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

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.

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

“…By contrast, the Danish PsTs did not apply it as a didactical technique. It becomes the main difference of didactical knowledge presented by both groups, and it is also what has been found in Putra's previous study [12].…”

“…The word "interpret", stated in the task, gives birth to the discussion that leads them to explain further their position. They suggest that pupils need to know how to find a common denominator, and the Indonesian pair focuses on presenting the algorithm for how to add and subtract two fractions, and this situation commonly appeared in some previous studies on Indonesian PsTs' collaborative work on fractions [6], [12].…”

“…In many cases, pre-service or in-service teachers provide various mathematical and didactical praxeologies during their collective work. The process of constructing knowledge in this study can be influenced by the interaction between pre-service teachers (PsTs) and HTTs during their collective work [12], [13].…”

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“…There are 7 courses related to mathematics education (18 credits/12.5%), namely: 1) Fundamental mathematics for elementary schools I, 2) Fundamental mathematics for elementary school II, 3) Mathematics education for the lower grades of elementary school, 4) Mathematics Education for the upper grades of elementary school, 5) Statistics for education, 6) Capita selecta of mathematics (e.g. problem solving and modeling in mathematics) and, and 7) Indonesian realistic mathematics education (Putra, 2019b). The last two courses are optional, but prospective elementary teachers have to choose one of these.…”

“…They add or subtract the numbers based on their position. The students do not realise that they need to change the technique as well as the theoretical lens from viewing a fraction as two distinct numbers into seeing it as a number as a whole, or what (Putra, 2019b) calls a praxeological change.…”