Physics students are expected to apply the mathematics learned in their mathematics courses to physics concepts and problems. Few PER studies have distinguished between difficulties students have with physics concepts and those with either mathematics concepts and their application or the representations used to connect the math and the physics. We are conducting empirical studies of student responses to mathematics questions dealing with graphical representations of (single-variable) integration. Reasoning in written responses could roughly be put into three major categories related to particular features of the graphs: area under the curve, position of the function, and shape of the curve. In subsequent individual interviews, we varied representational features to explore the depth and breadth of the contextual nature of student reasoning, with an emphasis on negative integrals. Results suggest an incomplete understanding of the criteria that determine the sign of a definite integral.

We present an empirical analysis of students' use of partial derivatives in the context of problem solving in upper-division thermodynamics. Task-based individual interviews were conducted with eight middledivision physics students. The interviews involved finding a partial derivative from information presented in a table and a contour plot. Using thematic analysis, we classified student problem-solving strategies into two principal categories: the analytical derivation strategy and the graphical analysis strategy. We developed a new flowchartlike analysis method: representational transformation. Our analysis of students' strategies using this method revealed three types of transformation phenomena: translation, consolidation, and dissociation. Students in this study did not seem to have much difficulty with the concepts underlying the partial derivative; instead, they seemed to have difficulty with the transformation phenomena, particularly the consolidation of multiple representations into a single representation.

Problem solving, which often involves multiple steps, is an integral part of physics learning and teaching. Using the perspective of the epistemic game, we documented a specific game that is commonly pursued by students while solving mathematically based physics problems: the analytical derivation game. This game involves deriving an equation through symbolic manipulations and routine mathematical operations, usually without any physical interpretation of the processes. This game often creates cognitive obstacles in students, preventing them from using alternative resources or better approaches during problem solving. We conducted hour-long, semi-structured, individual interviews with fourteen introductory physics students. Students were asked to solve four "pseudophysics" problems containing algebraic and graphical representations. The problems required the application of the fundamental theorem of calculus (FTC), which is one of the most frequently used mathematical concepts in physics problem solving. We show that the analytical derivation game is necessary, but not sufficient, to solve mathematically based physics problems, specifically those involving graphical representations.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts' interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation.

In a study of student understanding of negative definite integrals at two institutions, we administered a written survey and follow-up clinical interviews at one institution and found that “backward integrals”, where the integral was taken from right to left on the x-axis, were the most difficult for students to interpret. We then conducted additional interviews focused on backward integrals at a second institution. Our analysis uses the concept image framework and a recent categorical framework for mathematical sense making. We found that students were most successful using the Fundamental Theorem of Calculus to determine the sign of an integral when a symbolic expression was provided. When considering a definite integral in a graphical context, students often had difficulty if they viewed the integral as a spatial area, stating that area must always be positive. Some students were able to conceptualize $$\Delta x$$ Δ x or $$dx$$ dx as a difference or change and thus a signed quantity; when these students were able to view the area as a sum of smaller pieces, they were more successful in justifying a negative backward integral. Students who used amounts or similar images often had difficulty making sense of the negative sign. Even more progress was made when students either invoked or were asked explicitly about a specific physical context to be represented by the backward integral, other than spatial area. The context provided a meaning to the difference represented by $$\Delta x$$ Δ x or $$dx$$ dx and thus to the sign of that difference and the definite integral.