Volume 5 Number 2

May 1990

Math-Tech, Inc.

9155 North 70th Street

Milwaukee, WI 53223 USA

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

Chapman College

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

**ISGEm News
**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

Theoretical Perspectives

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:

Mathematics Department

Morehouse College

East Point, GA 30344 USA

(404) 681-2800, ExL 284

of Leaming Style and Cultural Environment on the Mathematical Experience of American Indians" and a subgroup suggested that they could collaborate in organizing a demonstration workshop on "Multicultural Classroom Activities" during the regular NCTM Program in New Orleans. Claudia Zaslavsky, Lawrence Shirley and Erica Voolich proposed participating in a panel at ICME-7 on "How to Incorporate Multicultural Perspectives into Math Classes." For further information on this SIG contact:

Towson State University

Towson, MD 21204-7079

For additional information on this SIG contact:

Education Department

St. Francis Xavier University

Antigonich, Nova Scotia B2G ICO CANADA

(902) 867-2254

Mathematics Department

Chapman College

Orange, CA 92666

(714) 997-6595

Complementarity and Bhutan

St' Francis Xavier University

*
*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

on.

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
**

Math-Tech, Inc.

*
*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:

- Becoming a mathematical problem solver

- Learning to communicate mathematically

- Learning to reason mathematically

The latter four goals may be necessary to achieve the first goal, but not sufficient. Valuing the discipline has more to do with one's sense of power and ownership in the products of that discipline.

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.

- Shift mathematics to an engaging activity

- Create and bolster public attitudes

- Change its focus from skills to power

- Increase its use of technology in problem solving

- Shift its focus from arbitrary rules to the science of patterns

I believe thata core curriculum that explores similarities and differences between cultures along the lines outlined above will make these shifts. Furthermore, I believe that in so doing, more students will learn to value mathematics.

**Have You Seen
**"Have You Seen" is a feature of the

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

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.

Math Education Meeting

Acapulco, Mexico, July 8-10, 1990

To receive further information write to:

CINVESTAV-IPN

Nicolas San Juan No.1421 Col Del Valle

Mexico DF 03100, MEXICO

S.A.E.M. THALES

Facultad de Matematicas

Apartado 1.1.60

41080-Sevilla, SPAIN

Miami, August 3-7, 1991

For further information write to:

College of Education

University of New Mexico

Albuquerque, NM 87131 USA

7th International Congress on

Math Education

Chair, IPC for ICME-7

Dept. ofMath & Statistics

Concordia University, Ioyola Campus