Developing students' rationality by constructing and exercising rational mathematical templates: The case of counter-examples

International audienceThe aim of this paper is to put into evidence two aspects of the production of counterexamples during students' approach to it and its development: the epistemic and the teleological (strategic) components of the specific process that brings to the production of a counterexample. By considering two 8 th grade classroom discussions (the first one, on counterexamples in arithmetic and the other, in geometry) I will discuss how those aspects are of general interest for the development of students' conscious dealing with the truth of statements and the effectiveness of strategies (i.e. with the development of students' rationality, according to Habermas). Also some affective aspects will be considered, as concerns their effects on the productivity of the discussion