A Comparative Study of Danish and Indonesian Pre-service Teachers’ Knowledge of Rational Numbers – Københavns Universitet

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A Comparative Study of Danish and Indonesian Pre-service Teachers’ Knowledge of Rational Numbers

Doctoral Candidate:

Zetra Putra 

Assessment Committee:

Associate Professor Ricardo Karam (Chair; IND, KU)

Professor Marianna Bosch, U. Ramon Llull, Spain

Professor Yves Chevallard, U. AIx-Marseille, France

Chair of the defense:

Associate Professor Frederik Voetmann Christiansen (IND, KU)

Supervisor:

Professor Carl Winsløw (IND, KU)

Abstract of thesis:

The arithmetic of rational numbers is not only a difficult topic for pupils to learn and for teachers to teach, but mastering this topic is also crucial for pupils to succeed in basic algebra and all of the more advanced topics which depend on algebra. It is a natural assumption that teachers’ knowledge of rational numbers is important to pupils’ success on this and other topics, and indeed much previous research focuses on investigating and comparing teachers’ individual knowledge using a large-scale study and written tests. Meanwhile, the present PhD thesis deals with the issue of studying and comparing teachers’ individual and collective mathematical and didactical knowledge of rational numbers. This study takes an approach from the Anthropological Theory of the Didactic (ATD), especially praxeological analysis, and an explicit model to study teachers’ individual and collective knowledge known as hypothetical teacher task (HTT). Five HTTs covering different aspects of rational numbers, representations, order structure, and arithmetic, were designed based on the didactic research literature on rational numbers. Each HTT provides an opportunity for teachers to show their individual mathematical praxeologies and to share and construct some potential didactical praxeologies during subsequent pair discussion. To enable a systematic and precise analysis of teachers’ mathematical and didactical knowledge, praxeological reference models in the sense of ATD are developed. The empirical case studies were conducted with 31 Danish PSTs and 32 Indonesian pre-service teachers. The studies in particular led to three overall results related to the theoretical and methodological choices, the comparison of Danish and Indonesian pre-service teachers’ praxeologies, and the new notion of praxeological change. As a first result, we argue that HTT could be a promising model to investigate pre-service teachers’ mathematical and didactical knowledge related to a specific teaching situation. Pupils’ possible misconceptions and challenges, as presented in each HTT, provide opportunities for the preservice teachers to share not only their practical knowledge but also their theoretical explanations, justifications and beliefs. The second result, from the comparative study, is that the Danish preservice teachers have more solid mathematical and didactical praxeologies for teaching rational numbers than the Indonesian pre-service teachers. The Danish pre-service teachers constructed their didactical praxeologies from pedagogical assumptions on teaching and they also draw on didactical theories developed and disseminated within Western societies. They propose didactical praxeologies based on real life situations and also based on explicit ideas about the mathematical construction involved. On the other hand, the Indonesian pre-service teachers draw their didactical praxeologies from an implicit yet consensual assumption that mathematical techniques should be conveyed through direct instruction and subsequent training in their use. The causes for this basic idea of direct teaching of mathematics as a set of techniques can be further explained by
 nstitutional conditions and constraints. The third result is that praxeological change is an important factor for pre-service teachers to succeed in supporting pupils to understand more subtle properties of rational numbers, such as density. Through this notion, one could describe the change of practices and logos from natural to rational numbers, and investigate to what extent the praxeological change among teachers is needed to deal with pupils’ difficulties on rational numbers. Finally, some perspectives generated from this study are related to the complexity of teachers’ knowledge and teaching as a profession.

The thesis can be downloaded here.