Schema development for the chain rule: A South African case

Abstract

This article reports on the use of APOS (Action-Process-Objects-Schema) as a theoretical framework to investigate first year, University of Technology students’ understanding of the chain rule, 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, 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 use and application of the chain rule, were designed to foster collaborative learning. A sample of thirty 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 stages of the Triad. It was found that students who had inadequate understanding of composition of functions, performed well in the application of the chain rule. The analysis led to a formulation of three techniques of the chain rule: 1) the straight form technique, 2) the link form techniques and 3) the Leibniz form technique.