Learning Mathematical Formalisms: Algebra in the Schools
The study of algebraic reasoning and its development is important for what it says about our understanding of cognitive development, but also for its role as a “gatekeeper” for economic opportunity and later education in the natural and social sciences. Algebraic reasoning involves a multifacted set of skills and knowledge. At its core, it extends earlier matheamtical reasoning because it allows one to reason about unknown quantities, reason with an array of abstract represenations such as equations and graphs, and reason in a relational manner. Two central themes emerge in this review. One is that students’ entry into algebraic reasoning is often supported through the use of pre-existing verbal reasoning abilities. I refer to this as the “verbal precedence model” of algebra development. Another is that students often exhibit reasoning about mathematical relations first as actions and operations. Only later do they exhibit reasoning about mathematical relations in terms of relations and structures. I refer to this as the “process prcedence model” of algebra development. In this talk I review findings from several areas of mathematics, including students’ understanding of the equal sign, the use of variables, symbol manipulation, graphing, and story problem solving. In each of these, alone or in combination, there is strong evidence for the verbal precedence model and the process precedence model. I then show how teachers use pedagogical methods through their speech and hand gestures to help students to make links between verbal language and mathematical symbols, and between actions and mathematical structures in order to foster the development of students’ algebraic reasoning.