Mathematical sense-making in quantum mechanics: An initial peek

Abstract

Mathematical sense-making-looking for coherence between the structure of the mathematical formalism and causal or functional relations in the world-is a core component of physics expertise. Some physics education research studies have explored what mathematical sense-making looks like at the introductory physics level, while some historians and "science studies" have explored how expert physicists engage in it. What is largely missing, with a few exceptions, is theoretical and empirical work at the intermediate level-upper division physics students-especially when they are learning difficult new mathematical formalism. In this paper, we present analysis of a segment of video-recorded discussion between two students grappling with a quantum mechanics question to illustrate what mathematical sensemaking can look like in quantum mechanics. We claim that mathematical sense-making is possible and productive for learning and problem solving in quantum mechanics. Mathematical sense-making in quantum mechanics is continuous in many ways with mathematical sense-making in introductory physics. However, in the context of quantum mechanics, the connections between formalism, intuitive conceptual schema, and the physical world become more compound (nested) and indirect. We illustrate these similarities and differences in part by proposing a new symbolic form, eigenvector eigenvalue, which is composed of multiple primitive symbolic forms.

Probing student reasoning in relating relative phase and quantum phenomena

“…It is worthwhile to mention that the tutorials provide students opportunities for mathematical sensemaking, which has become a recent focus in the physics education and science education research communities [8,18,33]. Gupta and Elby argue that mathematical sensemaking "involves looking for meaning and coherence within the mathematical formalism itself and between the math and the system it describes, if any" [33].…”

“…Previous studies on student understanding of quantum mechanics have tended to center on conceptual understanding [1][2][3][4][11][12][13][14][15][16][17]. Recently, there is a growing effort to research how students make sense of mathematical expressions in quantum mechanics [18,19]. However, little research has specifically focused on students' ability to relate the formalism to real-world quantum phenomena.…”

Categorizing mathematical sense making and an example of how physics understanding can support mathematical understanding

“…Situating our work in this general space of sense making, we consider the more specific topic of mathematical sense making (MSM). Recent work on MSM has used varied definitions of sense making; considering it to be the coordination of conceptual understanding with the use and interpretation of algebraic symbols [2] or "coherence seeking" between mathematical formalisms and functional relationships in the world [3]. We see both of these definitions as steps towards specifying the cognitive process strand in the Odden/Russ definition of sense making.…”

Student perceptions of math-physics interactions throughout spins-first quantum mechanics

“…Quantum mechanics is no exception, as it introduces complex mathematical formalisms that students have not seen in prior courses. This course has often been shown to be very challenging for students solely in terms of the new, abstract concepts that are introduced by the course [1][2][3][4][5][6].…”