Role of Graphs in Blending Physical and Mathematical Meaning of Partial Derivatives in the Context of the Heat Equation

From literature, we know that making the connections between mathematics and physics is not trivial for most students, even at the advanced level. In the specific context of partial derivatives in thermodynamics, research suggests that making explicit connections between the mathematics and the physics is necessary to foster student understanding. In this paper, we investigate how graphical reasoning can help undergraduate students in making connections between the partial derivatives of temperature with respect to position and to time and their respective physical meaning in the context of one-dimensional systems modeled by the heat equation. We conducted task-based, think aloud interviews with four pairs of undergraduate students majoring in physics or mathematics and use dynamic conceptual blending diagrams to analyze their reasoning processes and the role of the graphs therein. The results in this paper indicate that stimulating graphical reasoning is a promising way to promote the blending of mathematical and physical knowledge. Specifically, our data shows that constructing graphs can be a catalyst which helps students to give physical meaning to the partial derivatives ∂T/ ∂t and/or ∂T/ ∂x. Therefore, in teaching/learning activities, actively promoting graph construction and reasoning based on those graphs offers perspectives to support the blending of mathematics and physics.