Volume 6 Number 2
July 1991

Highlights from the ISGEm Business Meeting in New Orleans
State University, Maryland - new member-at-large for North
America; Henry Gore of Morehouse College, Georgia - new
program assistant; Jerome Turner, St Francis Xavier University, Nova Scotia/Canada - new NCTM representative; and Danny Martin of ARC Associates, Oakland, CA - new assistant editor.

Paulo Freire's Epistemology
By Marilyn Frankenstein
College of Public & Community Service,Univ. of Massachusetts
and Arthur B. Powell
Academic Foundations Department. Rutgers University
"...our task is not to teach students to think--they can already think; but to exchange our ways or thinking with each other and look together for better ways of approaching the decodification of an object. (Freire, 1982)"
In its connection with pedagogy, the key underlying assumption in the emerging field of ethnomathematics is that our students, through their daily activities, already think mathematically. To understand their ways of thinking mathematically, we need to reconsider what counts as mathematical knowledge. We need to learn about how culture - daily practice, language, and ideology-interacts with students' views of mathematics and their ways of thinking mathematically. Learning about these views and ways of thinking are opportunities to deepen our mathematical and pedagogical knowledge. We need to reclaim the hidden and distorted history of all cultures' contributions to mathematics. Further, to convince our students that they already think mathematically and that they can learn "school" or "acadentic" mathematics, we need to connect their mathematical understandings with an undistorted history of mathematics and with whatever "academic" mathematics they are studying.
We start with a discussion of Paulo Freire's theories about the nature of knowledge and introduce the range of intellectual traditions that inform ethnomathematics. We then proceed to argue that his epistemology informs the theoretical basis of ethnomathematics. We summarize the range of areas that are contributing to ethnomathematics and present reasons for and examples of curricular applications. In concluding, we indicate implications for further investigations of mathematical knowledge and its connections to cultural and political action.

We are offering a one year free membership to the winner of our ISGEm logo contest.


Become Involved
Our appeal to become involved in ISGEm was met by Evelyne Barbin of Saint Denis, France. She not only agreed to translate the statements about ISGEm into French and publish it in I.R.E.M. literature, but she also recruited Frederic Metin from I.R.E.M. to be a contact person in France to disseminate ISGEm newsletters! Kudos to Evelyn.
We still need volunteers to take on the duties of assistant newsletter editors, members for the Membership Committee and other areas. Please think about contributing your time and efforts.

Video Games for Math: A Case For "Kid Culture"
By Lawrence Shirley, Department of Mathematics
Towson State University. Towson, Maryland
Nintendo and other video games have gained a reputation (or notoriety) of occupying so much of children's time that their schoolwork may suffer. However, rather than fighting this invasion, I suggest that we exploit video games for mathematics education, drawing values from the thinking processes learned and used in playing the games. Recently, the Nintendo company made grants to research projects in education, so it appears the company is reaching out to education. Why don't we respond by seeing the educational value of video games?
There are several types of video games and hence different types of cognitive skills involved and different ways they could be used in mathematics. One of the older types is target shooting games, such as "Duck Hunt," which is even included in a popular basic package of the Nintendo control set. Of course, these kind of games depend a lot on hand-eye coordination and timing, but the player also gains practical experience with trajectories and speed-time relationships. Maze games, often with villains in the maze to chase you, date back to "Pac-Man" in the 1970s and remain popular today. An important skill in this type of game is to build a mental map of the maze and its dangers, leading to geometrical and topological thinking experience. A third type, often including a maze, is a quest game, such as "Castlevania," where the player has an assignment of saving the princess or finding the pot of gold after struggling through many obstacles. Like the mazes, these involve map building and topology, but also require developing strategies to fight the dangers and difficulties in the way. Other games involve sports activities, simulations of driving cars or flying planes, and other skills of varying relationships to mathematics.

Mathematical Values
Video games offer many different values to mathematics education. The most easy to see are in geometry. Game players need to develop a good sense of space and topological relationships. Teachers could use this by having children do projects of drawing maps of the paths they follow in maze and quest games, including special "warp zone" paths that jump from one region of the game's world to another. Planning and strategic thinking skills are a part of many games from chess to Mastermind and also are prominent in many video games. One must plan a course of action, gather necessary materials, and follow a smooth sequence of tactics to win the victory. This is also crucial to mathematical problem solving-Polya's second step is to plan a strategy for solution. Children can discuss the values of sequencing, and link this to seriation, the commutative law, the order of arithmetic operations, and flow-charting. Finding patterns is another mathematical skill of many video games--fitting pieces of a puzzle together, recognizing relationships--just as in finding mathematical patterns areas ranging from transformational geometry to abstract algebra to functional analysis.
It is easy to see how video game skills fit into the NCTM Standards. Problem solving is essential to success both in winning a video game and learning mathematics. Spatial sense and pattern finding are key ingredients of both. By encouraging children to talk about video game strategies and draw diagrams of game worlds, the teacher is strengthening mathematical communication. Of course by reaching out to "kid culture," the mathematics teacher is showing connections between mathematics and its applications in daily life. For best use of video games for mathematics, a teacher would need to become familiar with several of the popular games with an eye to where they fit into the curriculum. This may vary considerably from game to game and between different grade levels. However, as a sample we shall take a brief look at two popular Nintendo games, the Super Mario Brothers series and Tetris.
All of the Mario Brothers games involve traveling through various "levels" of several "worlds" in a quest activity. Each level of each world has its own environment with a series of obstacles and monsters blocking the way. One can avoid or fight the monsters, often collecting weapons along the way to throw at the monsters or to make the Mario figure larger or more powerful for jumping, running or even flying. To avoid having to go through all the worlds, there are several "warp zones" which are special hidden paths that allow jumping to advanced worlds instantaneously. The player must become familiar with the alternative paths, the locations of special tools, weapons, bonus coins, etc.
When kids talk about Mario games, the conversation often goes to secrets one has found that can be helpful to others in getting farther into the game without "dying." This is where mathematical thinking can be helpful and where a teacher could plug the game into the curriculum. The teacher could ask the students to draw maps of the various "worlds," marking the various paths, dangers, and bonuses along the way. Students would need a sense of order and sequence, a topological feel and spatial sense for the vertices of the paths, and an idea of scaling to transform the worlds from the monitor to paper. Other work could include discussion of the values of speeding through the worlds quickly versus stopping to collect the bonuses and tools in a more deliberate manner. While not directly linked to mathematical curricular topics, the game is an extended exercise in problem solving and strategic thinking.
My favorite of the game I've seen, especially from a mathematical vantage, is "Tetris." This is a game of packing geometrical shapes as efficiently and compactly as possible. The screen shows tetrominoes (shapes made of four squares arranged in various patterns so they always touch edge-to-edge) falling slowly from the top, eventually landing in ever-filling layers of squares. As the shapes fall, the player can move them from side-to-side and can rotate them a full 360 degrees, to try to make them fit into the rows at the bottom. However, it is necessary to work quickly to get the shapes positioned properly before they reach the bottom, for once they touch the piles of squares, they stick and cannot be further moved. If a solid line of squares is filled it is automatically cleaned away, keeping the pile from building too high; but as gaps develop the incomplete lines of squares stay and the pile grows toward the top. When it builds so high that the new pieces touch the pile immediately, the game is ended. Scoring comes from the number of lines successfully completed.
The packing task itself gives much experience in spatial sense and a feel for the relationship of the tetromino shapes. Transformational geometry is also experienced in the rotation and sliding operations applied to the falling shapes. Since some of the tetromonies are symsnetric and others are not, the play also recognizes these differences in how the shapes can and cannot be packed. More generally, the use of the tetrominoes provides a nice introduction to general polyominoes (including the famous puzzles of pentamonies), and indirectly to nets, tangrams, and the properties of other geometrical shapes.
Incidentally, the choice of which tetromino will fall is done randomly and a tally of the numbers of each piece is shown on the screen. This, of course, could be used as an example in statistics or probability topics. Also, graphs and averages of scores in a Tetris competition are further mathematical applications of the game. These represent only a beginning. The teacher should try out the games and use creativity to see their applications to math (or other) topics.

Ethnomathematics, like anthropology, sometimes has a flavor of the exotic about it. However, anthropologists now argue that their field really should be the study of human culture--all human culture, not just those previously assumed to be somehow "primitive." In African universities, such studies of culture are often deemed "sociology" to avoid any negative connotations of "anthropology." In the same way, ethnomathematics, by its original meaning, attempts to broaden the meaning of academic mathematics to look for mathematics in any and all cultures. That broad definition need not be limited to foreign or third world cultures. Right under our noses, our children have their own culture. Rather than dismiss it, we need to seek out "kid culture" and demonstrate that it too is mathematics.

UNESCO Mathematics
Curriculum in Guinea-Bissau
By Beatriz D'Ambrosio
The presentation focused on the process of curricular change in a developing country and the role of foreign aid in that process. The speaker described her experience as a consultant for UNESCO in Guinea-Bissau during the summer of 1990. The purpose of the project was to revise the curricula of the different subjects of the elementary grades throughout the country. The intent was that after revision the curricula would be comparable to that of any developed country allowing national graduates to attend secondary schools abroad.
Several conditions essential for effective change were raised. These included: an environment conducive to change; teachers' active participation in the reform initiative; in the specific case of mathematics, teachers' reconceptualization of their understanding of the nature of mathematics and the role of the teacher in the learning process; and time for experimentation and reconstruction of the curricula.
The difficulties encountered during the project were: an environment non-conducive to change; the project team's beliefs about the nature of mathematics, the learning of mathematics and the teaching of mathematics; the intellectual isolation of the members of the project team and of the individual teachers spread out throughout the country; the low expectations of teachers' and students' abilities by the members of the team; the fear of "falling behind" if curricula were catered to confronting national problems rather than adopting standard curricula used internationally; and finally, the speakers own personal beliefs about the change process were contradictory to those of the project team.
In conclusion, the speaker commended the national effort to focus on education as an imporant aspect of national development, however foresaw a need for a very intense level of in-service education with great participation of the teachers. The change process would require an elaborate support system for teachers and project staff. In contrast to the expectations of the curricula revision team, change would only occur over a long period of time.

Games Played Around the World
On Friday, April 19,1991, Claudia Zaslavsky spoke at the national NCTM conference on three-in-a-row games, one in a series of talks by ISGEm members on "Ethnomathematics and Games Played Around the World." The talk was based on her book "Tic Tac Toe and Other ThreeIn-A-Row Games, From Ancient Egypt to the Modern Computer" (Crowell, 1982). Besides their interest for students, games are valuable in the classroom in fostering critical thinking and cooperative learning. Children can be encouraged to help their opponents to improve their strategies, thus making the game more challenging for both players.
The familiar game of Tic-Tac-Toe was analyzed. Although the first two moves can be made in 9x8, or 72, different ways on a fixed board, the number can be reduced to 12 when one considers the symmetry of the square. As for strategy, provided neither player makes an error, the first player to go cannot lose and the second player cannot win.
Zaslavsky reviewed several versions of three-in-a-row games, beginning with the gameboard incised on the roofing slab of an ancient Egyptian temple about 3300 B.P. (before the present) - Tapatan in the Philippines, Shisima in Kenya, Tsoro Yematatu played on a triangular board in Zimbabwe, Noughts and Crosses in England, and the many European versions called "Mill," starting with the game introduced into Spain by Arabic-speaking Moors from northern Africa and recorded in the first European book of games about the year 1200.
The most complex versions require that each player use 12 counters. Girls in Sri Lanka play one variation, while another is popular in Lesotho, a tiny country surrounded by South Africa. An instructor in Lesotho found that middle grade students who were experienced in playing the game scored significantly higher in certain standardized geometric tasks than their non-playing colleagues.
It is interesting to note that among the computers designed by Charles Babbage, but never actually built, was a tic-tac-toe machine.
The session concluded with audience participation in comparing two versions of the game Picaria, played by the Pueblo Indians of the U.S. Southwest, and probably introduced to them by the Spanish Conquistadors. The participants could not agree on which was the better version, but all declared that it was fun! Alverna Champion of Grand Valley State University spoke on "Board Games of African Children." The games presented were Arrangements, Magic Squares, Networks, Achi, Kalan, N'Tchuba, Senat, Seega, along with one string puzzle. The audience had fun playing Arrangements. Champion provided suggestions for making inexpensive gameboards. The session was well-attended with many questions and answers interspersed throughtout.

The Philosophy of
Mathematics Education
By Dr. Paul Ernest
University of Exeter School of Education Exeter, United Kingdom
The philosophy of mathematics is in the midst of a Kuhnian revolution, with the absolutist paradigm increasingly under question. Publications by Lakatos, Davis and Hersh, Kitcher and Tymoczko, for example, are pointing towards a new fallibilist paradigm. At the same time develonments in the sociology of science, Imowledge and mathematics, and post-structuralist and post-modernist thought are looking towards social constructivist accounts of knowledge. These have important implications for mathematics and particularly for educational theory and practice.
In mathematics education there is an increasing awareness of the significance of epistemological and philosophical issues. Theories of learning, such as constructivism, are becoming epistemologically orientated. A growing number of areas of inquiry are drawing on the philosophy of mathematics and philosophical perspectives. These include problem solving and investigational pedagogies, curriculum theories, teacher education and development, teacher beliefs, applications of the Perry Theory, ethnomathematics, gender-fair and multicultural mathematics, and the sociology and the politics of mathematics education. In addition, researchers are becoming increasingly aware of the epistemological foundations of their methodologies and inquiries, and referring to them explicitly.
A Philosophy of Mathematics Education group has been formed to explore these and related issues. It is proposed to offer a Topic Group at the 7th International Congress of Mathematical Education, Quebec, August 16-23,1992, and a group at the British Congress of Mathematics Education, Loughborough, July 13-16, 1991. An international network with a newsletter has been established, and interested persons are invited to write in and to join the mailing list. A statement of related interests would be welcomed, but is not necessary.
The organning group includes Raffaella Borasi (USA), Leone Burton (UK), Paul Cobb (USA), Jere Confrey (USA), Kathryn Crawford (Australia), Phi1ip Davis (USA), Paul Ernest UK), Reuben Hersh (USA) Cluistine Keitel (FRG), Steve Irerman (UK), Marilyn Nickson (UK), Sal Restivo (USA), Leo Rogers (UK), Anna Sfard (Israel), Ole Skovsmose (Denmark) and John Volsninck (USA).
For more information or to join write to Dr. Paul Ernest, University of Exeter, School of Education, Exeter Exl 2LU, United Kingdom.

Recent Ethnomathematical Research in Mozambique

Most "mathematical" traditions that survived colonization and most "mathematical" activities in the daily life of the Mozambican people are not explicitly mathematical. The mathematics is "hidden." The first aim of the project "Ethnomathematics in Mozambique" is to "uncover" this "hidden" mathematics. As some traditions are nowadays rather obsolete, the "uncovering" often means also a tentative reconstruction of past knowledge.
In our study On the Awakening of Geometrical Thinking (1985) and our book Ethnogeometry: Cultural-Anthropological Contributions to the Genesis and Didactics of Geometry (concluded 1986, published 1990) some anthropological research methods were developed in order to "uncover" and reconstruct "hidden" mathematical thinking (cf. also [1]). The basic method then proposed for recognizing implicit mathematics may be characterized as follows: When analyzing the geometrical forms of traditional objects--like baskets, mats, pots, houses, fishtraps--the researcher poses the question: Why do these material products possess the form they have?
The researcher learns the usual production techniques and tries at each stage of the production process to vary the forms. Doing this, the researcher observes that the form generally represents many practical advantages and is, quite a lot of times, the only solution of a production problem. Applying this method in the period 1986-1990, new results have been obtained. Abdulcarimo Ismael (Department of Mathematics, Higher Pedagogical Institute, Maputo) did in 1989 fieldwork in the northern Mozambican province of Nampula. In his provisional report, he reveals interesting aspects of the (implicit) mathematical knowledge displayed by basketweavers. During our stay as a visiting professor at the State University of Sao Paulo (UNESP, Rio Claro, April-May 1988)--lecturing a postgraduate course on ethnomathematical research methods--we collected a series of Amer-Indian baskets and initiated their analysis.
It came out that to guarantee the beautiful, symmetrical wall ornamentation, the artisans had to use (and develop) arithmetical tools like multiplication and to know some of their properties like commutivity (see Chapter 3 in [2]. In two research papers 0n Ethnomathematical Research and Symnmetry (Chapter 2 in [2]) and Fivefold Symmetrv and (basket) Weaving in Various Cultures, we explain why basketweavers "prefer" certain symmetries.
As this method for recognizing "hidden" mathematics had been developed in the context of analyzing material production, like that of baskets, mats, pots, houses and fishtraps, the question of the possibility of extending the method to other spheres of production--such as artistic and/or symbolic production--had to be posed (objective 1), in view of the success of the method in the first field.
On analyzing, by the same method, spiral ornaments on the walls of old Egyptian tombs, it came out that ancient Egyptian artisans probably might have known how to construct a square equal in area to the sum of the areas of two given squares, which could have led to the discovery of the so-called Theorem of Pythagoras (see Chapter 4 in [2]).
Then we tried to apply the method to the analysis of traditional African and Asian designs, in particular the Tchokwe sand drawings [Angola, with relationship to the Luchazi (Zambia) and Makonde (Mozambique) graphic traditions] and the--from a technical point of view- related Tamil (South India) threshold designs. It came out that the aforementioned method for recognizing "hidden" mathematical thinking, as such, was not immediately applicable. The method had to be adapted and "refined." Instead of starting by posing the question why the (material) products possess the form they have, the researcher had first of all to ask "which are the cultural values that lay at the basis of the drawing tradition?" and only then, in view of these underlying cultural standards, to pose the question "why do these drawings possess the 'form' they have?".
Both Tchokwe and Tamil traditions are similar in the sense that the drawers use the same mnemonic device for the memorization of their standardized pictograms. After cleaning and smoothing the ground they first set out an orthogonal net of (equidistant) points. Then the curves are drawn in such a way that they surround the dots without touching them. Many such traditional Tamil threshold designs are "monolinear," i.e. made out of one closed, smooth line. In Reconstruction and Extension of Lost Svmmetries: Examples From the Tamil of South India (Chapter 6 in [2]), there is an investigation of a series of Tamil patterns which do not conform to their cultural standard, as they are composed of two, three or more superimposed closed paths. An analysis of possible construction errors shows that these "polylinear" designs are probably "degraded" versions of originally monolinear patterns.
Furthermore, it became possible to reconstruct these original patterns and to make explicit some of the geometrical knowledge of their inventors (transformation roles, geometrical algorithms, extension and generalization). The success obtained in developing the adapted and "refined" method (objective 2) for recognizing "hidden" mathematics and in applying it to the Tamil designs, stimulated its application in other contexts such as the Tchokwe sand drawings.
With the colonial penetration and occupation, the Tchokwe sand drawing tradition has been disappearing. Our analysis of the sand drawings that have been reported by missionaries and ethnographers, shows how symmetry and monolinearity played an important role as cultural values in this tradition. We succeeded (objective 1) in reconstructing classes of Tchokwe sand drawings that had been lost over time and in showing that the Tchokwe drawing experts had invested general construction rules and had discovered "theorems" about transformation rules, algorithms, dimensions and rules for the chaining of monolinear patterns to bigger monolinear patterns. The first results have been included in [2, p.120-189] and have been extended in [3] and [4].
It had been suggested by us that the origin of the mnemonic technique used in the Tchokwe and Tamil drawing tradition lies probably in weaving and as some of their designs may be characterized as plaited-strip-patterns ([3], p.7), we looked for such patterns in other cultural contexts. In Chapter 8 of our Ethnomathematical Studies [2], p. 190-209, (in German) we present the first results of this excursion:

.8.1: On snakes, plaited strip patterns and graphs in ancient Mesopotamia;

.8.3: On monolinear patterns form Northamerican Indians.

In On Culture, Geometrical Thinking and Mathematics Education (Chapter 9 in [2]) we summarized our experimentation (until 1987) with the incorporation of traditional African cultural elements into mathematics education (objective 3). The paper confronts a widespread prejudice about mathematical knowledge, that mathematics is "culture- free," by demonstrating alternative constructions of euclidean geometrical ideas developed from the traditional culture of Mozambique. As well as establishing the educational power of these constructions, the paper illustrates the methodology of "cultural conscientialization" in the context of teacher training.
In A Widespead Decorative Motif and the Pythagorean Theorem (Chapter 10 in [2]), we gave concrete examples of multi-culturalizing the mathematics curriculum, using a well-known African and also Scandinavian ornament motif as a starting point for doing and elaborating mathematics in the classroom. At the same time it is shown that there exists an infinity of (new) proofs for this theorem (cf. objective 4.1 and 4.4; see also our paper How Many Proofs of the Pythagorean Proposition do There Exist?, published in Sweden). In Chapter 11 of [2] we relate our first reflections on the possibilities of using the Tchokwe sand drawings in the mathematics classroom. The examples given in this paper range from the study of arithmetical relationships, symmetry, similarity, and Euler graphs to the determination of the greatest common divisor of two natural numbers.
Later on, a reflection on the results obtained in the historical reconstruction of the above-mentioned Tamil and Tchokwe designs and on the geometrical algorithms involved led to the formulation of a first series of geometric problems of the type Find the Missing Figures (Published also in the Swedish journal Namnaren).
In 1988 and 1989 we conducted further didactical experiments and concluded in early 1990 a book with problems of this type, entitled Lusona: Geometrical Recreations From Africa (English version [5] and Portuguese version [6]).
Many--both reported and reconstructed--Tchokwe drawings are aesthetically appealing and the analysis of the geometric algorithms involved stimulated their generalization and the invention of new patterns. In Examples of Algorithms and Monolinear Motifs Inspired by the Tchokwe Sona (Chapter 4 in [3], in [5] and in Pickover's The Pattern Book: Recipes of Beauty) we present some beautiful designs we found in this context.
The study of the mathematical potential (cf. objective 4.4) of the traditional Tchokwe designs and of their generalizations constitutes a new and attractive area of mathematical research. Already in 1987 we were stimulated by an analysis of a class of Tchokwe drawings to discover A Physical Model for the Determination of Prime Numbers (Chapter 14 in [2]). In 1988 we found that a whole class of Tchokwe ideograms satisfy a common construction principle. An analysis of all possible curves that satisfy the same construction principle, led to the discovery of some theorems, proved in 1989. The proofs are included in Chapter 15 of [2] and explained to a broader public in [7] and in Chapter 5 of [3]. Early 1990 we summarized our historical, educational and mathematical results in a manuscript for a book, entitled Geometry of the African Sona: History. Education. Recreation. Art Design ([3]). At the end of the introduction to this book, we summanze:
"The study of the Tchokwe drawing tradition, threatened with extinction during the colonial period, is not only interesting for historical reasons. The incorporation of this sona tradition in the curriculum, both in Africa and in other parts of the world, will contribute to the revival and valuing of the old practice of the sona experts, and will reinforce the comprehension of the value of the artistic and scientific heritage of Africa. It may contribute to the development of a more productive, more creative and multicultural mathematics education. Furthermore, an analysis of the Tchokwe patterns stimulates the development of new mathematical research areas."

[1] Paulus Gerdes, How to recognize hidden geometrical thinking? A contribution to the development of anthropological mathematics: For the Learning of Mathematics, Montreal, 1986, Vol.6, No.2, p.10-12, 17
[2] Paulus Gerdes, Ethnomathematische Studien Dr.Sc.nat.thesis, Leipzig, 1989, 360 p.
[3] Paulus Gerdes, Geometric of the African "sona": History, Education, Recreation, Art Design, 1990, 130p.
[4] Paulus Gerdes, Geometria dos "sona" africanos : historia, educacao, recreacao, desenho artistico, 1990, 130.
[5] Paulus Gerdes, Lusona : Geometrical Recreations from Africa,
[6] Paulus Gerdes, Lusona : Recreacoes geometricas de Africa, 1990, 120 p.
[7] Paulus Gerdes, On mathematical elements in the Tchokwe "sona" tradition to be published in :For the Learning of Mathematics, Montreal, 1990 11 p.

ICME-7 Congress
August 1992 in Quebec City
The Seventh International Congress on Mathematical Education (ICME-7) will be held at Universite Laval in Quebec City, Canada, from August 17 to 23, 1992. The Second Announcement is now available from: Congres ICME-7 Congress, Universite Laval, Quebec, QC, Phone: (418) 656-7592, Fax: (414) 656-2000, E-mail: ICME-7 @VM1.ULAVCAL.CA Itcontains information on all aspects of ICME7 including registration, accommodation and an application form to make a short presentation.
ICME-7 will provide participants with the opportunity to learn about recent develonments in mathematics education around the world and to be introduced to innovations and recent research on the learning and teaching of mathematics at all levels. The central feature of the scientific program is a set of 23 Working Groups each designed to involve participants in the active study of a selected aspect of mathematics education and to provide an international up-to-date context for study of that aspect. Each Working Group will meet for four 90-minute sessions.
Other activities will include several plenary talks, lectures, topic groups, study groups, national presentations, short presentations in the form of posters or videotapes or computer software, projects, workshops, films, as well as exhibitions of textbooks, software and other types of materials. A special half-day Miniconference on Calculators and Computers will be held at the beginning of the congress. Finally, various social and cultural events are planned for the duration of the congress.
Early registration is encouraged. The schedule of registration fees provides for significant savings for those who preregister by December 15, 1991. The deadline for those applying to make a short presentation is January 31,1992. Accommodation requests will be received up to July 1, 1992, although it is advisable to make reservations much earlier.
Full program details will be listed in the Third Announcement which will be available in April 1992 and will be sent to those whose registration forms and payment are received by June 15, 1992. Participants who register after this date will receive the program during on-site registration.

New Members
Norm Cote, Plymouth State College. Plymouth, New Hampshire 03264, USA
Rheta N. Rubenstein of the University of Windsor in Ontario, Canada is currently involved in curriculum development at grades 6-11 and is interested in broadening the multicultural aspects of the material.
Mary M. Thompson of the New Orleans Public Schools in Louisiana, USA, is a mathematics specialist with a concentration on cultural infusion in the mathematics curriculum and a focus on African Americans.
Anna Lothman of Enkoping, Sweden is a mathematics researcher with an interest in the history and pedagogy of mathematics and ethnomathematics.
Francisco Egger Moeliwald is a Brazilian mathematics education graduate student at Indiana University in Bloomington, Indiana.
Luisa Oliveras Contreras of the University of Grenada in Spain.
James Syta of Brasdell, New York, USA.
Hail Suryanto of the Department of Mathematics Education at Fpmipa, Ikip Yogyakarta in Karangmalang, Yogyakarta - Indonesia
H.W. Straley of Woodberry Forest, VA, USA is studying mathematics of the Mali Empire during the 11th to the 14th centuries.
Chris Klnsey of the Department of Mathematics at Canisus College in Buffalo, N.Y. USA.

Proposed Constitution and By-Laws of the ISGEm
The Advisory Board of ISGEm developed the Constitution and By-Laws which appear below. The membership should cut out and mail the absentee ballot on page 8 to Luis Ortiz-Franco by October 1, 1991. Direct all inquiries toLuis Ortiz-Franco, whose address is also listed on page 8.

Article I. Name. The name of this organization shall be the International Study Group on Ethnomathematics (IS GEm).
Article II. Purpose. The purpose of the organization shall be to encourage and maintain interest in the teaching and learning of mathematics in cultural contexts and to promote professional growth, fellowship and communication among its members.
Article III. Membership.
Section 1. Membership shall be open to all persons interested in ethnomathematics.
Section 2. (A) Members shall pay regular dues and be entitled to all privileges of the organization. (B) The dues shall be set by the Executive Board subject to approval of the membership. (C) At the discretion of the Executive Board, any person shall be granted an honorary membership upon request without payment of dues.
Section 3. The membership period coincides with the calendar year from January 1 to December 31.
Section 4. All members shall indicate the region to which they belong. The regions shall be: A. Africa; B. Asia (including the Middle East); C. South Pacific (including Australia, New Zealand and the Pacific Islands); D. Europe; E. The Americas (North, Central, South, and the Caribbean).
Article IV. Executive Board.
Section 1. The Executive Board shall consist of the officers and members-at-large, the NCTM representative, the editor of the newsletter, the immediate Past-President, the President-Elect, the Program Assistant and the Assistant Editor.
Section 2. The Executive Board shall attend to any business of the organization that may require attention in the interval between business meetings.
Article V. Officers. The officers of the organization shall be President, First Vice-President, Second Vice-President, Third Vice President, Recording Secretary, Corresponding Secretary and Treasurer.
Article VI. Duties and Election of Officers.
Section 1. The President shall preside at all meetings of the organization and shall be chairman, ex-officio, of the Executive Board, and shall appoint an NCTM representative, the editor of the newsletter and the Assistant Editor.
Section 2. The First Vice-President shall perform the duties of the President in the absence of the President and shall act as program chairman. The First Vice-President shall appoint as necessary a program committee and a Program Assistant or specify program representatives to promote presentations on Ethno-Mathematics at relevant professional meetings.
Section 3. The Second Vice-President shall perform the duties of the President in the absence of the President and the First Vice-President and shall act as membership officer.
Section 4. The Third Vice-President shall perform the duties of the President in the absence of the President, the First Vice-President and the Second Vice-President and shall act as coordinator of the Special Interest Groups (SIGs) in ISGEm and communicate with members-at- large concerning conferences relevant to ISGEm in their respective regions.
Section 5. The Secretary shall keep the minutes of the business meetings and shall pass these along to the newly elected secretary as a permanent record of the actions of the organization.
Section 6. The Treasurer shall receive and account for all monies of the organization, disburse all sums on order of the President, and render a financial report at the last meeting of the year. A yearly audit must be conducted by two members appointed by the Executive Board.
Article VII. Meetings. At least one business meeting shall be held during each calendar year. The time and place of these meetings shall be set by the Executive Board. All meetings are open to any memberof the Group.
Article VIII. Rules of Order. The organization shall be governed by Robert's Rules of Order except in matters otherwise provided for by the Constitution.
Article IX. Amendments. This Constitution may be amended at any meeting of the Group by a two-thirds majority vote of the members present and voting, provided notice of the proposed amendment has been given at the previous meeting.
Article X. Dissolution. If at any time the International Study Group on Ethnomathematics (ISGEm) shall cease to carry out the purposes herein stated, all assets held by it in trust or otherwise, shall, after the payment of its liabilities, be paid over to an organization selected by the final Executive Board of the International Study Group on Ethnomathematics which has similar purposes andhas established its tax-exempt status under Section 501 (c)(3) of the Internal Revenue Code of 1954 as now enacted or hereafter amended, and such assets shall be applied exclusively for such charitable, scientific, and educational programs.

Article I - Executive Board.
Section 1. Two of the members-at-large shall be elected from the South Pacific , three from Africa, three from Europe, three from Asia (including the Middle East). and three from the Americas.
Section 2. Additional members of the Executive Board shall include the Immediate Past-President, the President-Elect, the NCTM Representative. the Editor of the newsletter, the Assistant Editor, the Program Assistant, and the officers.
Article II - Election of Officers and Members-At-Large.
Section 1. The terms of office for all officers and members-at-large shall be four years with half the members-at-large elected every two years.
Section 2. All elections shall be held by ballot prior to the end of each even-numbered calendar year and shall be carried by a plurality vote of the ballots returned. Nominations for the officers and members- at-large shall be made by a Nominating Committee of five members, appointed by the President and approved by the Executive Board. The Nominating Committee shall recommend at least one candidate for each office to be filled. Other nominations shall be received as write-ins on the election ballot at the time of the election. The consent of each candidate, other than write-ins, must be obtained before the name is placed in nomination.
Section 3. Officers shall be elected in years divisible by four.
Section 4. Officers shall begin to serve two years after being elected.
Section 5. Members-at-large shall begin to serve on January 1 of the odd-numbered year immediately following election.
Section 6. Officers shall be elected by the entire membership.
Section 7. Members-at-large shall be elected by the members from their region.
Section 8. All officers and members-at-large can be re-elected.
Article III - Amendments.
These by-laws may be amended by written ballot by a majority vote of the ballots returned, provided notice of the proposed amendment has been given at the previous meeting.

Have You Seen
"Have You Seen" is a feature of the ISGEm Newsletter in
which works related to Edinomathematics can be reviewed.
We encourage all those interested to contribute to this column. Gloria Gilmer prepared "Have You Seen" for this issue.

Ascher, Marcia (1991). Ethnomathematics : A Multicultural View of Mathematical Ideas. Brooks/Cole Publishing Company, Pacific Grove, California, 93950, USA.
Ascher explores mathematical ideas of people in traditional cultures involving numbers, logic, spatial configuration, and the organization of these into systems and structures. These ideas are often omitted from discussions of mathematics. Donald Crowe says "essentially none of this has previously been presented in any depth in book form." Alvin White says, "The book demonstrates that mathematical ideas and applications arise in natural settings outside of the European and scientific traditions."

Kulm, Gerald (1990), Math Power in the Community. American Association for the Advancement of Science, 1333 H Street, NW, Washington, D.C. 20005, USA.
Kulm, Gerald (1990). Math Power in the Home. American Association for the Advancement of Science, 1333 H. Street, NW, Washing- ton, D.C. 20005, USA.
KuIm, Gerald (1990). Math Power in School. American Association for the Advancement of Science, 1333 H. Street, NW, Washington, D.C. 20005, USA.
The Math Power books are collections of learning activities, drawn from many sources, written by three separate writing teams and organized by Gerald Kulin. The power is in insisting that students look back and reflect on the learning experience and then draw some useful conclusions.

Moving Beyond Myths: Revitalizing Undergraduate Mathematics (1991). National Academy Press. 2101 Constitution Avenue, NW. Washington, D.C. 20418, USA
This final report of the Committee on the Mathematical Sciences in the Year 2000 of the National Research Council is mandatory reading for all mathematics educators. At least the mathematical community is facing up to its own responsibilities in the underpreparedness of many of the nations' learners at all levels. The problem is well defined and necessary action steps are clearly delineated. It is simply unclear who is going to take this bold action and when !

Counting on You. Actions Supporting Teaching Standards (1991) National Academy Press, 2101 Constitution Avenue, NW. Washington D.C. 20418, USA.
This thirty-six page document describes specific actions various publics can take to support the efforts of mathematics teachers to meet the standards for professional development and curriculum and evaluation. These publics include school boards, school administrators, parents, college and university faculties, policy makers, leaders in government, business and industry, members of the media and teachers themselves.

Gilmer, Gloria F. Developing African Americans in Mathematics: An Interview with Abdulalim Abdullah Shabazz. Math-Tech, Inc., 9155 N. 70th St. Milwaukee, WI 53223-2115, USA.
There are few accounts in the literature of those outstanding African-American scholars and teachers who account for upwards of half of the baccalaureate degrees in mathematics earned by African Americans. Dr. Shabazz is one such scholar and teacher and this interview is an account of his life, educational philosophy and achievements. It is possibly the only such extensive account of Dr. Shabazz and his contributions to the field of mathematics.

Gore, Henry and Gilmer, Gloria F. Effective Strategies for Teaching Calculus at the College Level: A survey Report (1990). Morehouse College, Atlanta, GA 30314.
This report is a descriptive sample survey of how calculus is actually taught today in approximately 150 colleges and universities in the United States and Canada. Some common practices are analyzed for their possible negative effects on learners. This study places Morehouse College at the frontier of calculus reforms aimed at nurturing African Americans.

Eighth Symposium of the
Southern Africa Mathematical Sciences Association
"The Education of Mathematics Teachers
in Ihe SADCC Region
Workshop: Mapping Theory
December 16-19, 1991
Higher Pedagogical Institute and
Eduardo Mondlane University
For further information write to SAMSA Organizing Committee
Abdulcarimo Ismael, Department of Mathematics
Higher Pedagogical Institute
P.O. Box 2923, Maputo, Mozambique
Tlepehone 420860, Telex 6-635, Telefax 430204

9th Interamerican Math Ed Conference
Miami, August 3-7, 1991
For further information, write to:
Rick Scott
College of Education
University of New Mexico
Albuquerque, NM 87131 USA

7th International Congress on Math Education
Quebec, CANADA August 1992
For further information write to:
David Wheeler, Chair IPC for ICME-7
Department of Math & Statistics
Concordia University, Loyola Campus
Montreal, Quebec CANADA H4B 1R6

ISGEm Executive Board

Gloria Gilmer, President
Math-Tech, Inc.
9155 North 70th Street
Milwaukee, WI 53223 USA

Ubiratan D'Ambrosio, First Vice President
Pro-Rector de Desenvolvimiento Univ.
Universidade Estadual de Campinas
Caixa Postal 6063
13081 Campinas, SP BRASIL

David Davison, Second Vice President
Dept of Cirruclum & Instruction
1500 N. 30th Street
Billings, MT 59101-0298 USA

Luis Ortiz-Franco,
Third Vice President
Department of Mathematics
Chapman College
Grange, CA 92666 USA

Claudia Zaslavsky, Secretary
45 Fairview Avenue #13-I
New York, NY 10040 USA

Anna Grosgalvis, Treasurer
Milwaukee Public Schools
3830 N. Humholdt Blvd.
Milwaukee, WI 53212 USA

Patrick (Rick) Scott, Editor
College of Education
University of New Mexico
Albuquerque, NM 78131 USA

San-Lin Tsang, Assistant Editor
Southwest Center for Educational Etluity
310 Eighth Street #305A
Oakland, CA 94607 USA