This study aims to evaluate Indonesian in-service elementary teachers’ knowledge on fraction by their responses to contextual problems on multiplication of a fraction by a whole number (O1) and multiplication of a fraction by a fraction (O2). Fifty Indonesian in-service elementary teachers from Elementary School for Teacher Education study program, University of Riau, participated in this research in 2015. This research is based on anthropological theory of the didactic (ATD), in particular praxeology, as a framework to analyse the data. The result showed that only 22% of Indonesian in-service elementary teachers have a satisfactory relation to O1 and 64% of them have succeeded on the problems related to O2.

This paper presents a framework of a PhD research of the first author about a comparative study of pre-service elementary teachers’ knowledge of rational numbers between Indonesia and Denmark. To obtain the data, the authors design a series of hypothetical teacher tasks (HTTs), inspired by a paper of Durand-Guerrier, Winsløw, and Yoshida (2010). Subjects in this research are pre-service elementary teachers from a selection of different University Colleges in Denmark and from the elementary school teacher education study program, Riau University, in Indonesia. The praxeological reference models and the levels of didactic codetermination are used as tools to analyse the result.

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.