Understanding, Motivation, and Teaching: Comment on Lampert's "Knowing, Doing, and Teaching Multiplication"

Abstract

It is a pleasure to comment on Magdalene Lampert's fine article, which uses an unusual method-the careful analysis of the author's classroom teaching and her students' learning- to study the complex and difficult topic of what it means to know mathematics. Lampert's article not only makes a contribution to the psychology of children's mathematical thinking, but also clarifies a central issue in mathematics education, namely, the misguided debate concerning calculation versus understanding. We are indebted to Lampert for raising some basic and neglected issues, including but not limited to the nature of mathematical understanding, the role motivation plays in it, and the nature of pedagogical techniques useful for fostering understanding. In this commentary, we discuss only a few of the many issues raised by Lampert's stimulating analysis. We argue (a) that an understanding of calculational procedures is more complex than Lampert acknowledges, (b) that motivation plays a central role in understanding, and (c) that effective teaching of arithmetic is a complex affair, the goals of which should be to foster the development of diverse areas of knowledge and to promote the connections among them, partly through the use of intermediary schemata.