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## December 06 : Ethnomathematics and pedagogy

**Report on the meeting of the REHSEIS ethnomathetmatics seminar held on 13/12/06 : **

**"Ethnomathematics and pedagogy : the question of string figures "**

This meeting was based on four articles :

Charles Moore. ` The Implication of string figures for American Indian Education’, Newspaper of American Indian Education, Volume 28 Number 1, October 1988

And three articles drawn from * Changing the faces of mathematics, Perspectives on indigenous people of North America*, University of Wisconsin-Madison, 2002 :

- The learning of geometry by the inuit, a problem of mathematical acculturation (Richard Pallasio, Richard Allaire, Louise Lafortune, Pierre Mongeau)
- Yup’ik culture and everyday experience as a base for school mathematics (Jerry Lipka)
- Teaching mathematics skills with string figures (Gelvin Stevenson, James R. Murphy)

The idea was to open a first window on the huge amount of literature whose topic is ethnomathematics and pedagogy, and to use this occasion to think of the kind of activities one could use in a class room on the basis of Eric Vandendriessche’s field work. Were present Eric Vandendriessche, Agathe Keller, Mitsuko Mizuno, Sophie Desrosiers, and a new arrival Catherine Nowak : a teacher of mathematics at high school and university level, who is also interested in what is the basis of mathematics, and in questions relating to numbers and anthropology.

The discussion starts by raising a first question : what is at stake in the pedagogy of string figures ; what does one want to teach with it ? Does one have to only focus on the mathematical dimension of this activity ? Should we also take into account its other properties, like its relaxing effect, or not ? Eric Vandendriessche characterizes string fugures as a "multi competenced" (perhaps "multi faceted") activity. We will regularly come back to this problem during the seminar.

First of all, we all agree that the problems are not the same if the public is that of children who practice string figures every day (as in Vanuatu) by contrast with children like those we have in French schools, who seldom use string figures.

In the first case, Sophie Desrosiers stresses that by focusing on their mathematical dimension, we may actually loose some elements which are attached to the comprehension of the object by the actors/children, and which are actually mathematical, but that we will overlook them because we have circumscribed in advance what interests us in this activity. Agathe Keller thinks that what is important is the link between the pupil’s knowledge and those of school mathematics, the important thing being to be able to come and go in between one and the other. The situations described in the article on Yup’ ik are particularly interesting in this regard ; they suppose a dialogue with 3 partners, the elders, the teacher and the pupils.

We first discuss Charles Moore’s article, that Sophie Desrosiers found disappointing, as if it was beating around the bush, without ever entering into the heart of the subject. Eric Vandendriessche appreciates it because it takes side with the creators of string figures. The aim of this article is clearly to show how the creation of string figures concerns mathematics. Agathe Keller stresses the historical dimension of the article (we learn that the Indians of the North of the United States practised string figures). Eric adds that while taking the point of view of how they were created, and by underlining how it rested on blocks of sub-procedures that one can arrange variously, it gives us an idea of the way in which these objects were actually produced. One can thus often bring back complex figures to simpler ones from which figures seem deduced by variation and exploration. Agathe Keller stresses that one should not confuse logical organization (rationnal) with the process of discovery. Sophie Desrosiers asks Eric Vandendriessche if he has met in his own corpora of string figures different modes of development or organization, some develloping in some kind of star process from an given figure, others progressing linearly. Eric Vandendriessche answers that he does not have enough elements yet to answer and that he did not see anything striking in one way or another. But that he often meets basic figures from which things are made. He quotes on this subject an article of D.Jenness, an anthropologist, on string figures practised in the islands of Entrecasteaux (New Guinea Nouvelle Guinea). He describes a figure which is called in vernacular language the "mother of all the figures". Agathe Keller stresses that perhaps this name indicates an organization of the corpus/knowledge of string figures by the actors, rather than the true testimony of the way the figures were generated, even if the idea of generation of various figures starting from a simple one seems for the moment a good idea.

We thus start to discuss Murphy’s article which adds the idea of iteration and reiteration in the description of the process of generation of new string figures. Murphy develops a whole set of arguments on the benefits of string figures beyond mathematics. Or perhaps at the limits of mathematics : does the concentration necessary at all the stages of a procedure, belong to mathematics ? Eric Vandendriessche explains that he sees an analogy with algebra. The problem being to know if this analogy has a meaning for his pupils or not.

For Agathe Keller, analogy does not have a heuristic value inevitably in pedagogy. Sophie Desrosiers and Catherine Nowak do not agree, insistent that analogy is a very strong thing and that it depends on culture. Eric Vandendriessche wonders how Murphy articulates the link between string figures and algebra : how can the study and analysis of some string figures, by analogy, help pupils in a mathematics class to overcome certain difficulties ?

While returning to the idea of system, Sophie Desrosiers is not sure however that the discovery of a system justifies the fact that it is taught in a mathematics class, because it could just as easily be useful for say grammar. For A. Keller when one has to deal with a system, and that one experiments the fact that it is a formal system, one is learning something that has to do with mathematics. The problem being to know if it is related to the school syllabus or not, and if so, how one establishes links between these two activities.

Sophie Desrosiers explains that it is important to develop an analytical spirit : it is necessary not only to see that it is a system but also understand how it works. For Eric Vandendriessche, it is indeed very important. He thinks however that many people, in Vanuatu for example, have an analytical comprehension of how such torsion of a string transforms in such a way a given figure. When he is on the field, he often starts by questioning the children and finishes by the elder ones. He has noted that certain ni-Vanuatu children have a good knowledge of the system of relations which exists between various string figures of the corpus.

He explains that he must make pedagogical cards in Vanuatu, that can be consequently used in regional schools. He has thought up two activities : one consists in working out ideas of symmetry by looking at how, when one reverses the roles of the small fingers and the thumbs in the algorithm which leads to a simple figure, one obtains a figure which changes by an axial symmetry. We try to reproduce the exercise of Eric Vandendriessche with greaty difficulty, but it is well understood that it is addressed to people who are accustomed to doing such activities. Sophie Desrosiers stresses that there are other forms of symmetries and asymmetries like that of the two hands used. Catherine Nowak finds this very complicated to introduce the simple idea of an axial symmetry in two dimensions. Eric Vandendriessche explains that according to his discussions with a teacher in Vanuatu, pupils had a hard time with plane geometry and were more at ease with the 3d, therefore it would be perhaps a manner of facilitating their access to it. Catherine Nowak evokes a work with pupils which consisted in making them cut fruits in two : there were different ways to cut one fruit and to observe the symmetries thus created. Another activity suggested by Eric consists in working with the symbolization created by Thomas Storer, who died regrettably in November before Eric and he could meet, and to see how it applies to the figures, in the way Murphy describes such activities. Agathe is convinced that that helps to understand the concrete and at the same time formal aspect of symbolic systems.