ACER: A Framework on the Use of Mathematics in Upper-division Physics

TL;DR

This paper introduces ACER, a theoretical framework designed to analyze students' difficulties with mathematical tools in upper-division physics, demonstrated through application to Taylor approximations.

Contribution

The paper presents a novel framework that links students' conceptual challenges to their mathematical tool usage in advanced physics courses.

Findings

Framework effectively connects student difficulties to mathematical concepts.

Application to Taylor approximations illustrates its practical utility.

Supports research and instruction in physics education.

Abstract

At the University of Colorado Boulder, as part of our broader efforts to transform middle- and upper-division physics courses, we research students' difficulties with particular concepts, methods, and tools in classical mechanics, electromagnetism, and quantum mechanics. Unsurprisingly, a number of difficulties are related to students' use of mathematical tools (e.g., approximation methods). Previous work has documented a number of challenges that students must overcome to use mathematical tools fluently in introductory physics (e.g., mapping meaning onto mathematical symbols). We have developed a theoretical framework to facilitate connecting students' difficulties to challenges with specific mathematical and physical concepts. In this paper, we motivate the need for this framework and demonstrate its utility for both researchers and course instructors by applying it to frame results from interview data on students' use of Taylor approximations.