“Mathematics is not a stalactite hanging over a stalagmite” (w. Kuyk) – the productive role of teaching

Resumen

Willem Kuyk had denounced with this image the widely shared view that mathematics education grows only by receiving some drops from above, figuring there the supreme instance for teaching. Bruno Belhoste, in his paper of 1998, had argued against this view and pleaded for re-assessing the role of teaching mathematics in the development of mathematics. Yet, he conceptualised its role in a polarised manner: while speaking of contributions of mathematical research as “invention”, as “production”, he juxtaposed teaching as “reproduction”, as “socialisation”. I am arguing, instead, that the teaching of mathematics can transgress this type of reproduction in a much more decisive manner – and effectively did so throughout the historical development of mathematics. I should like to discuss this productive role of teaching and in particular the methodological challenges for realising such analyses. Actually, historians of mathematics do not use to be very attached to methodological reflections, and, if ever dealing with them, use to reproduce the old dichotomy between internalism and externalism – quite contrary to historians of science who since quite a time overcame that dichotomy and use to study interactions between internal developments of ideas and broader cultural, social and political contexts. A key pattern for studying such interactions are institutions – i.e., institutions in which mathematicians are working. And, in general, the official major task there is teaching, forming new generations. Not only are such institutions at the crossroads of conceptual developments within the discipline and of contextual influences on the functioning of the institution, there also occurs the direct and concrete interaction between teaching and research. Not too rarely, it is the function of teaching which induces to innovations in mathematical concepts. The most paradigmatic patterns for this functioning were higher education institutions from the French Revolution: due to the establishment of systems of public education, the higher degrees of teaching intensity entailed incentives for systematic revisions of mathematical concepts and their foundations. But there are also revealing cases of innovations in the discipline induced by institutions in earlier periods. Particularly telling studies have been made by Christine Proust on the functioning of the scribal schools (edubba) in Old Babylonian times where the masters, forming the scribal apprentices, transgressed the routine tasks of teaching by validating procedures, by solving new classes of problems. Besides presenting and discussing pertinent cases for such interactions, the lecture will discuss another pertinent issue for interfaces between mathematical development and teaching: it is the notion of ‘element’ and of elementarisation of science. In fact, the notion of element connects the development of mathematics and the modes of teaching mathematics in a fundamental way. Since Euclid’s geometry textbook, the term ‘elements’ expresses the intention to give a systematic presentation of a mathematical theory, constructed from its basic components. While thus fixing the state of knowledge of mathematics or of one of its branches for a certain time and period, Felix Klein’s notion of elementarisation dynamises the notion, emphasising the various stages where the meanwhile accumulated new results and branches led to a restructuration of the bases, defining a renewed structure of elements – the new architecture achieved by Bourbaki constituting a well- known case for textbooks. It is incited by teaching and serving for teaching that textbooks are contributing to the progress of mathematics.