Volume 12 Number 2 May 1997

Minutes of the ISGEm Meeting

Minneapolis, MN, USA, April 19, 1997

Ubi D'Ambrosio calls session to order, and turns meeting over to Joanna Masingila, chair of the ad-hoc organizing committee.

Joanna asks for officers' reports:

Jim Barta give's treasury report -- we are well in the black.

Larry Shirley gives delegate report -- Our resolution for an NCTM committee on international affairs was defeated last year, but we may re-present it again next year depending on the NCTM's sub-committee discussion.

Rick Scott gives newsletter report -- newsletter is now available on-line at web.nmsu.edu/~pscott/isgem.htm

Ron Eglash gives web site report -- location is

www.cohums.ohio-state.edu/comp/isgem.htm

Luis Ortiz-Franco gives constitutional report -- last year's amendment allows for the creation of regional chapters.

Joanna asks for New Business:

A suggestion to put a list of members' activities on the web is discussed and found agreeable by all.

Activity reports are provided by Norma Presmeg, Gelsa Knijnik, María Luisa Oliveras, Marilyn Frankenstein, Amelie Presscot, Rick Silverman, David Mosimege, Bob London, and Joanna Masingila.

Meeting is adjourned.

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Minutes of North American Chapter of ISGEm

Minneapolis, MN, USA, April 19, 1997

Larry Shirley calls the meeting to order.

Nominations for the officers of the North American chapter of ISGEM from the nominating committee are presented and other nominations are solicited, and the following are voted in unopposed:

Joanna Masingila -- President

Rick Silverman -- VP

Ron Eglash -- secretary

Jim Barta -- treasurer

The ISGEm session for NCTM 98 is discussed. Larry Shirley agrees to work to get our business meeting on the program and reserve a room for us. A Native American emphasis is suggested, and Jim Barta agrees to help organize it. A brainstorming session on membership ensues.

The meeting is adjourned.

Prepared by Ron Eglash, The Ohio State University

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Report on Ethnomathematics Research

Joanna O. Masingila, Syracuse University

This column reports on current research in the area of ethnomathematics. If you know of researchers doing ethnomathematics research, please send Joanna Masingila this information: 215 Carnegie, Syracuse, NY 13244-1150 USA (jomasing@sued.syr.edu).

Mogege David Mosimege, from the University of the North in South Africa, is investigating the relationships between cultural games and the teaching and learning of mathematics. The main aim of his current work is to look at various cultural and traditional games that are found in different cultural settings, with a view toward making use of these games in the mathematics classroom. Three games that he has examined so far are "String Figures," "Morabaraba" and "Moruba." Mogege is investigating how these games are played and the strategies that players develop, and then looking at how these games might be used to teach children mathematical ideas.

Madalena Santos, from Faculdade de Ciencias da Universidade de Lisboa in Portugal, has been studying how students' mathematical knowledge is structured and developed through the interaction with their everyday activities in the context of the mathematics classroom. She has been working with a group of eighth grade students. Her work is grounded in Saxe's research framework, and she is focusing on the students' appropriation process of mathematical artifacts. She is examining the role of (a) social interactions with peers and the teacher, (b) the structure of practice, and (c) the students' individual goals while examining how students accomplish mathematical goals.

Ines María Gómez Chacón, from the Instituto de Estudios Pedagógicos Somosaguas in Spain, has been investigating connections between affective issues and cultural influences in the learning of mathematics for students in professional training programs. She and her colleagues have examined the different approaches to learning in the classroom and in the cabinet making workshop, and are studying students' thinking strategies both in the classroom and in the cabinet making workshop.

Frouke Buikema Draisma and Jan Draisma, from Universidade

Pedagógica, Delegação de Beira in Mozambique, have been examining the understandings that children in Mozambique have of operations such as subtraction and multiplication. Some of their work is focusing on the confrontation between subtraction verbalized in Portuguese and subtraction verbalized in Mozambican languages. They have already found that some Mozambican students' understanding of concepts of multiplication is influenced by their mother tongues, and their mental addition and subtraction habits are influenced by Bantu numeration systems.

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ISGEm at NCTM

Larry Shirley

ISGEm's NCTM Delegate

ISGEm continues to play a role at the level of the affiliated groups of the National Council of Teachers of Mathematics. We are finally (we hope) straightening our communication links with the NCTM Headquarters in Reston, Virginia.

For the second year, we offered a resolution to the Delegate Assembly of the representatives of all the affiliated groups. Most of the groups never submit resolutions, so that alone demonstrates our active role. As reported earlier, our resolution asked NCTM to reconstitute a committee on international affairs. We argued that there is much happening in math education outside of the USA. Mathematics educators in the USA have much to learn from increased contacts with the rest of the world. Also, such a committee would be able to assist international affiliates of NCTM such as ISGEm, the European Council of Teachers of Mathematics, the History and Pedagogy of Math group, and others.

The bad news was that our resolution did not come to the floor of the Assembly for consideration, but the good news is that it has been passed to the newly formed External Affairs Committee. There, a subcommittee including President Gail Burrill will c onsider it. Also, Jerry Becker, a member of the NCTM Board and an enthusiastic supporter of our resolution will be helping us at the level of the Board. We hope there may be action as soon as June 1997. Becker has suggested that if we still do not get favorable action, we should resubmit the resolution next year.

You, as ISGEm members, can help! Speak or write to any NCTM Board members that you may know or who are from your area. Point out the values of international contacts. Even the recent news of TIMSS results reflect our connections to the rest of the world. NCTM has formalized links with about a dozen other national mathematics education groups and we participate in international meetings such as the quadrennial International Congress on Mathematical Education. However, there is little coordination from NCTM. These are some of the areas an International Affairs Committee could handle.

Meanwhile, we found that much of the affiliated groups data base had out of date information on ISGEm and its officers. It is now updated and, as the NCTM Representative, I will try to keep Reston aware of our group, and keep you aware of news from NCTM.

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Ethnomathematics Production in Brazil

Gelsa Knijnik

Universidade do Vale do Rio dos Sinos - Brazil

This paper on Ethnomathematics production in Brazil was presented as part of a special session at the 1997 NCTM Annual Meeting in Minneapolis entitled A Bridge across the Americas.

The first name to be remembered when speaking of Brazilian Ethnomathematics is that of Ubiratan D'Ambrosio (1987, 1990, 1991, 1993, 1996, 1997), considered the most important theoretician in this field, both for the quality and range of his academic production, and for the role of leadership and dissemination of the ideas involved in Ethnomathematics, ideas which seek to establish a close relationship between mathematics, culture and society. I would like to be able to render him homage here, for all that we have learned from him regarding what it means to be an educator committed to his time.

The second name to be mentioned is that of Eduardo Sebastiani Ferreira (1987, 1991, 1993, 1994), a pioneer in field research in the Brazilian Ethnomathematics movement, who performed and guided investigations. His empirical research was/is developed in Native Brazilian communities in the central-west and northern regions of Brazil. Sebastiani Ferreira, based on his activities with inservice and preservice Native-Brazilian teachers, has contributed to furthering the theory of Native Brazilian education, specially focusing on the connections between what he calls "white man's mathematics" and "mother mathematics", an expression which he uses (as a homology to mother tongue) to "express the ethno knowledge of the child (...) [ which s/he] brings to school" (Ferreira, 1994, p.6).

Among the studies of Brazilian educators directly connected to Ethnomathematics, we should also mention those of Marcelo Borba (1990, 1992, 1993) with children of a slum, in the state of São Paulo; Nelson Carvalho (1991), with the Rikbaktsa tribe, who live in the central-west region of the country; Samuel Bello (1995), with the Guarani-Kaiowa tribe who live in the same region; Sergio Nobre (1989) about the "Animal Lottery, the most popular lottery in Brazil; Geraldo Pompeu (1992) about the influences on the changing of teachers' attitudes in a project he developed which attempted to introduce Ethnomathematics in the school curriculum; Adriana Leite about the interrelations between children's games and the learning of Mathematics; Sonia Clareto about children's notion of space in a small community of fishermen in the state of São Paulo; and the doctoral dissertation of Mariana Ferreira called From Human Origins to the Conquest of Writing: a Study of Indigenous Peoples and Schooling in Brazil, in which she discusses the construction of knowledge in a tribe of the North Region of the country.

From an ethonomathematical perspective there are Mathematics Educators developing work with one of the most important Brazilian social movements: Movimento dos Trabalhadores Rurais Sem-Terra, in English, the Landless People's Movement. It is a national organization, spread throughout 23 of 27 states of the country, involving about seven hundred thousand peasants who strive to achieve Land reform and social changes in a country with very deep social inequalities. Currently there are three ethnomathematics investigators working with the Landless People's Movement. Gelsa Knijnik (1993a, 1993b, 1994, 1996, 1997) is developing work in different projects of the Movement, such as Youth and Adult Numeracy and inservice Teacher Education. Her work is based in what she calls an Ethomathematics Approach: the investigation of the traditions, practices and mathematical concepts of a subordinated social group and the pedagogical work which was developed in order for the group to be able to interpret and decode its knowledge; to acquire the knowledge produced by academic Mathematics; and to establish comparisons between its knowledge and academic knowledge, thus being able to analyze the power relations involved in the use of both these kinds of knowledge (Knijnik, 1997).

The two other researchers working with the Landless People's Movement are Alexandrina Monteiro and Helena Doria. Monteiro, Sebastiani's doctoral student, is doing her research based on her work with a group of adults in a Numeracy Project. She is investigating their social practices which can be incorporated into the pedagogical process. Helena Doria, under Knijnik's supervision, is doing her Master's research in a settlement in the southernmost state of Brazil.

What we can say is that Ethnomathematics field in Brazil is linked with sectors of Brazilian society who have been systematically excluded from knowledge. It is in this sense that I consider the importance of Ethnomathematical thought, which problematizes scientism, the apparent neutrality of academic Mathematics, and brings to the scene "other" Mathematics, usually not mentioned at school, as a cultural production of non-hegemonic groups. It is in this sense that I consider the Ethnomathematics perspective particularly exemplary since, in the field of the curriculum, it counteracts the exclusion of the many and the citizenship of the few. In the concreteness of daily school work, it mitigates what sociologist Boaventura dos Santos called epistemicide - the destruction of the knowledge of a given social group - whose most radical form is the genocide, in which not only minds and hearts but also people's bodies are eliminated.

This is not, however, a mere attitude of "benevolence" toward the excluded. We educators, who from the ethical standpoint, are co-responsible for the great massacres which have been and are still being committed by mankind, are also participants in small daily massacres such as those practiced in our classrooms, on our everyday school life, when we exterminate the other knowledges which are not those of the dominant culture, when we pretend that those knowledges did not or do not even exist, and with our authorized voice as teachers value only erudite knowledge of white, male, middle class, urban western culture, not because it is in itself superior from the epistemological standpoint, but because it is the one practiced by the groups which are legitimated, in our society, as those which can/should/are able to produce science.

It is in this sense that I think that Ethnomathematics is contributing to a Brazilian society with more social justice.

References

Bello, Samuel Edmundo Lopez. Educação Matemática Indígena: Um Estudo Etnomatemático com os Indios Guarani-kaiowa do Mato Grosso do Sul. Curitiba: Universidade Federal do Paraná, 1995. Masters thesis.

Borba, Marcelo. Ethnomathematics and education. For the Learning of Mathematics, Vancouver, v.10, n.1, p. 39-43,1990.

___. Teaching mathematics: ethnomathematics, the voice of sociocultural groups. The Clearing House, v.65, n.3, p 134 - 135, 1992.

___. Etnomatemática e a cultura da sala de aula. A Educação Matemática em revista, Blumenau, v.1, n.1, p.43-58, 1993.

Carvalho, Nelson Luis Cardoso. Etnomatemática: O Conhecimento Matemático que Se Construi na Resistencia Cultural. Campinas: UNICAMP, 1991. Masters thesis.

Clareto, Sonia Maria. A Criança e seus Dois Mundos: A Representação do Mundo em Crianças de uma Comunidade Caiçara. Universidade Estadual Paulista. Doctoral Dissertation.

D'Ambrosio, Ubiratan. Reflections on Ethnomathematics. ISGEm Newsletter, v.3, n.1, p. 3-5, Sept. 1987.

___. Etnomatemática: Arte ou Técnica de Explicar ou Conhecer. São Paulo: Attica. 1990

___. Ethnomathematics and its Place in the History and Pedagogy of Mathematics. In: Harris, Mary (Ed.). Schools, Mathematics and Work. Hampshire: The Falmer Press, 1991. p.15-25.

___.Etnomatemática: um programa. Educação Matemática em Revista, Blumenau. v. 1, n.1, 1993.

___. Educação Matemática: Da Teoria a Prática. Campinas: Papyrus. 1996

___. Preface. In: Powell, Arthur & Frankenstein, Marilyn. Ethnomathematics: Challenging Eurocentrism in Mathematics Education. New York: Suny Press. 1997

Ferreira, Eduardo Sebastiani. The genetic principle and the ethnomathematics. In: Mathematics, Education and Society. Paris: UNESCO, 1989. p. 110-111. (Document Series 35).

___. Etnomatemática. In: Memorias del Primer Congreso Iberoamericano de Educación Matemática, Paris: UNESCO, 1991 p. 160-163. (Document Series 42).

___. Cidadania e Educação Matemática. A Educação Matemática em Revista, Blumenau, v.1, n.1, p. 12-18, 1993.

___. A importancia do conhecimento etnomatemático indígena na escola dos não-indios. Campinas: IMECC/UNICAMP. 1994. Texto digitado.

Ferreira, Mariana. . Da Origem dos Homens a Conquista da Escrita: Um Estudo sobre Povos Indigenas e Educação no Brasil. Universidade de São Paulo. Doctoral Dissertation.

Knijnik, Gelsa. The mathematics teaching laboratory: its repercussions in the preparation of the prospective secondary teachers. In: Proceedings of the International Congress of Mathematical Education, 6., Budapest, 1988. p. 160-164.

___. Culture, education and mathematics and the landless of southern Brazil. In: Julie, Ciryl; Angelis, Desi (Ed.) Political Dimensions of Mathematics Education. Johannesburg: Maskew Miller Longman, 1993a.

___. O saber popular e o saber académico na luta pela terra. Educação Matemática em Revista., Blumenau. v. 1, n.1, 1993b.

___ .Según para quien puede cambiar el para qué? Didáctica de las Matemáticas, Barcelona, n. 1, jul. 1994.

___. Exclusão e Resistencia: Educação Matemática e Legitimidade Cultural. Porto Alegre: Artes Médicas. 1996.

___. An ethnomathematical approach in mathematical education: a matter of political power. For the Learning of Mathematics, Vancouver, v.13, n.2, June 1993b. Reprinted In: Powell, Arthur & Frankenstein, Marilyn. Ethnomathematics: Challenging Eurocentrism in Mathematics Education. New York: SUNY Press. 1997.

Leite, Adriana. A Brinadeira é Coisa Seria: Estudos em Torno da Brincadeira, da Aprendizagem e da Matemática. Universidade Estadual. Doctoral Dissertation. 1995.

Nobre, Sergio Roberto. The ethnomathematics of the most popular lottery in Brazil: the "Animal Lottery". Mathematics, Education And Society. Paris: UNESCO, 1989. p. 175-177. (Document Series, 35)

Pompeu Jr, Geraldo. Bringing Ethnomathematics into the School Curriculum. Cambridge University. Doctoral Dissertation. 1992..

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Mathematics and Beadwork:

Notes from a Conversation with a Sioux Artist

Jim Barta and Rick Silverman

Mathematics occurs naturally in peoples' lives, as is evident from in-depth work, such as that by Masingila (1994) with carpet laying, Millroy (1992) on carpentry, Saxe (1988) with child candy sellers, and Nunes (1992) in a substantive overview of ethnomathematics. The literature also contains more informal reports. Silverman, for example, presented vignettes on mathematics that arose with youngsters on rollerblades (1994) and in a first grade child's curiosities about chapter books (Silverman, Fertig, and Loper, 1996), and Barta (1996) shared findings of mathematics gleaned from interviews with Seminoles. A. J. Bishop's (1988) categorization system unifies such examples into six area of human activity: counting, measuring, explaining, locating, playing, and designing.

Knowledge of this background puts one on alert for mathematics that arises in chance encounters in diverse settings, as when we happened to meet Cynthia Storer-Plays during our visit to the Institute of American Indian Arts in Santa Fe, New Mexico, in November, 1996. A Native American woman of Sioux extraction, she also goes by the name Buffalo Calf, named by a medicine man in the tradition of her people. She uses that name today when she creates her art.

The Sioux are also known by the name "Deneoh," which means "the people." They conceive the world to be like a large hoop, one that has, unfortunately, been broken by governed society. The Sioux, Cynthia commented, have a mission: to mend the broken hoop. Cynthia told us that because the medicine man had named her Buffalo Calf that she did have authority, conferred by virtue of a special presence, we inferred, to speak for "all Sioux..."

Cynthia was beading a tube for hair braids. She was encircling the tube with a thread on which she was stringing colored beads. The tube was covered first with a soft piece of leather. She said that suede would work, too. She looped the beads around the tube. Successive layers nested each new bead into the cavity between two beads she had previously placed one level back. The nesting made an isometric pattern like the cups in a Chinese checkers board.

When asked if there was any mathematical dimension to her activity, Cynthia replied that the beading connected her with time. "Sitting and beading produces an artifact that takes time to make", she stated. "Creating takes time; hands are at work to produce this craft. It takes time to string the beads on the thread", she added. The process of attaching beads to the circular cylinder, or any other object, is "time creating." Cynthia remarked that the beads indicate time intervals; that change in color, texture, and size of the beads indicate time intervals, not the beads specifically. It is the variation in the attributes from one bead to the next that evokes a sense of change in duration of time.

Cynthia told us that she was holding the design for this braid tube in her mind. The design has a basis in reality in that it may physically represent an image that has collective meaning but the design also evolves as she beads. (The design on the tube was to be a cross. Among the Sioux a cross stands for tobacco. Sioux folklore tells that the Great Spirit likes the taste of tobacco. Hence, the symbol of the cross is sacred.) She didn't really know what the final product would look like until she was finished.

She stated that the design may come from a spiritual center that she has the privilege of accessing because of blessings and namings she has received.

She showed us that she was working on another project as well, a hair band. For the hair band Cynthia had drawn a sketch of the design on a piece of leather. Cynthia was beading directly onto the leather strip that contained the sketch.

While the conversation was not a long one and because we were just visiting, this was our only opportunity to speak with her face to face. We have, however, been in touch with Cynthia by mail and edited our draft based on clarifications that she presented to us. We were both quite excited over this conversation and on driving home, we discussed the apparent mathematical implications.

The beading itself certainly fell within Bishop's (1988) category of designing and building. The attributes of the beads themselves denote classifications according to color, composition, size, texture, and the like. The beads go onto the string in linear fashion in such a way as to form a design in two dimensions through the coiling process by which the string of beads wraps around the tube. We noted these mathematics connotations by reflecting on Cynthia's activity through the lens of our own Western culture.

There is, however, evidence of meaning to Cynthia, to Buffalo Calf, as a member of the Sioux people. Her relating beading to construction in time and to changes in intervals of time, for example, are features of this activity that she derived based on her heritage, a connection we in the Western tradition did not draw. The relationship with time and nature is a somewhat spiritual, mystical one inherent to the culture of her people.

Here was a woman creating a clothing accessory, something very beautiful and useful, as well as personally meaningful to her as member of her cultural group. Mathematical application and thinking are embedded holistically in the beading process.

In this episode logical and spiritual qualities transform ideas into physical reality for adornment and for linkage with a transcendent realm in the mind of someone attuned to it as Cynthia is. She was a creator in an artistic and spiritual process as well as an active participant in selecting and using mathematics.

References

Barta, J. (1996). Mathematical thought and application in traditional Seminole culture. Newsletter of the International Study Group on Ethnomathematics, 11, 2, 4-5.

Bishop, A. J. (1988). Mathematical Enculturation: a Cultural Perspective on Mathematics Education. Boston: Kluwer Academic Publications.

Masingila, J. O. (1994). Mathematics practice in carpet laying. Anthropology & Education Quarterly. 25, 4, 430-462.

Millroy, W. L. (1992). An ethnographic study of the mathematical ideas of a group of carpenters. Journal for Research in Mathematics Education Monograph, 5. Reston, VA: National Council of Teachers of Mathematics.

Nunes, T. (1992). Ethnomathematics and everyday cognition. In Douglas A. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning, pp. 557-574, New York: Macmillan Publishing Company.

Saxe, G. B. (1988). Candy selling and math learning. Educational Research. 17, 6, 14-21.

Silverman, R. (F. L.) (1994). Mathematics and rollerblades. Newsletter of the International Study Group on Ethnomathematics, 10, 1, 1.

Silverman, F. L.; Fertig, G.; & Loper, C. (1996), Math: where you least expect it! Colorado Mathematics Teacher, 29, 3, 10-11.

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The World Counts

During spring semester, The Program in Mathematics Education in the Department of Scientific Foundations at Teachers College of Columbia University sponsored a symposium on the instructional implications of ethnomathematics. On each of four Saturdays, a principal speaker examined the field from a different point of view: Ubiratan D'Ambrosio reviewed its definition, role, tenets, and functions; Victor Katz discussed ways in which the history of mathematics informs ethnomathematics; Marcia Ascher attended to the mathematics of ethnomathematics; illustrating with his experiences in Mozambique, Paulus Gerdes spoke on ethnomathematics in practice.

Each Saturday there were complementary workshops, wide ranging in topics and issues and led by: Claudia Zaslavsky (African examples); Marilyn Frankenstein (political dimensions); Frederick Uy (applications in geometry); Daniel Ness (examples from music); Margit Echols (quilting); Gelsa Knijnik (contemporary Brazilian issues); Arthur Powell (language issues); Gloria Gilmer (links to students); Ron Eglash (applications in fractal geometry); and Rick Scott (Native American examples). In formal and informal discussions the entire participant group explored linkages into the classroom and into instructional programs. About 50 speakers and students gathered for each day's sessions. Those registered for credit subsequently prepared short papers on topics relating their classroom situation or professional experience to that of the symposium. There is no plan for preparing a formal report or proceedings from the symposium. However, consistent positive evaluation by registered students indicates the usefulness of such events. Joel Schneider (joels@columbia.edu) organized the symposium and welcomes any questions.

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Have You Seen

"Have You Seen" is a regular feature of the ISGEm Newsletter in which works related to Ethnomathematics can be reviewed. We encourage all those interested to contribute to this column.

Wood, Leign. (1997) Aboriginal and Other Counting Systems, in Petocz, P., et. al. Introductory Mathematics, Melbourne, Australia: Nelson, p. 163-166.

This section from a book coauthored by Leigh Wood highlights the base five numeration system of the Gomileroi people. It also shows the connection in their language between number words and body parts.

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Eglash, R. "Bamana Sand Divination: recursion in ethnomathematics." American Anthropologist, March 1997.

Eglash, R. "When math worlds collide: intention and invention in ethnomathematics." Science, Technology and Human Values , vol 22, no 1, pp. 79-97, Winter 1997.

Eglash, R. "The African heritage of Benjamin Banneker." Social Studies of Science, April 1997.

Eglash, R. "African influences in cybernetics." in The Cyborg Handbook,Chris Gray (ed), NY: Routledge 1995a.

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Arthur Powell and Marilyn Frankenstein (eds.). (1997) Ethnomathematics: Challenging Eurocentrism in Mathematics Education, Albany: State University of New York Press.

The following excerpts from Powell and Frankenstein's edited

book should give readers a good flavor of what it has to offer.

Perhaps the most telling point to mention in discussing an educational challenge to Eurocentrism is that

Geographically, Europe does not exist, since it is only a peninsula on the vast Eurasian continent . . . Europe has always been a political and cultural definition . . . Before the 19th century, geographers generally referred to it as "Christensom." When colonialism began to spread Western culture and religion to all corners of the globe, some British and German geographers began to delineate the eastern boundaries of a European continent. What they were actually doing was trying to draw for eastern limits of "western civilization" and the white race (Grossman, 1994, p. 39).

This is an important illustration of how false "facts" become part of our taken-for-granted knowledge of the world. That assumed "knowledge" extends beyond the mere creation of this fictious geographic entity to proclaiming Europe's centrality in the creation of knowledge and the development of "civilization." In the Eurocentric account, Europe (and "Europeanized" areas like the U.S.A.) has always been and currently is the superior Center from which knowledge, creativity, technology, culture, and so forth flow forth to the inferior Periphery, the so-called underdeveloped countries.

Of course, there are significant intellectual challenges to Eurocentrrism. Amin (1989) argues against this account by showing the central contributions of the Arab-Islamic cultures to world knowledge, and by showing how the Eurocentric version of "humanistic universalism . . . negates such universalism. For Eurocentrism has brought with it the destruction of peoples and civilizations who have resisted its spread" (p. 114). Diop (1991) demonstrates that the Greek foundations of European knowledge are themselves founded upon Black Egypti an civilization. Bernal (1987) illustrates how Eurocentrism developed in eighteenth-century Europe as the rationale for various forms of European slavery and imperialism. Blaut (1993) further shows that the successful conquest of the Americas and the spread of European colonialism, actions which were responsible for the selective development of Europe and the underdevelopment of Asia, Africa, and Latin America, "is not to be explained in terms of any internal characterisitcs of Europe, but instead reflects the mundane realities of location" (p. 2).

In spite of this scholarship, the Eurocentric myth persists and influences school curricula, even in a supposedly neutral discipline like mathematics. This book challenges the particular ways in which Eurocentrism premeates mathematics education: that the "academic" 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 "legitimate" 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. This book also attempts to organize the various intellectual currents in ethnomathematics, from an anti-Eurocentric, liberatory perspective. We are critically selective, not just interested, for example, in the mathematics 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 drawings.

This book is organized into sections that focus on specific challenges to Eurocentrism in mathematics education. Each section begins with an extensive introduction, followed by contributions we judge to be path-breaking to the development of that area of ethnomathematics. The first section, "Ethnomathematical knowledge," defines the field and points to other challenges to Eurocentrism. The second section, "Uncovering distorted and hidden history of mathematical knowledge," challenges the historiographic project of Eurocentrism. The third section, "Considering interactions between culture and mathematical knowledge," inquires into who does mathematics and how various practices influence mathematical activity. The fourth section, "Reconsidering what counts as mathematical knowledge," examines non-academic sources of mathematical knowledge. The fifth section, "Ethnomathematical praxis in the curriculum," discusses possibilities for incorporating broader notions of mathematics into traditional and nontraditional educational settings. Finally, section six, "Ethnomathematical research," analyzes research activity in the field and provides an example of a methodological approach that enables political challenges to the politics of silence and poverty.

A theme that emerges throughout these various directions of ethnomathematical thought concerns the need to reconsider the discrete categories common in academic thought. Asante (1987) argues that an underlying theoretical tenet of an Afrocentric perspective is that "oppositional dichotomies in real, every day experience do not exist," (p. 14) For Freire (1970, 1982) this means breaking down the dichotomy between subjectivity and objectivity, between action and reflection, betwween teaching and learning, and between knowledge and its application. For Fasheh (1989) and Adams (1983) this means that thought which is labeled "logic" and thought which is labeled "intuition" continuously and dialectically interact with each other. For D'Ambrosio (1987) this means that the notion that "there is only one underlying logic governing all thought" is too static. For Diop (1991) this means that the interactions between "logic" and "experience" change our definition of "logic" over time (p. 363). For Lave (1988) this means understanding how "activity-in-setting is seamlessly stretched across persons acting." For Diop (1991) this means that the distinctions between "Western," "Eastern," and "African" knowledge distort the human process of creating knowledge which results from interactions among humans and with the world. Throughout this book, we emphasize that underlying all these false dichotomies is the split between practical, everyday knowledge and abstract, theoretical knowledge. Understanding these dialetical interconnections, we believe, leads us to connect mathematics to all other disciplines, and to view mathematical knowledge as one aspect of humans trying to understand and act in the world. We see ethnomathematics as a powerful and insightful vehicle for conceptualizing these connections.

References

Adams III, H.H. (1983). African observers of the universe: The Sirius question. In I. Van Sertima (Ed.) Blacks in science: Ancient and modern (pp. 27-46). New Brunswick, NJ: Transaction.

Amin, S. (189). Eurocentirsm. New York: Montly Review.

Asante, M.K. (1987). The Afrocentric idea. Philadelphia: Temple University.

Bernal, M. (1987). Black Athena: The Afro-asiatic roots of classical civilization. Vol. 1. London: Free Association.

Blaut, J.M. (1993). The Colonizer's model of the world: Geographical diffusionism and Eurocentric history. New York: Guilford.

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ISGEm Home Page on the World Wide Web

Ron Eglash encourages you to browse the ISGEm Home Page. The URL is

http://www.cohums.ohio-state.edu/comp/isgem.htm

Some past issues of this Newsletter can be accessed through the above site or directly at

http://web.nmsu.edu/~pscott/isgem.htm

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ISGEm Distributors

The following individuals print and distribute the ISGEm Newsletter in their region. If you would be willing to distribute the ISGEm Newsletter please contact the Editor.

ARGENTINA, María Victoria Ponza, Fundación Cresinvio, Calle Javier de la Rosa 567, Prov de Santa Fe.

AUSTRALIA, Leigh Wood, PO Box 123, Broadway NSW 2007

BOLIVIA, Enrique Jemio, UNST-P, Casilla 5747, Cochabamba

BRAZIL, Geraldo Pompeu jr, Depto de Matemática, PUCCAMP, sn 112 km, Rodovia SP 340, 13100 Campinas SP

COSTA RICA, Leslie Villalobos, EARTH, Apartado 4 442-1000, San José

FRANCE, Frédéric Métin, IREM, Moulin de la Housse, 51100 Reims

GUADALOUPE, Jean Bichara, IREM Antilles - Guyane, BP 588, 97167 Pointe a Pitre, CEDEX

GUATEMALA, Leonel Morales Aldaña, 13 Avenida 5-43, Guatemala, Zona 2

ITALY, Franco Favilli, Dipartimento di Matematica, Universita di Pisa, 56100 Pisa

MEXICO, Elisa Bonilla, San Jerónimo 750-4, México DF 10200

NEW ZEALAND, Andy Begg, Centre for Science & Math Ed Research, U of Waikato, Private Bag 3105, Hamilton

NIGERIA, Caleb Bolaji, Institute of Education, Ahmadu Bello University, Zaria

NORTHERN IRELAND, School of Psychology, Queens University, Belfast BT7 INN

PERU, Martha Villavicencio, General Varela 598, Depto C, Miraflores, LIMA 18

PORTUGAL, Teresa Vergani, 16 Av. Bombeiros Vol., 2765 Estoril

SOUTH AFRICA, Mogege David Mosimege, University of the North, Private Bag 1106, Sovenga 0727

SPAIN, Maria Oliveras, Depto de Didáctica de Matemáticas, Campus Cartuja, U de Granada, 18071 Granada

UNITED KINGDOM, John Fauvel, Faculty of Math, The Open University, Walton Hall, Milton Keynes MK7 6AA

VENEZUELA, Julio Mosquera, CENAMEC, Arichuna con Cumaco, Edif. SVCN, El Marques - Caracas

ZIMBABWE, David Mtetwa, 14 Gotley Close, Marlborough, Harare

ISGEm Executive Board

Ubi D'Ambrosio, President

Rua Peixoto Gomide 1772 ap. 83

01409-002 São Paulo, SP BRAZIL

ubi@usp.br

Maria Luisa Oliveras Contreras, 1st VP

Depto de Didáctica de las Matemáticas

Campus Cartuja, Universidad de Granada

18071 Granada, SPAIN

oliveras@platon.ugr.es

Jolene Schillinger, 2nd Vice President

New England College BX 52

Henniker, NH 03242 USA

jus@nec2.nec.edu

Abdulcarimo Ismael, 3rd Vice President

Departamento de Matematica

Universidade Pedagogica Nacional

P.O. Box 4040

Maputo, MOZAMBIQUE

Gelsa Knijnik, Secretary

Rua Prof. Andre Puente 414 ap.301

90035-150 Porto Alegre, RS, BRAZIL

gelsa@portoweb.com.br

Jim Barta, Treasurer

Department of Elementary Education

Utah State University

Logan, Utah 84341 USA

Jbarta@cc.usu.edu

Patrick (Rick) Scott, Editor

College of Education

New Mexico State University

Las Cruces, NM 88003 USA

pscott@nmsu.edu

Lawrence Shirley, NCTM Representative

Dept of Mathematics

Towson State U

Towson, MD 21204-7079 USA

E7M2SHI@TOE.TOWSON.EDU

Gloria Gilmer, Past President

Math Tech, Inc.

9155 North 70 Street

Milwaukee, Wl 53223 USA

gilmer@cs.uwp.edu