The teacher's action, the researcher's conception in mathematics

Résumé En

Our point is to contrast both epistemological positions of teachers and researchers, by the means of their respective actions in a research process. Based on a threefold descriptive model of the teacher's action, our analyses examine the nature of the teaching techniques enacted about a given mathematical situation, the "Race to 20" (pupils aged 9-11 years) and the discourses of two teachers about this lesson. Our findings indicate that the teachers are primarily concerned with the educational coherence of the teaching process, using non specifically knowledge-related teaching techniques that researchers explain by some generic overdeterminations, from the didactical point of view. This gap has to be taken into account in further collaborative research, in order to make teacher's developing specific teaching techniques to foster the mathematical sense-making of the students. This paper investigates the relationship between two experienced teachers(T1 and T2) and a researchers' team, both involved in a particular research. This relationship is to be considered through the epistemological conceptions of each part, that we need to compare in order to understand the respective behaviors and discourses. In order to understand the teacher's action while he carries out mathematics lessons, we design a threefold model based upon didactical categories (Brousseau 1997 ; Chevallard, 1992). The first level of this theoretical frame comprises a set of micro teaching techniques, that were spotted several times among the interactions patterns concerning a given mathematical knowledge. These techniques should enable the researcher to describe in depth and precisely how a given piece of knowledge is handled by both the teacher and the students. A second level is also defined to gather the macro shifts that can be observed in the didactical contract, and therefore it sums up the main teacher's intentions about the knowledge. Then, a third level is to take into account the teachers' general practices and educational beliefs through behaviours observed during the lesson, and/or comments we may get from the teachers during interviews. Moving up the levels in the model shows a "gradient" in the teaching techniques : at the bottom, are the most knowledge-linked techniques