1Students conceptions: an introduction to a formal characterization

Abstract: We investigate in this paper the complexity of modeling students knowing of mathematics under the constraints of acknowledging both their possible lack of coherency and their local efficiency. For this purpose we propose a formalization of the notion of “conception ” as a possible tool to answer the epistemological problem we identify. We apply then this approach to the study of the possible conceptions of “function”, from an historical and then an epistemic point of view. We report the result of a case study in order to illustrate the benefit we expect from this approach. The notions of “conception”, “knowing ” and “concept ” are then related the one to the other within the model presented. 1.!!FROM BEHAVIOR TO MEANING The only indicators we have of the good or of the bad functioning of teaching are students ’ behaviors and productions, which are consequences of the knowing1 they have constructed and of their relationships to the content taught. But such evaluations are possible and their results are significant only in the case where one is able to establish a valid relationship between these observed behaviors and the considered knowledge itself. This relation between behaviors and knowing is crucial. It has been hidden as a result of the fight against behaviorism, but it has always been implicitly present in educational research at least at the methodological level. The key issue is that the meaning of a piece of knowledge cannot be reduced t