An analysis of Grade 12 learners’ mental constructions and difficulties of differentiation and integration: a case study of one school in Umlazi District.

Abstract

This study is an analysis of Grade 12 learners’ mental constructions and the difficulties they experienced with the calculus topics of differentiations and integration at a school in Umlazi District, South Africa. With the introduction of Technical Mathematics, which emphasises application, as a high school subject in South Africa in 2016, increasing attention has focussed on learners’ mental constructions. Consequently, learners’ understanding of differentiation and integration, which are key concepts taught in Grade 12, has come under scrutiny. It has therefore become imperative to investigate learners’ mental construction and the nature of the difficulties they encounter in learning the concepts of differentiation and integration. The study relied on the APOS (Action, Process, Object and Schema) theory of learning to explain the nature of learners’ mental constructions.A qualitative approach was used. Data was collected using activity sheets with all participants and interviews with five participants. A pilot study preceded the main study. Purposive sampling was used to select ten Grade 12 learners for the pilot study and ten others for the main study. Analysis of data was both deductive and inductive. The findings revealed that learners’ conception of differentiation and integration was mainly at the action stage (with reference to the APOS model), with few learners operating at the process stage. Weak foundational mathematical knowledge was found to hinder learners’ development of the necessary mental constructions 3.4. Concepts were overgeneralised from one domain to another and everyday language was used to explain mathematical concepts. Based on these findings, it is recommended that the teaching of differentiation and integration be integrated with algebraic concepts, which are the pre-requisite concepts needed for the schema development of differentiation and integration. It is also suggested that the genetic decomposition designed in this study be used by teachers as a tool to analyse their learners’ mental constructions in order to design and plan appropriate alternative instructional activities to improve learners’ development of differentiation and integration schema.