Abstract
The Action-Process-Object-Schema (APOS) theory is applied to study student understanding of implicit differentiation in the context of functions of one variable. The APOS notions of Schema and schema development in terms of the intra-, inter-, and trans-triad are used to analyze semi-structured interviews with 25 students who had just finished taking a single-variable calculus course.
Results suggest that the notions of chain rule and implicit function play a key role in the possibility of attaining implicit differentiation Schema coherence. For this, students need to construct at least a Process conception of implicit function and a chain rule Schema with coherence given by function composition.
Students also need to construct relations between implicit function and each one of three components of the implicit differentiation Schema: explicit function, derivative, and differentiation rules. The study shows that students taking an introductory calculus course can be expected to have difficulty understanding the main ideas of implicit differentiation unless special activities are designed to help them make the necessary connections between components of the implicit differentiation Schema.
The study suggests the need to further investigate the implementation of activities that foster the constructions proposed, in textbooks and classrooms.