Volume 8 Number 1, November 1992

Minutes from the ISGEm Meeting in Québec

A General Meeting of the International study Group on Ethnomathematics (ISGEm) was held on August 18, 1992, in Quebec at the Seventh International Congress on Mathematical Education.

An enthusiastic audience of about one hundred people attended.

The meeting was opened by the president, Gloria Gilmer, who welcomed the audience and presented the agenda.

Secretary Claudia Zaslavsky read the minutes of the open meeting in Budapest in August 1988 at ICME-6. The minutes were approved as read and accepted by those present.

Membership chair David Davison invited non-members to join. Thirty-eight new members joined and received membership cards. Ms. Chubbins Davison assisted her husband in collecting membership fees. He will submit a report on membership.

Editor Rick Scott presented the newly published Compendium of Newsletters and invited the audience to purchase the books at a price of $15 U.S. for non-members and $10 U.S. for members. Thirty-four books were sold for a total of $345. He also invited members to volunteer to translate, if necessary, and distribute the newsletter in their countries or regions. Twelve people volunteered their services, in addition to the six members who are already doing so.

Coordinators of the special interest groups (SIGs) invited the audience to meet with them at the end of the general meeting, based on their interests. Luis Ortiz-Franco, chair of the SIGs, will submit a report of these meetings.

Henry Gore invited members who had not yet done so to vote for the officers for the coming period. He will submit a report of the outcome.

Arthur Powell spoke about the Criticalmathematics Educators Group (CmEG). He announced the sale of Paulus Gerdes' book Lusona, to benefit the Higher Pedagogical Institute in Mozambique, which Gerdes heads. Marilyn Frankenstein introduced the CmEG newsletter and invited the audience to attend the joint ISGEm- CmEG meeting on Friday, August 21. John Volmink discussed the upcoming second meeting on the Political Dimensions in Mathematics Education in South Africa in April 1993, a sequel to the first meeting in London in April 1990.

Sunday Ajose volunteered to coordinate ISGEm's participation in ICME-8 in Spain in 1996 and appealed for presenters. We hope many members will answer the call.

The remaining time was taken with the SIG meetings, collection of dues, sale of Newsletter Compendia, and social interchange among people from many different countries. Then the meeting was adjourned.

The four SIG coordinators are listed below. Individuals interested in participating may contact them:

Lawrence Shirley
Curriculum and Classroom Applications
Department of Mathematics
Towson State University
Towson, MD 21204-7079 USA
E-mail: SHIRLEY-L@TOWSONVX

Ubiratan D'Ambrosio
Theoretical Perspectives
UNICAMP
Caixa Postal 6063
13081 Campinas, SP BRAZIL
E-mail: UBIRATAN@CCVAX.UNICAMP.ANSP.BR

Luis Ortiz-Franco
Researching Culturally Diverse Environments
Department of Mathematics
Chapman University
Orange, CA 92666 USA

Henry Gore
Out-of-School Applications
Department of Mathematics
Morehouse College
Atlanta, GA 30314 USA

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Africa Counts

Claudia Zaslavsky, author of the classic Africa Counts: Number and Pattern in African Culture, recently took part in a BBC Radio 5 broadcast called "Maths Miscellany". The episode was entitled "Africa Counts". Participating with Claudia and host John Fauvel were Lamin Mansaray from Sierra Leone and David Singmaster from South Bank University, London.

By the way, we note that Africa Counts available from Lawrence Hill Books, 230 Park Place - Suite 6A, Brooklyn, NY 11238, is catalogued by the Library of Congress under the new category "Ethnomathematics".

And Claudia will give a presentation entitled Multicultural Education in Mathematics: Involving All Students at the NCTM Annual Meeting in Seattle on Friday, April 2, at 10:30 AM.

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Crest of the Peacock

There has been some concern about the availability of George G. Joseph's Crest of the Peacock: Non-European Roots of Mathematics. It has now been published as a Penguin Books paperback. In the United States it is available from Viking Penguin, P.O. Box 120, Bergenfield, NJ 07621-0120 (telephone 1-800-526-0275). The price is $12.00 plus shipping and handling.

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Multicultural Mathematics at St. Peter's College

More than ninety different cultures are represented in the Northern New Jersey county where Saint Peter's College is located and from which the College draws the majority of its students. Teachers are challenged to develop teaching approaches which recognize cultural differences in the mathematical notions that children bring to class while building common bonds in problem-solving ability and abstract mathematical reasoning.

In 1989, Saint Peter's College established the Institute for the Advancement of Urban Education (IAUE) to develop and implement effective teaching strategies to meet these needs by cutting across the curriculum. In each discipline, research teams of teachers from local elementary and secondary schools and the college work in consultation with a national expert with them. Summer workshops introduce the outcomes to groups of teachers from participating schools. Participants communicate through an electronic bulletin board with computer technology provided by the Institute.

To foster learning, the mathematics team has been focusing on collaborative and cooperative learning, writing and speaking about mathematics, critical thinking, and classroom technology. Instructional materials acknowledge diverse cultural approaches and contributions to mathematics education. The results have been encouraging.

For more information contact:

Eileen L. Poiani

Math Department, Saint Peter's College

Jersey City, NJ 07306 USA

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ISGEm Communications Network Now Operational

The ISGEm Communications Network is now operational and available to anyone with an Internet connection. The network provides a forum for members of ISGEm to pose questions, offer opinions, further discussions, relay research results, and announce meetings and conferences on Ethnomathematics.

The ISGEm Communications Network operates like most electronic bulletin boards. Subscribers may post their message to the ISGEm e-mail address. The message is then relayed to all subscribers to the network. Special interest bulletin boards are a major factor in the rapid dissemination of information in a variety of fields. Ethnomathematics now has one of its own.

To subscribe to the ISGEm Communications Network send an e-mail message to:

isgem@mail.millikin.edu

No subject is needed. Your message should contain the word SUBSCRIBE and your name. Once you are on the network ISGEm messages will be automatically sent to you. You can send messages to the network using the ISGEm address: isgem@mail.millikin.edu

The ISGEm Communications Network is managed by ISGEm member James Rauff (Department of Mathematics, Millikin University, Decatur, IL 62522). Millikin University has donated the network facilities for ISGEm.

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Special Interest Group (SIG) on Research in Culturally Diverse Environments

Luis Ortiz-Franco

A total of ten people attended the meeting of the SIG on Research in Cultural Diverse Environments in Quebec, during ICME-7. Seven of these people were new members of ISGEm and the countries represented among them were Australia, England, Guatemala, Mozambique, South Africa, Sweden, and the United States. Each member introduced himself/herself to the group and gave a brief description of the project(s) in which they were currently working on. Those members who had an article to share with the group provided a copy to the SIG coordinator for distribution to the other members at a later time. Some of these papers are quite interesting. For example:

One article deals with the cultural heritage of the teaching of mathematics. The article frames the discussion of teaching the division and multiplication algorithm in an historical context by providing examples of the different ways that these algorithms are performed in different countries. The exploration of these issues was motivated by the fact that many immigrant students come from countries where these algorithms are performed quite differently from the way they are performed in the host country.

Another paper for distribution is a draft of a proposal to investigate the ethnomathematics of women subsistence farmers in Mali, West Africa. The author proposes to approach her research from three perspectives: the mathematics used by women farmers; the mathematics used by the Malian government and international agencies in their policies as they affect the lives of women farmers; and the mathematics used to develop an alternative economic model.

Two other people described some of their work dealing with preColumbian mathematics and the mathematics evident in the arts, crafts, and weaving among present day Mayan communities in Guatemala. There were descriptions of projects in other parts of the world but no written documents were made available for distribution.

In short, the meeting of the SIG on Research in Diverse Cultural Environments was very stimulating because it showed that there is empirical and bibliographic research in this area going on all over the world.

The SIG coordinator is communicating with all those people who attended the SIG meeting in Quebec. Copies of the articles and other literature given to the coordinator will be shared with all the SIG members.

In addition, the SIG on Research in Culturally Diverse Environments has proposed a session for the NCTM Annual Meeting in 1994 in Indianapolis. The topic of the proposed session is "Ethnomathematics and Teacher Education."

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Comment from SIG on Curriculum & Classroom Activities

Lawrence Shirley

Members may have noticed that many universities, school districts, individual schools, and other educational institutions have recently become increasingly concerned with strengthening the multicultural aspects of their programs. Many are establishing committees to help broaden the scope of their curricula. Often these efforts are directed toward social studies, literature, art, music, and other areas of the social sciences and humanities.

Many don't even think of culture interacting with math and science and hence overlook possible input from ethnomathematics and ethnoscience. Here is an opportunity for ISGEm members to both assert ourselves and serve our communities. We can let these committees know that ethnomathematics does indeed exist and can contribute both content and philosophy to multicultural programs. This is a real way ethnomathematicians can guide and influence important curricular decisions.

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ICME-7 Working Group 10 on Multicultural/Multilingual Classrooms

The following report on the activities of ICME-7 Working Group 10 will appear in the ICME-7 Proceedings to published soon.

Organizing Team: Patrick (Rick) Scott [USA], Elisa Bonilla [Mexico], Lloyd Dawes [Australia], Martha Villavicencio [Peru], Tom O'Shea [Canada].

The first session was plenary in which many participants presented posters describing their work in multicultural/multilingual classrooms, or materials designed for such classrooms. (Titles of posters, and names and addresses of presenters appear at the end of this article.) The next two sessions were discussions in the subgroups indicated below. The final session was plenary with reports from the subgroups and accompanying discussion.

Subgroup 1: Curriculum, Resources, and Materials for Multicultural/Multilingual Classrooms (Vera Preston [USA] - Presider)

How can different cultures in one classroom be recognized and taken into account and what role should Ethnomathematics play in multicultural and multilingual classrooms?

Students might write a paper about their cultural background, using topics from the history of mathematics about people from their culture, or by using the library or family resources. They may review movies related to mathematics (example: Stand and Deliver) or write about people from their own particular life experiences. By working across the curriculum using interdisciplinary procedures more material in the syllabus is covered and creative methods can be utilized. In educating student teachers have discussions with them about the problems they might encounter with students who are underprepared in mathematics. Also discuss with them ways in which they might teach underprepared students from different cultures. Other techniques which might be used in multicultural classrooms are the use of manipulatives, technology, collaborative or cooperative learning.

What is multicultural mathematics and how do you implement it?

An aspect of multicultural mathematics is the historical developments of mathematics in different cultures (e.g. the Mayan numeration system). Another aspect could be prominent people in different cultures who use mathematics (e.g. an African-American biologist, an Asian-American athlete). Mathematical applications can be made in cultural contexts (e.g. using fractions in food recipes from different cultures.) Social issues can be addressed via mathematics applications (e.g. use statistics to analyze demographic data.) Multicultural mathematics materials can be integrated into the regular instructional program, and personalized activities can be done that are related to different cultures and draw on students own experiences.

What are the issues to which we need to be sensitive?

Cultural-related differences in learning styles can present challenges. For example, Asian students are sometimes reluctant to ask a question in class and some American Indian students are reluctant to look teachers in the eye because it is considered rude. However, there should be caution in attempting to meet specific needs of a particular group of students as generalizations can be misleading and potentially damaging. There are wide ranges of ability and learning style within any group of people. Teachers need to be sensitive to the learning styles of all students in their classrooms regardless of their ethnic group. There is often teacher and student resistance to emphasizing cultural differences. Immigrant students may have learned different mathematics content or been in classrooms with different teaching styles. Teachers need to be aware that different algorithms are used for the basic arithmetic operations. Multilingual textbooks might present ideas in multiple languages or use icons (word-free symbols) for some kinds of problems and directions.

What multicultural mathematics is appropriate for majority culture classrooms?

While multicultural perspectives should be infused throughout the curriculum for all students, trivialization of both the multiculturalism and the mathematics must be avoided. Textbooks and curriculum materials need to reflect multicultural perspectives. And commercial materials will change only as teachers become persistent in requesting changes.

Subgroup 2: Teacher Education for Mathematics in Multicultural/Multilingual Classrooms (David Davison [USA] - Presider)

Multicultural perspectives are for all students.

One misconception addressed by the group was that African mathematics is just for black students, South American mathematics is just for Hispanic students, etc. In a pluralistic society, all students need to be exposed to multicultural aspects of mathematics as part of having them interact with students from a variety of cultures. Multicultural aspects of mathematics should be blended throughout the mathematics curriculum irrespective of minority culture. It was agreed that such an approach would lead to a broader view of the role of mathematics as a cultural phenomenon, and enhance the view of mathematics as a systematic body of knowledge.

What multicultural mathematics should be taught?

The group agreed that multicultural mathematics should not be taught as a form of mathematical oddity, or as a relief from the real mathematics of the curriculum. Topics such as symmetry and its application in a variety of cultures, the development of number in various cultures, were cited as examples that could be integrated into the regular curriculum.

Implications for Teacher Education

To accomplish the curricular goals identified above, changes are needed in teacher education programs. Multiculturalism should be embedded within the teacher education curriculum, and not taught as a separate strand. In this way, multiculturalism is not trivialized. Also, by using present-day real-life situations to reflect cultural diversity among students, mathematics is not trivialized. One approach to restructuring the mathematics curriculum is to focus on the diverse algorithmic techniques used around the world, for example, the different methods for subtracting whole numbers.

Subgroup 3: Multicultural/Multilingual Classrooms for the 21st Century (Jan Thomas [Australia] - Presider)

Acknowledgement of the cultural origins of mathematics.

A number of issues affecting outcomes were noted including racism, conflicting values and the role of parents. The gap between the third world and industrialized nations was increasing, not decreasing. The impact of global migration on all countries and the world-wide economic situation made defining concrete solutions of mathematics curriculum for the 21st Century a complex task. It was noted that Sweden and Germany were experimenting unprecedented immigration from non-native speakers for which the education systems were ill-prepared. The language support available in countries like Australia had been developed over a long time. The difficulties of poorer countries in dealing with the effects of migration - both incoming and sometimes the loss of talented people -are even greater.

The possible need to "reconstruct" mathematics if more equitable outcomes are to be achieved was discussed. The group accepted that while it may be helpful if some aspects of the formal mathematical culture did change (e.g. the language and values associated with mathematics), this will only happen from within. A model was developed that takes into account that students enter school with mathematical knowledge that is mainly informal. It can be considered ethnomathematics in that it is culturally and socially defined. That informal mathematics touches formal mathematics culture. Within the model, the minimum expectation (for life) should be that everyone can learn and appreciate the power of mathematics.

Goals for students include mathematics for everyday life, citizenship, employment and as a profession (love).

Only in the last case was it seen that the student needs to access fully formal mathematics culture, although it would be an objective that all students appreciate the power and the beauty of mathematics. The mathematics needed for everyday life is culturally and socially defined. It was noted that someone living in Papua New Guinea may be able to use traditional ways of measuring land if they stay in their village. Universal conventions may be needed away from the village or in dealing with visitors. Curriculum needs to take account of this. It was also noted that all aspects of mathematics curriculum must take account of calculators as they are beginning to impact on poorer nations as well as wealthier ones. Mathematics for citizenship was seen as that mathematics needed to be an effective participant in community and political decision making. The importance of statistics in this was noted. It was also argued that the mathematics curriculum needs to play a more active part in helping people make informed political decisions and that students need to be able to use mathematics in developing just and democratic societies. Mathematics for employment was seen as being defined by the type of employment. Curriculum for the 21st Century should be constructed in such a way as linguistic and cultural impediments do not prevent any students from life long learning that enables them to use mathematics personally, as a citizen and for employment. And those with a love and talent for mathematics should have pathways to the universal mathematics culture.

Need for effective leadership, respect for culture and authentic assessment.

Effective leadership was seen as essential in forming community and political views of education in establishing goals which foster mathematics learning. Account must be taken of cultural views of education including attitudes to achievement in mathematics, its perceived economic role, the effect of parental views of mathematics and the level of parental education. The knowledge the child brings to the classroom (mathematical, linguistic, social, etc.) is powerful and the mathematics curriculum must build on that existing base. Associated with learning is the need for accountability and authentic assessment which should support, but not drive, the curriculum.

Subgroup 4: Language and culture in the Mathematics Education of Indigenous Groups (Elisa Bonilla [Mexico] - Presider)

Need to preserve cultural values

There was general agreement on the need to preserve cultural values and the need to uncover one's cultural roots was stressed. One illustration of this was the learning of mathematics through the use of the ancient Bolivian flute, the zampoña, which serves as a means to uncover diverse mathematical relations that the construction and use of this instrument entails. Another illustration was an investigation of Bishop's six universal activities (counting, locating, measuring, designing, playing and explaining) in the highlands in the south of Mexico with the aim to use them in curriculum development for the primary school of one Mixe Community.

The importance of teaching and learning mathematics in the first language.

Participants explained about situations, past and present, where speaking the first language is a matter of embarrassment or an excuse for punishment. Hence the pressure to reinforce cultural values, particularly in regard to first language. The need to support teachers was a central issue. Often - it was said - teachers are resistant to change, they are particularly resistant to teach in the first language, and mostly their resistance seems to stem from the fact that thy did not do their schooling in their first language and thus do not see the necessity for it. On the other hand, the efforts made in some communities to teach in their first language right up through the tertiary level were put forward. Particularly members of the Welsh (U.K.) and Inuit (Canada) communities explained the need to establish teacher education programs in students' first language and how they have accomplished this. Although there was agreement on the need to promote bilingualism, there was not necessarily agreement on how to attain it. Mastering the first language and then going on to the second was one suggestion, with the learning of mathematics starting with the first language. However, learning both languages simultaneously and teaching mathematics simultaneously in both languages was also suggested.

Adding mathematical words to languages.

Much time was spent on the lack of certain mathematical words in particular languages. This strongly came out as another central issue for this subgroup. Various examples were offered and three methods were suggested for the incorporation of new words: 1) adapt existing words (which is acceptable as long as the mathematical meaning appears to be "close enough" to the everyday use), 2) borrow words from another language (like most languages have done for terms that refer to computer science), or 3) invent words (like French, in particular, has done for terms that refer to computer science). As to the question of how to make new words acceptable to teachers and the community, the benefits of engaging the elderly, who are often very knowledgeable about the language and widely respected, were highlighted. There are also some Maori (New Zealand) communities which have a language commission that helps in these matters.

An Annotated Bibliography of Multicultural Issues in Mathematics Education is available from Patricia S. Wilson, Mathematics Education, University of Georgia, 105 Aderhold Hall, Athens, GA 30602, USA.

Posters Presented to ICME-7 WG10

Teacher Preparation and the Multicultural Classroom, Sunday Ajose, Math Department, East Carolina U, Greenville, NC 27858 USA

Mathematics in the Mixe Language, Isaías Aldaz, M. Bravo No. 616-A, Morelos 1008, CP 68000, Oaxaca, OAX, MEXICO

Bridging the Language Gap in Multicultural Classrooms: An American Indian Perspective, David Davison, 1809 Sagebrush Road, Billings, MT 59105 USA

El Arte Nativo para una Enseñanza Efectiva - La Zampoña, Victor de la Fuente, Mayor Rocha 337, Cochabomba, BOLIVIA

Definition and Drawing of a Cube in Somalia, Franco Favilli and Vinicio Villani, Dipartimento de Matematica, Universita di Pisa, 56100 Pisa, ITALY

Mental Arithmetic and Language Structure for Teacher Education: A Research Proposal, Abdulcarimo Ismael, Departamento de Matematica, Instituto Superior Pedagogico, P.O. Box 2923, Maputo, MOZAMBIQUE

Children's Work on Games and Number Systems, Helen Jenner, 6 Driffield Road, London E3 5NF, UNITED KINGDOM

The Cultural Heritage of Teaching Mathematics, Kurt Paulsson, Department of Mathematics, Stockholm Institute of Education, Box 34103, 10026 Stockholm, SWEDEN

Cultural Diversity and Bonding in Mathematics, Eileen Poiani, St. Peter's College, 2641 Kennedy Blvd., Jersey City, NJ 07306 USA

A Book for Grade 5 in German, Croatian and Turkish, Fritz Schweiger, Institut für Didaktik der Naturwissenschaften, Universitat Salzburg, 5020 Salzburg, AUSTRIA

A Proposed Course Outline: Multicultural Perspectives on Mathematics and Mathematics Teaching, Patrick Scott, College of Education, University of New Mexico, Albuquerque, NM 87131 USA

African Mathematics Examples for the Middle School Classroom, Lawrence Shirley, Department of Mathematics, Towson State University, Towson, MD 21204 USA

Language Background and Student Participation in Tertiary Mathematics in Australia, Jan Thomas, Teacher Education, Victoria University of Technology, P.O. Box 64, Footscray, VIC 3011, AUSTRALIA

Bilingual Materials from Peru, Martha Villavicencio, Calle General Varela 598, Depto. C, 18 Miraflores, Lima, PERU

Multicultural Mathematics Materials from Addison-Wesley, Claudia Zaslavsky, 45 Fairview Ave, Apt 13-A, New York, NY 10040 USA

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ICME-7 Topic Group on Ethnomathematics & Math Education

Topic Group 2 "Ethnomathematics and Mathematics Education" met for two sessions at ICME-7 in Québec in August 1992. The first session of the Topic Group featured five reports on the situation around the world. The second session consisted of reports on the posters that presented information of relevance to Ethnomathematics. In addition to the report in the ICME-7 Proceedings, Ubiratan D'Ambrosio is coordinating the publication of a more complete report on the activities in Working Group 2.

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XIXth International Congress of History of Science

The XIXth International Congress of History of Science will be held from August 22-29, 1993 in Zaragoza, Spain. For further information contact:

Prof. Mariano Hormigón

Facultad de Ciencias (Matemáticas)

Ciudad Universitaria

50009 Zaragoza, SPAIN

E-mail: ICHS@cc.UNIZAR.ES

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Equity Studies

Gloria Gilmer prepared the following bibliography of Equity Studies for the 1992 NCTM Research Presession.

Au Kit-fong, Terry, and Harackiewicz, Judith M. (1986). The effects of perceived parental expectations on Chinese children's mathematical performance. Merrill-Palmer Ouarterly, 32, 383-392.

Band, Steven E. (1987). Children's self-attributions for success and failure on a mathematics achievement task. Dissertation Abstracts International, 47, 2950A. (University Microfilms No. 86-28203)

Bernal, Ernest M., Jr. (1977). Introduction: Perspectives on nondiscriminatory assessment. In Thomas Oakland (Ed.), Psychological and educational assessment of minority children (pp. xi-xiv). New York: Brunner/Mazel, Inc.

Bridges, Robert E. (1988, February). Black male child development: A broken model. Paper presented at the 120th annual convention of the American Association of School Administrators, Las Vegas, Nevada.

Buriel, Raymond, and Cardoza, Desdemona. (1988). Sociocultural correlates of achievement among three generations of Mexican American high school seniors. American Educational Research Journal, 25, 177-192.

Clark, Lesley A., and Halford, Graeme S. (1983). Does cognitive style account for cultural differences in scholastic achievement? Journal of Cross-Cultural Psychology, 14, 279-296.

Cooper, Kenneth J. (1989, December 19). Study foresees shortfall in technical skills. The Washington Post, p. A-6.

Corcoran, Thomas B., Lisa J. Walker and J. Lynne White. (1988). Working in urban schools. Washington, D.C.: Institute for Educational Leadership.

Crandall, J., Dale, T.C., Rhodes, N.C. and Spanos, G. (1987). English language skills for basic algebra. Englewood Cliffs, NJ: Prentice Hall.

Creswell, John L. (1983, Winter). Sex-related differences in the problem-solving abilities of rural back, anglo, and chicano adolescents. Texas Tech Journal of Education, 10(l), 29-33.

Creswell, John L. and Roxane H. Exezidis. (1982, Fall). Research brief - Sex and ethnic differences in mathematics achievement of black and mexican-american adolescents. Texas Tech Journal of Education, 9(3), 219-222.

Cummins, Jim. (1986). Empowering minority students: A framework for intervention. Harvard Educational Review, 56, 18-36.

Davis, Robert B. (1989). The culture of mathematics and the culture of schools. The Journal of Mathematical Behavior, 8, 143-160.

Duran, Richard P. (1988). Bilinguals' logical reasoning ability: A construct validity study. In Rodney R. Cocking and Jose P. Mestre (Eds.). Linguistic and cultural influences on learning mathematics (pp. 241-258). Hillsdale, N.J.: Erlbaum.

Elsholy, Richard, and Elsholy, Ellyin. (1989). The writing process: A model for problem solving. The Journal of Mathematical Behavior, 8, 161-166.

Espinosa, Ruben, and Ochoa, Alberto. (1986). Concentration of California Hispanic students in schools with low achievement: A research note. American Journal of Education, 95, 77-95.

Fauth, Gloria C., and Jacobs, Judith E. (1980). Equity in mathematics education: The educational leader's role. Educational Leadership, 37, 485-490.

Fennema, Elizabeth, and Sherman, Julia (1977). Sex-related differences in mathematics achievement, spatial visualization and affective factors. American Educational Research Journal, 14, 51-71.

Fischer, Florence E. (1988). The development of number concept using two curriculum approaches. Unpublished doctoral dissertation, University of Maryland at College Park, Maryland.

Fulton-Scott, Merle J., and Allen, Calvin D. (1983). Bilingual multi-cultural education vs. integrated and non-integrated ESL instruction. NABE: Journal for the National Association for the Bilingual Education, 7(3), 1-12.

Gerdes, Paulus. (1988). On culture, geometrical thinking and mathematics education. Educational Studies in Mathematics, 19, 137-162.

Ginsburg, Herbert. (1972). The myth of the deprived child: Poor children's intellect and education. Englewood Cliffs, NJ: Prentice-Hall.

Ginsburg, Herbert P., and Russell, Robert L. (1981). Social class and racial influences on early mathematical thinking. Monographs of the Society for Research in Child Development, 46 (6, Serial No. 193).

Goldstein, Amy. (1989, May 23). Study confirms drop in scores. The Washington Post, p. B-5.

Goldstein, Amy. (1990, January 10). Montgomery school study to analyze minority achievement. The Washington Post, p. C-4.

Hess, Robert D., Chik-Mei, Chang, and McDevitt, Teresa M. (1987). Cultural variations in family beliefs about children's performance in mathematics: Comparisons among people's Republic of China, Chinese-American and CaucasianAmerican families. Journal of Educational Psychology, 79,179-188.

Heyns, Barbara, and Catsambic, Sophia. (1986). Mother's employment and children's achievement: A critique. Sociology of Education, 59 (3),140-151.

Hollander, Sheila K. (1988). Teaching learning disabled students to read mathematics. School Science & Mathematics, 88, 509-515.

Holloway, Susan D. (1986). The relationship of mothers' beliefs to children's mathematics achievement: Some effects of sex differences. Merrill-Palmer Quarterly, 32, 231-250.

Jones, Lyle V. (1984). White-Black achievement differences: The narrowing gap. American Psychologist, 39, 1207-1213.

Kirk, Girvin E., Hunt, J. McVicker, and Volkmar, Fred (1975). Social class and preschool language skill: V. Cognitive and semantic mastery of number. Genetic Psychology Monographs, 92,131-153.

Kohr, Richard R. (1988, April). The influence of race, class and gender on the mathematics achievement and self-esteem for fifth eighth and eleventh grade students in Pennsylvania schools. Paper presented at the annual meeting of the American Educational Research Association. Washington, D.C.

Kroll, Diana, and Yabe, Toshiaki. (1987). A Japanese educator's perspective on teaching mathematics in the elementary school. Arithmetic Teacher, 35(2), 36-43.

Laosa, Luis M. (1977). Nonbiased assessment of children's abilities: Historical antecedents and current issues. In Thomas Oakland (Ed.), Psychological and educational assessment of minority children (pp. 1-20). New York: Brunner/Mazel, Inc.

Martin, Joanne Mitchell. (1986). The effects of a cooperative, competitive or combination goal structure on the mathematics performance of Black children from extended family backgrounds. Dissertation Abstracts International, 48, 2024A-2025A. (University Microfilms No. 87-19762)

Martinez, Diana I., Bernardo R. Ortiz de Montellano. (1988, March). Improving the science and mathematic achievement of Mexican American students through culturally relevant science. Las Cruces, New Mexico: ERIC Clearinghouse on Rural Education and Small Schools. (ERIC Document Reproduction Service no. EDO-RC-88-07)

Mathematical Sciences Education Board. (1990). Synthesized action plans from the six regional workshops. Making Mathematics Work for Minorities.

McLoughlin, James A., and Lewis, Rena B. (1990). Assessing special students. Columbus, OH: Merrill.

McNamee, G.D., Katz, L.L., and Bowman, B.T. (1981, April). Mathematical development in low income black preschool children. Paper presented at the meeting of the Society for Research in Child Development, Boston.

Mermelstein, Egon, and Shulman, Lee S. (1967). Lack of formal schooling and the acquisition of conservation. Child Development, 38, 39-52.

Milne, Ann M., Myers, David E., Rosenthal, Alvin S., and Ginsburg, Alan. (1986). Single parents, working mothers, and the educational achievement of school children. Sociology of Education, 59 (3),125-139.

Minato, Saburoh, and Yanase, Shyouchi. (1984). On the relationship between students attitudes towards school mathematics and their levels of intelligence. Educational Studies in Mathematics, 15, 313-320.

Minorities and Mathematics. (1984). Journal for Research in Mathematics Education (Theme Issue). 15.

Moore, Elsie, and Smith, A. Wade (1985). Mathematics aptitude effects of coursework, household language & ethnic differences. Urban Education, 20, 273-294.

Neves, Maria Apparecida Mameda and Fraga, Maria Lucia. (1987). Teaching of mathematics and development of operating cognitive structures - An experiment with Brazilian low-income children. In Jacques C. Bergeron, Nicolas Herscovics and Carolyn Kieran (Eds.), Proceedings of the 11th International Conference for the Psychology of Mathematics Education (Volume III) (pp. 17-22). Montreal: International Group for the Psychology of Mathematics Education.

Oakland, Thomas, and Matuszek, Paula. (1977). Using tests in nondiscriminatory assessment. In Thomas Oakland (Ed.), Psychological and educational assessment of minority children (pp. 52-69). New York: Brunner/Mazel, Inc.

Oar, Eleanor Wilson. (1987). Twice As Less: Black English and the performance of black students in mathematics and science. New York: Basic Books.

Pimp, David (1987). Speaking mathematically: Communication in the mathematics classroom. New York: Routledge & Kegan Paul.

Presmeg, Norma C. (1988). School mathematics in culture-conflict situations: Towards a mathematics curriculum for mutual understanding when diverse cultures come together in the same classroom. Educational Studies in Mathematics, 19, 163-177.

Ramsey, Patricia G. (1987). Teaching and learning in a diverse world: Multicultural education for young children. New York: Teachers College Press.

Reyes, Laurie Hart, and Stanic, George, M.A. (1988). Gender and race: Equity in primary and in middle school mathematics classrooms. Arithmetic Teacher, 35 (8), 46-48.

Reyes, Laurie Hart, and Stanic, George, M. A. (1988). Race, sex, socioeconomic status and mathematics. Journal for Research in Mathematics Education, 19, 26-43.

Research Advisory Committee of NCTM. (1988). NCTM Curriculum and Evaluation Standards for School Mathematics: Responses from the research community. Journal for Research in Mathematics Education, 19, 338-344.

Sagan, Carl. (1989, September 10). Why we need to understand science. Parade Magazine, p. 6-10.

Secada, Walter G. (1988) Diversity, equity and cognitive research. In Elizabeth Fennema, Thomas P. Carpenter and Susan J. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 20-58). Madison, WI: University of Wisconsin - Madison, National Center for Research in Mathematical Sciences Education.

Secada, Walter G. (1988a, May). The mathematics education of Hispanic students: Towards a research based vision of the possible. Paper presented at Symposium: Equity Issues in Mathematics and Science Achievement, Culver City, CA.

Secada, Walter G. (December, 1989). Parental involvement in a time of changing demographics. Arithmetic Teacher, 37(4), 33-35.

Secada, Walter G. (1989). Educational equity versus equality of education: An alternative conception. In Walter G. Secada (Ed.) Equity in Education, pp. 89-102, Pennsylvania, PA: The Falmer Press.

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Sowder, Judith and Sowder, Larry. (1988). The use of verbal explanation in Japanese and American classrooms. Arithmetic Teacher, 36 (2), 27-29.

Stedman, Lawrence C. (1987). It's time we changed the effective schools formula. Phi Delta Kappan, 68, 215-223.

Stenmark, Jean Kerr. (1989). What is mathematics equity? Arithmetic Teacher, 36 (5), 3.

Stigler, James W., and Perry, Michelle. (1988). Cross-cultural studies of mathematics teaching and learning recent findings and new directions. In Douglas A. Grouws & Thomas J. Cooney (Eds.)., Perspectives on research on effective mathematics teaching, (volume 1) (pp. 194-223). Reston, VA: National Council of Teachers of Mathematics.

Stodolsky, Susan S. (1985). Telling math: Origins of math aversion and anxiety. Educational Psychologist, 20, 125-133.

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Willig, Ann C., Harnisch, Delwyn L., Hill, Kennedy T., and Maehr, Martin L. (1983). Sociocultural and educational correlates of success-failure attributions and evaluation anxiety in the school setting for black, Hispanic and Anglo children. American Educational Research Journal, 20, 385-410.

Yando, R., Seitz, V., and Zigler, E. (1979). Intellectual and personality characteristics of children: Social class and ethnicgroup differences. Hillsdale, NJ: Erlbaum.

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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.

Schniedewind, Nancy and Davidson, Ellen. Open Minds to Equality: Learning Activities to Promote Race, Sex, Class, and Age Equity, Ellen Davidson, 30 Walnut St, Somerville, MA 02143, USA, 1992, US$20.00.

This is a sourcebook that enables educators working with elementary and middle school students to help them recognize inequalities based on sex, race, age, and class and move towards changing them. The book promotes academic and interpersonal equity among students -- making a classroom a more fair and cooperative learning environment.

The benefits to all students -- minorities and whites, females and males -- of changing inequities are actively examined. Young people gain new knowledge, explore their feelings and attitudes, and better understand the life experiences of others.

Learning activities are designed sequentially as students first develop trust, communication, and cooperation. Next, they learn to recognize stereotypes, discrimination and the isms, and explore the effects of discrimination on people's lives. Using the school, community, and media as laboratories to examine examples of possible institutional discrimination, students create change, gain self-confidence, and experience collective responsibility.

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Gilmer, Gloria; Soniat-Thompson, Mary; and Zaslavsky, Claudia. Building Bridges to Mathematics: Cultural Connections. Addison-Wesley Publishing Company, 1992.

Building Bridges to Mathematics: Cultural Connections is a collection of Multicultural Mathematics Activity Cards. The cards present situations and activities to student and then suggest projects for students to carry out in cooperative groups. The cards also give students suggestions to "Talk and Listen", "Plan and Act", and "Share Your Work". Extensions to the basic project are also provided.

Among the topics presented on the cards are: "Wampum Belts", "The Japanese Abacus", "Panpipes in the Andes", "Granville T. Woods, Inventor", "Pascal's Triangle in China" and "Art in the Southwest". Sets of cards are available to supplement mathematics instruction at various grade levels.

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Gerdes, Paulus. On Ethnomathematical Research and Symmetry, Symmetry: Culture and Science, vol. 1, no. 2, 1990, p. 154-170.

In this article Paulus Gerdus addresses three general questions:

Question 1: What is symmetry and why do symmetrical shapes appear so often in the world?

Question 2: What are some of the links between the ethnomathematical research on symmetry and other scientific and/or cultural spheres.

Question 3: Why do symmetries occur, why are they culturally valued, how can they be incorporated in the teaching of symmetry in particular and of geometry in general, and how can their mathematical potential be explored?

Each of these questions is explored with numerous examples from cultures throughout the world.

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Gerdes, Paulus, Fivefold Symmetry and (Basket) Weaving in Various Cultures, in I. Hargittai (ed.), Fivefold Symmetry, World Scientific, PO Box 128, Singapore 9128, 1992.

Although many have thought that "(fivefold) symmetry and other basic geometrical ideas arose in human culture as a blind copy of symmetry and physical form in nature", Gerdus suggests to the contrary that "regularity and symmetry of man-made objects are the result of creative human labor." To illustrate his point he discusses pentagonal-hexagonal baskets from Mozambique, pentagonal thimbles from Indonesia, semiregular pentagonal knots in handbags and shoulder bags from Mozambique, brooms from Mozambique, woven hats from Central Timor, ornamental pentagonal spiral patterns in Kenyan baskets, Chinese hats, Papago Indian baskets, and Japanese baskets.

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ISGEm Advisory Board

Gloria Gilmer, President
Math Tech, Inc.

Ubi D'Ambrosio, 1st Vice President
Universidade Estadual de Campinas

Alverna Champion, 2nd Vice President
Wyoming, MI 49509 USA

Luis Ortiz-Franco, 3rd Vice President
Chapman University

Claudia Zaslavsky, Secretary
New York, NY 10040 USA

Anna Grosgalvis, Treasurer
Milwaukee Public Schools

Patrick (Rick) Scott, Editor
College of Education, U of New Mexico

Henry A. Gore, Program Assistant
Dept of Mathematics, Morehouse College

David K. Mtetwa, Member-at-Large
Marlborough, Harare, ZIMBABWE

Lawrence Shirley, Member-at-Large
Dept of Mathematics, Towson State U