Students' understanding of the relation between tangent plane and directional derivatives of functions of two variables

APOS Theory is applied to study student understanding of directional derivatives of functions of two variables. A conjecture of mental constructions that students may do in order to come to understand the idea of a directional derivative is proposed and is tested by conducting semi-structured interviews with 26 students. The conjectured mental construction of directional derivative is largely based on the notion of slope. The interviews explored the specific conjectured constructions that student were able to do, the ones they had difficulty doing, as well as unexpected mental constructions that students seemed to do. The results of the empirical study suggest specific mental constructions that play a key role in the development of student understanding, common student difficulties typically overlooked in instruction, and ways to improve student understanding of this multivariable calculus topic. A refined version of the genetic decomposition for this concept is presented.