Volume 3 Number 1 September 1987
Gloria Gilmer, ISGEm Chair
The McKinley Center
2014 West McKinley Avenue
Milwaukee, WI 53205 USA
Gilbert J. Cuevas
School of Education & Allied Professions
University of Miami
P.O. Box 8065
Coral Gables, FL 33124 USA
Coordenador Geral dos Instiutos
Universidade Estadual de Campinas
Caixa Postal 1170
13100 Campinas, SP BRASIL
Patrick (Rick) Scott
College of Education
University of New Mexico
Albuquerque, NM 87131
Centro de Investigacion del IPN
Apartado Postal 14-740
Mexico, D.F., C.P.. 07000 MEXICO
45 Fairview Ave. #13-I
New York, NY 10040 USA
We welcome two new members to the ISGEm board: Claudia Zaslavsky and Elisa Bonilla. Claudia, author of Africa Counts, is now retired, but remains active writing and promoting socio-mathematics. She has been helping ISGEm seek funding for various projects. Elisa has recently returned to the Center for Research and Advanced Studies at Mexico's National Polytechnic Institute from graduate study in England. She has been helping us translate the ISGEm Newsletter into Spanish.
The ISGEm met at the Annual Meeting of the National Council of Teachers of Mathematics in Anaheim, California,
Reprinted with permission in 1992 by International
Study Group on Ethnomathematics.
in April of 1987. The meeting provided an opportunity for individuals with an interest in Ethnomathematics to meet and share their current concerns and interests. One concern that was discussed was a tendency for some individuals to think of Ethnomathematics as the mathematics of "primitive peoples" rather than the broader view of Ethnomathematics. As suggested at that meeting we are republishing below an article from the first ISGEm Newsletter entitled "Ethnomathematics: What Might It Be?" and are presenting comments on the matter from Ubiratan D'Ambrosio.
Another activity at the Anaheim meeting was a Research Section on Ethnomathematics. Gloria Gilmer, ISGEm Chair, presided as Ubiratan D'Ambrosio of UNICAMP in Brazil spoke on "Socio-Cultural Bases of Mathematics Education: Research Status Worldwide" and Marilyn Frankenstein of U-Mass, Boston, made a presentation on "Teaching Mathematics in a More Useful Way to Public and Community Service Workers."
What Might It Be? A Recap and Reconsideration on ISGEm's 2nd Anniversary
During its first two years the International Study Group on Ethnomathematics (ISGEm) has generated much interest and support among many educators and researchers. However, we have noticed that some may have felt that the "Ethno" in Ethnomathematics indicates that our only interest is in the mathematics used in "primitive" societies.
We have decided that it is time to take another look at what Ethnomathematics might be and to reaffirm that a broad conception of "Ethno" encompasses "identifiable cultural groups, such as national groups, children of a certain age bracket, professional classes, and so on." Therefore a Topologist has his or her own Ethnomathematics that might be different from that of an Algebraist,just as the Ethnomathematics of an engineer may be different from that of a carpenter, and that of a shaman different from that of a hunter/gatherer.
First we reprint our original article on "Ethnomathematics: What Might It Be?" from the first issue of the ISGEm Newsletter. We follow that article with some recent thoughts from Ubiratan D'Ambrosio on the subject.
Ethnomathematics: What Might It Be?
(reprinted from ISGEm Newsletter,1(1).
The coining of the term "Ethnomathematics" can probably be credited to Ubiratan D'Ambrosio. In recent speaking engagements and writings, Prof. D'Ambrosio has emphasized the influences of sociocultural factors on the teaching and learning of mathematics.
Ethnomathematics lies at the confluence of mathematics and cultural anthropology. At one level, it is what might be called "math in the environment" or "math in the community." At another related level, Ethnomathematics is the particular (and perhaps peculiar) way that specific cultural groups go about the tasks of classifying, ordering, counting and measuring.
Although Ethnomathematics has only recently received attention from mathematics educators, anthropologists (and before them world travelers) often commented on the peculiar uses of mathematics among indigenous groups. Various other branches of Ethnoscience such as Ethnobiology, Ethnobotany, Ethnochemistry and Ethnoastronomy gained acceptance around the turn of the century and have a history as recognized disciplines.
The formal development of Ethnomathematics may have been slowed by the pervasive view that somehow mathematics is universal and culture-free. However, recent research is revealing that much of the mathematics used in daily practice, as affected by distinctive modes of cognition, may be quite different from that which is taught in school.
Ethnomathematics suggests a broad conceptualization of mathematics and "ethno-." A broad view of mathematics includes ciphering, arithmetic, mensuration, classifying, ordering, inferring and modeling. "Ethno-" encompasses "identifiable cultural groups, such as national-tribal societies, labor groups, children of a certain age bracket, professional classes, and so on" and includes "their jargon, codes, symbols, myths and even specific ways a reasoning and inferring."
Prof. D'Ambrosio has suggested that the basic question concerning Ethnomathematics is: How "theoretical" can it be? Anthropological research has indicated that many culturally differentiated groups "know" mathematics in ways that are quite different from academic mathematics as taught in schools.
The tendency has been to consider these "ad hoc" mathematical practices as non-systematic and non-theoretical. In contrast, the study of Ethnomathematics approaches the "underlying structure of inquiry in ad hoc mathematical practices by considering the following questions:
1. How are ad hoc practices and solution of problems developed into methods?
2. How are methods developed into theories?
3. How are theories developed into scientific invention?
Along with answers to the above questions, examples of Ethnomathematics derived from culturally identifiable groups and related inferences about patterns of reasoning and modes of thought can lead to curriculum development projects that build on the intuitive understandings and practiced methods students bring with them to school.
Perhaps the most striking need for such curriculum development may be in Third World countries, yet there is mounting evidence that schools in general do not take advantage of their students' intuitive mathematical and scientific grasp of the world.
Reflections on Ethnomathematics
By Ubiratan D'Ambrosio
The History of Science and Mathematics tends to minimize,and in some cases ignore, the cultural atmosphere and motivation behind scientific advances. Societal and cultural factors which have determined the directions in which Science grew have not deserved enough attention when trying to understand and explain the process of scientific creativity. Indeed, the importance of these factors is even lessened when explaining intellectual productivity and creativity in Science and Mathematics.
Although it is recognized to be a practice among artists, writers and composers to travel to certain ambiences which are more favorable in generating the right and indeed necessary atmosphere for their production, in general it is regarded that scientific creativity is only related to the quality of labs and libraries, and improved by a higher level of discussions or seminars in which the scientist is engaged.
It is undeniable that these factors play an important role in the production of science and indeed they generate motivation for further advances. But motivation may come from different sources, such as the same natural and cultural environment which create the framework in which popular sageness finds its roots and grows to take shape as a body of knowledge.
This calls for a somewhat different way of looking into the History of Science and the epistemological foundations of scientific knowledge. It calls for an ethnological interpretation of mental processes and the recognition of different modes of thought, as well as different logics of explanation, which depend upon experiential background of the cultural group being considered. Thus we are led to disclaim the assertion that there is only one underlying logic governing all thought.
The "mathematical empiricism" proposed by Philip Kitcher (The Nature of Mathematical Knowledge, New York: Oxford University Press, 1984) represents a challenge to current epistemologies, as well as the cognitive approach to the history of science proposed by Richard H. Schlagel (From Myth to the Modern Mind, New York: Lang Publishing, 1985). In a similar vein and with specific goals for improving Mathematics and Science Education, we propose a research and pedagogical program centered on the concept of Ethnomathematics.
The first difficulty faced by Ethnomathematics resides in what is more an etymological barrier, which tends to see in the term "Ethnomathematics" a relation between mathematical behavior and race. Back in 1975 when I first used the term in discussing the role of time in the origins of Newton's ideas in Calculus, it was clear that, although race might be one of the factors intervening in the shaping of the concept and measurement of time, it was only part of the ethnomathematical practices which added up to the intellectual atmosphere where Newton's ideas flourished.
Although it should be clear that we are using the prefix "Ethno" in a much broader sense than merely race, it is still important to repeat and emphasize it. Our conception of "Ethno" encompasses all the ingredients that make up the cultural identity of a group: language, codes, values,jargon, beliefs,food and dress habits, physical traits, and so on.
A desire to understand and decipher the cosmic order, as well as to create and to gain knowledge, are universal drives, proper to the human species. The complementary roles of doing and knowing, which are essentially the techne and episteme which gave origin to what we nowadays call Western Science, are common to all civilizations and have been the main force behind every human action.
Recorded history reminds us that counting, measuring, and modes of inference and decision processes have been present in every civilization. In Greece, mathematics played a dominant role and from there we derive what might be called the Western mode of thought, which reserved for mathematics a prominent role in the educational system. This was carried on by the Romans and through the Middle Ages, until we now see, in our present school systems, the dominant role of mathematics over the other subjects.
In the past centuries the world has made enormous progress in Science and Technology and these advances, with both positive and negative effects, are dependent upon mathematics tools. We easily notice through an analysis of curricular trends in the last decades that cultural issues have only recently been regarded as playing any role in the discussion of mathematics curriculum in the developed countries. Among the many reasons for this we have the general epistemological attitude that mathematics knowledge is unchallengeable in its universality and that it is not culturally bound.
It has to be understood that the universality of the results and effects of scientific and technological advances speak in favor of this idea of an unchallenged universality of mathematics. Hence it is accepted that the only route to enter the modern world is the domain of mathematical knowledge.
On the other hand, efforts towards minimizing the gap between the haves and the have nots, both internally in societies and between nations, have strongly relied on schooling. It has been agreed, both by societies within nations and by entire nations through their governments, that the surest road to democracy and to development is via education. Enormous resources have been and continue to be poured into education and enormous efforts are made, by developing as well as developed countries, to advance their educational systems.
New industrial developments imply increasing automation, with a growing role for computers and modern managerial systems are largely dependent on decision processes involving complex manipulation of large data sets and rather elaborated simulation processes. What we see is that the level of creativity needed to overcome underdevelopment and underemployment depends largely on schooling with an important component of science and mathematics.
And yet, paradoxically, mathematics is the main school subject that strangles the process. In Third World countries mathematics is largely responsible for the high rate of early dropouts which are so frequent in those countries, and in industrialized nations with deprived minorities, the same occurs. Particularly in the USA, children of Third World immigrants, together with Blacks and American Indians, and to a certain extent women, show considerably lower achievement along the pipeline to access.
Similar situations are found in Canada, Australia and European countries with large numbers of immigrants. These results are confronted by substantial research findings which show a high creative potential in these same groups that fail in school mathematics. It is unthinkable that these groups would be less capable in going through mathematics than the others. On the other hand, mathematics is blocking access for these groups to the main careers in modern society.
Recently, it has been recognized by a few that mathematics has cultural roots and is indeed a cultural system. Cultural groups, children of a certain age range in a neighborhood, farmers cultivating wheat, engineers in car factories, and so on, have their own patterns of behavior, codes, symbols, modes of reasoning, ways of measuring, of classifying, of mathematizing. Particularly when we talk about school children we must remember that they come to school with a mathematics of their own.
At the elementary level, we recognize, in a certain cultural group of children, mathematical practices and ways of dealing with a certain situation which differ from those of other groups when dealing with the same situation. These different forms of mathematics which are proper to cultural groups we call Ethnomathematics.
There is an Ethnomathematics of a certain age group children in a certain neighborhood, as well as an Ethnomathematics of nuclear physicists, and so on. It is in the realm of one own Ethnomathematics that one's creativity will manifest and it is in the ground that has been laid by these practices that authentic creativity will emerge.
Once more we appeal to D.H. Lawrence: "Instead of life being drawn from the sun, it is the emanation of life itself, that from all the living plants and creatures which nourish the sun. With this metaphor we want to place the source of authentic mathematical and scientific creativity not formalized in mathematics and science, but in mathematics and science in the making, fed by the creative process itself.
Indeed, we are looking for a new paradigm which would bring us, through an undefined, unformalized and uncodified approach to Mathematics Education, closer to dealing with really real problems such as those posed to us by modern society. Ethnomathematics is about all this.
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
College of Education
University of New Mexico
Albuquerque,NM 87131 USA
Teaching of Geometry, vol.5 of Studies in Mathematics Education, edited by Robert Morris, UNESCO, 1986.
Contributors from more than 20 different countries present global picture of the present condition of geometry teaching in primary and secondary schools, and in teacher education.
Science and Mathematics Education in the General Secon- dary School in the Soviet Union, Document Series No.21, edited by V.G. Razumorskil, Division of Science Technical and Envi ronmental Education, UNESCO, 1986.
The efforts of the USSR to modernize the teaching of mathematics and the basic natural sciences are presented. The first chapter is on mathematics and is divided into three parts: an outline of the course content, the development of a scientific way of thinking through the study of mathematics, and a discussion of the polytechnical orientation of the mathematics courses. Examples of exercises directly related to practical applications of geometry are included. Two annotated bibliographies are also included: one for teachers and one for students.
"Bringing the World into the Math Class" by Claudia Zaslavsky, in Curriculum Review, vol.24 no.3, 1985.
This article provided a look at four sources of learning activities in the everyday lives of other societies. Numeration, measurement, games and architecture are used to integrate mathematics with the study of culture and history.
"Alternative Mathematics Worksheets" by Chris Bareen of the University of Capetown's Department of Education, 1986
This publication is a collection of antiracist mathematics worksheets and related journal articles selected from assignments submitted by Math Methods students at the University of Capetown. It contains background problems which require only a knowledge of basic arithmetic for their solution.Often problems require students to make decisions individually or in a group regarding apolitical situation. Copies may be obtained directly from the author at the following address:
University of Capetown
Department of Education
Randebusch 7700, SOUTH AFRICA
Discarded Minds: How Gender. Race and Class Biases
Prevent Young Women from Obtaining an Adequate Math and Science Education in New York City Public High Schools by Lisa Syron, Center for Public Advocacy Research, 1987.
This report presents data on the "disastrously" low math and science course participation in New York City's public high schools. It suggests that the basic structure of the city's high school programs contributes to racial, ethnic, class and gender disparities in math and science course participation. Copies of the report are available from the Center for Public Advocacy Research, 12 West 37th St., 8th Floor, New York, NY 10018 for $6.00.
"The Interaction of Mathematics Education with Culture"by Alan Bishop in Cultural Dynamics, vol.2 no.2, 1987.
This article emphasizes "an urgent need to 'multi-culturalise' the Mathematics curriculum." Bishop presents mathematics as a cultural phenomenon and illustrates with six "universal" environmental activities that are significant for mathematical development: counting, locating, measuring, designing, playing and explaining.He points out how in different cultures distinct symbolizations have arisen from the same environmental activities. He discusses the relationship between mathematical values and culture, and between education and mathematics/culture.
A Young Genius in Old Egypt and Senefer and Hapshepsut by Beatrice Lumpkin with illustrations by Peggy Lipschultz, Chicago: DuSable Museum Press, 1979 and 1983.
Tales for children with illustrations on the measuring and counting system of the Egyptians. Senefer and Hapshepsut recalls the Black origins of Egyptian culture.
The Politics of Mathematics Education by S. Mellin-Olsen, Holland: Reidel, 1987.
This book concerns the sociology and politics of mathematical knowledge, and examines the possibilities for politicizing mathematical knowledge in order to empower children. It is a direct intellectual attack on the problems of culture conflict and on the ownership of knowledge.
And More Additions to the
Alice, Margaret (1986), Hypatia's Heritage, Boston: Beacon Books.
Closs, Michael (ed.) (1986). Native American Mathematics,
Austin: University of Texas Press.
D'Ambrosio, Ubiratan (1986). Matematica per paesi ricchi e paesi poveri: anologie e differnze, L 'Educazione Matematica (Cagliari), 1(2), pp.187-197.
D'Ambrosio, Ubiratan (1986). Culture, cognition and science learning. In J.J. Gallager & G. Dawson (eds.), Science education and cultural environment in the Americas (pp.85- 92). Washington: NSTA/NSF/OAS.
D'Ambrosio, Ubiratan (1986). Some reflections on the western mode of thought. In Eiji Hat- tori (Ed.), Science and the boundaries of knowledge: The prologue of our cultural past (Final Report of Venice Symposium). Paris: UNESCO.
D'Ambrosio, Ubiratan (1985). A methodology for Ethnoscience: The need for alternative epistemologies. The~oria Segunda Epoca (San Sebastian), 1(3), pp.397-409.
Huygens, Cliristian (1986). The pendulum clock or geometrical demonstration concerning the motion of pendula as applied to clocks (J. Blackwell, Trans.). Ames: Univ. of Iowa Press.
Sjoo, Monica & Mor, Barbara (1987). The great cosmic Mother, San Francisco: Harper & Row.
Ethnomathematics at the VII Inter-American Conference
on Mathematics Education
The VII Inter-American Conference on Mathematics Education (VII IACME) was held in Santo Domingo, the Dominican Republic, from July 12-16, 1987, at the Universidad Ca Madre y Maestra.
While the term Ethnomathematics appeared in the title of only two presentations, the concept was often used in both and spirit. Two of the keynote speakers, Ubiratan D'Ambrosio of Brazil and Lelis Paez of Venezuela, both stressed the importance of the cultural dimensions of Mathematics Education.
A panel discussed the Integration of the Sociocultural text into the teaching of Mathematics. The panel on How Develop Student Problem Solving Abilities also had much to about the sociocultural context of problems and problem solving. A working group formed on Mathematics and Reality probably have much in common with Ethnomathematics.
ICME 6 to be Held in Budapest
The Sixth International Congress on Mathematics Education will be held from July 27 to August 3,1988, in Budapest, Hungary. For general Information you can write to:
Secretary of the OrganizingCommittee, Tibor Nemetz
ICME 6 Arrangements
Janos Bolyai Math Society
Budapest, POB 240, H-1368
Of special Ethnomathematical interest at ICME 6 will Theme 7- Curriculum: Towards the Year 2000. Subgroup 7.1. The mathematics curriculum in the Year 2000 and the changing character of social demands on, and need for, mathematics- looks particularly relevant to Ethnomathematics. Further infor mation on that subgroup is printed immediately below.
There is a growing pressure being applied to include various topics that will prepare students for life in the 21st century. In several of the socialist countries we observe a swing to and fro between a practical and a scientific orientation of subject matter, with an endeavor to use this swing in the sense of a dialectical advancement. With respect to developing countries, it is obvious that a more fundamental understanding (as a prerequisite for appropriate application) of the question of content is crucial. From the Western industrialized countries we know that an up-to-date survey and critical analysis of the following questions should provide a fresh approach to the complex matter of content:
Demands and qualifications
What skills, knowledge, metaknowledge and qualifications in mathematics will be required during the next 10-15 years and beyond?
What are the associated needs and demands and who defines them?
How should we match needs? Should we teach directly what is needed? Do we need more or less mathematics? Is there a common core of needs and responses independent of historical and social (cultural) change?
What can historical and empirical studies about the role and function of mathematics in specific areas in social life tell us?
Implicit mathematics in society: Mathematics 'frozen" in techniques, tools, instruments, conventions in specific social fields (e.g., the economy).
What are the topics that link "implicit mathematics" in social life and explicit mathematical application? Can they be developed and taught?
What are the relationships between "implicit mathematics" and explicit learning? How can explicit mathematics education prepare for situations in which normally no explicit mathematics is required?
The task of this Subgroup is to look at these questions and the curriculum in the broad sense (concepts and techniques, strategic and tactical skills for their deployment, contexts of application, learning activities, classroom roles) and to make suggestions for future development, with specific examples.
The Subgroup 7.1 Co-ordinator is Cristine Keitel, Technical University Berlin, Faculty of Mathematics, MA7-3, Strasse DES 17 Juni 136, D1OOO Berlin 12, F.R. Germany.
If you would like to participate in the work of Theme 7, please complete the attached FORM a and send it to:
Chief Organizer, Theme Group 7
Science and Math Education Centre
Curtin University of Technology
GPO Box U1987
Perth, Western Australia 60001
Also of special Ethnomathematical interest will be a 5th Day Special Program on Mathematics, Education and Society. Four time slots will be devoted to the following issues: Mathematics Education and Culture, Society and Institutionalized Mathematics Education, Educational Institutions and the Individual Learner, and Mathematics Education in the Global Village. Contact Dr. Peter Damerow, Max Planck Institute, Lentzeallee 94, D-1000 Berlin 33. F.R. GERMANY.
Ethnomathematics at MAA Meeting
At the MAA Meeting in Atlanta on January 9,1988, from 1:00 to 3:00 there will be an AAAS-AMS-MAA sponsored panel discussion on "How Does Ethnomathematics Make Sense at the College Level?"
2nd Latin American Congress on the History of Science and Technology to be held in Brazil
The 2nd Latin American Congress on the History of Science and Technology will be held in Brazil from June 30 to July 4,1988. For further information write to:
Caixa Postal 6063
13.081 Campinas - SP
18th International Congress of the History of Science to Meet in Germany
The 18th International Congress of the History of Science will take place in two cities in the Federal Republic of Germany: Hamburg from August 1-5, 1989, and Munich from August 6-9. The general theme of the Congress will be Science and Political Order. For the first announcement write to:
Prof. C.J. Scriba
Bundesstr, 55, IGN
Hamburg, F.R. GERMANY