Developing a coherent approach to multiplication and measurement

We examine opportunities and challenges of applying a single, explicit definition of multiplication when modeling situations across an important swathe of school mathematics. In so doing, we review two interrelated conversations within multiplication research. The first has to do with identifying and classifying situations that can be modeled by multiplication, and the second has to do with identifying what is consistently characteristic of the operation when considering nonnegative real numbers. We review seminal lines of research—including those of Vergnaud and Davydov—and highlight ways that these lines do not provide a thoroughly unified view of multiplication. Then we offer our own approach based in measurement. To underscore consequences of the approach we outline, we use rectangular area and division to illustrate that, as a field, we may need to adjust how we think about connections between multiplication equations and at least some problem situations. We close with a set of questions about unified approaches to topics related to multiplication.