Students’ understanding of two-variable calculus concepts in mathematics and physics contexts. I. The partial derivative and the directional derivative

This study investigated how students reason about the partial derivative and the directional derivative of a multivariable function at a given point, using different graphical representations for the function in the problem statement. Questions were formulated to be as isomorphic as possible in both mathematics and physics contexts and were given as pen-and-paper tests to 190 mathematics, physics, engineering, and bioengineering students in their second semester. Our first research question asked how students reason about partial derivatives and directional derivatives with different function representations and in different contexts. Our second research question investigated whether a physics context hinders or improves students’ understanding of these concepts. Performance on partial derivative questions varied with both the representations students used in their answers and the function representations used in the question. Students performed better when they provided algebraic-symbolic answers and when they answered Cartesian graph questions. For directional derivatives, students struggled with correct interpretations, often misinterpreting steepness as average steepness between points rather than as steepness at a specific point. The context did not significantly affect overall performance, but it did influence the representations students used in their answers, with a tendency toward linguistic descriptions in physics.