### i. State-of-the-art and objectives

Ethnomathematics investigates mathematical activities and practices that are not produced in the conventional institutions where it is taught and practiced(1) . Its focus is as much on the mathematics of the slums of Rio, as on the practices of aboriginal tribes in Australia(2). The vitality of this quite new field can be measured by the number of papers that were presented in February 2006 at the latest International Conference on Ethnomathematics(3) . Few European researchers presented their work there. Indeed, no team devoted only to ethnomathematics exists today, to our knowledge, in Europe. The first innovation of my team is thus its location, Europe. Most of the recent research has focused on deriving pedagogical material from second hand anthropological literature, or giving mathematical interpretations to folk activities. The second innovation of this project will be to do ethnographical fieldwork. Furthermore, ethnomathematics concentrates on mathematical objects and processes of different kinds. Some of these are easily identified as belonging to “mathematics”, such as computational activities. But ethnomathematics also analyzes practices like Sand Drawings or musical patterns(4) , controversially underlining that they have a “mathematical nature”. The third innovation consists then in taking both into account in a same *fieldwork*(5). Part of the problem with determining the “mathematical nature” of these activities has to do with the history and sociology of mathematics : what are the processes by which an activity is labelled “mathematics” ? And before that, how and when, within communities, do such activities become elements of recognized knowledge ? Our studies will take into account practitioner’s points of view on the subject, a fourth innovation. As a historian of mathematics, I will further attempt to articulate how these practices can be perceived in their relation to history. The use of history of mathematics is the fourth innovation. Finally, consistent with the origins of
ethnomathematics, the team should produce pedagogical material, which could be used in schools and by associations or museums active in science popularization.

### ii. Methodology

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Each member of the team will conduct specific fieldwork in India, the Paraguayan Chaco, the Andean area, Madagascar and in several Pacific Islands, concentrating on String Figures, Sand Drawings, Weaving techniques and Divination on one hand, and the different modalities of numbering and measuring on the other. Significant of our intricate collaboration, we hope to invite each other on our respective fields, as a way of actively putting to play our different points of view. This might help us understand also what separates our different areas of work as well. These fieldworks should produce several research papers, which we might collect in a book during the fifth year. I will animate a regular monthly seminar, continuing the one existing today in REHSEIS(6) . We will discuss in it the most recent literature and the progress of our respective investigations. Pedagogical ressources deriving from our research will be produced. This would include films, which could popularize such mathematical activities to a broader public. It will also involve the maintaining of the existing web-site(7) , notably developing online pedagogical activities and a free and open photographic data-base. We will also animate activities with associations and museums, in the line of what has already been done with INUKSUK(8) during the “Anthropology of Mathematics” ACI(9).

Ethnomathematics requires a specific methodology on the field, according to the object it intends to concentrate on. Furthermore, in some areas, such as String Figures or Divination, our research is already quite developed, while in other areas, such as weaving, the project should enable crucial progress. I will thus specify here what we intend to investigate, practice by practice.

In order to make a String Figure you first need to knot the ends of a long string to make a loop ; the activity then consists of a succession of operations applied to the string. It can be carried out alone or with a partner, using fingers and sometimes one’s feet or mouth. Although based on algorithmic thoughts, using concepts such as operations and sub-procedures, iterations and transformations, this activity is also of a geometrical and topological nature since it is founded on the modifications of complex spatial configurations(10) . As part of the ACI project “Anthropology of Mathematics”, E. Vandendriessche has carried out several ethnographical field studies in Papua New-Guinea, Vanuatu, the Marquesas Islands and Paraguay. E. Vandendriessche will concentrate on Papua New-Guinea and learn the Trobriander’s language (*Kilivila*). This will enable him to study the possible connexions between String Figure algorithms and the local culture. He additionally plans to return to Paraguay, were String Figures are still actively practiced in Chaco. He will finally collaborate with Céline Petit, a doctoral student at Nanterre University ( Paris), who collected Inuit String Figures in the Canadian Arctic (Nuavut). Simultaneously, Eric Vandendriessche will continue his research on the mathematical understanding of String Figures(11). His aim is to simultaneously analyze string figures from a formal perspective and from the cognitive representations of the practitioners, and finally examine any connexions existing in between these two points of view. Furthermore, Eric Vandendriessche is currently making with Cyrille Benhamou and Frédéric Peugeot a movie on String Figures in the Trobriands. This documentary film, which has so far a very poor funding, could be properly edited and distributed through this project. It should be a first experience enabling the other members of the team to carry out a similar project as well.

Sand Drawings can be seen as an algorithmic activity based on repetition and transformation of some particular patterns(12). Eric Vandendriessche plans to do fieldworks in Vanuatu and in Australia (with Aboriginal people) where the practice of String Figures and Sand Drawings still coexist in the same places. The Cultural Centre of Vanuatu, interested in this project, has asked E. Vandendriessche to conceive with this corpus a "non-academic" mathematics curriculum for High Schools. In Tamil Nadu, India, there is also a tradition of Sand Drawings called *kolams*. These activities have already been studied and extensive databases, mathematical formalizations, and some ethnographical studies have been published(13) . However the way it articulates to other forms of mathematical knowledge in day-to-day activities has not been investigated yet. In February 2006, I began a database of *kolams*, learning how to make them in Tamil Nadu with women who are indeed those who mainly practice them. Further fieldwork will enable me to investigate the kind of mathematical knowledge that may have been developed by women of the Nagai area outside of schools. The birth of graph theory, in association with mathematical recreations and especially the analysis of labyrinth as studied by Mitsuko Mizuno, PhD candidate at REHSEIS, may help me articulate the link of these live practices with the history of the mathematical field that describes them. On a more general point of view, the collecting of data on such activities in culturally distant areas, will, as for String Figures, help us reflect on the nature of the object, and on the link that might be made between the practitioners’ technical representation of this activity and the mathematical theory we have of it.

Andean weaving is a complex three-dimensional topological object. Its practice requires a high apprehension of complex spatial configurations. Arithmetical skills and combinatorial logic are used in the Andes, from the beginning of our era until today, to create new weaves and to make two-faced textiles showing the same design and the same weave on each face(14). Sophie Desrosiers, an anthropologist working on Andean textiles, has recently demonstrated that the results of Andean textiles share something with the geometry of satin(15). Interested in the history of Number Theory in Europe in the nineteenth century, Anne-Marie Décaillot studied the mathematical work of the French mathematician Edouard Lucas on satin weaving geometry (16). Sophie Desrosiers and Anne-Marie Décaillot will thus explore the existing data trying to work out a formal way to consider Andean weaving corpus. Such formalization, probably based on tools derived from Number Theory, will enable them to create a new classification. It will also provide tools to study the links between such a representation of weaving, and that of the practitioners in Peru, Bolivia and Chile. Data on other weaving traditions may further be investigated if such an activity is found on their fields by the other researchers of the team.

Besides traditional graphical activities, others practices in oral societies may have strong relations to geometrical investigations in connexion with numbers. This is the case of a form of divination practiced in Madagascar, where seeds are placed on the ground and arranged in the form of tables, the elements of which are equal to one or two seeds. M. Chemillier has shown that a particular part of the table called *renin-tsikidy* is a matrix (the « mother matrix ») of four rows and four columns. He has studied the transformations, such as reflexions or rotations that are made on these tables during divination. Marc Chemillier will continue to make further investigations on this subject, focussing on links between the divinor’s representations and the mathematical theory of their activity(17) .

All the members of the project will also work on the conceptions of numbers and quantifying algorithms on their respective fieldworks. In India, Senthil Babu will continue his research in the agricultural villages of Tamil Nadu. The problem of how local measuring units are converted to fit the land-owner’s units in English standards (lbs, acres) and that of the Indian State (l, kg and m), a political and social issue, in which numeracy is of crucial importance struck S. Babu and me during a fieldwork in February 2006. The project should enable us to further explore this issue. S. Babu will also study the different methods used to measure either accurately or approximately land and volume. He will additionally research the articulation of the computations used in occupational activities (such as carpentry) with the mathematics taught at school. I will attempt to complete S. Babu’s fieldwork, concentrating on the kind of mathematical activities women are led to do. I will also try to find other Indian scholars willing to conduct ethnomathematical fieldwork in other parts of India. My main aim will be to investigate the continuity of the measuring units and interest computations from medieval Sanskrits texts to the present. The part of these scholarly Sanskrit texts devoted to this question has not been the focus of recent historical research. It appears as an important bridge between the world of live practices and that of scholarly mathematics. I hope to produce with S. Babu and other Indian researchers a documentary movie on the measuring activities in rural south India. Eric Vandendriessche will study the different numerical systems practised in the Trobriand and in neighbouring Islands(18). He will focus on how these non-base 10 systems coexist with the base 10 numerical system taught at school. He will also study the methods used to assess the number of piled cones of yam, found in many Trobriand gardens. Such computations bear striking resemblances with the *rāśi* (piling) problems found in the scholarly texts of India, and some of the practices of rural India. This research will thus open new perspective for historians of ancient mathematical counting practices, such as C. Proust and M. Ross, for which textual data is fragmentary.

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(1) Ascher, M (2002). *Mathematics Elsewhere : An exploration of ideas across cultures*, Princeton University Press.

(2) Eglash, R. (2000) *Anthropological perspectives on ethnomathematics*. In *Mathematics across cultures : the history of non-western mathematics*, ed. H Selin. Dordrecht ; Boston : Kluwer Academic.

(3) See http://www.math.auckland.ac.nz/Even...

(4) M. Chemillier, *Les Mathématiques naturelles*, Paris, Odile Jacob, 2007

(5) Thus, Marcia Ascher has studied both types of activities, but she used secondary literature.

(6) The minutes of the exchanges can be found in French and sometimes in English on our web site : Séminaires

(7) Accueil du site Ethnomathématiques

(8) The French association INUKSUK is currently carrying out exhibitions, conferences and workshops bringing Inuit culture towards a larger public in France.

(9) Indeed, E. Vandendriessche and C. Petit have in the last years, regularly animated sessions with children and adults, teaching them String Figures and reflecting with them on their mathematical nature. They have also put up photographic exhibits : the money earned by the sale of some of these pictures will be given to the villages where the pictures were taken.

(10)Vandendriessche, Eric (2007), *les jeux de ficelle : une activité mathématiques dans certaines sociétés traditionnelles*, Revue d’histoire des mathématiques, (In press).

(11) In 1 988, the American mathematician Dr. Thomas Frederick Storer (1938 - 2006) created the concept of the “Heart-Sequence” of a String Figure, which enables a topological interpretation of String Figures ( Thomas Storer (1988), *String Figures*, Bulletin of the String Figures Association). More recently a Japanese team of four engineers published an article which demonstrated the possible use of Knot Theory to describe a String Figure ( Yamada Masashi, Burdiato Rahmat, Itoh Hidenori et Seki Hirohisa (1997), *Topology of Cat’s Cradle Diagrams and its Characterization using Knots Polynomials*, Transaction of Information Processing Society of Japan, vol.38 n°8, p.1573-1582.). For his PhD, E. Vandendriessche is developing from these works an extensive mathematical representation of this object.

(12) Ascher, Marcia (1988), *Graphs in Cultures : A Study in ethnomathematics*, Historia Mathematica 15, p.201-227. Chemillier, Marc (2007), *op-cit*.

(13) See for instance : Gabrielle Allouche ; Jean-Paul Allouche ; Jeffrey Shallit (2006). *Kolam indiens, dessins sur le sable aux îles Vanuatu, courbe de Sierpinski et morphismes de monoïde.* Annales de l’institut Fourier, 56 no. 7, p. 2115-2130.

(14) For instance : Branko Grünbaum (1990), *Periodic Ornamentation of the Fabric Plane : Lessons from Peruvian Fabrics*. Symmetry 1(1), p. 45-68. Anne Paul (1997), *Color Patterns on Paracas Necrópolis Weavings : A Combinatorial Language on Ancient Cloth*. Techniques & culture 29, p. 113-153. Anne Paul (2004), *Symmetry Schemes on Paracas Necrópolis Textiles*, p. 58-80, in : D.K. Washburn (ed), *Embedded Symmetries, Natural and Cultural*. Albuquerque, University of New Mexico Press.

(15) Sophie Desrosiers (1997), *Logicas textiles et logicas culturales en los Andes*, p.325- 349 *in :Saberes y memorias en los Andes*, s.l.d. de Th. Bouysse-Cassagne, Lima : Institut Français d’Etudes Andeans / Paris : Institut des Hautes Etudes sur l’Amérique Latine.

(16) Décaillot, Anne-Marie (2002), *Géométrie des tissus, Mosaïques. Echiquiers. Mathématiques curieuses et utiles*. Revue d’Histoire des mathématiques, p. 145-206.

(17) M.Chemillier, D. Jacquet, V. Randerianary, M. Zabalia (2007), *Aspects mathématiques et cognitifs de la divination Sikidy à Madagascar*, L’Homme n°181, p. 7-40.

(18) Lean, Glendon (1992), *Counting Systems of Papua New-Guinea and Oceania***, **Thesis. And, online : http://www.uog.ac.pg/glec/about_glec/about.htm.