LEARNING TO THINK MATHEMATICALLY

The choice of the phrase "learning to think mathematically" in this title is deliberately broad. Although the original charter for this chapter was to review the literature on problem solving and metacognition, those two literature's themselves are 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 (e.g. knowledge about one's thought processes, self-regulation during problem solving) which make it difficult to use as a concept. The 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 1980's. 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 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 view 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. He has, elsewhere, characterized the mathematical enterprise as follows. 

Mathematics is an inherently social activity, in which a community of trained practitioners (mathematical scientists) engages in the science of patterns — systematic attempts, based on observation, study, and experimentation, to determine the nature or principles of regularities in systems defined axiomatically or theoretically ("pure mathematics") or models of systems abstracted from real world objects ("applied mathematics"). The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation. However, being trained in the use of these tools no more means that one thinks mathematically than knowing how to use shop tools makes one a craftsman. Learning to think mathematically means 

(a) developing a mathematical point of view — valuing the processes of mathematization and abstraction and having the predilection to apply them, and 

(b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure — mathematical sense-making. (Schoenfeld, forthcoming)

This notion of mathematics has gained increasing currency as the mathematical community has grappled, in recent years, with issues of what it means to know mathematics and to be mathematically prepared for an increasingly technological world. The following quotation from Everybody Counts typifies the view, echoing themes in the NCTM Standards (NCTM, 1989) and Reshaping School Mathematics (National Research Council, 1990a).

Mathematics is a living subject which seeks to understand patterns that permeate both the world around us and the mind within us. Although the language of mathematics is based on rules that must be learned, it is important for motivation that students move beyond rules to be able to express things in the language of mathematics. This transformation suggests changes both in curricular content and instructional style. It involves renewed effort to focus on:

• Seeking solutions, not just memorizing procedures;

• Exploring patterns, not just memorizing formulas;

• Formulating conjectures, not just doing exercises.

As teaching begins to reflect these emphases, students will have opportunities to study mathematics as an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed body of laws to be memorized. They will be encouraged to see mathematics as a science, not as a canon, and to recognize that mathematics is really about patterns and not merely about numbers. (National Research Council, 1989, p. 84)

From this perspective, learning mathematics is empowering. Mathematically powerful students are quantitatively literate. They are capable of interpreting the vast amounts of quantitative data they encounter on a daily basis, and of making balanced judgments on the basis of those interpretations. They use mathematics in practical ways, from simple applications such as using proportional reasoning for recipes or scale models, to complex budget projections, statistical analyses, and computer modeling. They are flexible thinkers with a broad repertoire of techniques and perspectives for dealing with novel problems and situations. They are analytically, both in thinking issues through themselves and in examining the arguments put forth by others.