Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

Zetra Hainul Putra

doi:10.29333/ejmste/105867

Abstract

The present study aims to introduce the notion of praxeological change, developed based on the Anthropological Theory of the Didactic, to describe a necessity of changing mathematical praxeologies when passing from natural to rational numbers. It is applied to study and compare Danish and Indonesian pre-service teachers’ (PSTs) knowledge of the density of rational numbers. They work in pairs to solve and discuss a hypothetical teacher task, which involves both mathematical and didactical tasks, related to the density of rational numbers. The findings highlight significant differences of the mathematical and didactical knowledge which are shared by the Danish and Indonesian PSTs. In particular, the Danish PSTs are more successful than the Indonesian PSTs in proposing didactical praxeologies to support pupils’ praxeological change. They use the mathematical idea of converting fractions into decimals or vice versa and representing fractions and decimals on the same number line, while the Indonesian pairs tend to suggest pupils to order fractions and decimals based on the ordering properties of natural numbers.

Highlights

An extensive amount of research on teachers’ knowledge about rational numbers exists (An, Kulm, & Wu, 2004; Depaepe et al, 2015; Ma, 1999; Zhou, Peverly, & Xin, 2006)
The present study aims to introduce the notion of praxeological change, developed based on the Anthropological Theory of the Didactic, to describe a necessity of changing mathematical praxeologies when passing from natural to rational numbers
This study introduces the notion of praxeological change as a theoretical model developed within the anthropological theory of the didactic to describe the change of human practice and theory from natural to rational numbers

Summary

Introduction

An extensive amount of research on teachers’ knowledge about rational numbers exists (An, Kulm, & Wu, 2004; Depaepe et al, 2015; Ma, 1999; Zhou, Peverly, & Xin, 2006). Many previous studies (McMullen, Laakkonen, Hannula-Sormunen & Lehtinen; 2015; Prediger, 2008; Vamvakoussi, Christou, Mertens & Van Dooren, 2011; Vamvakoussi & Vosniadou, 2004, 2010) have argued that there is conceptual change involved in the process of passing from natural to rational numbers. This means that learning rational numbers requires one to change one’s prior conceptions of something, like numbers, in order to be compatible with a new mathematical situation – they cannot be adapted but need more fundamental revision. They learn that fractions can always be further broken down, such as dividing a “pizza” diagram in smaller units, but this is not sufficient for them to grasp the mathematical phenomenon of the density of rational numbers

Objectives

Methods

Results

Discussion

Conclusion