ISGEm Advisory Board
Gloria Gilmer, ISGEm Chair
2001 West Vliet Street
Milwaukee, WI 53205 USA
Gilbert J. Cuevas
School of Education & Allied Professions
University of Miami
P.O. Box 8065
Coral Gables, FL 33124 USA
Centro de Investigacion del IPN
Apartado Postal 14-740
Mexico, D.F., C.P. 07000 MEXICO
Pro-Rector de Desenvolvimiento U
Universidade Estadual de Campinas
Caixa Postal 1170
13100 Campinas, SP BRASIL
Patrick (Rick) Scott, Editor
College of Education
University of New Mexico
Albuquerque, NM 87131 USA
45 Fairview Ave. #13-I
New York, NY 10040 USA
Minutes of the meeting of the ISGEm in Chicago,
Illinois, on April 7, 1988.
Nineteen people attended the meeting, held in conjunction with the annual meeting of the National Council of Teachers of Mathematics. The following is the order of business as reported by Claudia Zaslavsky:
2. Ubiratan D'Ambrosio discussed the implications of the term "Ethnomathematics."
3. Discussion of spots on the programs of future conferences:
a. NCTM, April 1989, Orlando, Florida. Gloria Gilmer
will request our participation in the research
presession and in the regular session, as well as a
business meeting. Affiliation with NCTM was
discussed, but considered inappropriate, since
afffilates were local chapters. Subsequently, it was
noted that Women in Mathematics Education is an
b. AERA, April 1989. Luis Ortiz--Franco will arrange. c. MAA, January 1989. Gloria Gilmer will arrange.
4. ICME-6 (see on Page 2)
Reprinted with permission in 1992 by International Study Group on Ethnomathematics.
5. Funds. Gloria Gilmer announced $305.23 in the treasury, but very few members have paid current dues. A dues notice will be included in the Newsletter, and possibly an expiration date on the mailing labels. The English version is sent to about 200 people (65% in the U.S.) and the Spanish edition (translated by Elisa Bonilla) to about 200, mainly in Latin America. Altogether, it is mailed to about 45 countries. Ubi sends copies to folks in Brazil.
Suggestion that U.S. members contribute in addition to the $5 annual dues to support members in developing countries.
Discussion of ID number for IRS, complicated procedures.
6. Rick Scott invited contributions.
7. Suggestion that we read the work of Sylvia Scribner, Jean Lave, David Lancy, Geoffrey Saxe.
8. Several new people joined the group. Meeting was adjourned.
August Meeting of ISGEm Advisory Board
Meeting of ISGEm Advisory Board on August 1, 1988. Report prepared by Claudia Zaslavsky.
Present: Gloria, Ubi, Rick and Claudia
I. Four categories of interest:
1. Research project
2. Classroom applications
3. Out-of-school applications
4. Theoretical foundations
5. Other (?)
II. Plan for ICME-7 in Quebec, 1992. Coordinator for each category listed above. Plan a symposium for each category. Finalize in April 1989.
III. A draft of Constitution and By-Laws have been submitted by Luis Ortiz-Franco. Proposed regional subdivisions need discussion. The main purpose for a Constitution is to facilitate tax-exempt status in the U.S.
ISGEm Advisory Meeting in Budapest, Hungary
Minutes of the meeting of the ISGEm in Budapest, Hungary, on August 1, 1988.
Over forty people attended the meeting held in conjunction with the Sixth International Conference on Mathematics Education (ICME). The following is the order of business as reported by Claudia Zaslavsky.
The Chair, Gloria Gilmer, announced the agenda:
I. Thrusts of the ISGEm.
II. Geographical regions.
III. Plans for ICME-7 in Quebec in 1992.
IV. Plans for NCTM Annual Meeting in 1989.
V. Announcements, etc.
I.Thrusts of the ISGEm:
1. Research in culturally diversified projects.
2. Curriculum development projects and classroom applications.
3. Out-of-school applications.
4. Conceptual and theoretical foundations.
a. It was suggested that the thrusts could be used for themes for presentations at ICME-7. One whole day (such as the "Mathematics, Education and Societv" Special Fifth Day in Budapest) or as part of previously established Theme or Topic Groups or create our own Theme or Topic group and/ or some areas in the Special Day Program.
b. There was much discussion concerning the name of the organization. Although it is intended that Ethnomathematics apply to all groups in all countries, the prefix "Ethno" is usually associated with ethnic groups. People do not see the diversity implicit in this area of study. Numerous alternatives were suggested: Meaningful Mathematics, Mathematics (what ISGEm intends) vs. Fossil Mathematics (what is traditionally done in schools), Environmental Mathematics, Ethical Mathematics, Sociomathematics or Sociocultural Mathematics, Real Mathematics, Out-of-School Mathematics or Contextual Mathematics. John Volmink remarked that Ethnomathematics is associated with "primitive cultures" in Western minds. Rick Scott invited the audience to send contributions on any of the above topics,including the name of the organization.
c. Richard Noss suggested that a fifth thrust could be "Sociology and Politics of Mathematics."
II. Possible geographicals were proposed: Africa, Asia (and the Middle East?), Australasia and the Pacific, Europe, Latin America and the Caribbean, USA and Canada. The purpose of the subdivisions would be for general representation, and for translation and distribution of the Newsletter.
Avoid divisions based on "homogeneous regions." Objections were raised to:
a.Lumping the Middle East and Asia together.
b. Having Canada and the USA as a separate region because of their connections with Native Americans and Africa.
c. Africa as a single region (it could be subdivided into North of the Sahara, Francophone Africa, Anglophone Africa).
III. ICME-7. Proposal for four symposia based on the four thrusts listed in I. above. The program should be finalized by April 1989. ISGem expects to affiliate with the International Commission on Mathematics Instruction (ICMI) at the 1992 Congress in Quebec.
IV. NCTM Annual Meeting. Martin Johnson discussed possibilities for the Research Presession.
V. Announcements, etc.
1. Dr. Denes Nagy called for papers, workshop topics, and exhibits for the forthcoming meeting, "Symmetry or Structure: An interdisciplinary Symposium" to be held in Budapest from August 13-19, 1989.
2. Richard Noss introduced a resolution expressing opposition to apartheid in South Africa (which had been drafted by a group of ICME participants). He invited people to show their support by signing the resolution, and by writing comments in appropriate newsletters and to the ICMI members and planners for ICME-7 to urge that a position be taken with regard to the situation in South Africa in contrast to the present "no position." A copy of the resolution had been submitted to the ICME-6 New Sheet, but it was not published. It was noted that the Mathematical Association of America (MAA) has divested of holdings in the Nicholas Fund, which has major investments in South Africa, and that this fact should appear in the ISGEm Newsletter.
3 Katherine Crawford recommended that ICMI should give travel grants to people who will need financial aid to attend ICME-7.
The meeting was adjourned due to expiration of time for use of the room.
Proposal Accepted for NCTM Presession
ISGEm Proposal Accepted for the Research Presession at the Annual Meeting of the National Council of Teachers of Mathematics (NCTM) to be held in Orlando, Florida on April 10-12, 1989.
The ISGEm presentation on "Ethnomathematics: Theoretical Foundations and Research Methodologies" will be introduced and moderated by Ubiratan D'Ambrosio. Papers will be presented by Gloria Gilmer, Pat Rogers and Rick Scott. Discussions of the papers will be led by Gil Cuevas and Luis Ortiz-Franco.
During the regular program there will be a general meeting of the ISGEm. If you will be in Orlando please plan to attend.
Have You Seen
"Have You Seen" is a feature of the ISGEm Newsletter in which works related to Ethnomathematics can be reviewed. We encourage all those interested to contribute to this column. Contributions can be sent to: Rick Scott, Editor, ISGEm Newsletter, College of Education. University of New Mexico, Albuquerque, NM 87131 USA
Foundations of Eurocentrism in Mathematics
Joseph, George Ghevarughese, "Foundations of
Eurocentrism in Mathematics," Race and Class, XXVII,
Joseph suggests that "there exists a widespread Eurocentric bias in the production, dissemination and evaluation of scientific knowledge." He diagrams the "classical" Eurocentric approach as follows:
Joseph claims that this Eurocentric approach served as
a "comforting rationale for an imperialist/racist
ideology of dominance" and has remained strong despite
evidence that there was significant mathematical
development in Mesopotamia, Egypt, China, pre-Columbian America, India and Arabia, and that Greek
mathematics owed a significant debt to the mathematics
of most of those cultures.
A somewhat grudging acceptance of the debts owed to Greek mathematics and to Arabic contributions led some mathematical historians to accept "the 'modified' Eurocentric trajectory":
The modified trajectory still does not take into account the contributions of India and China, nor does it indicate the route through which Hellenistic, Chinese, Indian and Arabic mathematical translations, refinements, syntheses and augmentations arrived in Western Europe. Joseph therefore suggests the following "alternative trajectory" (from 8th to 15th century):
Among the interesting history presented by Joseph is that the earliest known general proof of the Theorem of Pythagorus is contained the Sulbasutras (circa 600-800 B.C.) from India, that "there is no evidence that Pythagorus had either stated or proved the theorem," that Arabic geometers laid the foundations for Saccheri's work in non-Euclidean geometry, that Spain and Sicily were the main points of contact for dissemination of mathematical knowledge to Western Europe, and that "practically all topics taught in school mathematics today are directly derived from the work of mathematicians originating outside Western Europe before the twelfth century A.D."
Joseph refutes the suggestion that pre-Greek mathematics lacked the concept of proof and insists that criticism of Egyptian and Babylonian mathematics as "more a practical tool than an intellectual pursuit" is symptomatic of Western intellectual elitism and racism. Joseph urges the "countering of Eurocentrism in the classroom." His concluding paragraph appears to be a strong statement of support for Ethnomathematics in the classroom and is reproduced below in its entirety:
"Finally, if we accept the principle that teaching should be tailored to children's experience of the social and physical environment in which they live, mathematics should also draw on these experiences, which would include in contemporary Britain the presence of different ethnic minorities with their own mathematical heritage. Drawing on the mathematical traditions of these groups, indicating that these cultures are recognized and valued, would also help to counter the entrenched historical devaluation of them. Again, by promoting such an approach, mathematics is brought into contact with a wide range of disciplines,including art and design, history and social studies, which it conventionally ignores. Such a holistic approach would serve to augment, rather than fragment, a child's understanding and imagination.
A Research Program in the History of Ideas and in Cognition
By Ubiralan D'Ambrosio
This research program, with clear pedagogical implications, has its origins in our first attempts to teach a course in "History of Mathematics" in the so-called third world. A frequent option is to follow the practice of teaching History of Mathematics as a mere collection of results dispasosed in a chronological order and of names associated with them plus some anecdotal remarks, which is indeed a History of European Mathematics.
The mere identification of native practitioners of Mathematics in the academic registers and in publications--local or from Europe, in colonial times and in the early years of independence up to current days--does not change the eurocentric character of what is called Mathematics. Without the need of any adjective, by Mathematics it is understood to be the mode of thought which took shape in Greece some 2500 years ago and which was shaped through medieval and renaissance Europe into its current forms.
The overall objectives of this mode of thought
are, as an etymological analysis would reveal, an art
or technique (techne = tics) of understanding,
explaining, learning about, coping with and managing
the natural, social and political environment. The
divinatory, hence mystical, nature of these objectives
are undeniable, and other of such arts or techniques
were well developed in the same Greece, in the
civilizations of Egypt and Africa, in the Near East
and the Far East, and in the trans Adantic and trans
Pacific civilizations. Other cultural systems were
also looking for their own art or technique of understanding, explaining, learning about,coping with and managing the natural, social and political environment, and the dinivatory nature, hence mysticism, associated with these objectives are again undeniable.
In particular, many of these techniques rely on processes like counting, measuring, sorting, ordering and inferring. The search, which continued throughout History, has been, and continues to be, the essential motivation of well-identified cultural groups for building up corpora of knowledge which came to be called Religion, Art, Philosophy and Science. When we say well- identified cultural groups we mean a group of people who share common and distinctive civilization characteristics, such as jargon, codes of behavior, hopes and fears, or summing up, language and culture in their broad sense. We might say ethnic group in the broad acception of the root ethnos, which has been abusively associated, in the colonial minds, exclusively with race.
We call Ethnomathematics the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering and inferring which result from well-identified cultural groups. In the case of the Greeks the divinatory nature of these objectives are undeniable, and this was done through techniques, learned from Egypt, Babylonia and elsewhere, of counting, ordering, measuring, inferring, among others, which were in competition with oracle practices. These arts or techniques were called by different names, among them geometry, arithmetic, ars magna. The same is true with the advances of this form of thought in Islam, and among them one was called al-iabr, the other al-mucabala.
The same is with the development of sacred geometries and number mysticism in Medieval Christian Europe. No one would then use the word mathematics, and even less Ethnomathematics, to describe such practices. It is also clear that the exposure of different cultural groups one to another brings about inevitable cultural changes. These cultural dynamics result in intense and frequent modifications of arts, techniques and the full range of manifestations of intellectual behavior, obviously including Ethnomathematics.
The successful European enterprise of bringing "civilization" to the entire globe, successfully undertaken in the more span of the 16th century, brought with it the mode of thought then beginning to be called Mathematics, carrying with it the sense of rationality, precision, efficiency and truth. This form of thought has since then been claimed as the essence of man's rationality, disregarding any of the resulting modifications which result from cultural dynamics. More than religion, art, philosophy and the sciences in general, which have been subjected to the basic principles of cultural dynamics, Mathematics has imposed itself as an eminently eurocentric mode of thought, originating from the Mediterranean and incorporating Islamic traditions, absolute in its codes and paradigms. So absolute was this imposition that most of its codes were kept, imposing themselves (through a mechanism of insertion) into languages of non-European origin. Some histories of mathematics bring examples of developments in China, India, Japan and even the Andean civilizations and try to match a few of their results and practices with European similars. The general tone has been "see how good they were! They knew the zero and they even knew a form of Pythagoras as theorem!"
Even the references of Egyptian mathematics are limited to the showing that they were able to solve a few problems which resemble manipulation of fractions. The very essence of the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering, inferring or their equlvalent among the Egyptians, or the Chinese, or the Aztecs, or the Bambaras, has never been mentioned in current histories of mathematics. Indeed, what is usually called history of mathematics should be called History of European Mathematics.
Much research is needed to increase scholarship in Ethnomathematics. We need some categorization of this research in order to draw from several projects going on all over the world, under different names, but which satisfy our conceptualization of Ethnomathematics and, consequently, contribute by adding to the as yet limited knowledge of it. The categories which we use to synthesize relevant research in Ethnomathematics are:
I.Research in culturally diversified environments.
II.Curriculum development projects and classroom applications.
IV.Conceptual and theoretical foundations.
Closely related to this is the research program in the history of mathematics, which can be identified with the very conceptuaization of Ethnomathematics as described above, and takes into account cultural dynamics, which undeniably underlines the evolution of cognitive processes and places history of mathematics in the broader framework of the history of ideas and the even broader vision of general history. Clearly, all these stages of historical analysis must be faced by both the vision of the winners--in the case of Mathematics, it is European (or Academic) Mathematics--and of the losers.
In the case of Mathematics, this means investigating precolonial practices, as identified through monuments, artifacts, documents and preserved practices among communities with strong cultural roots. The program ends with a critical analysis of the transfer, as seen in the institutionalization and in the academic productivity both quantitative and qualitative, of Mathematics to peripherical nations.
In Latin America, since the mid-1970s we have been stressing a research program in the following general direction:
1. Epistemological foundations: Ethnomathe-matics.
2. Socio-cultural bases of European Mathematics: an historical approach.
3. Specificities of Iberic Science in the Middle Ages: Math of the discoveries and of early colonial period.
4. Pre-Columbian Mathematics: a historical approach.
5. Late colonial period: efforts towards introduction of Modern Mathematics in Spain and Portugal and reflection in the colonies.
6. Independence movements, modern ideas and European Mathematics in Latin America in 19th century: institutional aspects.
7. History of Native, Popular and Professional Mathematics (Mathematics in everyday use, Rural Mathematics, Commercial Mathematics, Mathematics of Engineers and Scientists): a socio-cultural approach.
8. Late 19th and 20th century introduction and production of Mathematics in Latin America: quantitative and qualitative analysis.
This is the table of contents of a book in preparation, and most of these topics have been partially presented in a series of papers:
(1) Ubiratan D'Ambrosio; History of Ibero-American Mathematics, Historica Mathematica vol.6,1980, pp.452-453.
(2) Ubiratan D'Ambrosio: L'adaptation de la structure de l'enseignement aux besoins des pays en voic de developement, Impact of Science in Society, vol.25, n. 1, 1975, pp.100-101.
(3) Ubiratan D'Ambrosio: Objectives and Goals of Mathematics Education, Proceedings of the 3rd International Congress of Mathematics Education, Karlsruhe, 1976 (LINES CO, Paris, 1979).
(4) Ubiratan D'Ambrosio: Science and Technology in Latin America during its discovery in Impact of Science on Society, vol.27, n. 3.1977, pp.267-274.
(5) Ubiratan D'Ambrosio: Knowledge Transfer and the Universities: A Policy Dilemma Impact of Science on Society, vol.29, n. 3, 1979, pp.233-240.
(6) Ubiratan D'Ambrosio: Mathematics and Society: Some Historical and Pedagogical Implications, International Journal of Mathematics Education in Science and Tech- nology, vol.11, n. 4, 1980, pp.479-488.
(7) Ubiratan D'Ambrosio: Mathematical Education in a Cultural Setting, International Journal of Mathematics Education in Science and Technolo~v, vol.16, n. 4,1985, pp. 469-477.
(8)Ubiratan D'Ambrosio: Socio-Cultural Bases for Mathematics Education, UNICAP, Campinas, 1955.
(9) Ubiratan D'Ambrosio; Da Realidades a Acao: reflexoes sobre Educacao C Matematica, Summus Editorial, Sao Paulo, 1986 (2a. edicao 1988).
(10) Ubiratan D'Ambrosio: A Methodology for Ethnoscience: The need for Alternative Epistemologies THEORIA Segunda Epoca, n. 2, 1986.
(11) Ubiratan D'Ambrosio: Socio-Cultural Influences in the Transmission of Scientific Knowledge and Alternative Methodologies, in Cross Cultural Diffusion of Science. Latin America ed. Juan Jose Saldana, Cuadernos de Quipu n.2, Sociedad Latinoamericana de Historiade las Ciencias y la Tecnologia, Mexico, 1988; pp.125-133.
With some modifications, mainly affecting chapters 4 through 8, the same program may be adapted to other regions of the world.
ICME-6 Special Program on Mathematics, Education and Society
The Fifth Day Special Program on Mathematics, Education and Society at ICME-6 was organized by Christine Keitel, Alan Bishop, Peter Damerow and Paulus Gerdes, and included almost 90 contributors from over forty countries. Ethnomathematics was a particularly strong theme during the Special Program. The report for the Conference Proceedings states that "there is a growing awareness of the importance of ethnomathematical activities as a means to overcome eurocentrism and cultural oppression in mathematical learning." Many of the panels that day dealt implicitly and explicitly with Ethnomathematics. Below we present abstracts from that report on three panels that focused quite directly on Ethnomathematics.
Ethnomathematics and Schools
Mathematical knowledge of a different kind from that which is usually dealt within the school curriculum was considered in this topic. Sources for ethnomathematical ideas and their significance in schools was discussed by the panel members. Gloria Gilmer, USA, gave a survey report of research activities as reported to the newsletter of the International Study Group on Ethnomathematics. Randall Souviney, USA, discussed the role of the Indigenous Mathematics Project in Papua New Guinea. Eduardo Sebastiani Ferreira, Brazil, showed by a lot of examples from history of mathematics and ethnomathematics that the generic principle and ethnomathematical methods are linked, and can be part of the same method of teach mg mathematics in school.
How can the mathematics learning situation be structured so as to provide for the acceptance of the child's ethnomathematical knowledge? The panel members offered various examples of ethnomathematics. Salimata Doumbia, Ivory Coast, spoke about the mathematics in some traditional African games. Sergio R. Nobre, Brazil, described the mathematics involved in the most popular, but illegal, lottery in Brazil.
Nigel Langdon, Ghana, presented various ethnomathematical activities like mathematics of workmanship, crafts and economy as cultural starting points for the learning of mathematics.
Hearing: What Can We Expect from Ethnomathematics?
Questioner Ubiratan D'Ambrosio (Campinas State
University, Brazil) asked the panelists why each of
them came to ethnomathematics. Mary Harris (University
of London, Great Britain) contrasted the low
achievement of girls in mathematics at schools with
their capacity to do complicated needlework, sewing,
etc. The problem lies in who defines what mathematics
is, and who defines which are the standards. In a male
dominated society, women's mathematics is not
Munir Fasheh (Birzeit University, Palestine) compared the monopoly of Western mathematics with that of Coca Cola: instead of drinking the pure water in their environment people drink Coca Cola thus reinforcing their economic dependency. (Editor's note: The absurdity of the situation was emphasized by the presence of a Pepsi Cola bottle beside him as he made his remarks.) Western Mathematics is the worst fundamentalism there is as it leaves no choice to believe in it or not. It justifies itself by arguments of universalism and objectivity. What we need is meaningful mathematics.
Paulus Gerdes (Eduardo Mondiane University, Maputo, Mozambique) explained how racial and colonial ideology negated the capacity of Africans to do mathematics. In order to build up the economy of an independent Mozambique and to defend the country against South African aggression, the people need to know mathematics in order to master as quickly as possible the necessary mathematics, and the formerly negated mathematical practices may serve as a starting point. Patrick Scott (University of New Mexico, USA) described the activities of the International Study Group on Ethnomathematics. He pointed out that there seems to be three principle conceptualizations of ethnomathematics: the D'Ambrosio/Gerdus model of ethnomathematics for cultural reaffirmation, the Claudia Zaslavsky model of Ethnomathematics for "bringing the world into the mathematics classroom" and the Marcia Ascher model of ethnomathematics as the study of the mathematics of nonliterate peoples.
The contributions of the panelists were interrupted by applause from the audience. The available time did not allow for either more contributions from the panelists or for questions from the floor.