Continuous Directed Scaling: How Could Dynamic Multiplication and Division Diagrams Be Used to Cross Mathematical Borders?

Abstract

We consider how dynamic diagrams can be used to model multiplication and division. This chapter begins with a review of how multiplication is conceptualized in elementary mathematics education. We consider the affordances of familiar representations of multiplication (e.g., repeated addition, the area representation) and highlight aspects of arithmetic (e.g., multiplication with signed numbers) that are difficult for them to represent. Next, we develop geometric models of multiplication and division by adapting compass-and-straightedge procedures to dynamic geometry. We argue that dynamic diagrams can model multiplication and division as continuous scaling operations on directed lengths, and we consider how dynamic diagrams could be used as virtual manipulatives in pre-service teacher education.