Volume 5 Number 2
David Davison, Second Vice President
Dept. of Curriculum & Instruction
Eastern Montana University
1500 N. 30th Street
Billings, MT 49101-0298 USA
Claudia Zaslavsky, Secretary
45 Fairview Avenue, #1 3-I
New York, NY 10040 USA
Patrick (Rick) Scott, Editor
College of Education
University of New Mexico
Albuquerque, NM 87131 USA
Sau-Lin Tsang, Member-at-Large
Southwest Center for Educational Equity
310 Eighth Steeet, #305-A
Oakland, CA 94607 USA
Ubiratan D'Ambrosio, First Vice President
Pro-Rector de Desenvolvimiento Univ.
Universidade Estadual de Carnpinas
Caixa Postal 6063
13081 Campinas, SP BRASIL
Luis Ortiz-Franco, Third Vice President
Department of Mathematics
Orange, CA 92666 USA
Anna Grosgalvis, Treasurer
Milwaukee Public Schools
3830 N. Humboldt Blvd.
Milwaukee, WI 53212 USA
Elisa Bonilla, Assistant Editor
Centro de Investigacion del IPN
Apartado Postal 14-740
Mexico, D.F., C.P. 07000 MEXICO3
A business meeting of the ISGEm was held on April19,1990, during the Annual Meeting of the National Council of Teachers of Mathematics in Salt Lake City. About 40 individuals attended the meeting chaired by Gloria Gilrner.
1. Claudia Zaslavsky read the minutes of the meeting from April 13, 1989, in Orlando.
Reprinted with permission in 1992 by International Study Group on Ethnomathematics.
2. Ubiratan D'Ambrosio discussed the meaning of Ethnomathematics. He pointed out that most Mathematics as taught in schools is eurocentric. There is usually the assumption that the mathematical practice of various cultures lacks a theoretical basis and is non-academic. ISGEm is trying to correct those assumptions. The "Ethno" refers to any identifiable cultural group, "math" is a way of understanding reality and "tics" means a technique. Therefore, Ethnomathematics is a technique for understanding reality used by a cultural group. Greek-based mathematics is but one strand.
3. Gloria Gilmer introduced the officers.
4. Luis Ortiz-Franco reported on the effort to affiliate with NCTM. We have a Constitution and the application has been sent it.
5. Ubiratan D'Ambrosio described the process of affiliation with the International Congress on Mathematics Instruction (ICMI). A letter has been sent to Geoffrey Howson. ICMI seems to be favorably disposed to our affiliation with them. It is hoped that they will act upon it in their 1992 meeting.
6. David Davidson reported on the membership situation. Only 39 members paid dues last year, although over 200 remain on the mailing list. Individuals are encouraged to pay for up to three years at $5 per year.
7. Rick Scott indicated that the Newsletter should be published in May and November. He appealed for contributions for the November issue to be in by October 30.
8. The leaders of the four Special Interest Groups described the activities of their groups.
9. Claudia Zaslavsky reported on the following meetings:
a. Popularization of Mathematics Conference, Leeds, England, September 1989.
b. Political Dimension of Mathematics Education, London, April 1990.
c. History in Mathematics Education.
10. Those in attendance meet in four Special Interest Groups (SIGs). Below is a report on those meetings.
Out of School Application
Curriculum and Classroom Applications
Research in Culturally Diverse Environments
The following isa summary of the business transacted in the SIG meetings:
Out of School Applications.
Alverna Champion proposed to speak at ICME-7 on "The Mathematics of African Games." Gloria Gilmer and Henry Gore suggested a presentation of a "Survey of Effective Strategies in Undergraduate Mathematics." For further information on the SIG contact:
Ladies and gentleman, via a slide presentation, I would like to present a summary of the ethnomathematical research that I conducted in the country of Bhutan from 1986 - 1988.
At its grassroots level, this ethnomathematics research dealt with the teaching of elementary mathematics through the games, songs and play activities of Bhutanese children. At its theoretical level, I conclude that this form of pedagogy is an actualization of the principle of complementarity and further conclude that complementarity can be used as a theoretical foundation for the concept of ethnomathematics.
This research has made an original contribution to lcnowledge by extending and applying the Principle of Complementarity to human behavior. By definition, the Principle of Complementarity states that two descriptions or sets of concepts, though mutually exclusive, are nonetheless both necessary for a complete description of the situation. The physicist and Nobel Laureate, Niels Bohr, firmly believed that the Principle of Complementarity had wide application outside the realm of physics and declared that one day complementarity would be taught in schools and become part of public education.
The most recent research on hemispheric specialization of the human brain has extended the validity of the concept of complementarity as a theoretical explanation of human behavior. Together they reveal that there are two complementary modes of knowing: the intuitive and the rational.
This slide portrays a summary of the predominant
functions of the hemispheres of the human brain. The
predominant functions of the right hemisphere are
complementary to the predominant functions of the left
hemisphere. The non-verbal is complementary to the verbal,
the holistic is complementary to the analytic, the
viso-spatial is complementary to the rational, and so
The second major contribution to knowledge of this research derives from the study of ethnomathematics. The Brazilian mathematics educator Ubiratan D'Ambrosio has been credited with coining the term ethnomathematics, and in this research it related to how counting, ordering, sorting, measuring and weighing were inherent within the games, songs and play activities of Bhutanese children.
There are two major reasons why the Far East, Buddhist nation of Bhutan was an ideal setting for this investigation. First, Niels Bohr has stated: "for a parallel to the lessons of atomic theory we must in fact turn to the kind of epistemological problems that a thinker like Buddha has confronted."
As the Bhutanese religious historian Rigzin Dorji has revealed, Buddhism permeates the everyday life of Bhutan's people. From the religious songs sung during the construction of a house, to the spiritual songs of celebration performed in honor of his majesty's birthday, to a Buddhist prayer recited by primary school children prior to the beginning of a school day, from the national sport of archery, to the painting of murals, Buddhist philosophy and beliefs significantly influence each day in the life of a Bhutanese citizen.
The second reason why Bhutan was an ideal setting for this investigation centers upon the concept of ethnomathematics. This concept was originally created in the context of developing countries who are struggling for a more meaningful way to help their children learn mathematics. Ninety-five percent of the Bhutanese people are agrarian and are involved in subsistence farming. The per capita income of this Third World national is $116.00 U.S. dollars per year and it is classified by the United Nations as one of the 31 least developed countries of the world. Consequently, Bhutan was an ideal setting to investigate ethnomathematics and to establish a theoretical framework for this concept.
During the school year of 1987, at the Teachers' Training Centre and Demonstration School, Paro, Bhutan, a case study was conducted involving two lower primary classes and their teachers. Also participating in this study were teacher trainees and the investigator as participant observer. This case study focused on a pedagogical process involving the complementary relationship between the world of play, and mathematics within the world of school, of Bhutanese children.
The analysis of data presented the analytical term: propositional theme. In the context of this investigation, it was defined as a recurring behavioral pattern which denoted a fundamental truth that was shown through demonstration to be an actualization of complementarity. Three such propositional themes were derived from the data; they were:
Propositional Theme I - Complementarity: The Voices of Bhutanese Children.
Propositional Theme II - Complementarity: The Games of Bhutanese Children.
Propositional Theme III - Complementarity: The Play Activities of Bhutanese Children.
This research concluded that the teaching of elementary school mathematics through the games, songs, and play activities of Bhutanese children could be considered to be actualization of complementarity; and that complementarity could be regarded as a theoretical foundation for ethnomathematics.
Lastly, implications of this research are drawn for the education of the whole child; that is, to educate children in a holistic manner we must give equal emphasis to art, dance, drama, music, and physical education along with the 3Rs. Further, complementarity can become a theoretical structure for ethnomathematics.
Bhutanese people are involved in the ethnomathematical practices of counting, ordering, sorting, measuring, and weighing during their weekly market day, indicating that these ethnomathematical practices are a part of many cultural activities of the Bhutanese.
Another implication of this research is that Bhutan's NAPW Project can be effective, and can create a positive attitude in the learner towards acquiring knowledge.
In conclusion, this research implies that children can learn the necessary numeracy and literacy skills through their songs, games and play activities that will help them to adininister UNICEF's GOBI project.
An Ethnomath Approach to Curriculum Development
Ethnomathematics - Concepts and Dimensions
In the United States, as in many other countries, there is "widespread recognition of the need to re-evaluate the total school experience in the face of the educational failure of many children from ethnic minority communities." In particular, "pressures are mounting for schools in many countries to reflect in their curriculum the multicultural nature of their societies."Mathematics curricula, though, have been slow to change, due partly to a failure to separate the universality of truth of mathematical ideas (the sum of the angles of a triangle is 180 degrees the world over) with the cultural basis of that knowledge. Ethnomathematics views school mathematics as the process of inducting young people into mathematical aspects of their culture.
Recently research evidence from anthropological and cross- cultural studies demonstrate convincingly that the mathematics that we know is a culture bound phenomena, and that other cultures have created ideas which are clearly "other mathematics." One can cite the work of Zaslavsky who has shown in her book AfricaCounts the range of mathematical ideas existing in indigenous African cultures. On other continents, the research of Lacy, Lean and Bishop in Papua, New Guinea, Lewis in Aboriginal Australia, and Pinxten with the Navajo Indians in North America have also shown evidence which points conclusively to the fact that other cultures have created other mathematics. Now there is an urgent need to multiculturize the mathematics curriculum.
Goals for Students Across Grades K-12 - The Standards.
In The Standards, the following five goals are listed for students across grades K- 12:
Curriculum Development Using a Cultural Base
Ethnomathematicians, "The Ethnos," view mathematics as a cultural product which developed as a result of at least six environmental activities suggested by Bishop: counting, locating, measuring, designing, playing and explaining.
Counting. Studies reveal that there are not just two systems of numbers--civilized and primitive--but a rich variety of systems in all societies varying according to environmental needs. For example, in Papua, New Guinea, Lacy classified 225 different counting systems in four types: (1) a body part tally system with the number of body parts varying from 12 to 68, (2) a tally system using counter-like sticks with bases between two and five, (3) mixed bases of 5 and 20 using compound number names like two hands and a foot to mean 15 and (4) base ten systems with several discrete rather than compound number names.
Locating. This term characterizes activities relating to finding one's way around knowing one's home area, traveling without getting lost, and relating objects to each other. All societies have developed different ways to code and symbolize their spatial environment, and different societies find different aspects to be of significance. Mapping, navigation, and the spatial organization of objects develop important mathematical ideas in all cultures. Compass points are almost universal, and the stars may be the same, and the activity of locating might well be universal, but the resulting conceptualizations and explanations may differ from culture to culture. Pinxten looks in detail at the Navajo Indians of North America's way of conceptualizing large-scale space.
Measuring. Measuring is another universally significant activity for the development of mathematical ideas. Measurement is concerned with comparing, with ordering and with valuing, and all societies value certain things. Precision, though, and the systems of units develop in relation to particular environmental needs and in particular societal contexts. For example, in Papua, New Guinea, Jones collected information about quantities and measures which included statements like, "The local unit of distance is a day's travel." Zaslavsky refers to a basket holding about ten pounds, a package of coffee beans, and a bundle of sweet potatoes as standard measures to local people in Ghanda or Uganda. These measures have an element of inaccuracy which allows for social and commercial negotiation! So accuracy is not necessarily to be valued highly, it depends on the purpose and importance of measuring. But all societies engage in plenty of measuring activities.
Designing. Another universal and important source of mathematical ideas are the many aspects of designing pursued by all cultures. The activities of designing concern all the manmade objects and artifacts which cultures create for their home life, trade, adornment, warfare, games and religious purposes. In addition, there are designs on a larger scale, such as houses, villages, gardens, fields, roads and towns. What is important mathematically is the plan structure, imagined shape, perceived spatial relationship between object and purpose, abstract form, and the abstracted process. The designed object often serves as the representation of the design by which other objects can be constructed. Drawings in the sand, construction models, drawings on paper and on electronic screens are all developments created by the need to consider aspects of the designed form without having actually to make the object. These in turn have developed important mathematical ideas concerning shape, size, scale,ratio, proportions and many other geometric concepts.
Playing, Playing may seem to be a rather strange activity to include in a collection of cultural activities relevant to the development of mathematical ideas, until one realizes just how many games have mathematical connections. All cultures play, and what is more important, they take their play very seriously. Certainly games, their description, analysis and roles, feature widely in the anthropological literature. Although characteristics of play can be seen in descriptions of games, the notion of game is more restricted than that of play. Playing is the activity, and the idea of game is the formalization of playing. Once the play form itself becomes the focus and the game develops, then the rules, procedures, tasks and criteria become formalized and ritualized. Games are often valued by mathematicians because their rule- governed behavior is like mathematics itself. It is not too difficult to imagine how the role-governed criteria of mathematics have developed from the pleasures and the satisfaction of the rule- governed behavior in games.
Explaining. The universal activity called explaining lifts human cognition above the level associated merely by experiencing in the environment. Explaining is the activity of exposing connections between phenomena. The quest for explanatory theory is basically the quest for unity underlying apparent diversity, for simplicity underlying apparent complexity, for order underlying apparent disorder, for regularity underlying apparent anomaly. It is the security of things familiar which probably makes us seek sameness or similarity. Here, the fundamental and universal representation is the "story." In thinking about mathematics in culture, its most interesting feature is the ability of the language to connect discourse in rich and varied ways. In research terms, attention has focused on the logical connectives in Ianguage which allow propositions to be combined, opposed, extended, restricted, exemplified, elaborated, etc. From these, the ideas of proof have developed along with criteria of consistency, elegance and conviction.
Summary. The symbolizations which have evolved through these six activities and reflections on them are what we call mathematics. In short, all cultures develop their own mathematics.
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, ISGEm Newsletter Editor
College of Education, University of New Mexico
Albuquerque, NM 87121 USA The following is the Introduction to Paulus Gerdes' soon-to- be-published book entitled Geometry of the African Sona.
Sona is what the Tchokwe people of northeast Angola call their standardized drawing in the sand. These sona are beautiful and interesting from many points of view.
With the colonial penetration and occupation, the sona tradition has been disappearing. "What we find today - second half of the twentieth century - are only the remnants, becoming more and more obsolete, of a once amazingly rich and varied repertoire of symbols" (Kubik, 1987). Following a description of the drawing tradition of the Tchokwe people, I obtained some results in the reconstruction of the mathematical knowledge that has been involved in the invention of sona which are presented in the first Chapter of the book.
In Chapter 2 some possible uses of the Tchokwe drawings in the mathematics classroom are suggested. The examples given range from the study of arithmetical relationships, progressions, symmetry, and Euler graphs, to the (geometrical) determination of the greatest common divisor of two natural numbers. As a variation on the well-known theme of arithmetical problems of the type "Find the missing number," a series of geometrical problems and recreations "Find the missing figures" is presented in Chapter 3. The objective of these problems is to develop a sense for geometric algorithms, generalization and symmetry.
Many sona are aesthetically appealing. They may be used for instance in textile design. By filming a monolinear pattern (made out of one line), starting the curve at one point, one sees a geometrical algorithm at work. In Chapter 4 I present some new algorithms and monolinear motifs inspired by the style of the Tchokwe sona.
The study of the mathematical properties of sona and their variations constitutes a new and attractive research field. In Chapter5 some interesting properties of a whole class of Tchokwe patterns are illustrated.
The study of the Tchokwe drawing tradition, threatened with extinction during the colonial period, is not only interesting for historical reasons. The incorporation of this sona tradition in the curriculum, both in Africa and in other parts of the world, will contribute to the revival and valuing of the old practice of the sona experts, and will reinforce the comprehension of the value of the artistic and scientific heritage of Africa. It may contribute to the development of a more productive, more creative and multicultural mathematics education. Furthermore, an analysis of the Tchokwe patterns stimulates the development of new mathematical research areas.