The teaching and learning of function has received a great deal of focus in mathematics education, but until now results have focused on functions of one variable. Understanding how students generalize their thinking to functions of two or more variables is critical in curriculum and reform efforts for the calculus sequence.
In this talk, I describe the results of a teaching experiment focused on developing working models of student thinking as they participated in an instructional sequence focused on thinking about the process of generating two-variable functions and reasoning about directional derivatives. I describe two major frameworks that emerged from these analyses. I will focus heavily on the framework focused on students, thinking about directional derivative, and discuss implications of these frameworks for the teaching and learning of undergraduate mathematics.