Analytic framework for students’ use of mathematics in upper-division physics

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.

The use of ACER to develop and analyze student responses to expectation value problems

“…In this section we describe the ACER framework [3] and how it was used in this research. We also discuss the context of our study, including an overview of the two questions students were asked to solve.…”

“…The use of math in physics is vastly different than its use in mathematics [2]. ACER was designed as an analytic framework to describe how students use various mathematical tools in advanced physics contexts [3]. In this study we use ACER to both design and analyze mathematically rich questions in the context of quantum mechanics.…”

Mathematization and the ‘Boas course’

“…For upper-division physics courses, Wilcox et al have proposed the ACER framework 'to guide and structure investigations of students' difficulties with the sophisticated mathematical tools used in their physics classes.' [11] In this framework, students must activate the appropriate mathematical tool in addition to constructing a model, executing the mathematics, and then reflecting on results.…”

Analytical derivation: An epistemic game for solving mathematically based physics problems

“…Thus, students are expected to have a good grasp of at least basic mathematical skills in order for them to be able to apply these skills during problem solving. There has been little research on the mathematical aspect of physics problem solving [5,[9][10][11]. Previous studies have shown a positive correlation between student difficulties with physics concepts and those with either the mathematics concepts, application of those concepts, or the representations used to connect the mathematics and the physics.…”

“…The four steps include (i) modeling the physical system to mathematical representation, (ii) processing the mathematical operations, (iii) interpreting the results of the mathematical operations in terms of the physical system, and (iv) evaluating the results to justify the initial modeling. Wilcox and colleagues refined Redish's model to explain students'use of mathematics in upper-division physics [10]. Their framework also contains four elements in the problem-solving process including activation of mathematical tools, construction of mathematical models, execution of the mathematics, and reflection on the results (ACER).…”