ISGEm 12.2



Volume 13 Number 2 May 1998

Report on Ethnomathematics Research

Joanna O. Masingila, Syracuse University
jomasing@sued.syr.edu

This column reports on current research in the area of Ethnomathematics. If you know of researchers doing Ethnomathematics research, please send me this information either by mail (215 Carnegie, Syracuse, NY 13244-1150 USA) or email (jomasing@sued.syr.edu).

Samson Muthwii, from Kenyatta University in Kenya, and William Rosen, from Exeter University in the U.K., have been investigating how the home culture of various Kenyan communities influences how primary school children from those communities understand concepts of measurement. Their research has involved observing in a number of different primary school classrooms as well as interviewing teachers in the schools.

Muthwii is also working with Joanna Masingila, currently at Kenyatta University as a Fulbright scholar, in examining upper primary school children's perceptions of their out-of-school science and mathematics practice. They are working with children in an urban school and a rural school, and are collecting data via interviews and logs kept by the children. Besides analyzing the data from this study in its own right, Muthwii and Masingila will be comparing these Kenyan children's perceptions with those of American children of the same age from a similar study carried out by Masingila and Saouma Boujaoude, from American University in Beirut.

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1st International Congress on Ethnomathematics
to be held in Granada, Spain

The First International Congress on Ethnomathematics will be held in Granada, Spain, September 2-5, 1998. The site of the meeting will be the University of Granada that was founded in 1431. A variety of hotels are offering special rates for Congress participants and a variety of excursions have been planned

Plenary addresses by Eduardo Sebastiani (Brazil), Martha Villavicencio (Peru), Martha Ascher (USA), Bill Barton (New Zealand), Jama Muse Jama (Somalia), Teresa Vergani (Portugal), and Paulus Gerdes (Mozambique) are planned. There will also be many other presentations and poster/video sessions.

Among the cultural activities that have been arragned are visits to the Albaicín district and the Alhambra, and a dinner with Flamenco music and dancing.

For further information contact:

María Luisa Oliveras
Depto. de Didáctica de la Matemática
Universidad de Granada
18071 Granada, SPAIN

fax: (345)824-6359

email: oliveras@platon.ugr.es

http:// www.ugr.es/~oliveras

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ISGEm Home Page

Ron Eglash, ISGEm WebMaster, encourages you to browse at:

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

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ISGEm at Meeting at NCTM in Washington
April 4, 1998



Ubi D'Ambrosio, International ISGEm President, called the meeting to order. It was suggested that the first international meeting should focus on getting Ethnomathematics into curricula.

The meeting was turned over to Joanna Masingila, North American Chapter President.

Have the North American chapter conduct contact with NCTM was discussed.

Luis Ortiz-Franco suggested an amendment such that all regional chapters should be affiliates for their nation. Larry will check on NCTM implications.

Amilee Preston suggested we consider alternatives to Robert's Rules of Order.

Marilyn Frankenstein called for an agenda. Joanna Masingila said we already had one and just had not followed it. Marilyn suggested a listserv; Rick Scott said he would look into setting one up.

Treasurer Report: Jim Barta reported a balance of $855.97. We discussed better reminders for expired membership and more recruitment.

Web report: Ron Eglash showed screens, noted we were awarded web site of the month from Learning in Motion.
Program committee: Rick Silverman collected names.

Newsletter report: Rick Scott asked about a new compendium. Proposal to create a 2 volume compendium passes.

NCTM rep report: Larry Shirley said that the delegate assembly finally passed our resolution to reconstitute a standing committee on international affairs. Proposal to have ISGEm write a letter of support passed.

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Summer Course at Southern Oregon U on Ethnomathematics

Richard Montgomery
Rmontgomery@sou.edu



Math 481/581 Ethnomathematics: A Multicultural View of Mathematical Ideas and Methods will be taught at Southern Oregon University in Ashland for 4 weeks (June 22 - July 16 ) for 4 credit hours on M-T-W-Th from 9:00 to 11:30 a.m. This course will examine the mathematical thought and practice of other cultures. It will look at the mathematics found in other traditions and consider how the mathematics reflects that culture's heritage and world view. It will make comparisons to the Western tradition. Mathematical topics include: numbering and accounting systems, and geometry and spacial organization. There will be readings, seminar style discussions, classroom activities, and written reports.

Prerequisites: Upper-division standing. Math 581 has the additional prerequisite of senior standing or higher. Enrollment limited to 20.

There will be a mix of activities (e.g. making and understanding the complexity and use of an Inca quipu) and readings from brief case studies (such as the Sikidy system of divination practiced in Madagascar) to more comprehensive studies (such as the book, The Social Life of Numbers: A Quechua Ontology and Philosophy of Arithmetic by anthropologist Gary Urton). For culturally determined organization of space, we will study Marcia Ascher's discussion of the structure and modeling of space (found in her book Ethnomathematics ). In addition, we will examine our own western way of structuring mathematics so that we might better appreciate that of other cultures.

Please consider joining us this summer. This should be an enjoyable and worthwhile venture for all. Please call or email if you would like to know more about the course.

Dick Montgomery (541) 552-6580 or 1-800-552-7672 (Oregon), or Rmontgomery@sou.edu

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Resolution from ISGEm, Approved by the
NCTM Delegate Assembly, April 2, 1998



RESOLVED, that the Board of Directors form a committee to coordinate with international mathematics organizations on issues of mathematics education, including but not limited to matters of curriculum design, instructional practices, achievement assessment, technology, and research.

This committee would also be responsible for formulating NCTM policy and public statements relating to international mathematics.

Please note that although this resolution passed the Delegate Assembly unanimously, it is sent to the NCTM Board of Directors as advisory only. ISGEm members are encouraged to lobby any Board members they may know to urge them to support the resolution.

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ALM5 Conference to be Held in
Utrecht, Netherlands, 1-2-3 July 1998



The 5th Conference on Adults Learning Mathematics (ALM5) will be held July 1-3 in Utrecht, Netherlands. One keynote address entitled "Empowerment and Numeracy Development: Research Challenges" will be given by Iddo Gal. A second keynote address will be "Everyday Mathematics and Adult Mathematics Education" given by Analucia Schliemann.



For futher information contact:

Mieke van Groenestijn
Hogeschool van Utrecht
Institute of Higher Education
Faculty of Education
P.O. Box 14007
3508 SB Utrecht, THE NETHERLANDS
email: Mieke.v.Groenestijn@feo.hvo.nl
http://www.euronet.nl/~groenest/alm5

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Two Mathematics Education Conferences
in South America in July 1998



Two important mathematics education conferences will be held in South America during the month of July. At both meeting most presentations will be in Spanish.

12th Latin American Math Ed Meeting
RELME-12
National University of Colombia
Bogotá, Colombia
July 6-10, 1998

For further information contact:

RELME-12
Universidad Nacional de Colombia
Apartado Aéreo 5997
Santafé de Bogotá, D.C., Colombia
Fax: (571) 368 08 66
macevedo@matematicas.unal.edu.co
http://www.cinvestav.mx/clame/



3rd Iberoamerican Congress on Math Education
III CIBEM

Central University of Venezuela
Caracas, Venezuela
July 26-31, 1998

For further information contact:

Cipriano Cruz
Depto. de Matemática Aplicada
Universidade Central de Venezuela,
Ciudad Universitaria, Los Chaguaramos
Caracas, Venezuela
Fax: (582) 693-06-29
email: ccruz@sagi.ucv.edu.ve
http://proem.pucsp.br/eventos/cibemesp.htm

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Symmetry Patterns of Ute Beadwork

Cathy A. Barkley
Mesa State College, Grand Junction, Colorado USA



Funding for this study was provided by a grant from the Council Of Chairs Research Grant, Mesa State College.

The "day-to-day" activities of people, both past and present, involve a large amount of mathematical applications. Most of the applications have not been considered as mathematics because they do not involve lengthy calculations or formulas. However, six universal activities have been identified (Bishop, 1988) as mathematical practices by any culture; these activities are counting, measuring, designing, locating, explaining, and playing.

Designing, or organizing shapes, patterns, and colors, on surfaces is one way that people have used one of these basic mathematical practices. The practicality of an item is not enhanced by the creation of a design on the item. A basket, for example, is no more or less useful for carrying if it is a plain, utilitarian item or if it has an intricately patterned design woven into it. But all people seem to share a need for, and love of, creative designs on their items used for everyday living purposes.

The group of indigenous peoples in the Rocky Mountains beyond Denver and extending into central Utah were know as the Blue Sky people or the Yuuttaa (Utes). Their origin is unclear but Spanish records first mention the Utes in 1626. Their native language belongs to a language group known as Uto-Aztecan; an ancient version of this language is still spoken by some Aztecs in central Mexico, so perhaps their ancestors fled from the warfare and disruption of the Aztec world. The Utes were considered a relatively small tribe, never numbering more than 5,000 to 10,000 at any single time. They were fiercely independent and lived in relative isolation by preference.

Every household item, including utensils, clothing, and weapons, had to be made by hand or obtained by trading. The Utes learned that leather was a good trade commodity and they soon became known for their fine quality tanned deer and elk hides. Leather trade goods from the Utes included moccasins, buckskin shirts, leggings, arm gauntlets, pipe bags, and quivers. Spanish and American settlers provided a ready market for the finely tanned garments made by the Utes. Settlers needed clothing and shelter materials that could be fashioned from the tanned hides. Some of the leather products were even shipped to Europe.

Beautiful and intricate beadwork was incorporated into the leather products made by the Utes. The Utes were a nomadic group of people and all household/living items had to be moved from place to place. Thus, their need for artistic decoration and expression manifested itself through their everyday articles of use. Decoration and pattern were found in carrying pouches, knife cases, moccasins, tobacco pouches, etc. Early decorations were done with quills from the porcupine.

After glass trade beads arrived in the New World, the Iroquois Indians became specialists in intricate patterns and ornamental designs on clothing. The beadwork was a natural adaptation of the quill work produced earlier by Native Americans. Large areas of solid quill work resembling a separate fabric were replaced by solid areas of beading to achieve the same effect. Thin bands of quill work gave way to delicate bands of beading and stitching.

Glass beads brought by the Europeans were not valued for their monetary or exchange value, but rather for their symbolic value related to the existing religious and ideological frameworks of the natives. Many of the Iroquois nation words for glass, mirror, and metal are linked with words for seeing, divining, and the soul. So the combination of the glass beads with totem and ritual passage designs on clothing produced many items of exquisite beadwork. Artifacts can be dated in part by the type of beads used on the articles. During the early l800's, the Venetian "pony beads" (so-called because they were packed into the back country by ponies) were the common beads imported into the New World. They were primarily light colors of white and sky blue. Other light colored beads were available but not in such large quantities. Most of the beadwork done during this time had a light or white background and the tradition continued even as other beads became more popular. Around 1840, the smaller, more delicate "seed beads" were available for trade and more delicate beadwork became popular. This type of beadwork was very popular with the European settlers and beadwork as trade experienced a huge increase. By 1860, beading was extremely popular and the Bohemian beads with their darker colors and bluish tinge were imported and traded to the Indians for their trade beadwork. By 1870, translucent beads were available, and by 1885, glass beads colored silver or gold were brought from Europe. The influx of new types of beads continued to fuel the trade beadwork market. However, like all trends, beadwork was waning, and by 1900, the great beadwork period was finished. Beading continued into the early part of the century, but it never experienced the great popularity it had during the second half of the 1800's.

Plains tribes learned the beadwork craft from trade with the Indians of the northeastern United States, particularly the Iroquois nation. The Plains tribes then traded with the Utes of Colorado and they began to make beautifully beaded trade goods. Earliest evidence of Ute beadwork is 1860; by 1930, beadwork had gone out of fashion among Europeans and Ute beadwork was to become almost a lost art. Much of the Ute beadwork found in museums and private collections was made strictly for the purpose of trade and was never intended to be used by the Utes themselves. Unlike the eastern tribes, both the Plains tribes and the Utes used straight line geometry in their beadwork designs. All floral, irregular designs were done strictly for trade to the Europeans. The evolution of beadwork from quill work is evident in the straight even designs used by the Utes. Backgrounds are always a light color, usually white. Early simple forms include blocks, crosses, and triangles. Triangles, especially equilateral, isosceles, and right angle triangles, are all popular designs. Tall, elegant triangles, and congruent triangles that reflect one another, are found over and over again in the beadwork. Delicate designs using a basic rectangle were often embellished with forks and terraces to form new, similar designs. As the beads replaced the use of quills, designs evolved that used more rounded edges and circular patterns. Many rose floral patterns are found in Ute beadwork designs from the turn of the century.

This study is a brief examination of Ute beadwork and designs. Many fine examples of Ute beadwork are in existence today and there are several items in the collection of the Denver Art Museum, Denver, Colorado. This study examined Ute beadwork articles in two locations: The Museum of Western Colorado, Grand Junction, Colorado, and The Ute Indian Museum, Montrose, Colorado. It is not meant to be an exhaustive study, and with the small number of items examined (119), no conclusions can be drawn about the frequency of patterns used throughout the Ute nations. We were interested in comparing the number of symmetry patterns found in Ute beadwork, and their frequency, with the previously done study of Woodland Indian bead designs.

The discipline of mathematics includes the study of patterns. Patterns can be found everywhere in nature. Often these patterns are copied and adapted by humans to enhance their world. Nowhere is this more evident than in the study of designs by indigenous peoples. The study of mathematical ideas of native or indigenous peoples is referred to as ethnomathematics. One particular aspect of mathematics evident in all cultures is the use of symmetry in strip patterns. These patterns are evident in the use of beading designs done by the Ute Indians.

The structure or balance of a design is described by mathematicians as an isometry or rigid motion. For various figures, there are different motions that will move the figures about the plane but are unchanged except for the new orientation. Because the space in the plane is limited (strip design), there are a limited number of motions that are possible. There are exactly four types of rigid motions possible: a) translation or slide, moving the figure forward repeatedly; b) reflection, mirror image in a vertical, horizontal, or combination line; c) rotation, movement around a fixed center point through 180 degrees; d) glide reflection, a movement that translates the figure and then reflects it. Every strip pattern can be made from one of these four types of rigid motion or a combination of them. But the resulting patterns are limited in number.

The different types of isometries that result are called symmetry groups. For any strip pattern, its symmetries will be classified into one of seven distinctive groups. To refer to the groups, a system for classification was developed by crystallographers who identified three-dimensional patterns found in crystals. The classification system uses a four-character symbol for each strip pattern. The classification system uses this coding scheme:



Character One: If the group contains a translation (which they all do), code p.

Character Two: This indicates a vertical reflection symmetry.
Code m if it has a line of vertical reflection symmetry.
Code 1 if it does not.

Character Three: This indicates a horizontal/glide symmetry.
Code m if it has a horizontal line of symmetry.
Code a if it has a glide reflection but not horizontal line.
Otherwise, code 1.

Character Four: This indicates a 180 degree rotation.
Code 2 if it has a point of 180 degree rotation symmetry.
Code 1 if it does not.

This four-character classification system gives complete information about the symmetry groups of any strip pattern. Using this classification system, we examined and coded several articles from the museums that were examples of Ute beadwork strip patterns. We were comparing the seven types of symmetry strip patterns and their frequency among the beadwork designs. We have examined 33 items from the Museum of Western Colorado in Grand Junction, Colorado. Of these items, 24 represented true strip beadwork specimens. From the Ute Museum in Montrose, Colorado, we examined 86 items, with 72 showing examples of strip patterns. Some of the items, such as the headstall owned by Chipeta, had more than one type of strip pattern represented.



Table 1 summarizes the frequency of symmetry patterns found by Nishimoto and Berken in the analysis of Wisconsin Indian beadwork (1996).

Symmetry Type Frequency
1 pmm2 77
2 pma2 10
3 pm11 14
4 p1m1 21
5 p1a1 7
6 p112 6
7 p111 13


Table 2 summarizes the frequencies of patterns found in the current Ute beadwork study by Barkley and Osborn (1997).



Symmetry Type Frequency
1 pmm2 55
2 pma2 2
3 pm11 35
4 p1m1 2
5 p1a1 1
6 p112 3
7 p111 28

The most common pattern found in both studies is the pmm2, the pattern that exhibits the most symmetrically balanced elements of all the patterns. This pattern has horizontal, vertical, and rotational symmetry. For this pattern to occur, the geometric figures used in the design must have specific symmetries. With the small sample used in the Ute study, it is difficult to detect any other similarities to the Woodland study.

The mathematics imbedded within the artwork of the Ute beadwork is representative of the area of Ethnomathematics study. These Native American artisans had no formal training in classical Euclidean geometry, but it is clear that they had an understanding of basic geometry elements. These beautiful beaded treasures are clearly not done by random design, but rather by people who carefully followed some specific guidelines of geometric principles. It is hoped by the researchers that more study of the Ute beadwork designs can be done during Summer 1998. More examples of this authentic art may enable more specific patterns to emerge.

References

Bishop, A. (1988). Mathematical Enculturation, Dordrecht, North Holland: Kluwer.
Nishimoto, K. & Berken, B. (1996). Symmetry patterns of the Wisconsin Woodland Indians. ISGEm Newsletter, 12(1), 6-8.

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Hairstyles Talk a Hit at NCTM!

Gloria Gilmer and Mary Porter

Gloria Gilmer and Ron Eglash presented the 1998 ISGEm talk at the NCTM Annual Conference to an overflow crowd at the Convention Center in Washington, DC. Even ISGEm's editor, Rick Scott, was turned away from the crowded room. The popular talk was on Hairstyles in African American Communities. In an Oprah Winfrey-like fashion, Gilmer highlighted styles worn by participants in the audience, many of whom were African American. She noted that these styles are seen in schools and communities but not in mathematics textbooks, and this must be changed.

In preparation for the talk, Dr. Gilmer and Ms. Mary Porter, the presider, interviewed a master hairbraider from Nigeria and one of her American-born students in their respective shops. Their customers and operators were also interviewed. The idea was to determine what the hairbraiding and hairweaving enterprise can contribute to mathematics teaching and learning and what mathematics can contribute to the enterprise.

The observations and interviews provided insight into some of the cultural values that form the basis of hairbraiding and weaving traditions in African American communities and the creative use of geometrical patterns in the design of hairstyles. For generations, African Americans were told that "nappy" hair was bad and were made to feel that the only way to attain "good" hair was to straighten it. The chemicals and heat treatments used to straighten the hair often resulted in damaged, unhealthy hair that would not grow. The customers we interviewed, however, felt good about having a beautiful hairstyle without altering the natural texture of their hair. We were told that some styles required sixteen or more hours to complete. Often this is done in a single sitting. Beyond beauty, the enterprise is also an important source of income for African Americans. At the age of eleven, one stylist said she was the neighborhood braider and could always make money. Therefore, concepts of time use and value, cost of supplies and equipment, and establishing workers' salaries and customer fees are important sources of mathematical problems for the classroom.

Gilmer displayed a sequence of hairstyles using male and female models. Some styles involved tessellations of the scalp with triangles, rectangles, and pentagons. Cornrows, both in concentric circles and in spirals, were seen on the scalp; spirals were also seen in planes orthogonal to the scalp. Self-similarity, symmetry, and fractal patterns were also shown. Eglash showed that these aspects of the hair styles are part of a much broader fractal design theme in African material culture, ranging from architecture and craft production to symbolic and quantitative systems. The storyboard for a computer-based math lab showed how students can use simulations of these designs to support learning in geometry, algebra and trigonometry.

The two presentations demonstrated the first two steps in Gilmer's Five Step Model for Concept Development. These steps examined the importance of the concept intuitively and its occurrence in nature or in culture. Hence, they drew heavily upon the visual aspects of mathematics and one's own experiences. The last three steps, which treat the concept with mathematical rigor, were alluded to. This model was used successfully to engage students in mathematical studies at all levels. We used it to motivate the study of Euclidean and fractal geometry. In this way, the relevance of the study was made clear.

Feedback from the session has been quite encouraging. A Harcourt representative wrote of her interest in including photos similar to those shown in the presentation in their math textbooks. Jim Barta, ISGEm treasurer, commented, "The feeling in that room was so wonderful!" Shirley Burkes, a mathematics and science support teacher in the Baltimore Public Schools, said, "Your part on the program was excellent! I find myself looking at everybody's hair and thinking of how I can use the design in class because all the girls braid each other's hair." Another participant said, "I heard people talking about this session on the train. I was here forty minutes before the session began, and the room was already full."

Our experience suggests that if we are to increase African American appreciation for mathematics, as the Standards state, then the curriculum content must be extended to include a range of activities that African American communities engage in naturally and from which mathematics is derived. Our experience also suggests that it is beneficial for mathematics educators to subject a familiar thing to detailed study for this is, after all, one of the outstanding educational values of mathematics.

Next year, Gilmer and Eglash hope to demonstrate how the quality of learning can be enhanced by manipulating computer simulations of these hairstyles. They also hope to extend the time of the session and maybe even get a larger room.



References

Sagay, Esi. (1983). African Hairstyles, Portsmouth NH: Heinemann Educational Books.
Oliveras Contreras, Maria Luisa. (1997). Mathematics and crafts in Andalusia. ISGEm Newsletter; 13(1), 3-5.
Gerdes, Paulus. (1997). On Culture, Geometrical Thinking and Mathematics Education in Ethnomathematics: Challenging Eurocentrism in Mathematics Education. Arthur B. Powell and Marilyn Frankenstein (editors), State University of New York Press.

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ISGEm Inaugurates Electronic Discussion Group



ISGEm has established isgem@nmsu.edu, an electronic discussion that all those interested in Ethnomathematics are invited to join. It had been announced at the last ISGEm meeting that those already on the ISGEm distribution list would be automatically subscribed to the new list, but it was later decided not to do so.

Technically, isgem@nmsu.edu, uses "listproc" software that is very similar to the, perhaps more widely known, "listserv".

To join the ISGEm discussion group send an email message (without Subject) to

listproc@nmsu.edu

The only message you need include is

SUBSCRIBE isgem your-name

Of course, you substitute your real name for your-name. If you have any difficulties, please send a message to pscott@nmsu.edu

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ISGEm Held "Conference Within a Conference"
at NCTM Annual Meeting



The picture below shows some of the participants at the day-long Conference Within a Conference that was held in Washington DC on April 12. Details on the presentations can be found in an article by Kay Gilliland in the April 1998 issue of the NCSM Newsletter.

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Ubi D'Ambrosio Honored at
Joint MAA/AMS Meeting



ISGEm and the International Study Group on the Relations between History and Pedagogy of Mathematics (HPM) jointly sponsored a conference in honor of the 65th birthday of Ubi D'Ambrosio, President of ISGEm, in Baltimore, USA, on January 6, 1998. The picture below shows some of the many well-wishers.

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Have You Seen?

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

Zaslavsky, Claudia. (1998). Ethnomathematics and multicultural mathematics education. Teaching Children Mathematics, 4(9), 502-503.

The article is the lead editorial, "In My Opinion", in the May 1998 issue of NCTM's Teaching Children Mathematics.

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Zaslavsky, Claudia. (1998). Math Games and Activities from around the World. Independent Publishers Group, ISBN 1-55652-287-8, $14.95, (312)337-0747 or (800)-888-4741.

More than seventy math games, puzzles, and projects from all over the world encourage kids to hone their math skills as they use geometry to design board games, probability to analyze the outcomes of games of chance, and logical thinking to devise strategies for games.

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Lumpkin, Bea. (1998). Algebra Activities from Many Cultures. J. Weston Walch, P.O. Box 658, 321 Valley St., Portland, ME 04104-0658, ISBN 0-32843-023, $17.95+$3.95 shipping, tel: (800)341-6094, fax: (207)772-3105, www.walch.com

From Aztec numerals to Zuni games of chance, these examples of algebra's applications are from different cultures and historical eras.

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Lumpkin, Bea. (1998). Geometry Activities from Many Cultures. J. Weston Walch, P.O. Box 658, 321 Valley St., Portland, ME 04104-0658, ISBN 0-32851-023, $17.95+$3.95 shipping, tel: (800)341-6094, fax: (207)772-3105, www.walch.com

This book's array of topics ranges from ancient geometry to its use in modern times. As in the algebra collection it is "designed with the NCTM Standards in mind".

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Ascher, Marcia. (1997). Malagasy Sikidy: A case in Ethnomathematics. Historia Mathematica, 24(1197), 376-395.

This article presents and examines a system of divination that plays a significant role in the lives of people of Madagascar. In sikidy, formal algebraic algorithms are applied to initial random data, and knowledge of the internal logic of the resulting array enables the diviner to check for and detect errors. In the article, sikidy and the mathematical ideas within it are placed in their cultural and historical contexts.

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Gerdes, Paulus. (1998). Molecular Modeling of Fullerenes with Hexastrips. The Chemical Intelligencer, January 1998, 40-45.

Ideas with their origin in the geometry of hexagonal basket weaving are developed to formulate hypotheses about the existence of certain isomers of fullerenes. (This paper will be reproduced in The Mathematical Intelligencer).

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Gerdes, Paulus. On Culture and Mathematics Teacher Education, Journal of Mathematics Teacher Education, 1(1), 33-53.

Many countries are multicultural. Mathematics teacher education classes often are composed of students with varied cultural and linguistic backgrounds, and mathematics teachers may be employed in regions from which they do not originate. Formal (mathematics) education is a process of cultural interaction, and every child and teacher may experience some degree of social and cultural conflict in that process. This paper describes and reflects on some of these tensions, as they exist in mathematics teacher education in Mozambique. The process of developing among future mathematics teachers an awareness of the social and cultural bases of mathematics is the object of reflection.

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Gerdes, Paulus. (1998). Culture and the Awakening of Geometrical Thinking. Anthropological, Historical, and Philosophical Considerations. An Ethnomathematical Study [Preface by Dirk J. Struik]. MEP Press, c/o School of Physics, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455-0112, USA (Tel: (612) 922-7993; E-mail: marqu002@tc.umn.edu)

This translation should be available in June 1998. Earlier versions of this book were published in German (Franzbecker Verlag, Bad Salzdetfurth, 1990; preface by Peter Damerow) and in Portuguese (both in Mozambique and in Brazil). The Brazilian edition (Federal University of Parana', Curitiba, 1992) has a preface by Ubiratan D'Ambrosio.

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Gerdes, Paulus. (1998). Women, Art and Geometry in Southern Africa. Africa World Press, 11-D Princess Road, Lawrenceville, NJ 08648-2319, USA (Tel: [609] 844-9583; Fax: 844-0198]; www.africanworld.com) / Asmara (Eritrea) / Addis Abada (Ethiopia), ISBN / Price: ISBN 0-86543-601-0 Cloth; $79.95; ISBN 0-86543-602-9, Paper $21.95.

This is a new edition of the award winning book Women and Geometry in Southern Africa (Pedagogical University, Maputo, Mozambique, 1995) extended with an appendix by Salimo Saide on pottery decoration among the Yao in northern Mozambique



Gerdes, Paulus. (1998) Geometrical and Educational Explorations Inspired by African Cultural Activities, [Preface by Arthur B.Powell, Rutgers University, Newark, NJ], Mathematical Association of America (MAA), P.O.Box 91112, Washington D.C. 20090-1112 (Tel: 1-800-331-1622; Fax: [301] 206-9789).

This book of activities should be available from the MAA during the second half of 1998.

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Urton, Gary (with the collaboration of Primitivo Nina Llanos). (1997). The Social Life of Numbers: A Quechua Ontology of Numbers and Philosophy of Arithmetic. University of Texas Press, P.O. Box 7819, Austin, TX 78713-7819, ISBN 0-292-78533-X, $35.00, hardcover, ISBN 0-292-78534-8, $17.95, paperback.
www.utexas.edu/utpress/books/urton.html

Written by an anthropologist in collaboration with a professor of Quechua and based on extensive fieldwork, the book discusses the meaning and significance of numbers and the philosophical principles underlying the practice of arithmetic among the Quechua-speaking peoples of the Andes.

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Eglash, Ron. Geometry in Mangbetu design. Mathematics Teacher, 91(5), 376-381.

"This article introduces a few examples of Mangbetu designs and examines their underlying structures. The author encourages teachers and students to join him in discovering the geometric basis for these beautiful patterns.



Eglash, R., Diatta, C., Badiane, N. (1994). Fractal structure in jola material culture, 61(368/369), 367-371.



Eglash, R. (1997) Appropriating Technology in Technology and Democracy, University of Oslo, 65-74.

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Have You Seen on The Web?

www.dm.unipi.it/~jama/ethno/
Jama Musse Jama, a Somali at the Univeristy of Pisa, maintains a web page with commentary on the role of Ethnomathematics in Mathematics Education, links to many ethnomathematical references, and an online indexed database of references. The stated purpose of the site is to "be of help especially for the young researchers and the students who are approaching for the first time research in ethnomathematics and relating topics (both mathematical and non mathematical topics)."

www.ldc.lu.se/~latinam/virtual/virtual5.htm
This site contains online versions of "Ethnomathematics as Revisionism?" by Ubiratan D'Ambrosio and "Ethnogeometria" by Oscar Pacheco Ríos.

www.math.buffalo.edu/mad/amu_chma_announce.html
Scott Williams (Mathematics Department, State University of New York at Buffalo) has installed a webpage with information on the African Mathematical Union Commission on the History of Mathematics in Africa(AMUCHMA). The web page includes the 19 issues of the AMUCHMA-Newsletter published so far.

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ISGEm Distributors

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

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

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

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

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

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

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ISGEm Executive Board

Ubi D'Ambrosio, President
Rua Peixoto Gomide 1772 ap. 83
01409-002 São Paulo, SP BRAZIL
ubi@usp.br

Maria Luisa Oliveras Contreras, 1st VP
Depto de Didáctica de las Matemáticas
Campus Cartuja, Universidad de Granada
18071 Granada, SPAIN
oliveras@platon.ugr.es

Jolene Schillinger, 2nd Vice President
New England College BX 52
Henniker, NH 03242 USA
jus@nec2.nec.edu

Abdulcarimo Ismael, 3rd Vice President
Departamento de Matemática
Universidade Pedagógica
P.O. Box 4040
Maputo, MOZAMBIQUE
ismael@fcnmup.uem.mz

Gelsa Knijnik, Secretary
Rua Prof. Andre Puente 414 ap.301
90035-150 Porto Alegre, RS, BRAZIL
gelsa@portoweb.com.br

Jim Barta, Treasurer
Department of Elementary Education
Utah State University
Logan, Utah 84341 USA
Jbarta@cc.usu.edu

Patrick (Rick) Scott, Editor
College of Education
New Mexico State University
Las Cruces, NM 88003 USA
pscott@nmsu.edu

Lawrence Shirley, NCTM Representative
Dept of Mathematics
Towson State U
Towson, MD 21204-7079 USA
shirley@saber.towson.edu

Gloria Gilmer, Past President
Math Tech, Inc.
9155 North 70 Street
Milwaukee, Wl 53223 USA
ggilme@aol.com