Students’ understanding of two-variable calculus concepts in mathematics and physics contexts. II. The gradient and the Laplacian - Maria Al Dehaybes; Johan Deprez; Paul van Kampen; Mieke De Cock (Physical Review Physics Education Research)
When learning physics, students need more than just an understanding of mathematical and physical concepts. Integrating the two fields is crucial, as research indicates that students often struggle even when they have a strong grasp of both. In this paper, we use the heat equation as an example from higher education. Given the importance of the gradient and the Laplacian in the heat equation, as well as fluency and flexibility in using multiple external representations, this study investigates students’ reasoning about the gradient and the Laplacian of a function within different function representations in mathematics and physics contexts. To do so, we designed isomorphic open-response questions, administered as paper-and-pencil tests to 190 first-year students who were taking calculus-based introductory physics courses and multivariable calculus courses during their second semester. These students were enrolled in physics, mathematics, engineering, and bioengineering programs. We found that students face conceptual challenges with both the gradient and the Laplacian. Student difficulties with interpreting the gradient may be related to an incomplete understanding of steepness in graphical representations of functions of two independent variables. Moreover, their understanding of the Laplacian is limited. While context did not significantly affect students’ reasoning in terms of the types of difficulties they encountered, the proportion of students facing these difficulties, and their tendency to link or discuss more than one aspect of a concept, we did find context-specific prevalence of graphical, algebraic-symbolic, or linguistic descriptions. Students used linguistic descriptions more frequently in a physics context. We observed varying abilities to express the meaning of concepts in different answer representations. Linguistic descriptions of concepts proved particularly challenging for both concepts and contexts. Our findings provide insight into students’ understanding of two concepts related to the 2D heat equation.