Students’ arguments about the growth of a two-variable function

Calculus is a central part of the curriculum for tertiary educations in mathematics, science, and technology. At its core lies the concept of derivative, which is known to be problematic for many students. As the corresponding multi-variable concepts of partial derivative, gradient, and directional derivative are not mathematically equivalent, it is essential for students to learn their relations and what they represent geometrically. In this paper, 20 students’ written solutions to an exam problem about the growth of a two-variable function are studied. The warrants they present for their claims are characterized in terms of which representations, concepts, connections, and calculations they use. The findings indicate that students who solve the problem by calculation of directional derivatives are less explicit with their warrants than students who rely on properties of the gradient vector. While the first group only uses algebraic representations, the second combines algebraic and graphical representations.