DEVELOPING THE DEFINITE INTEGRAL AND ACCUMULATION FUNCTION THROUGH ADDING UP PIECES: A HYPOTHETICAL LEARNING TRAJECTORY

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DEVELOPING THE DEFINITE INTEGRAL AND ACCUMULATION FUNCTION THROUGH ADDING UP PIECES: A HYPOTHETICAL LEARNING TRAJECTORY

Abstract

Integration is a core concept of calculus. As such, significant work has been done on understanding how students come to reason about integrals, including both the definite integral and the accumulation function. A path towards understanding the accumulation function first, then the definite integral as a single point on the accumulation function has been presented in the literature. However, there seems to be an accessible path that begins first with understanding the definite integral through an Adding Up Pieces (AUP) perspective and extending that understanding to the accumulation function. This study provides a viable hypothetical learning trajectory (HLT) for beginning instruction with an AUP perspective of the definite integral and extending this understanding to accumulation functions. This HLT was implemented in a small-scale teaching experiment that provides empirical data for the type of student reasoning that can occur through the various learning activities. The HLT also appears to be a promising springboard into developing the Fundamental Theorem of Calculus. Additionally, this study offers a systematic framework for understanding the process- and object-level thinking that occurs at different layers of integration.

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