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Etnomatemática

Concepção etnoantropológica de matemática

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by Ubiratan D’Ambrosio

In 1976, in the Third International Congress of Mathematics Education, which took place in Karlsruhe, Germany, I was in charge of the group dealing with "Objectives and Goals of Mathematics Education". The preparation of the papers was an interesting exercise. It took several steps: 1) to write a proposal for the paper, discussed by the entire group of organizers; 2) to elaborate the paper; 3) to send the paper to a number of mathematicians and mathematics educators all over the World, asking for comments; 4) to re-elaborate the paper, taking into account the comments received; 5) to conduct an open discussion of the paper to participants of the discussions of the group during ICME 3; 6) to write a final version of the paper, for publication in New Trends in Mathematics Education IV, by UNESCO. [1] The entire process took about two years.

I was particularly pleased with the outcome. The paper "Why teach mathematics? – Objectives and Goals of Mathematics Education" was the starting point to develop my ideas on the socio-cultural roots of mathematics education. The great opportunity to present my views came when I was invited to deliver the opening plenary lecture for the Fifth International Congress of Mathematics Education, in 1984, in Adelaide, Australia.

There has been much talking about the word ethnomathematics. I will elaborate on how I did start using this word and how it developed into the meaning I give to it.

I had not seen the word ethnomathematics in print before I first used it, but I would not be surprised if a search in the literature, mainly on Anthropology, may reveal earlier users of it. I recently learned from Claudia Zaslavsky that Otto Raum wrote, in a review of her book, published in African Studies (1976):"(This Mathematics) might perhaps be most suitably called ethno-maths on the analogy of ethno-music, ethno-semantics, etc." And Wilbur Mellerna, in a letter to Gloria Gilmer, published in the NEWSLETTER of the ISGEM (vol.6,n.1,November 1990), says that he had invented the word ethnomathematics in 1967 and that he gave a talk in 1971 using it.

I was aware that the word ethnohistory had been introduced in the 1940s, as the history of non-literate people, and that in 1955, an International Society of Ethnomusicology was founded. Also, I knew the field of ethnopsychiatry, focusing on "exotic societies". When I lectured at the Linguistic Institute at SUNY at Buffalo, in the Summer of 1971, the word ethnolinguistics was current. Ethnopsychology, ethnobotany, ethnomedicine were also frequent to refer to the study of practices of different racial groups. And the sociologist Harold Garfinkel coined the word ethnomethodology, in 1967, to express his interest in how social interactions and practices are related.

My use of the word ethnomathematics has a history of its own. I see three moments in building-up my current views on ethnomathematics.

In ICME 3, in Karlsruhe, in 1976, I proposed, in the section "Why Teach Mathematics", a broader view of why mathematics should be in school curricula. I proposed that this should include a discussion of the nature of mathematical knowledge, with special attention given History, Philosophy and cognition in the broad sense, not specifically dealing with the philosophy and the history of mathematics and the theories about learning mathematics. Why this? Because much of the history and philosophy of mathematics, as well as the mathematical cognition are redundant and biased. I see great advances in the History of Mathematics, in the sense that we have a much better grasp of why and how people have acquired mathematical notions and norms, and we are only at the beginning of this voyage of discovery. This is mainly because scientific explanations of how minds work have got incredibly better in the last few decades.

Thus, much to the maze of my colleagues, I proposed an unusual bibliography, including Nietsche, Spengler, Mead and other authors apparently unrelated to the theme. I also insisted on the fact that there are other ways of doing mathematics, proper to different cultures. And I found the reference to the now classic book by Claudia Zaslavsky, Africa Counts, very important to my paper.

The expanded version of the paper became the first of a set of small books which, together, represent my basic thinking about Science and Mathematics Education. The first one was a booklet, produced in the university, and widely distributed: Overall Goals and Objectives for Mathematical Education, UNICAMP, Campinas, 1976.

In the early 80’s, these ideas developed into a more ambitious research program, going beyond Science and Mathematics Education. My overall motif is to understand the human condition, based on the triad individual⇔other(s) ⇔reality, and the struggle of humans for survival and transcendence. These ideas emerged as a result of my reflections about Science and Mathematics and Education. If we look for a transcultural perception of the nature and the history of mathematical knowledge, we need to adopt a transdisciplinarian view. Transcultural and transdisciplinarian knowledge characterizes my approach to knowledge. But let me return to the early moments of my involvement with ethnomathematics.

In the Congress of Karlsruhe, it did not occur to me that ethnomathematics would be a good name for the mathematics of other cultural environments. I was familiar with ethnobotany, ethnomusicology, ethnopsychiatry, and other ethno-knowledges. It is easy to understand why the prefix "ethno" did not come to my mind in the first stages of my reflections about this. The work of botanists, musicians, psychiatrists and others are mainly an ethnographical approach. No doubt, this is important also in mathematics. We have to learn the mathematics of other cultural environments. Indeed, this has been very interesting and helpful in classrooms. But it can easily get into a folkloristic view of how other cultures do their counting, measuring, etc. with a total lack of respect for the complexity of their cultural specificity. I still see the reliance on curiosities and anecdotes on how other cultures deal with numbers and figures as an equivocated way of doing ethnomathematics. Unless properly situated in an ample cultural scenario, this approach to ethnomathematics is prone to reinforce eurocentrism.

Much of the motivation behind the views expressed in the Karlsruhe paper derived from three major experiences: my work with minorities while graduate chairman of the Department of Mathematics at State University of New York/SUNY, at Buffalo, from 1968 through 1971; my work in Mali, West Africa, in the doctoral program at the Centre Pédagogique Supérieur de Bamako, sponsored by UNESCO, from 1970 through 1978; and by coordinating the interdisciplinarian Multinational Master’s Degree Program of Science and Mathematics Education, at the Universidade Estadual de Campinas/UNICAMP, sponsored by the Organization of the American States and the Brazilian Ministry of Education, covering all the countries of Latin America, from 1975 through 1982.

The three projects gave me an unique opportunity of involvement with socio-cultural environments, respectively in the Afro-American community in the USA, in Africa, and in Latin America and the Caribbean. I was strongly motivated to understand how knowledge, particularly mathematical knowledge, was generated, intellectually and socially organized, and diffused in these different socio-cultural environments. Thus grew a research program, basically interdisciplinarian, relying mainly on studies of the mind and cognition, anthropology, linguistics, history, epistemology, politics, education. In the late-seventies I began to refer to ethnomathematics as the underlying framework behind architecture, calendrical systems, measurement, particularly weighing, in the traditional cultures of sub-Sahara Africa, and in the Andean and the Amazonian native populations.

My interest was -– and continues to be –- to understand two basic points: 1. the nature of knowledge, in particular of mathematical knowledge, in different cultures; 2. the reasons why a relatively small periphery of the Eurasian continent, which sprang out of Mediterranean civilizations, became a dominant force in the entire planet. [2]

These questions were essential in designing my 1976 paper and I relied on the recognition that mathematics is part of broad cultural contexts, having everything to do with religion, the arts, economics, politics and the organization of societies. At that time, I had not yet fully formulated the research program focused on the generation, the intellectual and social organization, and the diffusion of knowledge, which became the backbone of the Ethnomathematics Program.

In February 1978, during the Annual Meeting of the American Association for the Advancement of Science, Rayna Green organized a section on "Native American Science". In my paper, which was never published, I used the words ethnoscience and ethnomathematics to designate scientific and the mathematical knowledge and practice of the native American cultures. These words were mainly focusing extant practices of peoples marginalized by the colonial process.

I was driving into questioning the reasons for investing human and material resources in mathematics research in poor countries. Can we conciliate urgent social needs with the cost of research in mathematics? This is a question which needs a broader look into the history of mathematics. Somewhat motivated by this question, UNESCO organized, in February 1978, a meeting in Khartoum entitled "Developing Mathematics in Third World Countries", to some extent exploring the ideas behind the Project of the Centre Pédagogique Supérieur of Bamako. The Proceedings, published as a book, pointed to the importance of choosing appropriate lines of research which should call for the concentrated efforts with focus on development. [3] These were my earlier incursions in the social and political dimensions of research mathematics.

This was a fruitful year for the development of ethnomathematics. Bernhelm Booss and Mogens Niss organized a meeting on "Mathematics and the Real World", in the new Roskilde University, in Denmark, which was soon after followed by a memorable satellite conference on "Mathematics and Society", preceding the International Congress of Mathematicians/ICM 78, in Helsinki. Many requests to the IPC of the congress to include this conference as part of the program met with an incredulous reaction. The argument was, as expected, that this theme had nothing to do with mathematics as such. But the concession of a space for the satellite conference and the large number of attendants were big achievements. This was important in drawing the interest of the mathematical establishment to broader societal issues.

All these events contributed to building-up my views on the nature of mathematical knowledge. Why did our species developed such a thing as mathematics? And how does mathematics evolve?

My interest in the history of the evolution of academic mathematics in Europe led into examining the cultural dynamics in the development of Mediterranean civilizations and in the expansion of Christianity. Particularly interesting is the role of paganism in this process.

In the visits to Denmark and Finland, I got very much impressed by the Pagan cultures of Scandinavia. The Vikings, with their ship-building, their navigation instruments and their symbols were, and still are, intriguing. I was impressed by the long length of the days and by the fact that in the Northern part of the region, which I visited later, the days lasted six months!

How is the cosmovision of these cultures? Considering that space and time, as tied together in the cosmic scenario, had such an important role in the development of mathematical ideas, how could these people make sense of their own experiences and of the cultural interactions during the almost thousand years of conquest of Southern Europe? The conqueror absorbed much of the culture of the conquered. This was a dual situation of what I had seen in the cultural dynamics which occurred in the Americas and in Africa during the conquest and colonial era.

The crux is to understand how individuals and cultural groups respond to the drives for survival [proper of every living structure] and transcendence [specific of human beings], which are intrinsic of human nature. The response to these drives are ways, styles and techniques of doing, and the search of explanations, understanding and learning. Systems of knowledge are the complexes of the responses to the drives for survival and transcendence. Each culture has its own response to these drives.

While in Finland, it was a good exercise to ask how would the Finns express their ways of satisfying their needs of explaining, understanding, learning, intrinsic to the human drives toward survival and transcendence. I like to play with dictionaries. After all, as the philosopher Humberto Maturana says, language organizes, synthesizes and relates our behavior. When going to different countries, I usually buy a small dictionary and use much of my leisure time in browsing through it. How would the Finns refer to the cultural bases of their techniques of explaining? A world that Finns might use is alusta-sivistyksellinen-tapas-selitys! Or, making it a little less frightening, alustapasivistykselitys. Free time in congresses are always very stimulating for fantasying! This was the fantasy of someone that likes to play with dictionaries, with an underlying curiosity of how different people of different traditions respond to the basic impulse of explaining facts and phenomena.

I realized that the abusive fantasy of playing with the Finnish language would be less shocking if I were playing with Greek roots. Why not ethno-techné-mathemata or, as it would sound much better, ethno-mathema-tics.

Each culture developed its own ways, styles and techniques of doing, and responses to the search of explanations, understanding and learning. These are the systems of knowledge. All these systems use inference, quantification, comparison, classification, representation, measuring. Of course, Western mathematics is such a system of knowledge, as a broad view of its history shows. But other cultures developed, also, other systems of knowledge with the same aims. That is, other "mathematics", using different ways of inferring, quantifying, comparing, classifying, representing, measuring. All these systems of knowledge might well be called ethnomathematics. They are "mathematics" of different natural and cultural environments, all motivated by the drives for survival and transcendence. Mathematics, basically, respond both to "How" and "Why".

I had already used ethnomathematics in the narrower sense of representing real facts, counting, measuring, classifying, comparing, inferring, etc. in different cultural environments. But to understand the nature of ethnomathematics was still a challenge for me.

The breadth of my research program can not be appreciated unless we move beyond Mathematics and its Education. Indeed, I believe that much of the troubles and the declining presence of mathematics in educational systems all over the world, are the result of looking into Mathematics Education as a limited area of research. It is not merely a technical problem, dealt with improving teacher training, curricula and methodologies. The problem with mathematics education is with mathematics and the essence of its relation to human and social behavior.

I began to look to the history of Western Mathematics in the broader sense of responses to the needs of survival and transcendence, mainly the practical and mystic motivations in its development. I considered exploring the idea of systems of knowledge which respond to the drives for survival and transcendence the right track to be followed. It was clear to me, examining several different cultures, that the word ethno-mathema-tics, the result of an etymological playing, carried in it the synthesis of my research program: the generation, intellectual and social organization, and diffusion of knowledge.

This great opportunity to put together these views and present my ideas to the community of mathematics educators came with the invitation to deliver the opening plenary lecture in ICME 5, in Adelaide, Australia. In the lecture I showed many mathematical practices [ethnomathematics] from different cultures to back some theoretical reflections on how mathematical knowledge is generated, organized and diffused. This was an early version of the theoretical framework for ethnomathematics. These ideas and further developments were published in a series of five small books: Socio-cultural bases for Mathematics education, UNICAMP, Campinas, 1985; Da Realidade à Ação. Reflexões sobre Educação (e) Matemática, Summus Editorial, São Paulo, 1986; ETNOMATEMÁTICA. A Arte ou Técnica de Explicar e Conhecer, Editora Ática, São Paulo, 1990 [translated as: ETHNOMATHEMATICS: The Art or Technique of Explaining and Knowing, transl. by Patrick B. Scott, ISGEm/NMSU, Las Cruces, 1998]; Several Dimensions of Science Education. A Latin American Perspective, CIDE/REDUC, Santiago, 1991. Together, they represent theory and practice of ethnomathematics.

In 1979, I was surprised by an invitation to attend the 29^th Annual Pugwash Conference on Science and World Affairs, in Mexico City. The organization, founded in 1955, by Albert Einstein, Bertrand Russell and several other nobelists, is mainly concerned with the responsibility of scientists with the implications of scientific and technological development for the human condition. The priority is Peace and Disarmament. Soon, it became clear to me that knowledge systems, particularly mathematical knowledge, and political structure and war, are closely related. [4]

My interest in the nature, in the history and in the implications, or the use, of scientific, in particular mathematical knowledge, became part of a single concern. This requires to look into the cognitive sciences, anthropology, history, sociology, politics and, naturally, education, in an integrated way. These ideas were put together when the idea of transdisciplinarity took form, after an important meeting organized by UNESCO in Venice, in 1986, as part of its Forum on Science and Culture. The Symposium was a challenge to C.P. Snow’s concept of two cultures. Scientists got together with philosophers and artists. The participants came from different parts of the World and distinct traditions. Much of these views are in a book which I published in 1997. [5]

The research program which I call Ethnomathematics goes much beyong mathematical knowledge. It is a culmination of different lines of thought, built up in the five moments of my academic life briefly sketched above. The program asks for understanding and explaining the generation, intellectual and social organization, and diffusion of knowledge. I understand knowledge in the broad sense of the ways human beings deal, in their specific cultural and natural environment, with their basic needs of survival and transcendence.

This very general and ambitious program has many subsidiary programs. One of the many subsidiary research programs refers to mathematical knowledge. Thus, ethnomathematics emerged as a research program on the History and Epistemology of Mathematics with Pedagogical Implications. [6]

The pedagogical implications result from a broad understanding of diffusion of knowledge, which is key for education. While the cycle of knowledge must be understood in its integrality, education has been, in every moment of history, regarded with strong emphases in the transmission of knowledge. Education has been conducted following strategies, which is called curriculum. History of education reveals that the curriculum, since the Roman trivium, is subordinated to the prevailing model and goals of society. The current curriculum, which continues to give priority for reading, writing and calculations, is insufficient both for the production models and for increasing demand for a new socio-political model, which rejects inequity, arrogance and bigotry. In the present, we are witnessing un-precedent possibilities in the technologies of information and communication. The curriculum must emphasize the acquisition of the instruments which allow for a dignified and productive life, which are the instruments to communicate, to analyze situations and to use, adequately, the technology available, instead of the pure transmission of information and abilites. [7]

Another subsidiary research program refers, specifically, to the cultural dynamics of knowledge transfer, particularly of mathematical knowledge. This affects particularly the History of Mathematics of countries that were parts of the colonial process. [8] In other words, it leads to the study of the cultural dynamics of colonialism. My views are synthesized in what I call the "basin metaphor". [9] This view of history reveals how mathematics was used as an instrument of the conqueror and the colonizer. Indeed, mathematics is the most important instrument of those in power. Politics is essential in discussions of education and the political dimension of the Program Ethnomathematics is one of its most intense components.

Independence did not redeem the effects of the colonial era. About 80% of the human population, living both in the so-called developed and undeveloped countries, live in poverty, in fear and in undignified moral and material conditions. Of course, Mathematics is not responsible for this. But mathematics is the instrument responsible for the status quo: economic mal-distribution, irresponsible production. At the same time, there is agreement that mathematics is the instrument which is needed to revert this situation. [10]

how to incorporate ethics in mathematical practice, both in teaching and research?

Movimento

Etnopedagogia

 1 Concepção

Célestin Freinet 7

 2 Pensamento

Paulo Freire 8

 3 Estruturação

Ubiratan D`Ambrosio 9

 4 Paradigmas

Edgar Morin 10

 5 Vivências

Pessoas & Livros 11

 6 Processo

E-pombo @ Correio 12

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