Volume 2 Number 1, September 1986

ISGEm Advisory Board

Gloria Gilmer, ISGEm Chair
Milwaukee, WI USA

Ubiratan D'Ambrosio
13100 Campinas, SP BRASIL

Gilbert J. Cuevas
Coral Gables, FL USA

Patrick (Rick) Scott
Albuquerque, NM USA

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Report on ISGEm Business Meeting

A business meeting of ISGEm was held on April 2, 1986 at the Annual Meeting of the National Council of Teachers of Mathematics in Washington DC with twelve persons in attendance.

Gloria Gilmer reported that approximately sixty memberships had been received from eleven countries. It was decided to publish the Newsletter twice a year. Attempts will be made to identify editors by countries who will be responsible for duplicating and mailing in their own country, and translating into their own language(s) if necessary. The Newsletter will include a section called "Have You Seen" in which to review the literature in the field.

It was suggested that attempts should be made to affiliate with the International Congress on Mathematical Instruction (ICMI) and with the NCTM.

Plans to create a Document Center for Ethnomathematics include possibilities at UNESCO in Paris or ERIC in Ohio.

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HAVE YOU SEEN

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

Rick Scott, Editor
ISGEm Newsletter
College of Education
University of New Mexico
Albuquerque, NM 87131 USA

Elementos de Análisis en Matemáticas Quichua y Castellano (Elements of Analysis in Quichua and Spanish) by Consuelo Yánez Cossío and Augustín Jerez of the Pontificia Universidad Católica in Quito, Ecuador.

The authors start from the premise that different languages not only have differences in basic mathematical vocabulary, but also in morphological structure, operational mechanisms and situations in which mathematics is used. In societies without written communications calculations are made to handle various social and commercial situations, but the development of those mathematical systems is often retarded by the imposition of the system used by the socially and politically dominant culture.

In traditional Quichua culture in Ecuador children learn to handle mathematical concepts and operations at an early age as a natural part of their participation in the system of production. Currently not everyone who began the traditional learning process has finished it. Some received interference from mathematics of the Spanish language culture, some have participated in activities that do not require knowledge of traditional Quichua mathematics.

One important difference between Quichua and Spanish mathematics is that learning of Quichua mathematics is always bound to concrete social applications while the learning of Spanish mathematics is very often devoid of such real life applications.

Another important difference is that the number names in Quichua relate clearly to the base ten numeration system. For example the Quichua expression for 222 is ishcai patsac ishcai chunca ishcai which can be translated directly as two hundred two ten two. The Spanish expression for 222, doscientos veintidos, does not relate so directly to its meaning in base ten. This lack of direct and obvious relation with the structure of base ten may be partly responsible for memoristic rather than meaningful learning.

Quichua culture has a space-time system that is spiral rather than linear. The authors hypothesize that their mathematical systems are also spiral in nature. They see years as cycles related to the crop cycle rather than a linear array of 365 days. Another manifestation of the spiral/circular nature of their mathematical system is a device made of bone called a "huari" that is analogous to a die, but is more circular than cubical.

Operations within the Quichuan mathematical system are made at three levels: the first level is concrete and called "graneo", on a second level words are used to express quantities and at the third level there are mental calculations without overt words or symbols. The fundamental arithmetical processes tend to be based on tens and fives. For instance to add 266 and 288 the process can be related more or less as follows:

266 = 200 + 50 + 10 + 5 + 1

+288 = 200 + 50 + 30 + 5 + 3

400 + 100 = 500 (picha patsac)

40 + 10 = 50 (picha chunca)

3 + 1 = 4 (chusca)

554

(picha patsac picha chunca chusca)

The term "graneo" refers to the use of grains of corn, beans, seeds, pebbles, etc. either directly or with an abacus-like device called a "Contador del Cañar" (Cañar Counter).

Mental calculations can be quite sophisticated. One informant expressed the proces of mental calculation as follows:

In the mind there's a kind of path with spaces for numbers; there's a space divided in 10, then spaces that indicate each of the tens until 90, then the hundreds, the thousands and like that all the rest. Everyone has their path and there they mark the quantities they need to add and subtract. Each ten has its units and each hundred its tens.

The "paths" that the informant indicated tend to be circular with "nodes" that indicate the tens, hundreds, thousands, etc.

Work still needs to be done to translate what is know about Quichua mathematics into the school curriculum so that it can help facilitate learning of mathematics among Quichua children.

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Learning, Aboriginal World View, and Ethnomathematics by Robert P. Hunting of the Western Australia Institute of Technology, 1985.

Hunting stressed that "functional mathematics knowledge" is a "most valuable commodity in Australia" and that Aboriginal children have a right to that knowledge. To help facilitate the Aboriginal child's knowledge of mathematics he suggested that we must under-stand very clearly the nature of learning and mathematics, as well as the world view of Aboriginal people.

He outlined Western conceptions of know-ledge and reality, and emphasized the distinction between children's mathematical concepts and the mathematical knowledge of teachers. He noted that there is some consensus among cognitive psychologists that effective teaching demands that prior knowledge of the learner be identified. He presented a discussion of aspects of learning that are relevant to an understanding of how children learn mathematics: learning as imitation, as remembering, as problem solving, as executing, as restructuring, and as reconstructing. In addition to prior cognitive knowledge it is necessary to consider social and affective forces and barriers.

If we are to address teaching mathematics across cultures, Hunting stressed the importance of attending to tacit assumptions about life and existence. In the case of the Australian Aborigine those tacit assumptions can be very different from those of Western culture.

Included in assumptions about life and existence are the ways in which the data of experience are organized and classified. Hunting reviewed the work of Rudder which indicates that the Yolnu Aborigines use number concepts that are largely cardinal without a parallel ordinal meaning. Thus a physical collection of five objects is not usually considered to contain, at the same time, a collection of four, which in turn has a collection of three, etc.

Therefore one may ponder if English and Aboriginal number words really have the same meaning, and just how whole number addition and subtraction can be meaningfully understood. Aboriginal children will have access to the power of mathematics in modern Australian if "trouble is taken to fit that mathematics sensitively onto and around the beliefs, values, thinking patterns and problem solving processes contained in Aboriginal culture."

Hopkins proposed a research program that addresses the following questions in an attempt to identify Aboriginal activities and processes that "have potential for connections with mathematical concepts, techniques, and procedures":

1. What problems arise in traditional environments which require application of mathematics knowledge for their solution?

2. What is the nature of the mathematical processes used to solve those problems?



3. How does the mathematics of a culture or community change in response to changes brought about by contact with a different culture or community?

Hunting suggested that the data arising from an attempt to answer those questions could be called Ethnomathematics.

He further defined Ethnomathematics as "mathematics used by a defined cultural group in the course of dealing with environmental problems and activities." He expected that the research program would focus on "the identification of possible platforms for establishing number and measurement concepts," because such concepts are "basic to Western economic and technological knowledge." In addition to looking at traditional quantifying, sorting, grouping, and sharing, the highly developed visualization and geometric abilities in Aboriginal cultures may serve as links to number concepts.

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An Experience with Mathematics Testing Bilingually in Guatemala

The National Bilingual Education Program (PRONEBI) in Guatemala stresses the use of Mayan languages for basic mathematics instruction in preprimary, first and second grades. For those grade levels the mathematics textbooks are written in Mayan languages. In third grade instruction and accompanying textbooks are offered parallelly in Mayan and Spanish. Testing for program evaluation in first grade has been done with the testers reading the test items in the Mayan language as children respond on test sheets with appropriate graphics and symbols. In second, third and fourth grade the Math tests have been written tests in Spanish.

In control schools all instruction is in Spanish except at the preprimary level. Testing done at the end of the 1984 school year indicated the that students in the second grade in the experimental bilingual schools had significantly higher level of math achievement despite written tests in Spanish while a Mayan language was used for most mathematics instruction. Testing at the beginning of the 1986 school year in third and fourth grade using second and third grade tests respectively also gave some evidence that the students in the experimental bilingual schools have somewhat higher achievement levels.

Since the majority of the instruction uses the Mayan language it was decided to try to discover if achievement levels would be even higher if the testing was done in Mayan language. Although PRONEBI concentrates on instruction in the four major Mayan languages of Guatemala, this testing experience was conducted for only one of them. Mathematics testing was conducted in April in Spanish and in June in the Mayan language. The results for samples of students from second and third grades in five bilingual schools are shown in Tables 1 and 2.

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Table 1

Mathematics Achievement in Second Grade as Measured
by Tests Written in Spanish and Mayan



Mean in Spanish SD Mean in Mayan SD n
School 1 38.0% 13.1% 44.7% 9.7% 11
School 2 33.6% 8.8% 31.3% 9.9% 7
School 3 31.4% 13.9% 37.1% 9.6% 19
School 4 23.2% 13.3% 33.0% 9.8% 8
School 5 21.7% 10.3 25.0% 7.3% 15
Total 29.4% 13.3% 34.3% 11.1% 60

The difference in the two totals is significant at the .002 level.

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Table 2

Mathematics Achievement in Third Grade as Measured
by Tests Written in Spanish and Mayan



Mean in Spanish SD Mean in Mayan SD n
School 1 41.5% 14.5% 37.9% 11.6% 6
School 2 37.2% 8.5% 32.5% 10.8% 10
School 3 31.0% 11.9% 39.4% 6.9% 13
School 4 28.4% 12.9% 18.9% 6.6% 7
School 5 24.4% 6.8% 35.9% 7.6% 11
Total 31.7% 11.85 33.9% 10.7% 47

The difference in the two totals is not significant.

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The differences in achievement levels do not seem to be of any practical significance particularly considering the two months in between the tests in which the students had the opportunity to learn more of the material. Achievement levels in general are quite low. It may be noteworthy that the differences that are observed on the total mean scores are not consistent in all the schools. It may well be that the Mayan and Spanish languages are not used in all the schools in Mathematics instruction in the way suggested by the program.

The most reading was required on word problems. The second grade test had eight word problems. For the Spanish language version the score on the word problems subscale was 44% and it was 51% on the Mayan language version. The third grade test had nine word problems. The score was 30% on the word problems subscale for each version. These results seem to parallel the overall results and suggest that perhaps in second grade the students perform better in Mathematics in their maternal language, but that by third grade they per-form equally in their maternal language and Spanish.

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Vii Inter-American Conference on Mathematics Education

The VII Inter-American Conference on Mathematics Education (VII IACME) will be held in Santo Domingo, the Dominican Republic, from July 12-16, 1987, at the Universidad Católica Madre y Maestra.

Throughout the VII IACME, plenary sessions and panel discussions will be held. Work and discussion groups will be organized to study certain issues of mathematics teaching relevant to the American continent, with emphasis on the particular difficulties of each level. There will also be poster sessions and displays of materials related to mathematics teaching.

The official languages of the Conference will be Spanish, English and Portuguese. The papers will be mainly presented in Spanish and English.



During VII IACME, three plenary sessions will be given by internationally known specialists in the field of mathematics education.

Four panel discussions will be held on the following topics:

A. Integration of the sociocultural context in mathematics teaching.

B. How to develop problem solving skills in students.

C. Innovational uses of calculators and computers in mathematics teaching in Latin America.

D. How to improve geometry teaching in elementary and secondary schools.

You may submit a proposal to participate in one of the features of the VII IACME. Each of the main conference activities is briefly described below. Select the feature you wish to participate in, fill out the PROPOSAL FORM and ABSTRACT, and send them to the following address before October 30, 1986:

VII IACME
Centro de Investigaciones
Universidad Católica Madre y Maestra
Apartado Postal 822
Santiago de los Caballeros
Dominican Republic

Panel Discussions: These are long panel sessions where several people express their opinions concerning one of the topics A, B, C, or D mentioned above. Each panel will meet twice, for two hours per session. In the first meeting they will present their opinions and in the second session there will be questions, comments and a summary of the panel discussion.

Paper Sessions: These will be short sessions of about 15 minutes each, where one person or a group of people will present their work in mathematics education.

Discussion Groups: You may request the formation of a discussion group on a current topic in mathematics education. In the second announcement of this conference, a list will be given of the topics already selected so that other people interested in the same topic may be included.

Work groups: You may propose the formation of a work group where you present your experiences in a project carried out in mathematics education. In this case, you will be the leader of the group. In the second announcement of VII IACME a list will be given of the work groups already proposed, so that other people interested in the same kind of work may be included.

Poster Sessions: In a poster session, the display will be made graphically on a board specially prepared by the presenters and displayed in the exhibition area. The presenters should stay near their posters as long as possible to answer any questions that may arise.

Exhibitions: Displays of mathematics teaching materials that you or a group of people have prepared. (It includes software).