At the center of the goals proposed by the NCTM standards is the desire to create a curriculum that will increase students' understanding of mathematical concepts and procedures. My objective in writing the two chapters in this section is to provide a theoretical context for thinking about learning with understanding. The chapters, therefore, contain examples of theories of learning, reasoning, and problem solving that have been formulated by cognitive psychologists. Chapter 2 looks at theories that emphasize rule learning, whereas chapter 3 focuses on the organization of conceptual knowledge.
In order to solve problems, we have to carry out a series of actions such as constructing and solving an equation. These actions can be theoretically represented as a set of rules. Our focus in this chapter is on a rule-based theory of learning called ACT* ( J. R. Anderson, 1983). There are two reasons why this theory is important. First, it represents a tremendous theoretical accomplishment. It is the most detailed and extensively tested theory of learning that has been developed by cognitive psychologists. Second, the theory has been used to develop computer-based tutors for geometry, algebra, and a computer programming language called LISP, giving us the opportunity to see how theoretical ideas can be applied to the classroom ( Anderson, Corbett, Koedinger, & Pelletier, 1995).
In spite of these successes, there is a concern among many mathematics educators that theories such as ACT* place too great an emphasis on the efficient learning of procedures, while neglecting conceptual understanding. For example, we have all learned when it is necessary to "borrow" when subtracting numbers, but do we have a good conceptual understanding of what borrowing signifies? It is a concern with these kinds of issues that motivated the NCTM standards.