Het samenbrengen van wiskunde en fysica in gevorderde opleidingsonderdelen: partiële differentiaalvergelijkingen als voorbeeld

The story of mathematics-physics interplay is one of historical genesis. However, education does not fully capture this interplay, where mathematics is often treated merely as a tool for physics and physics is often considered merely as a context for mathematics. Students are often expected to develop an understanding of this interplay independently when learning new concepts, which proves to be challenging. Therefore, curricula that focus on the relationship between the two disciplines are needed to support students' understanding. Students' recognition of the fundamental role of mathematics in physics is crucial for them to expand the use of mathematics in physics beyond the application of formulas. To help students realize the organic interaction between the two disciplines, there has been growing emphasis on interdisciplinary research, particularly in its efforts to design teaching and learning practices that help students develop key competencies across disciplines. In this dissertation, we investigate students' understanding of the interplay between mathematics and physics in the context of the 2D heat equation. We chose the 2D heat equation as a topic for this investigation, for it is a partial differential equation. Partial differential equations are central to many areas in mathematics, physics, and engineering. In addition, understanding the 2D heat equation requires individuals to engage in the mathematics-physics interplay. The 2D heat equation models how the temperature distribution in a 2D plate evolves over time due to heat conduction. The conceptual understanding of the 2D heat equation involves students' understanding of a whole network of concepts, their graphical representations, and linguistic interpretations. In our work, we used the APOS framework and extended it into physics education to describe the mathematics-physics interplay within our topic of choice, the 2D heat equation. APOS—an acronym for Action, Process, Object, and Schema—is a theoretical framework used in mathematics education research. The central claim of this theory is that students' understanding of mathematical concepts evolves through identifiable mental constructions (Action, Process, Object, and Schema). A central methodological tool in APOS is the Genetic Decomposition. A genetic decomposition is a model of mental constructions a generic students need to learn a certain concept. This research had two aims. The first was to investigate students' reasoning with the interplay of mathematics and physics in the context of the 2D heat equation. The second was to help students overcome difficulties and to support them in understanding the interplay of mathematics and physics in the context of the 2D heat equation. As a first step towards these aims, this research was started by investigating students' understanding of and difficulties with four concepts: partial derivative, directional derivative, gradient, and Laplacian (chapters 2 and 3). We designed isomorphic questions in mathematics and physics contexts. In each context, we designed two questions per concept that varied based on the function representation, a total of 16 questions. We administered paper-and-pencil tests to 190 mathematics, physics, and engineering students who have covered multivariable calculus courses. Each student solved four questions on four concepts in one context. Our findings indicate that students' understanding of concepts varies with function representation and answer representations. Students struggle with steepness, partly due to overgeneralizing from single-variable contexts and misinterpreting contour plots, and they show limited understanding of the Laplacian, often confusing it with first partial derivatives or the gradient. Students' performance did not significantly depend on the context; however, the types of representations that students used in their answer affected their performance. In addition, there was a tendency to use linguistic descriptions in the physics context. After investigating students' understanding of two-variable calculus concepts, we used APOS to investigate students' conceptual understanding of the mathematical formulation of the 2D heat equation (chapter 4). In chapter 4, we propose extending APOS to also include physics concepts, using the learning of the 2D heat equation as an example where mathematics and physics are deeply intertwined. We proposed a hypothetical model of mental constructions for the 2D heat equation. For this purpose, we used the main tool from the APOS theoretical framework, the preliminary genetic decomposition. In our approach, we emphasized the structural integration of mathematics and physics in different conceptions—not only in the schema conception, but also in the object and even the process conception. We also considered the key concepts and the necessary depth of students' pre-requisite knowledge of the chosen context, the challenges students commonly face with these concepts, and the use of multiple external representations. Following APOS' extension, we validate and review the preliminary genetic decomposition (chapter 5). We carry out interviews with eight students in the second year of their B.Sc. enrolled in either engineering, physics, or twin (mathematics and physics) majors. In the interviews, we used questions precisely designed to probe specific predicted mental constructions from the preliminary genetic decomposition. Our findings indicate that many predicted mental constructions of the genetic decomposition were validated. In particular, coordination and encapsulation of two process conceptions of the Laplacian of temperature improves understanding but it is challenging. Meanwhile, a few other mental constructions of the genetic decomposition require a revision. These include mental constructions related to temperature distribution function, heat flow, and temperature gradient. In this thesis, how students reason about the interplay of mathematics and physics in the context of the 2D heat equation is explored. We extended APOS to investigate students' understanding of the 2D heat equation where we proposed, validated, and reviewed a preliminary genetic decomposition. We identified specific student difficulties and outlined a learning path. Therefore, we have set important steps forward, but we also highlighted that there is still much to discover, learn, and refine. Therefore, possible directions for future research are proposed.