Archives 1996–2012 de l’ancienne équipe REHSEIS
Laboratoire SPHERE – UMR 7219

The seminar’s aim was to discuss the following articles :

Englash, Ron. ’Anthropological prospects one Ethnomathematics’. In Mathematics across cultures : the history of not-western mathematics, ED. H Selin. Dordrecht ; Boston : Kluwer Academic : 2000.

Urbitan D’Ambrosio :

The article published in the special series of « Pour la Science » (The French edition of Scientific American) called "Exotic Mathematics" of April 2006

And the articles on his Internet web site : http://vello.sites.uol.com.br/ubi.htm

And finally to browse and look at the abstracts and talks posted on the web site of the last international conference on Ethnomathematics : http://www.math.auckland.ac.nz/ poisard/ICEm3/ICEm3.html

Were present in addition to Cyrille Benhamou, Eric Vandendriessche, Mistuko Mitsuno and Agathe Keller (on which you can find details of why they are interested in Ethnomathematics in the reports/minutes of the preceding meetings) Michel Paty, Marie-Josée Durand Richard and Martin Zerner. We explain to each other our interests in this subject : Michel Paty is an emeritus researcher of CNRS. He works on the philosophy and history of physics and sciences in general, and is interested in the various systems of rationality which he would like to compare, and for him, mathematics is a privileged place for the exploration of these forms of rationality. See in particular : « The question of rationality in front of the diversity of knowledge practices », in D’Ambrosio, Ubiratan (ed.), Cultural Diversity : New Perspectives in the History of Sciences, in Saldaña, Juan José (ed.), Science and Cultural Diversity. Proceedings of the xxi st International Congress of History of Science (Mexico, 2001), Universidad Autónoma de México & Sociedad Mexicana de Historia de la Ciencia y de la Tecnologia, CD-Rom, México, 2005, vol. 42 , p. 3261-3281.). In addition, he knows Ubiratan D’Ambrosio personally. Marie-Josée Durand-Richard explains that she is interested, as a historian of algebra in the England of the XVIIIth and XIXth century, where the question of the transmissions of knowledge from one community to another is important. She organized in REHSEIS, a few years ago, a one-day workshop on Ethnomathematics. She discovered D’Ambrosio’s work at the 1st European summer school on "History and Epistemology in mathematical education". He had discussed the question of circulation of knowledge using the basin of circulation metaphor, (see distributed text : "Ethnomathematics, history of mathematics and the basin metaphor", Acts of this summer school, edition IREM of Montpellier, 1995). This metaphor appears suitable to be applied at the same time to the circulation of knowledge from one community to another, and from a culture to another. Martin Zerner is interested in Ethnomathematics as a mathematician, teacher and a historian of mathematics. He asks that we summarize D’Ambrosio ideas before discussing them.

Agathe Keller starts by summarizing what she retains of his work : U D’Ambrosio is the person who created the term « Ethnomathematics ». He understands the discipline on two different levels. One is that of the genesis of mathematics, which is produced by communities living in a given natural and social settings, the question being why and how mathematics appear. Once mathematics exists as a body of knowledge, how does it interact with other types of knowledge and how does it circulate in the world. In particular he is interested in the political dimension of mathematics, especially the politics of how this discipline is taught, as shown in his work on Ethnomathematics and peace. Agathe Keller explains that she understands d’Ambrosio as a dreamer in Bachelard’s sense of a daydreamer, d’Ambrosio daydreaming on the question of the universality of mathematics. This means that we have to do with a sprawling body of texts and themes, somewhat multiform and confused, where the author advances generally by analogy rather than by a logico-deductive reasoning. Teaching is at the heart of his thought. Eric Vandendriessche compares d’Ambrosio‘s definition of Ethnomathematics with that of Marcia Ascher’s, which is more or less contemporary to his. She considers rationality in traditional communities, rationality which one can describe with a mathematical language.

MJDR and MP explain that what they like in his work is that he tries to think and establish the universality of mathematics without denying the various forms which it can take. For MJDR it shows how universality was built and how it must still be built. MP notices that our humanity is used as a basis to build this universality. For him any rationality is mathematical. Martin Zerner wonders how does one identify that such activity is mathematical or not. Agathe Keller explains to him that this question is the topic of our ACI. However she points out that the question does not arise for what concerns quantification, arithmetic. Whereas it is problematic for the object on which Eric works (string figures), which would belong to geometry. For MJDR, this comes from our own history of mathematics which considers that arithmetic belongs to material practices and that geometry is theoretical and universal. Martin Zerner states the importance of case studies. Eric points out how little today Ethnomathematics is devoted to such kind of specific field work:out of 30 presentations given at the ICEM 3, 6 were related to case studies. The others were general and raised questions about teaching and Ethnomathematics. AK circulates papers which were proposed at the ICEM3 but not presented.

Alberti, M. & Gorgorió, N. Cultural tools as mediators of mathematical cognition : Iron and bamboo compasses of the Torajan woodcarvers of Sulawesi.

Alberti, M. The Kira-kira method of the Torajan woodcarvers of Sulawesi to divide a segment into equal parts.

Karam, R. & Liblik A. Skilled calculus from Brazilian´s northeastern farming region.

Others can be found on the web site.

We return to the political dimension of d’Ambrosio’s work. There is first of all the questions raised by the fact that he considers that a specific knowledge is born from natural conditions which influence it. AK evokes the work of Vithal, Renuka & Skovsmose, ole, ‘The end of innoncence : A critique of ’ethnomathematic’s inEducational Studies in Mathematics (34), 131-158,1997, which parallels such speeches with the educational programs of South Africa under apartheid. Of course d’Ambrosio is not racist, but one should be aware that way of linking nature and culture un-specifically does have undertones. What needs to be unraveled is the way that nature and culture articulate themselves in mathematics, which is of course problematic. For MJDR knowledge is born from experiment, but the operations of the mind are innate, and thus universal. The problem remains to determine what are the "simple ideas" on which the mind operates....

AK raises the question of the dates of d’Ambrosio’s work. She finds them somewhat outdated. D’Ambrosio defends the idea that there is a "politically correct" manner (read : leftwing manner of showing that mathematics is plural, in becoming etc) to teach mathematics which would make it possible in a univocal way to make the world better. In parallel thus, there would be a right-wing manner (eg monolithic and closed on itself) of teaching mathematics which would produce and is the product of a fascistic and capitalist society. This point of view is at the same time simplistic and supposes that human beings are always coherent in their activities with their political ideas. For MJDR the fact remains that a good number of our colleagues who launched out in the history of the mathematics in the 1970’s shared the idea that there was something wrong in teaching mathematics as a theoretical building without any bond with the reality or lives of the pupils. It was catastrophic, and if not reactionary at least élitiste, insofar as it is understood only by those students whose social back ground give them access to this kind of language. The problem remains complete today in France, around the question of the existence of a single « egalitarian » junior high school : it is not enough to teach the same body of knowledge to everyone, since the pupils arrive from different backgrounds.

Agathe Keller thinks one should use the distinction between mathematical language to describe objects and the point of view of the speakers. MJDR gives the example of the work of Bernard Jaulin on the bell ringers of the XVIIth century : the succession of the various manners of sounding bells formed a group. For MJDR what counts is : it is operational ? Do the actors actually have the sense of a closure, that they have formed an homogenous set ? Michel Paty quotes « La pensée sauvage » (The wild thought ?) of Levi- Strauss and the fact that systems of family relationships in the cultures of "people without history writing" can be formulized as a group. Eric Vandendriessche points out that the mathematisation of the object came from Andre Weil and that Marcia Ascher treates this example in one of her articles ; All this would testify to non-explicit mathematical activities certainly but no less effective (MP). Agathe Keller evokes the fact of speaking a language : one is not always conscious that it forms a system. For MJDR and MP, the problem is that a language is not a system, its complexities are higher than that of a formal system, particularly of mathematical objects.

Martin Zerner raises the question of field methodology. We evoke the fact that it is necessary to use ones own representation of mathematics to account for the object which one observes. We thus evoke the experiments of Marc Chemillier and the way in which he seeks the "saturation of logical spaces".

Eric Vandendriessche explains that in his work he does not raise any more the question to know if string-figures are mathematics or not. He has answered the question, to some extent. However he wonders how he can translated this into some teaching material. He returns thus to the problems mentioned in the last seminar : the material which he must produce for Vanuatu teachers, which is the condition given by the authorities of this country to grant him a researcher’s visa. It is thus him, somebody from the outside, who will create a teaching document. But in the name of what ? The universal idea of symmetry ?

Agathe Keller deviates evoking how the string figures seem to us, who do not practise them, an object complicated to teach symmetry. MJDR underlines on this subject that, in our heritage, so rationalist (Descartes), for empirists (Locke), the "simple ones" combine to make a "complicated one". Silence is made in general on the way in which "simple ones" are extracted and recognized from the « complicated one » they form. MP recalls that d’Alembert (in a Cartesian vein, although in the line of locke) on the contrary explains how it is necessary to abstract the simple one from the complicated facts. The theorization and the rebuilding of simple imply that thought, rationality is applied onto the complexity of facts. Eric Vandendriessche underlines here how simplicity seems to vary from culture to culture. Martin Zerner explains that that has a consequence on the order in which one carries out a lesson. We reconsider the question of Eric. MZ evokes the figure of a teacher fropm Chiapas studying his own language and its numeration to be able to teach it. A story of Ethnomathematics lived from the inside. Everything changes if the desire comes from within the community. AK evokes the communities which on the contrary are not interested in their "devalued" traditional knowledge, what should we do then ?

V evokes discussions on the question of the translations of terms according to the languages, evoked with the ICEM3 and which is a part of these questions. Cyrille Benhamou mentions the question of the context which makes it possible to give a meaning to what is taught. It is a question of passing from practice to knowledge. AK stresses that the question between practice and knowledge and how one passes from the one to the other is one of the topics of our working group. CB adds that to teach is to enable one to leave a particular framework where an activity is practised to allow a general use of this object (which thus has become a knowledge). AK recalls the work of Senthil Babu on the traditional teaching of mathematics in South India. The problems and the knowledge transmitted within the school framework are taken again and developed within the family framework and the vicinity. The transmission of knowledge is collective, and is not limited in a place and to a person. CB evokes the case of emigrant’s children who by losing the language of their parents also lose their access to French knowledge. AK wonders the share of cultural and the share of social in these problems (if one comes from a family of intellectuals whatever his/her culture one is at ease with knowledge...). MP mentions the question of psychology in teaching.