ABSTRACT
The purpose of this chapter is to offer information on what the analysis of cognitive abilities can say about the nature of mathematical thinking. Mathematical thinking takes numerous forms, depending on the nature of the mathematical task. Mathematical tasks differ over the various branches of mathematics—arithmetic, algebra, geometry, calculus, symbolic logic, topology, number theory, and so on. Even within one of these branches, however, like arithmetic, tasks (problems) can differ in their structure and their difficulty. For example, some tasks posed in arithmetic are purely formal, such as adding, subtracting, multiplying, or dividing given numbers, whereas others are presented as “word problems,” stating real-world situations in which the respondent must determine how the given numbers are correctly handled to yield an answer. Presumably, the analysis of any mathematical task, in any branch of mathematics, should yield some insight into the nature of the mathematical thinking required to perform it. This chapter, however, is not immediately concerned with such analyses; doubtless they are addressed in other chapters of this volume.
