The relational nature of rational numbers

Bruce Brown

doi:10.4102/pythagoras.v36i1.273

Abstract

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

Highlights

The 21st century has seen a resurgence of research, in both mathematics education and cognitive psychology, into the teaching, learning and understanding of rational numbers
Some emphasise the unity between whole and rational numbers, based on the observation that both represent magnitudes: quantities that can be represented by a point on a number line (Siegler, Thompson & Schneider, 2011). They advocate ways of teaching that build on this unifying property. Others, such as privileged domain theories (Gelman & Williams, 1998) and conceptual change theories (Ni & Zhou, 2005), emphasise the discontinuity between rational number concepts and whole number concepts and investigate the conflicts these introduce into rational number learning
That is, working with whole or rational numbers as magnitudes uniquely placed on the number line will require a conceptual transition from numbers as absolute counts, to numbers as relational, in this case relative to the chosen reference unit

Summary

Introduction

The 21st century has seen a resurgence of research, in both mathematics education and cognitive psychology, into the teaching, learning and understanding of rational numbers. The interference of whole number properties in rational number learning derives from the fundamental change in the nature of the concept of quantity, from absolute count to relative comparison. Having successfully completing the two-container task, they were all able to confidently and quickly count how many scoops were needed in different numbers of cones or cups, showing an effective use of the 1–2 and 1–3 relation.

Results

Conclusion