Teaching Toward the Big Ideas of Algebra

Abstract

IN WHAT WAYS DO WE WANT MIDDLE-GRADES STUDENTS TO UNDERSTAND ALGEBRA? Hiebert and Carpenter (1992) describe the need for students to gain both procedural knowledge and broadly connected conceptual knowledge to understand mathematics. A knowledge of rules and procedures provides students with tools for efficient problem solving. However, in learning the procedures of algebraic manipulation, for example, students often develop what Skemp (1978) calls an “instrumental understanding” of algebra. He explains, “It is what I have in the past described as ‘rules without reasons,’ without realizing that for many pupils… the possession of such a rule, and the ability to use it, was what they meant by ‘understanding’ ” (p. 9). Skemp contrasts instrumental understanding with “relational understanding,” which “consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point” (p. 14).