Stoltenberg, 1986), research on the relationship between self-efficacy and academic performance in areas such as mathematics is still limited (Bouffard-Bouchard, 1989). Studies of math self-efficacy have been largely correlational, and researchers have emphasized the need to construct causal models with which to conceptualize and test hypothesized relationships (Hackett & Betz, 1989; Meece, Wigfield, & Eccles, 1990). When causal modeling has been used, most models have excluded key variables identified as influencing math performance (most notably, self-concept), or the theoretical framework used to hypothesize relationships was not based on social cognitive theory. Thus, results have added little to a better understanding of self-efficacy's influence. Bandura (1986) hypothesized that self-efficacy beliefs mediate the effect of other determinants of performance such as gender and prior experience on subsequent perfor