Mercredi 17 octobre 2018; 3h 35min. 44sec 



Dominique Blanc  ETHNOMATHEMATICS 

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ETHNOMATHEMATICS
DECONSTRUCTING A PARADOX
by Dominique BLANC
Ecole des Hautes Etudes en Sciences Sociales
LISST  Centre d'Anthropologie Sociale
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[Texte de la communication prĂ©sentĂ©e au colloque de l'EASA Ă Bristol en Septembre 2006]
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First of all, let me explain in a few words, how and why I came across ethnomathematics. I am not a mathematician. I do not teach mathematics. Like the majority of us, I suppose, itâs sometimes very difficult to me to cope with an environment which requires some mathematical knowledge. So, why did I suggest speaking about this strange sort of Mathematics called "ethnomathematics"? At the Centre dâAnthropologie, in Toulouse, we were interested during several years in subjects like the use of reading and writing in ordinary practices and in everyday life. We tried to elaborate on these issues calling upon what was coined the anthropology of literacy. We studied housewives everyday writings, young girlâs diaries [my contribution], family albums written by young mothers on the occasion of the birth o their first child, etc. The main issue was the study not only of literacy in itself but rather of various forms of situated literacies. Actually, we were little interested in the issue of what is often called ânumeracyâ. Even an essay about the everyday practices of computer specialists was not at all dealing with numeracy.
Nevertheless, it became necessary to us to look behind, to consider the past. Then, when we became interested in the history of literacy, it was quite easy to find out a clearcut difference between literates and illiterates when the practice of reading and writing was at stake but it was very difficult to determine who exactly knew what about counting, reckoning and numbers at large. In various archives, biographies and autobiographies and in the memory of old informants who had known some of the last native illiterates in south Europe, the discourse was quite the same: âtheyâ didnât know how to read, they didnât know how to write but for counting and measuring and reckoning they were the best, even better than those who had attended schools, even better than the schoolteacher himself! I collected various narratives and various anecdotes on this matter. And I collected them because I was interested in the admixture of facts and fantasy embedded in such assertions and because I was interested in disentangling the facts from the fantasy.
Concerning the facts, it is true that numerous farmers, craftsmen or even illiterate storekeepers had to cope with an environment which required measuring, quantifying or giving a price. And it is a fact that, long before they were supposed to benefit from some kind of informal or formal education, those illiterates were managing to achieve useful operations in their everyday lives. As a consequence, measuring, quantifying and reckoning may appear to the majority of peasants living in remote countries, but also to some observers coming from outside, as skilful operations, possibly due to an elaborated and effective system (supposedly an arithmetic one), totally different from school mathematics. Nevertheless, the users couldnât explain how their specific systems of reckoning were functioning and the observers from outside didnât try hard to know how the concrete operations were carried out. Thus, we soon leave the reality to enter a world of fantasy: a glance is enough, as it is said, to evaluate precisely a volume or a weight. Indeed, a peasant used to evaluating some quantity of wood can announce a quite precise volume and weight after viewing it during a few seconds. But, most of the time, when we are given an explanation, it refers to some mysterious workings of the mind and we are often told, by some folklorists for instance, that marginalized groups, like countrymen in the margin of urbanisation and industrialisation, have invented by themselves and for themselves, an effective mathematics. Thus, when mathematics (most of the time only a simple arithmetic) is at stake, we are beyond the Great Divide between the savage and the civilised and between literates and illiterates and there are two main reasons for this: 1) various activities, like reckoning, measuring, etc. are mathematical ones but are also involved in everyday activities of the ordinary folk; and 2) because these activities do not require imperatively some level of literacy.
This was concerning social or professional groups of illiterates but we must go a step beyond. The best performers were seen as some kind of prodigious individuals. And while more and more children were entering primary schools to learn reading and writing, during the 19th century, some mysterious young boys appeared, all over Europe, who could neither read nor write but, nevertheless, were very learned in counting and reckoning. This time, what was at stake was not some operation of quantifying or measuring in connection with a social or a professional necessity. They performed amazing calculations by dealing with pure sets of numbers, some kind of calculations that students in higher grades couldnât carry out. Even school teachers, when some of them were required to challenge such amazing performances, failed pitifully. These children had some noteworthy characteristics: they were boys who generally lived in a remote farm and were employed in their very young days as shepherds. They had lived a long period of time alone and out of all kind of human communication. To put it in other words: they had lived very close to Nature or to a âsavage worldâ and very far from Civilisation.
One day, the mental calculator is discovered accidentally, typically by a schoolteacher and then a wonderful story begin: the completely illiterate young boy when asked some question of arithmetic, answers easily while demonstrating the use of a system totally unknown and apparently entirely invented by his own imagination when he was let alone with his lambs or his cows. Struck by such a prodigy, the schoolteacher leaves his job and, henceforth, he dedicates his entire life to this strange student, trying to understand, how the socalled ânatural calculatorâ is proceeding. Generally, this strange couple starts a longterm travel all over Europe before ending up destitute, after having known a very short period of fame, at least some of them. But the point is that during this short period of fame, various scientists and prestigious members of scientific academies were interested in their performances. We see some great mathematicians fascinated by what they call ânatural mathematicsâ, claiming that they have found out a sort of savage mind reinventing mathematics outside schools, without any education. Some scholars claim that God is the author of such a prodigy, of course, but most of them assume that, when it must cope with a natural or an hostile environment, the human mind can reinvent on his own the basis of science, and specifically the basis of mathematics. My aim is not to discuss here the fallacy of such apparently prodigious calculations (which actually are a mix of a genuine mental calculation improved by a constant and often obsessive training and then mixed with some tricks wellknown by the connoisseurs of numbers). My aim is neither to make fun of the naĂŻve enthusiasm of great mathematicians who knew nothing about some quite simple astute processes used traditionally by mental calculators. My aim is to point out that they are mixing up true stories and amazing legends, common representations and scholarly assumptions. While studying the life of these calculating prodigies and why they were drawing such attention to their supposed knowledge and amazing performances, I came across ethnomatematics, a field in which, surprisingly, I recognized some of the issues I had dealt with.
Actually, I was interested in Thomas Fuller, an African shipped to America in 1724 as a slave at the age of 14. His legend is very similar to the one of my European young prodigies. It encompasses all the stereotypic exploits I have found out in the narratives I have studied. But I was somewhat surprised when I discovered that some abolitionists had mention him for demonstrating that blacks were not mentally inferior to whites and I was much more surprised when I discovered that he was mentioned in current publications as possibly the first African American mathematician and that his amazing knowledge was due to his former African culture. We read in a publication from the University of Buffalo Mathematicians of the African Diaspora: âOur present understanding of mathematics in Africa at that time allows us to claim that when Thomas Fuller arrived in 1724 Virginia, he had already developed his calculation abilities, his learning of number words, a numeration system, of arithmetical operations, of riddles and mathematics games, etc.â Then, the paper elaborates on the mathematical abilities of the inhabitants of Fida, on the coast of Benin where Fuller was supposed to come from. In an essay addressed to American schoolteachers, which title is, significantly: Women and Minorities in Mathematics. Incorporating their mathematical achievements into school classrooms, we read that âtoday no one knows exactly how Thomas Fuller performed his calculations. However, the algorithms he used were probably based on traditional African counting systems. The people of the Yoruba area of southwest Nigeria have a complex counting system with very high numbers that probably dates back to Fullerâs timeâ.
These two publications are part of what is labeled ethnomathematics. Apparently, we are here in the same admixture of facts and fantasy I mentioned earlier when speaking about the young European calculating prodigies. But we must try to go beyond this first impression of âdĂ©jĂ vuâ. We are now in a postcolonial context and we must consider the political and ideological side of all assumption about culture and knowledge if we want to understand and discuss even the issue of science and ethnoscience. One of the aims of ethnomathematics is âChallenging eurocentrism in mathematics educationâ, according to a collection of essays edited by Marilyn Frankenstein and Arthur Powell at the University of NewYork Press in 1997. According to them âthe Eurocentric myth persists and influences school curricula, even in a supposedly neutral discipline like mathematics. [We must] challenge the particular ways in which Eurocentrism permeates mathematics education: that the âacademicâ mathematics taught in schools worldwide was created solely by European males and diffused to the Periphery; that mathematical knowledge exists outside of, and unaffected by, culture; and that only a narrow part of human activity is mathematical and, moreover, worthy of serious contemplation as âlegitimateâ mathematics. This challenge has brought together knowledge from mathematics, mathematics education, history, anthropology, cognitive psychology, feminist studies, and studies of the Americas, Asia, Africa, White America, Native America, and African America to create a new discipline: ethnomathematics. [The collection of essays] also attempts to organize the various intellectual currents in ethnomathematics, from an antiEurocentric, liberatory perspective. [The authors] are critically selective, not just interested, for example, in the âmathematicsâ of Angolan sand drawings, but also in the politics of imperialism that arrested the development of this cultural tradition, and in the politics of cultural imperialism that discounts the mathematical activity involved in creating Angolan sand drawingsâ. This radicalism is not shared by all the âethnomathematiciansâ, even by those who contributed to the mentioned collection. But there is a consensus for a somewhat different way of looking into the History of Science and for a somewhat different way of teaching mathematics by introducing âthe world cultures in the mathematics classâ.
From my point of view there are three quite different issues. First of all, as we have seen, we are dealing with an ideological and sometimes political movement which intents to rehabilitate marginalized human groups and societies, challenging the legacy of the colonial period. The argument could be: âThe more you were (or you are) marginalized, the more you deserve attention and you must be rehabilitated in all domains, including scienceâ. Concerning the origin of mathematics, for example, the argument is: Westerners (white and male Westerners, of course...) say that the only true Mathematics was invented by Greek mathematicians living in Alexandria. We know that Alexandria is situated in Egypt. Now, we know that Ancient Egyptians were good mathematicians constructing sophisticated pyramids before the Greeks supposedly invented Mathematics, so we can say that the Greeks living in Egypt have borrowed mathematics from Egyptian scribes. To go further, we know that Egypt is part of Africa and we know now that the ancient Egyptians were probably black Africans. So, it is not unbelievable to think and to say that Mathematics was invented by black people. This theory was (and is) currently maintained, particularly by some Africans Americans scholars after the success of bestsellers like Martin Bernalâs book: Black Athena: the AfroAsiatic Roots of Classical Civilization (1987) and George G. Josephâs book: The Crest of the Peacock: NonEuropean Roots of Mathematics (1992). The assumptions reported here are ideological consequences of a real fact: mathematicians and historians of Mathematics have paid very little attention to nonEuropean traditions in Mathematics: Egyptian, Indian and Chinese mathematics are not wellknown in the West and the importance of Arabic transmission of ancient theory and their creativity in this domain is often ignored. So the world strikes back, using an admixture of facts and fantasy.
The second issue is education. To summarise: âthe curricular praxis of ethnomathematics can be developed by investigating the ethnomathematics of a culture to construct curricula with people from that culture, and by exploring the ethnomathematics of other cultures to create curricula so that peopleâs knowledge of mathematics will be enriched.â According to Munir Fasheh (from her essay: âMathematics, culture and authorityâ): âTeaching maths without a cultural context by claming that it is absolute, abstract and universal, is the main reason (âŠ) for the alienation and failure of the vast majority of students in the subjectâ. This argument is twofold: on one hand it deals with culture as a whole, on the other hand, it deals with culture as the current knowledge of the schoolchildren. To put it in the words of the famous Brazilian educator Paulo Freire: âThe opposite of manipulation is not illusory neutrality, neither is it an illusory spontaneity. The opposite of being directive is not being nondirective  that likewise an illusion. The opposite both of manipulation and spontaneity is critical and democratic participation by the learners in the act of knowing, of which they are the subjectsâ. Thus, we are dealing with an interesting mix of culturalism and constructivism I have no time to discuss here. Let me just quote how some students of Paulus Gerdes, the promoter of the Ethnomatematics Project of the Universidade Pedagogica in Mozambique, reacted when âthey uncoveredâ the Pythagorean theorem while analysing the geometric construction of a traditional woven basket button: âHad Pythagoras  or somebody else before him  not discovered this theorem, we would have discovered it!... By not only making explicit the geometrical thinking âculturally frozenâ in the squarewoven buttons, butâŠ by revealing its full potential, one stimulates the development ofâŠ culturalmathematical (self) confidenceâŠ The debate starts. âCould our ancestors have discovered the Theorem of Pythagoras? Did they? Why donât we know it? ... Slavery, colonialismâŠ By defrosting âfrozen mathematical thinkingâ one stimulates a reflection on the impact of colonialism, on the historical and political dimensions of mathematics [education]â, Paulus Gerdes claims.
The third issue is the one the two others issues depends on and that is crucial for anthropology. To put it very simply: what is the meaning of âethnoâ and what is the meaning of âmathematicsâ in âethnomathematicsâ? Since we return to the beginning, we must return to the beginner. Ubiratan dâAmbrosio, a Brazilian mathematician, is generally recognized as the inventor of this term in the 1980s. He defines âethnoâ in the following way: âOur conception of âethnoâ encompasses all the ingredients that make up the cultural identity of a group: language, codes, values, jargon, beliefs, food and dress habits, physical traitsâŠ Cultural groups, children of a certain age range in a neighbourhood, farmers cultivating wheat, engineers in car factories, and so on.â As you can see, the spectrum is very wide and it almost encompasses all human groups. Concerning mathematics, he states: âAll [of these groups] have their own patterns of behaviour, codes, symbols, modes of reasoning, ways of measuring, of classifying, of mathematizingâ. The spectrum is very wide, once more, and it almost concerns all human activities. And thatâs the point: can we name âmathematicsâ, as Alan Bishop does, for instance, universal activities like locating because its emphasis is on the topographical and cartographical feature of the environment  how space is conceptualized and how people and objects are positioned in the spatial environment or can we name âmathematicsâ designing which refers to the conceptualization of objects that are manufactured for use in our homes and leads to the idea of âshapeâ? According to Bishop, both activities fall into the category of spatial structuring which has been of great significance in the development of mathematical ideas and that makes sense with the DâAmbrosio conception of a continuum between ad hoc practices and scientific invention. We are âtrying (DâAmbrosio says) to find an underlying structure of inquiry in these ad hoc practices. In other terms, we have to pose the followings questions: 1) How are ad hoc practices and solution of problems developed into methods; 2) How are methods developed into theories? And 3) How are theories developed into scientific invention?â and later in his paper âEthnomatematics and the history of mathematicsâ, he states: âWe claim a status for these practices, ethnomathematics, which do not reach the level of mathematization in the usual, traditional [he means: western academic] senseâ. As a consequence, he argues that for Third World countries the distinction between Pure and Applied mathematics is highly artificial and ideologically dangerous. Why âfor Third World countryâ only? I would ask him. What is dangerous, it seems to me, is to maintain a misleading confusion between an elaborated body of knowledge with its own speculative and reflexive rules, namely what we call mathematics, which, of course, is not culturefree and admits an historical, social and cultural diversity (Mathematics is an Indian, Chinese, Arabic and European science) on one hand and, on the other hand, the existence of simple or complex number and counting systems, even in illiterate societies, which may be used only for simple reckoning or may be present in some sophisticated native cosmologies, for example. But it is the cosmology that is sophisticated and not the numerical system per se. Unfortunately I canât elaborate on this point. I just can argue that the role of the anthropologist is not, it seems to me, to uncover what, besides, is said to be only OUR mathematics supposedly unconsciously embedded in THEIR conception of the universe: that is the paradox of ethnomathematics. All societies measure and classify and people have, of course, more than a purely utilitarian relationship with things as symbolic anthropology and âCognition in practiceâ studies show it. But all societies and all human groups have not either consciously or unconsciously invented mathematics.
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