Modeling Teachers' Collective Knowledge of Rational Numbers through Dialectic between Questions and Answers: A Case of Denmark and Indonesia

Abstract

This paper aims at presenting a method for comparing teachers' mathematical and didactical knowledge of rational numbers. The method is developed based on the anthropological theory of the didactic (ATD) using a specific notion of dialectic between questions and answers. The initial questions and the pupils' answers are integrated into hypothetical teacher tasks (HTTs) which have been tested to Indonesian and Danish pre-service teachers (PsTs) who mostly work in pairs. In this particular study, we focus the comparative analysis on a specific case of two pairs, working on the addition and subtraction of fractions, to present the potential of the method to study teachers' collective knowledge. The findings show that the analysis through the dialectic between questions and answers provides a general picture of how the two pairs differ from formulating collective mathematical and didactical knowledge in teaching addition and subtraction of fractions. The analysis shows what directions their discussions take and how the two pairs link between their mathematical and didactical knowledge. The Indonesian pair focused their discussion exclusively on the standard procedures for adding and subtracting fractions. In contrast, the Danish pair discussed both the reasons for the pupils' error and tried to find a visual representation for teaching addition and subtraction of fraction. In addition, the findings also show that neither the Danish nor Indonesian pair considers to change fractions to decimals or to use an instrumental technique such as using a calculator to support pupils' learning of adding and subtracting fractions. An implication from this study is that the method gives a detailed picture of how the mathematical and didactical knowledge is shared and developed between two PsTs during their collective work.