Investigating students’ use of mathematical tools and representations in undergraduate physics problem-solving

Abstract

One of the major focuses in the PER community has been on understanding student learning in order to improve how we teach undergraduate physics courses. Problem-solving receives significant emphasis across the undergraduate physics curriculum, and students’ ability to solve physics problems has been researched extensively. Studying student problem- solving processes in upper-division physics courses is a growing area of research. The research presented in this dissertation is done in an effort to understand student problem-solving processes at the upper-division level. In particular, we investigate students’ use of mathematical tools across the undergraduate physics curriculum, in particular, at intermediate mechanics students’ written homework solutions in order to better understand how students use these tools as they solve traditional mechanics problems. We use a modified version of the ACER (Activation-Construction-Execution-Reflection) framework to analyze students’ solutions and to identify patterns of mathematical skill use on traditional problems. We apply techniques borrowed from network analysis and the Resources Framework to build a “fingerprint” of students’ mathematical tool use. In this study, we present preliminary findings on patterns that we have identified in students’ problem-solving. Then we shift our focus to investigate how students use multiple representations as they solve problems. First, we study how students translate among different representations. Data for this study is drawn from an upper-division Electromagnetism I course, where students engaged in individual oral exams. We do a moment-by-moment analysis of students’ problem solving to see how they translate among durable representations (diagrams, written mathematical equations) with the help of evanescent representations (gestures, words), and how students build up durable representations where they can “stand fast” later. In this study, we present the case of Larry as an exemplary case for problem-solving. Larry starts from a durable representation (a diagram). He then builds from there, using evanescent representations (gestures and words) and standing fast on the diagram. He later translates to a different kind of durable representation (mathematics), where he reasons and answers the original problem. In our investigation to understand how students use multiple representations while solving problems, we further investigate how students construct spontaneous representations and also make connections among these representations. We use a social semiotic perspective to sketch a theoretical framework that accounts for how semiotic resources might be combined to solve problems. We identify the semiotic and conceptual resources that Larry uses, and we use a resource-graph representation to show Larry’s coordination of resources in his problem- solving activity. Larry’s case exemplifies coordination between multiple semiotic resources with different disciplinary affordances to build up compound representations. Our analysis of this case illustrates a novel way to think about how students use semiotic resources as they solve physics problems. After that, we revise our theoretical framework to better describe students connecting multimodal representations. The revised framework focuses on the semiotic modes and it describes how students coordinate among different semiotic resources with different modes. In this study, we use multiple cases of students solving oral-exam problems to generalize the approach.