Distinguishing between “change” and “amount” infinitesimals in first-semester calculus-based physics

Abstract

From the perspective of an introductory calculus course, an integral is simply a Riemann sum: a particular limit of a sum of small quantities. However, students connect those mathematical quantities to physical representations in different ways. For example, integrals that add up mass and integrals that add up displacement use infinitesimals differently. Students who are not cognizant of these differences may not understand what they are doing when they integrate. Further, they may not understand how to set up an integral. We propose a framework for scaffolding students' knowledge of integrals using a distinction between “change” and “amount” infinitesimals. In support of the framework, we present results from two qualitative studies about student understanding of integration.