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.
Highlights
Massive studies on pre-service and in-service teachers’ mathematical knowledge lead us to some questions; to what extent do we perceive teachers’ knowledge? And how could a method, a model, or an approach be used to investigate teachers’ knowledge? Some answers have been presented to both questions, and most common studies are to present what mathematical knowledge teachers have and how they teach pupils about that knowledge [1], [2], under a practice-based theory of content knowledge and pedagogical content knowledge for teaching [3]
We seek this answer from the perspective of the anthropological theory of the didactic (ATD)
It is to have the data to be more comparable to the Indonesian counterparts because they are prepared to teach pupils from grade 1 to 6, and secondly, they have already completed the course of learning mathematics, numbers, and arithmetic
Summary
Massive studies on pre-service and in-service teachers’ mathematical knowledge lead us to some questions; to what extent do we perceive teachers’ knowledge? And how could a method, a model, or an approach be used to investigate teachers’ knowledge? Some answers have been presented to both questions, and most common studies are to present what mathematical knowledge teachers have and how they teach pupils about that knowledge [1], [2], under a practice-based theory of content knowledge and pedagogical content knowledge for teaching [3]. MKT has been used to study teachers’ knowledge, and it provides a general picture of teachers’ mathematical and pedagogical knowledge This model does not highlight teachers’ knowledge as a collective construction that teachers should build together in a learning and teaching situation [6]. It is needed to develop a model that could provide a detailed picture of how teachers build their mathematical and didactical knowledge during their interaction on a particular mathematical problem. We seek this answer from the perspective of the anthropological theory of the didactic (ATD)
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