Notes for the teaching of school mathematics under Wittgensteinian inspiration

Abstract

The text brings together some reflections on the nature of mathematics and its teaching in the school context, anchored in the therapeutic results obtained in the second phase of the philosopher Ludwig Wittgenstein's thinking on the foundations of mathematics. Among them, the idea of ​​autonomy of mathematical propositions in relation to the empirical that results from his criticism of the referential model of language present in the realist and idealist conceptions of mathematics, thus relativizing dogmatic beliefs about the nature of its contents, with immediate implications for pedagogical practices. The central hypothesis is that when the normative (and not descriptive) function of mathematical statements is made explicit, several mistakes and confusions can be avoided in the teaching of the discipline, which occur, in most cases, when characteristic methods of natural sciences are introduced that would supposedly lead to the discovery of mathematical contents in general. It is observed that, when disregarding the distinct role played by mathematical propositions in relation to that of empirical propositions, one starts to believe that mathematical conjectures would be hypotheses to be tested through empirical experiments, relegating mathematical proofs to second plan, or even discarding them, as if the formality of mathematics were an obstacle to their learning in the school context. In opposition to this belief, it is proposed that the paradigmatic function of formal demonstrations is made explicit in teacher education, since they not only produce new meanings, but also instruct the student to use the theorems that are part of the school curriculum. The text concludes by proposing guidelines to prevent confusion in the classroom arising from pedagogies permeated by philosophical conceptions about the foundations of mathematics which, in turn, are linked to a referential conception of mathematical language.