Addison-Wesley hosted our first reception in
celebration of ISGEm's affiliation with NCTM. A great time
was had by all. Kudos to Addison-Wesley!!
We are offering a one year free membership to the winner
of our ISGEm logo contest.
- A bound volume of Past Newsletters will be available at
ICME-7.
- Anna Grosgalvis accepted the NCTM certificate of
membership for ISGEm at the full Delegate Assembly in New
Orleans April 17, 1991.
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.
Epilogue
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;
- 8.2: On Celtic know ornaments;
.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."
References
[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.
Constitution
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.
By-Laws
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.
UPCOMING EVENTS
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
Maput--Mozambique
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
