Mathematicians’ Conceptualizations of Differentials in Calculus and Differential Equations

Abstract

The symbol \(dx\) is one example of a differential, a calculus symbol that is found in multiple settings and expressions. Literature suggests that students have both many and varied conceptualizations of these differentials. For example, is \(dx\) a very small quantity? How small does it have to be? Is it merely notation? For our study, we interviewed ten mathematicians to determine how experts conceptualize differentials presented in multiple mathematical contexts, such as differentiation, integration, and differential equations. Using thematic analysis, we analyzed their responses and found that our interview subjects’ conceptualizations were likewise many and varied, with many different interpretations offered for the same differentials. In this paper, we present results from our study and the initial classification system that emerged from this data, while understanding that ongoing and future research may expand and deepen our initial classification system.