Abstract
Many students in upper-division physics courses struggle with the mathematically sophisticated tools and techniques that are required for advanced physics content. We have developed an analytical framework to assist instructors and researchers in characterizing students' difficulties with specific mathematical tools when solving the long and complex problems that are characteristic of upper-division. In this paper, we present this framework, including its motivation and development. We also describe an application of the framework to investigations of student difficulties with direct integration in electricity and magnetism (i.e., Coulomb's Law) and approximation methods in classical mechanics (i.e., Taylor series). These investigations provide examples of the types of difficulties encountered by advanced physics students, as well as the utility of the framework for both researchers and instructors.
Highlights
Previous research has identified a considerable number of students’ conceptual and mathematical difficulties, at the introductory level
Rather than stemming primarily from a failure to recall Eq (2), we argue below that this difficulty likely originated from a failure to reject these other methods
We have presented an analytic framework, ACER, that is targeted towards characterizing student difficulties with mathematics in upper-division physics
Summary
Previous research has identified a considerable number of students’ conceptual and mathematical difficulties, at the introductory level (see Ref. [1] for a review). In addition to the significant work at the introductory level, researchers have recently begun to characterize students’ conceptual knowledge in more advanced physics courses [3,4,5,6,7,8,9,10]. A small but growing body of research suggests that upper-division students continue to struggle to make sense of the mathematics necessary to solve problems in physics [11,12,13]. Upper-division physics content requires students to manipulate sophisticated mathematical tools (e.g., multivariable integration, approximation methods, special techniques for solving partial differential equations, etc.). Students are taught these tools in their mathematics courses and use them to solve numerous abstract mathematical exercises. Upper-division instructors face significant pressure to cover new content, a task made more difficult by constantly having to review the relevant
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