Volume 13 Number 1 November 1997
Report on Ethnomathematics Research
Joanna O. Masingila, Syracuse University
This column reports on current research in the area of Ethnomathematics. If you know of researchers doing Ethnomathematics research, please send Joanna Masingila this information: 215 Carnegie, Syracuse, NY 13244-1150 USA (email@example.com).
Judit Moschkovich, from the Institute for Research on Learning, and colleagues have been investigating the mathematical activity of people in a variety of work situations. One recent study involved the mathematical activities in the work of insurance agents. Moschkovich and her colleagues found that these practices do not involve only arithmetic computation but also estimating, using heuristics, explaining complex relationships between quantities, and describing diagrams.
Jack Smith, from Michigan State University, is examining the mathematical demands of "blue collar" work in various workplaces involved in automobile manufacturing. His work has been oriented by current concerns that U.S. high school graduates are often ill-prepared for high-skill, high-wage work in a variety of industries. Smith has studied what sort of mathematical knowledge and skills high school graduates need to perform competently in the automobile industry. Two main findings emerged from this research: (1) the demands of high-volume assembly work appear well within the current content of the K-12 curriculum, while (2) more challenging and rewarding work, like computer-mediated machine tool operation, requires extensive spatial and geometric competence that the U.S. curriculum does not strongly support.
Norma Presmeg, from Florida State University, is working on developing a theoretical framework, drawing on literature from semiotics and Ethnomathematics, to address the ways in which real experiences and cultural practices of students may be connected with mathematics classroom pedagogy.
José Fonseca, from the University of Arizona, is involved in a project that is using children's out-of-school knowledge to teach school mathematics. As part of an earlier study, Fonseca and colleagues found that 60% of the students at a bilingual middle school had some knowledge of and skills related to construction. Fonseca designed an instructional unit in which students develop model houses (designing, drawing, and constructing) and learn mathematical concepts and process through this context. He is examining how the students' out-of-school experiences were connected to learning academic mathematics.
ISGEm at NCTM in Washington
ISGEm's NCTM Delegate
I have had a reply from my request for room, time, and program listing for the NCTM April Conference in Washington. They confirmed that they will list us in the program, and that we will get a room that Saturday evening. However, since the conference is not going on into Sunday, the regular sessions may run later into Saturday evening. We probably will have to start about 5:00 or 5:30, after the last Saturday afternoon sessions are over. They could not confirm an exact room or time yet and probably won't until January. Hence, at the time all we can report is that we will be meeting that Saturday evening, and tell readers to check in the NCTM program book for the exact time and place.
Meanwhile, they have noted that our resolution has been received. The committee will be looking at all of the submitted resolutions over the next month or so, to determine their status.
Special Meeting to Honor Ubiratan D'Ambrosio
The International Study Group on the Relations between History and Pedagogy of Mathematics (HPM) and the International Study Group on Ethnomathematics (ISGEm) will jointly sponsor a conference in the honor of the 65th birthday of Ubiratan D'Ambrosio, professor of mathematics emeritus at the University of Campinas in Brazil, to take place on Tuesday, January 6, l998 at the Omni Hotel in Baltimore, Md. USA. This is the day preceding the opening of the annual joint meeting of the American Mathematical Society and the Mathematical Association of America. We welcome the attendance of all mathematicians and mathematics educators who wish to honor Professor D'Ambrosio. Most of us are aware of Professor D'Ambrosio's influence in the areas of Ethnomathematics, the history of mathematics and mathematics education.
To register for the conference, send your name, addresses (mail and e-mail) and phone numbers, along with a check for US$50 to Karen Dee Michalowicz, Treasurer Americas Section, 5855 Glen Forest Drive, Falls Church, VA 22041 USA.
The check should be payable to HPM. The fee is to primarily cover the cost of a festive birthday dinner in the evening.
An Investigation of Muskogee Creek Indian Counting Words
Southern Connecticut State University
About the Source
Except where noted, the main source for this paper was a Muskogee Creek Indian named Cheneya. She provided historical information about her tribe, as well as comments about their language. Cheneya provided me with a lengthy list of Muskogee Creek counting words, which we analyzed together.
This paper was originally prepared as part of the World Counts symposium that was organized at Teachers College, Columbia University by Joel Schneider.
Background of the Tribe and Language
The Muskogee Creek Indians originated in Georgia and Alabama, and are related to the Seminoles. Approximately 30,000 are listed on the tribal rolls, including those who are not full-blooded Creek. Less than 7,000 still speak the language (Census Bureau, 1990), which contains dialects and a ceremonial language as well. The written language was developed in 1853. Prior to that, pictures were used to express words. Seven letters are not used in their alphabet: b, d, g, j, q, x and z. The Creek use the letters 'k' and 'v' frequently; 'v' sounds like 'u' in 'put'. Hence, one spelling of the tribe would be Mvskogee, but there are others. In this paper, we will use 'Muskogee' for simplicity, but all other words will be spelled as they appear in the language.
Forms of the First 10 Counting Words
In 1775, James Adair published his History of the American Indians, in which he lists the first ten counting words in the Muskogee Creek language. Below we compare them with the modern list supplied by Cheneya :
|Adair (1775)||Cheneya (1997)|
There are similarities between most of the words. Of course, we must remember Adair wrote down what he heard, not what he read.
"I am sorry that I have not sufficient skill in the Mushohge dialect, to make up useful observations on this head," Adair wrote (Williams, 1986)
The Muskogee employ a base ten counting system, as can be seen by examining numbers above ten. For example, 'eleven' is palen-hvmken-tvlaken (hyphens are mine). The words for 'ten' and 'one' are clearly evident. Tvlaken is a form of 'great', hence 'eleven' is translated as 'one greater than ten'. Interestingly, the idea of 'greater than' is discontinued with numbers from 'twelve' to 'nineteen', and a more colorful expression is used. 'Twelve' is palen-hokkol-ohkaken, which contains the words for 'ten' and 'two', and thirteen is palen-tutcen-ohkaken, which contains the words for 'ten' and 'three'. All the words from 'twelve' to 'nineteen' end with oskaken, which means 'sit upon'. Hence, thirteen is translated as 'three sit upon ten'.
Multiples of ten are formed by using pale with the appropriate counting word from 'one' to 'nine', hence, 'twenty' becomes pale-hokkolen, or literally, 'two tens', as the words are read from right to left. 'Thirty' is pale-tutcenen, and so on. 'Twenty-two' becomes pale-hokkolen-hokkol-ohkaken, or literally, 'two sit upon two tens'.
The word for 'one hundred' is cukpe-hvmken. Since cukpe means 'a hundred' and hvmken means 'one', the complete word is translated as 'one a hundred'. Similarly, 'two hundred' is written and translated as 'two a hundred'.
For 'one thousand', cukpe-rakko-hvmken, a new root word is introduced. The words for 'one hundred' and 'one' are evident, but I wondered if rakko had some significance. Cheneya supplied the answer, translating rakko as 'large'; hence, 'one thousand' is essentially 'one large hundred'. 'Ten thousand' is cukpe-rakko-palen, that is, 'ten large hundreds'. 'One hundred thousand' is cukpe-rakko-cukpe-hvmken, which is 'one hundred large hundreds'.
When we reach 'one million', which is cukpe-rakko-vcule-hvmken, we can see 'one hundred', 'large' and 'one'. Again, Cheneya provided a translation for the word vcule: 'old' or 'aged'. Thus 'one million' is translated as 'one old large hundred'. Using the word 'old' or 'aged' to amplify the size of a number is not unique: in the Cherokee language, Adair translated 'one thousand' as 'the old one's hundred' (Williams, 1986).
Unanswered Questions and Conclusions
The words 'six' through 'nine' contain paken, which could suggest a form of counting up from 'five' or down from 'ten'. I asked Cheneya if the word had some significance (since it is similar to 'ten', palen), but she had no idea. Also, it was not clear to Cheneya why the word for 'eleven' uses the idea of 'one greater than' while 'twelve' through 'nineteen' use the idea of 'sit upon'.
This was an interesting collaboration between two cultures. As I looked for patterns in the number words, Cheneya reflected on her own language and was able to view her words in a different light often mentioning she had never thought of the words 'that way' before. Without her help, I would not have been able to give the appropriate meaning to many of the number words.
Williams, S. (ed.). (1986). Adair's History of the American Indians, New York: Promontory Press.
Mathematics and Crafts in Andalusia: An Anthropological-Didactic Study
María Luisa Oliveras Contreras
In this paper I shall attempt to set out the work plan and the methodology from an anthropological study in which I explored the uses of concepts, properties, theorems, etc. of different aspects of Mathematics in performing typical crafts in Andalusia, Spain. I used an ethnomathematical approach for the theoretical foundations, and an interpretative-qualitative methodology.
From the point of view of positivist research, it is very difficult to try to explain the plan of work, the methodology, and the results of any ethnographical research on the Didactics of Mathematics. Positivist research presupposes a structured design in which the hypotheses and techniques are determined a priori. This is in contrast to the flexibility which is appropriate for an interpretative and qualitative methodology which proceeds with very open and interactive plans, and emerges as the first phase of the field work is being carried out.
I shall attempt to set out the general questions which motivated me to start this work, as well as the characteristics of the field work and the scenarios which form it. The origin of this work was my desire to discover the degree of the social use of concepts, properties, relationships, theorems, etc. corresponding to intuitive Geometry in the performance of the tasks which constitute the production process of certain hand-made products which have been in Andalusia in a traditional way for many centuries. This curiosity was inspired by my personal involvement with Didactics of Mathematics from my work on methodology, as well as from my interest in the social and cultural manifestations in the environment in which I live.
Objectives and Scenarios for the Research
It is clear that the "teaching of Mathematics" should give way to "mathematical education or enculturation", because it does not merely involve isolated cognitive variables, but the intricate reality of a human being. A person's education is completed within a society, in the different institutions which form that society: school, family, etc. These subgroups which make up a person's environment may have a defined curriculum or may have aims and messages which emanate from the cultural heritage. Many of the conflicts within teaching arise from the conflict between planned studies and real life experiences.
Also, in Mathematics, it may be stated that there are specific creations in each cultural environment, within the generalization that belongs to scientific truths, in spite of the fact that the process of "removing facts from their original context" (abstraction), makes us forget the social and cultural bases which precede abstract ideas. When we delve into the processes which surround personal construction of mathematical knowledge, we have no options but to go back to the cultural roots of the social groups of which that person is the result. To a large extent we must accept an approach to these didactical questions that is much more anthropological than psychological.
Therefore, I am attempting to shed some light onto the unclear relationship between school mathematics education and popular mathematics culture in Andalusia. In order to accomplish that, the general objectives of this work are:
A. To discover the mathematics contained within the context of certain cultural products, which have been kept alive up to our days, handed down orally and by experience. Accomplishment of this first objective should contribute to the knowledge of the:
a-1 most hidden facts of our popular culture, and
a-2 social requirements for a mathematical preparation suitable for certain fields of work.
B. Explore the various didactic implications of this knowledge through an analysis of:
b-1 the relationship between the school curriculum and popular knowledge and awareness of the divergence between culture and education related to Mathematics,
b-2 the didactic relationship underlying the learning process,
b-3 the sources of information and the utility of education, using case studies of the professional histories of the informants, and
b-4 the learning involved and its application to the preparation of teachers and resultant models of teacher preparation.
The field to be studied is made up of a selection of "scenarios" or craft work which I considered to be representative of our culture and which will allow us, a priori, to select those processes in which mathematics is applied. Table 1 contains a list of the crafts that were considered.
Crafts that Were Studied
1. Marquetry (inlaid work)
2. Musical instruments, particularly guitars
3. Landscaping, particularly with stone laying
4. Granada pottery
5. Carpentry, particularly hand-made and designer furniture (bargueños, jamudas)
6. Dressmaking, preparation of patterns
7. Sculpting and carving in marble and wood
8. Hand-woven carpets
9. Stained glass and lamps
10. Crochet work, lacemaking
11. Bronze and glass craftsmanship, specifically in Granada lamps
12. Iron and copper forging, railings and decorations
13. Hand-made Alpujarra textiles, design and production
14. Goldsmithing and jewelry making
15. Embroidery using tule and gold
16. Graphics arts, sign painting, illustrations, design
17. Other products of construction work: plastering, from carpentry
This list, although it is not totally exhaustive, is a good representation of the work and cultural traditions which were at their cultural zenith during our parents' generation and are much less evident today.
By way of an hypothesis, with respect to objective A, it was expected that we would observe geometrical, topological and measurement therein. With respect to objective B, it was expected that there would be little influence of the school curriculum on their mathematical knowledge prior to specific training or apprenticeship.
Theoretical Foundations and Work Plan
I had a firm conviction that the interpretative methodological approach involved a paradigm that was suitable for many studies in the Didactics of Mathematics and that it cast doubt on the relevance of the positivist approach for explaining current problems. Consequently, this research has emerged from a "context" or natural situation, i.e. the performance of everyday tasks by craftsman when they make their products. The researcher and her collaborators collected the data with the intention of perceiving meaning and widespread relationships in the observed phenomena. This approach, according to Lincoln and Guba (1985), has the advantages of adaptability, immediate processing, holistic capacity, and possibilities for clarifying responses and detecting those which have unusual or idiosyncratic characteristics.
I used direct qualitative techniques for collecting data: participatory and non-participatory observation, interviews, and abbreviated professional case studies. I began by working out an outline of the aspects to be borne in mind during the observation and for predicting the contents to be included, worked out on the basis of the theoretical knowledge of the scenarios to be observed. The semi-structured interviews of the informants, who were the most expert or experienced craftsmen, contained the request that they should tell their own professional story, emphasizing the initial training period and the stages of greatest difficulty. The final interviews, since there were initially informal contacts, were audio or video recorded, so that they could be later analyzed. These audiovisual techniques may be considered to be included under the heading of direct techniques, since they were obtained directly by the researcher. The work plan covered two phases, which were developed over four academic years (1989-1993), according to a qualitative time series design. The sample was divided into two smaller samples, the first in those professions numbered 1 to 10 in Table 1, and the second involved those numbered 11 to 17, and 1, 3 and 8 for more profound study.
The inductive analysis of the data, as a first step in processing, was by means of a plan suggested by Miles and Huberman (1984).
The next, highly time consuming step, was that of reducing these in accordance with the most important objectives of the study. The third step was the display of the abridged or simplified data.
In order to break down the data I chose to select and simplify them by making the initial reduction by centering my attention on one topic: mathematics. Therefore, all other information was ignored. The relationships between the different sequences of the whole process, as well as the processing of the data are cyclical and not linear as in positivist frameworks, which causes a partial advance in aspects which at the same time shall reinforce those that follow.
There may be many ways of displaying the information: matrices, double entry tables, diagrams,
etc. I chose double entry tables as most appropriate in this case.
A Presentation of Some Results
For reasons of space, I cannot go into detailed descriptions of each of the crafts that was analyzed. I shall present a table (Table 2) with the relationships between the mathematical contents discovered and the craft from the first phase of the study, using for this the numbers from Table 1. The mathematical contents discovered are labeled as follows:
Relationship between the Crafts and the Mathematical Content
It may be observed that craft No. 3, landscaping and stone laying, is the one which has the highest number of relationships with the mathematical contents. No. 6, dressmaking, follows it with five. As may be observed, the geometrical shapes and symmetry are concepts applied in all the crafts, whereas only in crafts 1 and 10 is there any specific symbolization. However, the most interesting point is hidden when the summarized data is presented. For example, the different uses of symmetry and tessellation which include an incredible scope, ranging from the designs and patterns for stonework and sculptures to controlling the weaving process for carpets, which is carried out by means of copying the work of another more experienced person, who follows the design. The inlays for the inlaid work, using millimetric measurements, are astounding given their great accuracy, whereas in stonework the mastering of large areas is required.
The use of symbolization by means of areas with different designs and the interpretation of plans, as well as the creations of designs, used in landscaping, carpentry and brickwork, are crafts with a symbolization and interpretation which I consider to be a form of Ethnomathematics. About 90% of the craftsmen were not aware of their use of geometrical elements or of calculations and measurements which are related to Mathematics. They think that they do not need any training in this field and believe that they only need practical training with a master craftsman. The teaching relationship involved is based on authority and uses as its principal teaching techniques simple observation and the copying of actions.
Lincoln, V. and Guba, E. (1985). Naturalistic Inquiry. Beverly Hills: Sage.
Miles, M. and Huberman, A. (1984). Drawing valid meaning from qualitative data: Toward a shared craft. Educational Researcher, 13, pp. 20-30.
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.
Barton, Bill. (1996) Making Sense of Ethno-mathematics: Ethnomathematics Is Making Sense. Educational Studies in Mathematics; v31 n1-2 p201-33.
Proposes a framework to review the literature of the culture of mathematics, specifically in the use of the term Ethnomathematics. Derives a definition of Ethnomathematics; reviews two examples as a test of the power of the definition and the resultant description of Ethnomathematics.
Moore, Charles. (1994) Research in Native American Mathematics Education. For the Learning of Mathematics; v14 n2 p9-14.
Discusses past research involving Piagetian conservation concepts in Native American students; the relation of language to mathematics education; holism in mathematics learning; mathematics and culture; the Outdoor World Science and Mathematics Project, which developed learning modules involving Native Americans; and mentorship in an atmosphere of cultural diversity.
Ferreira, Mariana K. L. (1997). When 1+12. Making Mathematics in Central Brazil. American Ethnologist, 24 (1): 132-147.
This ethnographic account of mathematical activity among the Juruna, Kayabi, and Suya of Central Brazil shows arithmetic practices being fashioned in specific social setting. Values and symbolic properties of both the gift exchange and capitalist economics structure arithmetic dilemmas in the Xingu Indian Park. Within a broad social field that transcends the boundaries of the park to include prospecting sites and cattle ranches, economic calculations are extended to all kinds of goods, material and symbolic. The distribution and circulation of these different forms of capital are discussed in view of the constitution of particular arenas of exchange. Practice theories (Bourdieu, 1991; Lave, 1988) highlight the ways in which mathematical knowledge is constituted in everyday activities, challenging functional assumptions about cognition and schooling. By articulating principles of the gift with those of capitalist exchanges, mathematics is construed by the Juruna, Kayabi and Suya as a product of social work and symbolic fashioning.
Ascher, Marcia and Ascher, Robert. (1997). Mathematics of the Incas: Code of the Quipu, New York: Dover Publications.
This is a new edition and retitling of the now classic 1981 Code of the Quipu: A Study in Media, Mathematics, and Culture.
Sebastiani Ferreira, Eduardo. (1997) Etnomatemática: Uma Proposta Metodológica (Ethnomathematics: A Methodological Proposal, Rio de Janeiro: Mestrado en Educação/ Universidade Santa Ursula.
Sebastiani has perhaps been most well known for his ethnomathematical work with indigenous groups of Northern and Northeastern Brazil. This current volume, available only in Portuguese, is part of the Series on Reflections in Mathematics Education published by Santa Ursula University in Rio de Janeiro. It is the result of the discussions in a seminar on Ethnomathematics held at that university. Sebastiani outlines a course in Ethnomathematics, gives some of the background of Ethnomathematics, and presents Ethnomathematics as a pedagogical model. The volume also includes summaries of student projects, and an appendix with three additional papers by Sebastiani:
O professor de matemática como pesquisador
(The teacher as researcher)
Da oralidade a numeramento
(From "Orality" to "Numeracy")
A importância do conocimento etnomatemático indígena na escola dos não-indios
(The importance of indigenous ethnomathematical knowledge in schools for non-Indians)
Have You Seen on The Web?
Lesley Sisco, a High School teacher in Winnipeg, Manitoba, in an effort to "value her students interests and also connect them to ideas and traditions growing out of centuries of mathematical exploration and invention" has developed a web page that includes lesson plans on area measurement that use, among other units for area, the Aztec quahuitl.
A bibliography on Ethnomathematics is maintained at Beloit College's Department of Mathematics and Computer Science. The following criteria were used to sort the bibliography:
1) If a source is about one region of the world, it is put in that sub-heading.
2) Within world regions, references are grouped by topic (corresponding largely to the chapters of the text) where possible. Sources in that region that cannot be easily classified are then grouped in that region, sub-heading "Other".
3) After the "Regions" headings, additional sources are grouped according to the mathematical topic; again by chapters of the text.
4) Finally, other sources that cannot be classified easily (general, unclear, or where I cannot be sure of the English translation) are collected at the end as "Unclassified".
Another Ethnomathematics bibliography is maintained by Todd Shockey at the University of Virginia.
Yet another useful bibliography available on line is The Annotated Bibliography of Multicultural Issues in Mathematics Education, which is the product of work at the University of Georgia directed by Patricia Wilson.
ISGEm Home Page on "Top 10 List"
The "International Study Group on Ethnomathematics" site is featured in the November issue of the "Top 10 Educational Sites on the World-Wide Web." Learning in Motion" publishes the Monthly Top 10 List for educators and students who are interested in integrating the Internet with their schoolwork.
Ron Eglash, who maintains the ISGEm Home Page, encourages you to browse it at:
New Journal on Multicultural Math to Be Published in Brazil
Mariana K. Leal Ferreira is currently editing a volume on multicultural mathematics in Portuguese, Matemática na Aldeia, featuring experiences with the teaching of mathematics among native populations. She will consider proposals for the publication of articles and review articles on books and other materials that would be translated into Portuguese by either herself or one of her colleagues at the Department of Anthropology, Universidade de São Paulo, Brasil. Proposals can be sent to:
Mariana K. Leal Ferreira
MARI-Grupo de Educação Indigena
Departamento de Antropologia-FFLCH
05508-900 São Paulo, S.P.
Ethnomathematics Conference and Speaker Series at U-mass
On November 3rd and 4th , the College of Public and Community Service (CPCS) at the University of Massachusetts-Boston sponsored an Ethnomathematics conference as part of its 25th anniversary celebration. To add to the celebration, there was also a book party for the new book Ethnomathematics: Challenging Eurocentrism in Mathematics Education edited by Arthur B. Powell and Marilyn Frankenstein.
The keynote speakers were Dirk Struik and Lee Lorch both of whom are internationally acclaimed mathematicians. Both were cited as scholar-activists whose work exemplifies the commitment of CPCS to unite theory and practice in the struggle for justice. Dirk Struik spoke on Ethnomathematics in the History of Mathematics and Lee Lorch spoke on The Painful Path Toward Inclusiveness: The Struggle Against Racism in Mathematical Circles. On the first day, Dirk Struik's talk was followed by a panel which included, Paulus Gerdes, Munir Fasheh, Gloria Gilmer, Arthur Powell and Marilyn Frankenstein all of whom were contributors to the new publication.
On January 12, Gelsa Knijnik will speak on Mathematics Education and the Landless People Movement's Struggle for Justice. In April, Martin Bernal will speak on Ancient Egyptian Science: Fact or Fabrication?
For more information on the speaker series, contact Marilyn Frankenstein at 617-287-7144 or e-mail:firstname.lastname@example.org
ISGEm's Tribute to Dr. Gloria Gilmer
In a very beautiful setting of candle light, flowers and dinner music at the Minneapolis Marriott Center City Hotel, officers and members of the ISGEm paid a tribute to Dr. Gloria Gilmer, their president of 11 years. Patrick Scott, newsletter editor, presided over the program. Maria Reid gave the Occasion. Tributes were presented by Lawrence Shirley, Luis Ortiz-Franco, and Jolene Schillinger, read a letter from Claudia Zaslavsky, who could not attend. Ubiratan A'mbrosio, the new president of ISGEm, presented a plaque to Dr. Gilmer on behalf of the Study Group and a special gift of a book on Brazil. The arrangements committee was Henry Gore and Maria Reid. Others in attendance included: Jim Barta, Ron Eglash, Marilyn Frankenstein, Ed Jacobsen, Gelsa Knijnik, Beatrice Lumpkin, Joanne Masingila, Clo Mingo, María Luisa Oliveras, Marjorie Palmer and Norma Presmeg.
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.
AUSTRALIA, Leigh Wood, PO Box 123, Broadway NSW 2007
BOLIVIA, Enrique Jemio, UNST-P, Casilla 5747, Cochabamba
BRAZIL, Geraldo Pompeu jr, Depto de Matemática, PUCCAMP, sn 112 km, Rodovia SP 340, 13100 Campinas SP
COSTA RICA, Leslie Villalobos, EARTH, Apartado 4 442-1000, San José
FRANCE, Frédéric Métin, IREM, Moulin de la Housse, 51100 Reims
GUADALOUPE, Jean Bichara, IREM Antilles - Guyane, BP 588, 97167 Pointe a Pitre, CEDEX
GUATEMALA, Leonel Morales Aldaña, 13 Avenida 5-43, Guatemala, Zona 2
ITALY, Franco Favilli, Dipartimento di Matematica, Universita di Pisa, 56100 Pisa
MEXICO, Elisa Bonilla, San Jerónimo 750-4, México DF 10200
NEW ZEALAND, Andy Begg, Centre for Science & Math Ed Research, U of Waikato, Private Bag 3105, Hamilton
NIGERIA, Caleb Bolaji, Institute of Education, Ahmadu Bello University, Zaria
NORTHERN IRELAND, School of Psychology, Queens University, Belfast BT7 INN
PERU, Martha Villavicencio, General Varela 598, Depto C, Miraflores, LIMA 18
PORTUGAL, Teresa Vergani, 16 Av. Bombeiros Vol., 2765 Estoril
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
VENEZUELA, Julio Mosquera, CENAMEC, Arichuna con Cumaco, Edif. SVCN, El Marques - Caracas
ZIMBABWE, David Mtetwa, 14 Gotley Close, Marlborough, Harare
ISGEm Executive Board
Ubi D'Ambrosio, President
Rua Peixoto Gomide 1772 ap. 83
01409-002 São Paulo, SP BRAZIL
Maria Luisa Oliveras Contreras, 1st VP
Depto de Didáctica de las Matemáticas
Campus Cartuja, Universidad de Granada
18071 Granada, SPAIN
Jolene Schillinger, 2nd Vice President
New England College BX 52
Henniker, NH 03242 USA
Abdulcarimo Ismael, 3rd Vice President
Departamento de Matematica
Universidade Pedagogica Nacional
P.O. Box 4040
Gelsa Knijnik, Secretary
Rua Prof. Andre Puente 414 ap.301
90035-150 Porto Alegre, RS, BRAZIL
Jim Barta, Treasurer
Department of Elementary Education
Utah State University
Logan, Utah 84341 USA
Patrick (Rick) Scott, Editor
College of Education
New Mexico State University
Las Cruces, NM 88003 USA
Lawrence Shirley, NCTM Representative
Dept of Mathematics
Towson State U
Towson, MD 21204-7079 USA
Gloria Gilmer, Past President
Math Tech, Inc.
9155 North 70 Street
Milwaukee, Wl 53223 USA