The relationship between teachers' mathematical knowledge and their classroom practices: a case study on the role of manipulatives in South African primary schools
Maboya, Mantlhake Julia
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The use of manipulatives to enhance conceptual understanding of mathematics is a critical component of the primary school mathematics curriculum in South Africa. Manipulatives are concrete or visual objects that are specifically designed to represent mathematical ideas, concepts and/or procedures. Whether or not manipulatives are used in the teaching of mathematics in the primary school classroom, and how they are used, if at all, depends on the teachers’ knowledge and understanding of mathematics and their conceptions about classroom practice and the role of manipulatives therein. In this study, teachers’ mathematical knowledge is defined as knowledge of both content and pedagogy, whilst classroom practice refers to the interaction among teachers, students and content. The present study therefore explores the use of manipulatives in the teaching of mathematics in primary school classrooms. The study examines the role of manipulatives in shaping both the teachers’ knowledge of primary school mathematics and their classroom (pedagogical) practices. Critical theory is used as the underlying theoretical framework for the study and helps to frame the key constructs of the study, namely; teacher knowledge, mathematical manipulatives and classroom practice. The study uses a multiple-case study qualitative approach designed with unstructured interviews (employing the free attitude interview technique) with four grade six mathematics teachers from each of four primary schools as the main data collection tools. Additional data were gathered through observation of lessons conducted in three of the four primary schools, group discussions and curriculum documents analysis. A Participatory Action Research (PAR) approach was chosen to collect data in respect of teachers’ own knowledge, experiences and thinking about their mathematical knowledge, classroom practice and the use of manipulatives. The study employs the socio-cognitive approach to discourse analysis as a strategy to analyse the data obtained. The study’s main findings suggest that teacher knowledge of mathematics is more crucial in the effective use of manipulatives than perhaps any other single teacher attribute. Effective use of manipulatives is essentially characterised as the abstraction of mathematical concepts and relationships embedded in those manipulatives. To successfully do this highly cognitive mathematical task teachers are forced to draw heavily on their own knowledge of mathematics. Any other factors such as teacher beliefs, teacher pedagogy, etc. can only serve as a support base for teacher knowledge. The study concludes that teachers can only abstract mathematical concepts and make connections between them effectively if they themselves have sufficient knowledge of those mathematical concepts and their relationship. Furthermore, the study suggests that over time and with relevant professional teacher development support, the use of manipulatives may have the potential to shape/reshape teachers’ mathematical knowledge. This study concludes that the influence of manipulatives on teachers’ mathematical knowledge and their classroom practices can be explained and understood within the context of the tensions and opportunities that arise in and from a teaching practice where teachers use manipulatives. Based on the findings, the study then recommends a comprehensive professional teacher development programme for primary school teachers that provides hands-on experiences with manipulatives and promotes the reorientation of classroom practice through reflection and co-learning by the teachers alongside their learners.