Schema development for the chain rule: A South African case study

Abstract

This article reports on the use of the Action-Process-Objects-Schema (APOS) theory as a theoretical framework to investigate first-year students’ understanding of the chain rule at a University of Technology in South Africa. Instructional design as part of APOS, based on the genetic decomposition, was used to teach the first-year students differentiation involving the concept of the chain rule and its use. A questionnaire based on functions, the composition of functions, derivatives and the structure of the integrand was used to monitor the development of the chain rule schema. Later, worksheets based on the use and application of the chain rule were designed to foster collaborative learning. A sample of 30 students participated in the study. In this manner differentiation of each function in the composite function was accomplished. Students either operated in the Inter- or Trans-stage of the Triad. It was found that students who had inadequate understanding of the composition of functions, performed well in the application of the chain rule. The analysis led to a formulation of three techniques for using the chain rule, namely: (1) the straight form technique; (2) the link form technique; and (3) the Leibniz form technique.