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Blending of mathematics and physics: undergraduate students' reasoning in the context of the heat equation
Book - Dissertation
The role of mathematics in physics is multifaceted: for example, mathematics can function as a 'toolbox' (technical function), as a language (communicative function) or for building logical-deductive reasoning (structural function). Moreover, mathematics has not only been essential for the development of physics, but conversely, many mathematical concepts arose from a desire to describe nature. The intertwining of mathematics and physics is so strong that it is sometimes difficult to separate the two.Yet this strong connection is not always reflected in education. In physics education, mathematics is often reduced to its technical function of describing relationships and making calculations. In mathematics education, physics is often reduced to a context in which abstract mathematical ideas are illustrated and applied. This dichotomy makes it difficult for pupils and students to integrate their knowledge of both disciplines. The question of how best to bring the two together to support learners' learning remains largely unanswered, despite the amount of discipline specific educational research on learning and teaching mathematics and physics separately. Recent research on the interplay of mathematics and physics in students' reasoning shows that this often remains a challenge even for more advanced students (further into the bachelor's and master's degrees). These students are usually proficient in performing calculations (which refers to the technical function of mathematics), but even for them the structural role of mathematics in physics proves difficult to grasp. How education can be designed to better support this is not yet very clear.In this thesis, for a specific topic (partial differential equations), we examine how students bring mathematics and physics together in their reasoning, what the specific difficulties they face in doing so are, and ultimately how we can respond to them in designing educational learning activities.Partial differential equations are typically covered in undergraduate courses in physics and mathematics, and both disciplines play an important role in this course. For example, the heat equation describes heat transport in a rod and thus provides a prime example of the interplay between the two disciplines. To fully describe the physical system, an initial condition and boundary conditions are required in addition to the differential equation. The initial condition establishes the initial temperature distribution and the boundary conditions impose conditions on the temperature or heat flow at the two ends of the rod. For example, the rod may be kept at a constant temperature at the edges or it may be isolated there. When solving a problem, students must bring together knowledge from physics and mathematics to set up or interpret the boundary conditions. Therefore, these boundary conditions became the focus of our research.To analyze students' reasoning processes, we use the framework of conceptual blending. This framework has its origins in linguistics, where Fauconnier and Turner (1998) developed it to describe how people make meaning by selectively bringing together information from two or more previous experiences/other domains. The framework has been used previously within physics education research to describe the bringing together, or blending, of mathematics and physics.As a first step (Chapter 2), we studied what difficulties students have in mathematically describing the boundary conditions of a physical system. We conducted in-depth interviews with twelve students in which we asked them to solve some problems while thinking aloud. The goal was to identify difficulties and characterize them in terms of their position in the conceptual blending framework, i.e., difficulties of a physical or mathematical nature, or difficulties related to blending both. Our analysis shows that conceptual blending provides a good framework for describing students' difficulties: we were able to identify both difficulties in mathematics and physics separately, but also found that blending mathematics and physics can go wrong in different ways. This supports the idea that knowledge of mathematics and physics separately is insufficient to bring both disciplines together when describing a physical phenomenon mathematically.In addition to understanding the difficulties, we also wanted to gain a better understanding of students' reasoning process when they combine mathematical and physical knowledge so that we could develop better educational learning activities that support students in that blending process. Therefore, in a second study, we developed a new method of analysis: the dynamic blending diagram (DBD) (Chapter 3). When constructing a DBD, we first categorize the elements of the student's reasoning as physics, mathematics, or blended. On top of that, we represent the order of those elements by numbering them and we connect elements that were related by the student. In this way, students' reasoning and blending process is visually represented.We conducted in-depth interviews with four pairs of students. The data was analyzed using DBDs, which showed that graphs can be important to support the blending of mathematics and physics (Chapter 4). Students who analyzed graphs from the problem statement and/or constructed their own graphs were often better at formulating the correct boundary condition. We found that constructing graphs can help make the connection between the physical phenomenon of heat flow and its mathematical description as a partial derivative of temperature with respect to position, a connection that appeared to be not obvious to many students. However, we also found that graphical reasoning alone was often not enough. Based on graphs, students were able to eliminate incorrect options for the mathematical description, but only one pair managed to provide a proper and complete justification for the correct relationship.In a final study (Chapter 5), we combined the findings from the previous studies with findings from the research literature to develop educational materials that encourage and support the blending of mathematics and physics. We developed a tutorial to help students make the connection between the physical phenomenon of heat flow and its mathematical description as a partial derivative of temperature with respect to position, since the previous interview series showed that this is difficult for most students. The design of the tutorial is based on three design principles: (1) explicitly focusing on both the mathematical and physical aspects of the intended reasoning, (2) encouraging graphical reasoning to stimulate blending of mathematics and physics, and (3) the principle of blended encapsulation, developing the concept of derivative step by step while linking it to physical meaning.The concept of (partial) derivative is crucial for understanding the heat equation, its boundary conditions, and the relationship between the partial derivative of temperature with respect to position and heat flow. Therefore, we start from the layered model for the concept of derivative as developed in mathematics didactics by Zandieh (2000). We extend this framework using the conceptual blending framework and our knowledge of the beneficial effect of graphical reasoning to represent the relationship between the mathematical concept of the partial derivative T/x and the physical concept of heat flow. This results in our so-called blended partial derivative framework, which provides the theoretical basis for the approach in the tutorial. This framework consists of several layers and columns, which provides opportunities to encourage the blend between mathematics and physics in smaller steps. The step-by-step construction of the layers was named encapsulation in the context of mathematics education research. Because of the explicit link to physics that we make at each layer, we speak here of blended encapsulation.The developed tutorial was tested with three groups of three students in teaching-learning interviews. From the results, we conclude that the blended encapsulation approach can help students in recognizing how temperature differences lead to heat flow and how this can be formulated mathematically. We also make some recommendations to optimize the current design, such as encouraging students to reason about heat flow through a specific point in the system rather than the system as a whole.In this thesis, for a specific topic (boundary conditions for the heat equation), we explored in detail how students bring mathematics and physics together in their reasoning, and the difficulties they face in doing so. We drew on the framework of conceptual blending and used this framework to describe the difficulties and to design materials that can encourage and support this blending process. The three studies in this dissertation use qualitative data and analysis methods, and examine the reasoning of a small number of students. In follow-up research, we recommend that this be expanded on a larger scale. Our findings are context-specific, but they also have implications for other topics within physics where mathematical reasoning plays an important role.