Exploring pre-service teachers' mental constructions of matrix algebra concepts : a South African case study.
At the university where the study was conducted matrix algebra is one of the first advanced mathematics courses that pre-service teachers encounter. The transfer of knowledge from a primarily procedural or algorithmic school approach to formal presentation of concepts is a priority for conceptualisation of matrix algebra concepts. However, it seems to be creating many difficulties for many pre-service teachers. This is due to the fact that many of them are barely coping with procedural aspects of mathematical concepts. The aim of conducting the study was to explore the pre-service teachers’ mental constructions when learning matrix algebra. The study was guided by the belief that understanding the mental constructions the pre-service teachers made when learning mathematical concepts leads to improved instructional methods. The study is underpinned by APOS theory (Action, Process, Object and Schema) and uses APOS theory to describe the nature of mental constructions displayed by pre-service teachers when learning matrix algebra concepts. To understand and explain the mental constructions made or not made, the preliminary genetic decompositions for matrix algebra concepts was used to analyse the nature of mental constructions made by these pre-service teachers together with triad mechanism which originates from Piaget’s work of reflective abstraction. APOS theory is an extension of reflective abstractions so using these two tools to analyse pre-service teachers’ mental constructions strengthen the trustworthiness of this study. As part of this research project several case studies were conducted where groups of first and second year students were exposed to teaching and learning of some of matrix algebra concepts. These concepts explored are the ones that these students learn under matrix algebra at this university. These concepts were first taught to students and students were expected to express their thinking through solving matrix algebra related problems during tutorials and taking part in the interviews. Analysis of written work and interviews from ten pre-service teachers provided insight into their mental constructions, revealing ways in which they understood the concepts. In explaining and synthesising the results major themes emerged from which conclusions were drawn about the mental constructions that were or not made in the learning of matrix concepts. Several themes emerged which were categorised in certain headings in order to identify patterns that emerged from all tasks. What mostly transpired across all tasks was that background knowledge and understanding of notation are important aspects for students to understand in order to conceptualise the concepts in matrix algebra. It was noted that those students who had a weak schema of basic algebra were not able to make the necessary mental constructions or vice versa. Also, it was noted that students often made nonstandard notation and linguistic distinctions. For example, students use A11 when referring to entries of a matrix or use |𝐴| while determining the determinant of matrix C. Moreover, evidence from their responses revealed that many pre- service teachers had limited knowledge constructed of the taught concepts. This was observed as they struggle to represent the solutions of a system geometrically, recognise concepts in different registers and unable to link major concepts. Findings from this study revealed that the mental constructions made by pre-service teachers in most cases concur with the preliminary genetic decompositions. In terms of APOS theory students responses revealed that many were mainly operating at an action and process stages, with few pre-service teachers operating at an object stage. Since difficulties with the learning of linear algebra by average students are universally acknowledged, this study provided a modified itemised genetic decomposition which is anticipated to help in the teaching and learning of matrix algebra concepts. The aim of providing the modified genetic decomposition is to contribute in the teaching and learning of advanced mathematics as lectures could use the modified genetic decomposition to analyse the mental constructions of their students when learning matrix algebra concepts. Besides making a contribution to the teaching and learning of some mathematical concepts, the modified genetic decomposition is a contribution to APOS theory as it is shown it can be used in other mathematical concepts in different context.