Building on slope: results of a second research cycle on the differential calculus of two-variable functions

Abstract

This study applies Action-Process-Object-Schema (APOS) theory in a second research cycle investigating how an approach based on the explicit consideration of slope in three dimensions and local linearity contributes to students' learning of the differential calculus of two-variable functions. We compare the problem-solving tendency of students from two sections that were taught differently. We find thatstudents' conceptualizations of slope were different in the two sections and that their understanding of slope was reflected in their problem-solving and the justifications of the relations that they constructed between different basic notions of the differentialcalculus. Overall, we show that the students who based their constructions on slope and local linearity obtained a deeper understanding of the differential calculus.