The Role of Multiplicative Objects in Students’ Reasoning With Formulas

Researchers have identified both the affordances of engaging students in symbolization activities and the difficulties students have in constructing meaningful representations of contexts through algebraic expressions and formulas. This article includes clinical interviews and teaching experiments with preservice secondary mathematics teachers focused on engaging them with dynamic geometric contexts to understand their representational activity with formulas. I introduce the notion of a multiplicative object in the context of formulas and express its importance in the construction of productive meanings for constructing formulas via covariational reasoning. Specifically, I provide a framework for the different types of multiplicative objects for formulas and connect students’ constructions of a multiplicative object with their covariational reasoning within and across different geometric contexts.