For an ethnomathematical questioning
on the teaching of proof

When one browses the International Handbook for Mathematics Education, one discovers that the chapter dedicated to Proof is inserted in between the chapters on epistemology and on ethnomathematics. Should one see there simply a coincidence, an evidence of chance in the academic organization of the book? Maybe not if one considers the introduction by Ken Clements who tries to give some coherence to the part of the book that assembles these three chapters. Whatever the case be, I propose that we engage in a reflection on this proximity: To begin, in this note I will affirm that an ethnomathematical questioning is as necessary for research on the teaching and the learning of proof in mathematics as is the epistemological questioning that is more familiar to us (and that actually is a classic point of departure in this domain of research).
   To situate the ethnomathematical perspective, I take the characterization proposed by one of its founders, Ubiratán D'Ambrosio: to study the mathematical practices of culturally identifiable groups in contrast with academic or scholarly mathematics (but I will leave the discussion open with respect to the choice to use the term mathematics to design the practices at stake). One can also consider the following characterization -- probably less committed to an epistemological project -- suggested by Paulus Gerdes, the author of the chapter cited above: to study the mathematics (or the mathematical ideas) in their relations to that which constitutes globally social life and culture (ibid. p. 916). It is remarkable that in his chapter, where he lists numerous works on ethnomathematics, Gerdes does not mention a single one that would have mathematical proof as its focus, whereas there are many mentions to works on geometric drawings, number, or computation. Yet, it is difficult to think of a kind of teaching committed to the initiation on a mathematical rationality that would not take into account the rationality external to the class. This statement seems to go without saying if one judges it on the basis of the ongoing discussion on the relations between proof and argumentation, but my intention is to go a farther than that:
   I suggest that, in fact, any didactic transposition of proof in mathematics takes into account the rationality that is dominant in the society and the culture within which this transposition unfolds. The object of teaching constituted by proof is marked not only by an epistemological conception carried (and protected) by the scientific community, but also by a cultural conception of the relationship to truth and to refutationóa conception that is proper to the society and the culture where the didactic system works. An indication that this conjecture is tenable, also showing its force, is the great diversity in the vocabulary used by diverse languages and in different circumstances to speak about proof, about what is true or valid, and about refutation. This vocabulary can change along the course of the school years, or between the official programmes and their effective teaching (we will see about that in the following months by way of examples that come from Hungary and Japan). These variations, to which a mathematician is usually insensible, can entail serious difficulties in international mathematics education exchanges which imply translations (a classical example is that of the distinction between preuve and d monstration, that romance languages permit to make and that is barely tolerated in English ). But this is not just a matter of words, it has important theoretical implications! These difficulties cannot be overcome without an ethnomathematical questioning enabling us to understand the origin of this diversity and what does this diversity mean with respect to the relationship between school mathematics and its social and cultural context. In a book on cognitive development which takes into account social context, Barbara Rogoff notes, apropos of a field observation, that what separates the observed from the observer is not so much each oneís ìlogicî as the possibility to reach an agreement on what could be accepted as true (ibid. p. 30). What is at stake is the recognition and the taking into account of the ìinstitutionalî character of the decision and control rules that are related to the use of representations, computational techniques, and all the things that Alan Bishop calls symbolic technologies of mathematics. How does school take into account these rules and the ìformalizedî (but usually implicit) practices that accompany their usage?

I solicit those who are interested by these questions to contribute to initiate this ethnomathematical questioning by responding some questions that I formulate below or by soliciting the contribution of others who may not have access to the web but could provide some information:

• What are the words that are used in your language (and within your culture) to translate d monstration and preuve? What aspects are distinguished by the various possibilities? What terms are used in teaching (according to the levels of schooling)?
• What are the words used in your language (and within your culture) to translate contre-exemple and r futation? What aspects are distinguished by the various possibilities? What terms are used in teaching (according to the levels of schooling)?
• In your culture or society, how does one express the fact that one is certain on the validity of a statement or that one is confident on its truth? Are there different linguistic or pragmatic ways? What happens in school?
• How does one express a disagreement in your culture or in your society? Are there different linguistic or pragmatic ways? What happens in school?

Bishop A. J. et al. (eds.) (1996) International Handbook of Mathematical Education (esp. Ch. 22, 23 & 24). Dordrecht: Kluwer Academic Publishers.
Bishop A. J. (1988) Mathematical Enculturation. Dordrecht: Kluwer Ac. Pub.
d'Ambrosio U. (1993) Etnomatemática. São Paulo: Editora Atica S. A.
Gerdes P. (1996) Ethnomatematics and Mathematics. In: Bishop A. J. et al.(eds.) International Handbook of Mathematical Education (pp.909-943). Dordrecht: Kluwer Academic Publishers.
Rogoff B. (1990) Apprenticeship in thinking. Oxford University Press.

NT. A way to maintain some of the distinction in English is to translate preuve as proof and d monstration as mathematical proof.

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