Approaches to teaching primary level mathematics

Caroline Long,Tim Dunne

doi:10.4102/sajce.v4i2.208

Abstract

In this article we explore approaches to curriculum in the primary school in order to map and manage the omissions implicit in the current unfolding of the Curriculum and Assessment Policy Statement for mathematics. The focus of school-based research has been on curriculum coverage and cognitive depth. To address the challenges of teaching mathematics from the perspective of the learner, we ask whether the learners engage with the subject in such a way that they build foundations for more advanced mathematics. We firstly discuss three approaches that inform the teaching of mathematics in the primary school and which may be taken singly or in conjunction into organising the curriculum: the topics approach, the process approach, and the conceptual fields approach. Each of the approaches is described and evaluated by presenting both their advantages and disadvantages. We then expand on the conceptual fields approach by means of an illustrative example. The planning of an instructional design integrates both a topics and a process approach into a conceptual fields approach. To address conceptual depth within this approach, we draw on five dimensions required for understanding a mathematical concept. In conclusion, we reflect on an approach to curriculum development that draws on the integrated theory of conceptual fields to support teachers and learners in the quest for improved teaching and learning.

Highlights

The theoretical question explored here is how the particular approach taken to teaching mathematics in the primary school impacts on the effective learning of mathematics.1 The focus of school-based research has generally been on whether the ‘curriculum has been covered’, and whether this coverage has been achieved to the appropriate ‘cognitive depth’ (Reeves & Muller 2005, among others)
A teacher may well have covered the curriculum in that the ninety or so topics in the Intermediate Phase curriculum2 have been addressed in class, but the important question is essentially whether the learners have engaged with the underlying mathematical structures in such a way that they build the foundations for more advanced mathematics, or whether, in contrast, the concepts as acquired are likely to lead to a frustrating outcome, such as the inability to make the transition to advanced mathematics
We propose that the two major transitions required in the Intermediate Phase are firstly acquiring an understanding of multiplication and its inverse in division, and of several other mathematical concepts introduced during this phase which build on an understanding of multiplication and division

Summary

Introduction

The theoretical question explored here is how the particular approach taken to teaching mathematics in the primary school impacts on the effective learning of mathematics. The focus of school-based research has generally been on whether the ‘curriculum has been covered’, and whether this coverage has been achieved to the appropriate ‘cognitive depth’ (Reeves & Muller 2005, among others). Though the problems within the multiplicative conceptual field appear to be infinitely variable, the essential structure underlying the problems can be encapsulated by identifying distinct measure spaces (dimensions) and identifying the unknown, as illustrated in Table 1 (see Long 2011, drawing on Vergnaud 1983) Note that in this particular example the variables are the number of carriages and the number of people. In order to ensure that multiple facets of the concepts, for example ‘adding two-digit numbers’ or ‘calculations involving hours and minutes’, are covered, we propose drawing on the five dimensions required to fully understand a mathematical concept (Usiskin 2012) Usiskin labels these elements the skills-algorithm dimension, property-proof dimension, use-application dimension, representation-metaphor dimension, and history-culture dimension of understanding. The five dimensions are overlaid or embedded in a conceptual fields approach, of which only some essential features have been presented here. Greater elaboration of this approach may be found in Long (2011)

Findings

Discussion

Conclusion