First-year engineering students' concept development of integral calculus at a South African university of technology.

Abstract

This thesis reports on a study to explore the development of the concept of integration among the first year engineering students at a South African university of technology. The study focused on concept definitions that were evoked through symbolic as well as visualisation of integrals. It further explored various concept images evoked the techniques of integration. A framework combining the Action-Process-Object-Schema (APOS) and the Three-Worlds of Mathematics (TWM) theories was adopted as a tool to analyse students’ concept formation of an integral. This was a qualitative case study that consisted of two phases. Firstly, a pilot phase was introduced as Phase 1 of the study to uncover issues that could be probed more deeply when the study was rolled out to a larger group of students. The activity sheet was administered and interviews were conducted with seven students who were willing to participate in the study. Secondly, as Phase 2 of the study, the modified activity sheet was then administered to 22 first year students who also volunteered to be in the study. The intention was to provide comprehensive investigation of concept development of integral calculus. Students were also organised into focus groups in order to explore emerging mental constructions during the discussions among the students. The findings of the research indicated that students operated mainly at an action level of cognition for integral calculus. Their definition of an integral was restricted to the notion finding an integral with no association to the area below the graph of a function. Students mainly conceptualised an integral as an anti-derivative. With regard to techniques of integration, students relied on rules and algorithms without reflecting on objects and processes embedded within the rules. Cases of inadequate perquisite schemas for integral calculus such as basic algebra, inverse trigonometric functions and some aspects of differentiation were also noted. Although there were notable strengths in skills such as completing a square and resolving fraction into partial fractions, there was little understanding of the underlying concepts. This study contributed by presenting a genetic decomposition for integration that is premised on APOS and TWM theories. While the action level of APOS was dominant, the proceptual-symbolic was the main prevalent world of mathematics learning.