On students’ understanding of the differential calculus of functions of two variables

• A GD of main ideas of differential calculus of two-variable functions is proposed.
• Semi-structured student interviews reveal many interesting mental constructions.
• A nontraditional construction of plane helps students’ inter-relating of ideas.
• A nontraditional construction of directional derivative improves understanding.
• Initial results suggest the genetic decomposition (GD) can help student learning.

APOS Theory is applied to study student understanding of the differential calculus of functions of two variables, meaning by that, the concepts of partial derivative, tangent plane, the differential, directional derivative, and their interrelationship. A genetic decomposition largely based on the idea of a directional slope in three dimensions is proposed and tested by conducting semi-structured interviews with 26 students who had just taken a course in multivariable calculus. The interviews explored the mental constructions of the genetic decomposition they can do or have difficulty doing. Results give evidence of those mental constructions that seem to play an important role in the understanding of these important concepts.