ISGEm 12.2



Volume 11 Number 1 December 1995



ISGEm Panel at the NCTM Meeting in San Diego



A Panel on Making Connections between Ethnomathe-matics and the History of Mathematics is scheduled for April 27, 9-10am at NCTM's Annual Meeting in San Diego.

The scheduled Panelists are:

Gloria Gilmer (Math-Tech, Inc.) "Contemporary Aspects of Ethnomathematics"

Felix Browder (Rutgers University)

Karen Dee Michalowicz (The Langley School) "Uses of the Mathematical and Scientific Ideas of Traditional Peoples, Specifically Native Americans, in the Pre-College (Secondary & Middle School) Classroom"

James Rauff (Millikin University) "Ethnoalgebra of the Warlpiri"

Jolene Schillinger (New England College) "Women and Ethnomathematics"

Ubiratan D'Ambrosio (State U of Campinas) "Recent Research in Ethnomath: Opening Areas of Action"

Other sessions related to Ethnomathematics include:

Ron Eglash speaking on Fractal Structures in Traditional African Culture on Friday from 10:30 to 11:30. Eglash will report on a project which began with the visual observation that aeial photos of traditional African settlements tend to have a fractal structure (scalingin steet branching, recursive rectangular enclosures, circles of circular dwellings, etc.)

Jim Barta and Gloria Gilmer doing a joint presentation "Cultural Bridges in Math Classes."

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ISGEm Annual Business Meeting in San Diego



The ISGEm Annual Business Meeting will be held from 4:30 to 6:00 on Saturday, April 27, during the NCTM Annual Meeting in San Diego. Information on the location of the meeting will be available in San Diego.

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Ethnomathematics at the 1996 AERA Meeting

Ethnomathematics: Examples Derived from Yup'ik, Navajo and Yoruba Cultures is the titl e of a poster session and symposium to be presented in Section 2 of Division K (Teaching and Teacher Education) at the Annual Meeting of the American Educational Research Association (AERA), April 8-12, 1996, in New York City.

The principal presenter, Jerry Lipka, of the University of Alaska at Fairbanks, is director of a project to bring Yup'ik mathematics and science into the curriculum of the Yup'ik schools of southwest Alaska, a model for the integration of indigenous perspectives on schooling. He will discuss:

The Centrality of the Body in Yup'ik Conceptions of Numeration, Orientation and Patterns.

Claudia Zaslavsky, author of Africa Counts and The Multicultural Classroom, will elaborate on Numeration Systems Based on Grouping by Twenties, with examples drawn from Yup'ik, Maya and Yoruba languages.

University of Alaska professor Claudette Bradley will discuss Symmetry and Tessellations in American Indian and Alaska Native Designs.

Yup'ik teacher educator Esther Ilutsik will illustrate applications in the classroom in her talk Yup'ik Patterns and Teaching Elementary School Children.

For time and place of meeting contact

Susan Melnick
(517)353-0690
susanm@msu.edu

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ICME-8

Seville, Spain, July 14-21, 1996



The 8th International Conference on Mathematics Education will be held in Seville, Spain, July 14-21, 1996. For the 2nd Announcement write:

ICME-8
Apartado de Correos 4172
E-41080 Sevilla
SPAIN

e-mail: icme8@obelix.cica.es
And now a World Wide Web Site
http://icme8.us.es/icme8.html

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ICME News

Sunday Ajose



For those of you planning to attend ICME8 in Seville next July, the Working Group (WG) 21 has a program that should be of interest to all ISGEm members. WG 21 - The Teaching of Mathematics in Different Cultures - has four 90- minute sessions.

Session 1: Two plenary lectures by distinguished speakers. Expect scholarly discourse on the state-of-the-art in mathematics and culture.



Sessions 2 & 3: Participants will break up into four subgroups focussing on:
a) The influence of culture on mathematics teaching and curriculum;
b) Preparing teachers to teach to diversity;
c) Research; crosscultural research, critical issues, questions, methodology, &
d) Mathematics learned outside of school.

Session 4: Reports on the work of the subgroups, an overview of the work of WG 21, future directions.

Your participation in the deliberations of WG 21 is encouraged. If you wish to share a paper at the conference, send the paper, its summary or abstract to:

Professor Jerry P. Becker
Curriculum and Instruction
Southern Illinois University
Carbondale, Illinois 62901-4610 USA

Fax: (618) 453-4244
Phone: (618) 453-4241
E-mail: jbecker@siu.edu

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Report on Ethnomathematics Research

Joanna O. Masingila, Syracuse University
jomasing@sued.syr.edu

This column reports on current research in the area of ethnomathematics. If you know of researchers doing ethnomathematics research, please send me this information either by mail (215 Carnegie, Syracuse, NY 13244-1150 USA) or by email (jomasing@sued.syr.edu).

Paolo Boero, from the Universita di Genova in Italy, and colleagues have been investigating some cognitive and didactic issues regarding the relationship between mathematics and culture in teaching and learning mathematics in compulsory school. They have focused on (1) how everyday culture may be used within school to build up mathematical concepts and skills, (2) the contribution that school mathematics may give to everyday culture to allow (and spread) a scientific interpretation of natural and social phenomena, and (3) teaching mathematics as part of the scientific culture.

David Carraher, from Universidade Federal de Pernambuco in Brazil, and colleagues have been examining ways in which knowledge learned in one situation might be transferred to other situations. Their work has convinced Carraher and his colleagues that the ways in which students learn to deal with new specific situations involves remarkable use of previous knowledge as analogies, categorizations, comparisons across situations, search for correspondences between different settings, as well as generalizations, and that to recognize them as such one needs to set aside stereotyped/formal views about what transfer and use of previous knowledge are.

Venus Dawson, from the University of California at Los Angeles, is examining the mathematics practice of basketball players in an inner city basketball league. She is particularly focusing on how the players make sense of and use concepts related to statistics. In a second phase of her study, she is investigating how some basketball players and some non-basketball players make sense of problems involving statistics when the problems are similar to those found in school textbooks and when the problems are framed in a basketball context.

Guida de Abreu, from the University of Luton in the UK, has been investigating how children growing up in a rural sugar-cane farming community in the state of Pernambuco, Brazil, experience the relationship between their home and school mathematics. When engaged in the practices of sugar-cane farming, people in this community make use of an indigenous mathematics that differs markedly from the mathematics taught in the local schools. As part of this study, Abreu studied two teachers who taught in the primary school in this rural community. Both case studies illustrate that to come to terms with the situation (i.e., the wide gap between the students' home and school mathematics), teachers develop representations of mathematics that enable them to: (1) understand and explain the situation and also justify their teaching practices, and (2) locate themselves and the children in the social structure of the farming community.

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RECENT THESES & DISSERTATIONS ON ETHNOMATHEMATICS

Ubiratan D'Ambrosio



A number of dissertations on Ethnomathematics are reported in different parts of the world. A number have been submitted in Brazil and in Spain. These are effective research contributions which add significantly to the area. This is a report on a number of them.

On May 1995, Maria Luiza Oliveras Contreras presented a doctoral dissertation at the University of Granada, Spain, with title Etnomatemáticas en Trabajos de Artesanía Andaluza: Su Integración en un Modelo para la Formación de Profesores y en la Innovación del Currículo Matemático Escolar [Ethnomathematics in the Artisanal Work in Andalusia. Its Integration in a Model for Preservice Teacher Training and in Innovation of School Mathematics Curricula]. This important work is the result of more than 10 years of research on the Mathematics identified in artistic artifacts typical of Granada. Three kinds of these were chosen for the research: empedrados (stone pavement), taraceas (marquetry) and alfombras (carpets). A very original ethnography is proposed by the author to identify the Mathematical contents of these beautiful handworks. An ethnomathematical theoretical framework allowed the recognition of important styles of doing Mathematics which would be unrecognizable with the prevailing views of academic mathematics. An important aspect of the theses is researching the way the techniques of work are transmitted among artisans, the masters and the apprentices. This was very appropriately called "ethnodidactics" by the author. And the methods there observed were important in proposing a structure of teacher training through projects. We recognize there a model of training teachers to act as researchers. This important contribution to Ethnomathematics will probably become a book in the series published by the Department of Didactics of Mathematics at the University of Granada.

In March 1995 Gelsa Knijnik submitted to the Faculty of Education of the Federal University of Rio Grande do Sul, in Porto Alegre, Brazil, a thesis under the title Cultura, Matemática, Educação na Luta pela Terra [Culture, Mathematics, Education in the Struggle for Land] . This very important work is the result of several years of research among teachers of the so-called "Movimento dos Sem-Terra". This is a political action with the objective of occupying the lands which, according to Brazilian constitution, are subjected to expropriation for land reform. The effective possession of these large tracts of land after the occupation implies several legal démarches which may take years, normally about five years. Meanwhile, those occupying the lands are confined to these areas and have to develop their own social structures: schools, medical assistance and production. They cannot leave the territory and the support they receive is nonpermanent, obeying humanitarian demands. In this period of confinement they have to rely on their own resources. These rural populations have a minimal education and have to run their own surveying and land demarcation practices, and the production system, as well as their schooling. There is so much mathematics in all these activities. The ethnographic research of Gelsa Knijnik focused on identifying the Ethnomathematics of these processes and giving the supporting instruments to integrate these practices in a school mathematics curriculum relevant for their immediate needs and allowing the transition to the official school system after overcoming the legal obstacles. How to conduct the teacher training for this parallel educational systems, relying, of course, on the human resources provided by uneducated confined population, is a major challenge. The thesis of Knijnik presents a socio-political and pedagogical study of these issues, always stressing the Mathematical content in every step of the process. The theoretical framework includes a thorough discussion of conceptual aspects of Ethnomathematics.

In April 1995 Adriana Isler P. Leite presented a dissertation to the Programa de Pos-Graduação de Educação Matemática of the Universidade Estadual Paulista/UNESP at Rio Claro, under the title A Brinadeira é Coisa Seria: Estudos em Torno da Brincadeira, da Aprendizagem e da Matemática [Playing is serious: Studies about playing, learning and Mathematics] . The dissertation was the result of an extended ethnographic research over three years involving children aged between 5 and 8 years old. The focus was understanding the way children play spontaneously and recognizing the Mathematics contents of these activities. The theoretical framework was Ethnomathe-matics and the ethnography adopted, with the analysis of about 60 hours of video taping, lead to an important contribution to understanding the formation of mathematical concepts in early childhood. It is of much importance to the conceptual discussion of the nature of Ethnomathematics in view of the theories of cognition and learning, particularly of Vygotski.

Marianna Kawall Leal Ferreira submitted a dissertation to the University of São Paulo on Da Origem dos Homens a Conquista da Escrita: Um Estudo sobre Povos Indigenas e Educação Escolar no Brasil [From the Origin of Men to the Conquest of Writing: A Study of Indian Peoples and School Education in Brazil] dealing with the construction of knowledge in an Amazonian tribe. Very careful research was conducted among a number of different tribes of the Parque Indígena do Xingú. A variety of cultures provided the author with the opportunity to understand the historical and philosophical ground upon which these tribes build their knowledge. Several aspects of Indian culture, as seen in the schools of the tribe, are analyzed, focusing on the educational process which give emphasis on the transmission of "official" knowledge and values.

Sônia Maria Clareto worked in a small fishing community on the seashore (caiçara) in the State of São Paulo. The dissertation was an ethnographic study of the space perception of school children after taking classes of Geography. Specifically, what was the perception of the child as "standing upside down" after being exposed to a terrestrial globe. A most interesting dissertation entitled A criança e seus dois mundos: A representação do Mundo em crianças de uma comunidade caiçara [The child and its two worlds: The representation of the World by children of a "caiçara" community] based on this research was submitted to the Universidade Estadual Paulista/UNESP at Rio Claro.

Samuel Lopez Bello submitted a dissertation on Educação Matemática Indígena -- Un Estudo Etnomatemático dos Indios Guarani-Kaiová do Mato Grosso do Sul [Indigenous Mathematical Education -- An Ethnomathematical Study of the Guarani-Kaiovaa Indians in the State of Southern Mato Grosso] . The dissertation refers essentially to questions about Education, particularly Mathematical Education, among Indian communities in a somewhat remote State in Western Brazil. The main objectives were to identify and recognize different ways of explaining and knowing in the Guarani culture and to relate these with the strategies of formal schooling. The ethnographic research gave origin to new methodologies and techniques on participant observation. New interpretations of cognitive models among indigenous cultures resulted from the research. An important result was the recognition of the role of the history of the individuals and of the communities in the cognitive processes. Among the variety of topics discussed, particularly important were questions about shapes, measures and counting.



General comments

The thesis of Gelsa Knijnik was published, with slight modifications, as a book with title Exclusão e Resistencia: Educação Matemática e Legitimidade Cultural [Exclusion and Resistance: Mathematics Education and Cultural Legitimacy], Artes Médicas, Porto Alegre, 1995. The thesis of Maria Luiza Oliveras Contreras will also appear as a book. The mathematical part of the dissertation of Mariana K. Leal Ferreira became a booklet: Com quantos paus se faz uma canoa! A Matemática na vida cotidiana e na experiência escolar indígena [With how many logs one can make a canoe! Mathematics in the daily life and in Indian school experiences], MEC/Assessoria de Educação Escolar Indígena, Brasilia, 1994. The others will appear only as papers presenting partially the important results. The fact that they are in Portuguese and Spanish limit, in a sense, the accessibility to these important contributions to Ethnomathematics. Indeed, a considerable amount of research in these fields comes out of research in Latin America, as well as in Lusophone countries in Africa and in Portugal and Spain, but language is still a barrier. Fortunately, much of the important work of Paulus Gerdes has been translated into English and French.

These works reveal the large scope of Ethnomathematics. Indeed, we can hardly classify these as Mathematical works. This is, in a sense, a sort of "epistemological aggression". The distinction between Ethnomathematics and Ethnoscience, Ethnohistory, Ethnomusicology, Ethnomedicine, Ethnopsychiatry, Ethnomethodology, becomes very artificial and difficult to establish. Even in the Mediterranean civilizations and as recently as the XV century, Mathematics and Religion, the Sciences and the Arts, are difficult to separate.

This leads us to look into different ways, styles, techniques of explaining, of understanding, of coping with the surrounding natural and cultural environments as the essence of the History of Ideas. In order to organize these studies, we have to coin a few words to express the above: different ways, styles, techniques [tics], of explaining, understanding, coping with [mathema] the surrounding natural and cultural environments [ethno]. Thus we have the word Ethnomathematics, which in this conception obviously incorporates Ethnoscience.

Ways, styles, techniques of explaining, of understanding, of coping with the surrounding natural and cultural environments have been developed and accumulated throughout the entire history of mankind in different cultures, with different objectives and following different patterns of thought. We can hardly fit knowledge recognized in a variety of cultural environments into the current academic classification of knowledge which comes from the civilizations around the Mediterranean. With the increasing attention to -- and respectful attitude towards -- different cultures, broader epistemologies are needed. There is a general acceptance of multiculturalism in Education. But this can be voided if we do not adopt a multicultural approach to the History of Ideas. The Program Ethnomathematics is an answer to this. The theses and dissertations mentioned above fit into this program.



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Ethnomathematics and Multicultural Mathematics

Bill Collins
Syracuse (NY) City School District



The June 1995 ISGEm Newsletter carried a report of the Curriculum and Classroom Activities SIG meeting at NCTM in Boston. While I was unable to attend that meeting, it's description is of great interest to me, not from the point of view of curriculum and classroom activities, rather the discussion of the relationship between ethnomathe-matics and multicultural mathematics which was reported. While I take no issue with the arguments listed which place multicultural math as a subset of ethnomathematics, I'd like to offer another point of view, that, if accepted, could lead to a different view of the relationship, perhaps one which shows an intersection, but no "set, subset" asso-ciation.

In my view of mathematics, presented in a classroom, from a multicultural perspective, ethnomathematics plays an important part. A good grasp of history and the role that mathematics has played in it, an acknowledgment of human differences which have led (and continue to lead) to varying approaches to solving problems, and a strong connection with the work environment, all speak of ethnomathematics and all contribute to breathing life into a subject which too often has been seen as lifeless. But my version of multicultural mathematics is broader than that:

It includes the acknowledgment of biases that have led to lesser expectations for some groups, and actively working to correct these wrongs.

It includes study of the research attempting to link race, gender or ethnicity to learning styles. Admittedly, this is dangerous ground on which to walk, since overemphasis on such data can lead one to stereotyping and the evils inherent in such behavior. But as we look at each student and attempt to ascertain what works best for him or her as we present mathematics, group tendencies, if they can be documented, should not be ignored.

It includes discrete mathematics which deals with honest, real life statistics. As we teach children to deal with data, real world numbers showing school, community, state or national statistics detailing differences by race, help to put a realism in our classrooms that already exists for our children when the day's last bell rings.

It strikes me that, just as I can define multicultural mathematics as broadly as is comfortable for me, another reader may do the same with ethnomathematics. That person could categorize my above examples as ethnomathe-matics and thereby come to a conclusion counter to my own. So be it. Professor Paul Pederson, of Syracuse University, in remarks made at a New York State-wide conference on multicultural education, stated that true multiculturalism allows two conflicting points of view to be allowed as true, as the point of view of each party must be taken into account.

Before reading the newsletter, I would have said that ethnomathematics is a subset of multicultural mathematics, for the reasons given above. The points addressed in the June '95 article have convinced me, that two intersecting figures, each of which containing a part of the other is the best model I can envision for the relationship between these two important and growing fields of information. While the typical figure used to represent this intersecting sets is a circle, the amorphous and ever evolving nature of these concepts makes a circle inappropriate. No figure should accompany this letter. It is up to the reader to envision the relationship and therefore supply the mental graphics.



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A Definition of Ethnomathematics

Gloria Gilmer, Math-Tech Inc.



The following definition of Ethnomathematics has been prepared for a dictionary of multicultural education:

(1) Ethnomathematics is the study of the mathematical practices of specific cultural groups in the course of dealing with their environmental problems and activities; For example, the manner in professional basketball players estimate angles and distances differs greatly from the corresponding manner used by truck drivers. Both professional basketball players and truck drivers are identifiable cultural groups that use mathematics in their daily work. They have their own language and specific ways of obtaining these estimates and ethnomathematicians study their techniques.

The prefix 'ethno' refers to identifiable cultural groups, such as national-tribal societies, labor groups, children of a certain age bracket, professional classes, etc. and includes their ideologies, language, daily practices, and their specific ways of reasoning and inferring.

'Mathema' here means to explain, understand and manage reality specifically by ciphering, counting, measuring, classifying, ordering, inferring and modeling patterns arising in the environment.

The suffix 'tics' means art or technique.

(2) Thus, ethnomathematics is the study of mathematical techniques used by identifiable cultural groups in understanding, explaining, and managing problems and activities arising in their own environment.

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Descartes and Ethnomathematical Ideas

Fernando Castro
Maturín, Venezuela



D'Ambrosio is credited with coining the term Ethnomathematics, which he has defined as:

the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering and inferring which result from well-identified cultural groups (D'Ambrosio, 1988).

Bishop (1988) includes playing, designing and locating as other environmental processes rich in Mathematical ideas.

It is interesting to point out that Descartes anticipated some mathematical ideas when in "Rules for the Direction of the Mind" he said:

The message of this rule is the at we must not take up the more difficult and arduous issues immediately, but must first tackle the simplest and least exalted arts, and specifically those in which order prevails - such as weaving and carpet-making, or the more feminine arts of embroidery, in which threads are interwoven in an infinitely varied pattern. Number games and any games involving arithmetic, and the like, belong here. (Descartes, 1985, p. 35)

In this way Descartes was also anticipating the modern studies on symmetry and its relationship with the notion of group (Bourbaki, 1972).

We can also think that if Descartes recognized spontaneous Mathematical ideas in some playing and designing activities, then he was conceiving an alternative way of generation of knowledge different from the "academic" one. So when he said: "everyone would be firmly convinced that the sciences, however, abstruse, are to be deduced only from matters which are easy and highly accessible, and not from those which are grand and obscure" (Descartes, 1985, p. 34), it seems that he was referring to environmental sources for sciences.



References

Bishop, A. (1988). "The Interaction of Mathematics Education with Culture", Cultural Dynamics, vol. 2, no. 2.

Bourbaki, N. (1972). Elementos de Historia de las Matemáticas. Alianza Editorial. Madrid.

D'Ambrosio, U. (1988). " A Research Program in the History of Ideas and Cognition", ISGEm Newsletter, vol. 4, no. 1.

Descartes, R. (1985). The Philosophical Writings of Descartes, vol. 1, translated by John Cottingham, Robert Stoothall and Donald Murdoch, Cambridge, Cambridge University Press.

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UW-Madison Lands National Science Education Institute



The University of Wisconsin - Madison has been selected by the National Science Foundation (NSF) as the site of a $10 million national science education institute.

The one-of-a-kind institute, to be funded at $2 million a year over five years, will be the nation's premiere center of research and development of issues of science, math and engineering education.

The new institute will be headed by Denice D. Denton, a UW-Madison professor of electrical and computer engineering, and Andrew C. Porter, a UW-Madison professor of educational psychology and director of the Wisconsin Center for Education Research. The National Center for Improving Science Education, based in Washington, D.C., will be UW-Madison's chief partner institute.

"This is an opportunity for us to play a role in providing national leadership in science, math and engineering education at a time when the national is looking very closely at those activities," said Porter.

The new institute will be funded through a cooperative agreement between UW-Madison and NSF. The overarching goal of the institute will be to lay a foundation for enduring science, math and engineering education reform from kindergarten through college.

"We are establishing the institute because of a need to examine more systematically the fundamental educational reforms under way in the United States," said Luther Williams, NSF's assistant director for education and human resources.

The state of science and math education in the United States has been the subject of sharp debate and concern. Many scientists and policy makers argue the nation's ability to field a scientifically literate public and modern technical workforce is at risk because science has been made inaccessible, inequitable and unpalatable for millions of Americans.

A critical task of the new institute will be to ensure access and equity for groups traditionally under represented in science, math and engineering, a condition reflected in very low numbers of women and minorities in those fields.

In part, the institute will be a national think tank for science education issues and strategies, and will attract visiting scholars from around the world. One of the earliest initiatives will focus on enhancing introductory science courses in two-year colleges.

Key aspects of the institute's work will be communicating with and disseminating information to elementary and secondary school science teachers nationwide, and creating professional development models for educators.

For further information contact

Denice D. Denton
(608)263-2354

Andrew C. Porter
(608)263-4200

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Have You Seen

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


Gerdes, Paulus, African Pythagoras: A Study in Culture and Mathematics Education, Universidade Pedagógica, Maputo, Mozambique, 1994. (Available from the author at P.O. Box 3276, Maputo, Mozambique).

This translation of a 1992 Portuguese version includes the following chapters:
Did ancient Egyptian artisans know how to find a square equal in area to two given squares?
From woven buttons to the Theorem of Pythagoras
From fourfold symmetry to 'Pythagoras'
'Pythagoras', similar triangles and the "elephants' defense" pattern of the Bakuba
Widespread decorative motif & Theorem of Pythagoras
From weaving patterns to Pythagoras & magic squares
A new proof by means of limits
A new proof related to an ancient Egyptian decoration technique



Gerdes, Paulus, Ethnomathematics and Education in Africa, Institute of Education, Stockholm University, 1995.

This report was published within the framework of the institutional cooperation that started in 1991 between the Institute of International Education (IIE), Stockholm University and what is now the Universidade Pedagógica in Maputo, Mozambique. It includes the following chapters:

Ethnomathematical research: preparing a response to a major challenge in mathematics education in Africa
On the concept of ethnomathematics
How to recognize hidden geometrical thinking: a contribution to the development of an anthropology of mathematics
On culture, geometrical thinking and mathematics education
A widespread decorative motif and the Theorem of Pythagoras
'Pythagoras', similar triangles and the "elephants' defense" pattern of the Bakuba
On possible uses of traditional Angolan sand drawings in the mathematics classroom
Exploration of the mathematical potential of 'SONA': an example of stimulating cultural awareness in mathematics teacher education
Technology, Art, Games and Mathematics Education: an example
On the history of mathematics in Africa south of the Sahara

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Villalobos, Leslie, Un Enfoque Humano de la Matemática (A Human Focus on Mathematics), Editorial EARTH, Apartado Postal 4442-1000, Guácimo, Limón, Costa Rica, 1995.

This book, written by a mathematics instructor at the Agricultural School for the Humid Tropical Region (Escuela de Agricultura de la Región Tropical Húmeda - EARTH), motivates the study of mathematical techniques by presenting problem situations such as "Sustainable Agriculture", "Learning to Produce", "War and Peace", "An Arab Legend", "Mathematical Bees" and "Heron Rides Again".

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Zaslavsky, Claudia, The Multicultural Classroom: Bringing tin the World, Heinemann, 361 Hanover St, Portsmouth, NH, 03801-3912, USA ($26.50), 1995.

Claudia Zaslavsky's new book introduces a multicultural perspective to the elementary and middle grade math curriculum, revealing how such a perspective can enrich the learning of all students, whatever their gender, ethnic/racial heritage, or socioeconomic status. Students learn that mathematics was created by real people attempting to solve real problems. They're asked to solve the same kinds of problems and to extend their problem solving to issues within their communities.

The chapters included are:

Overview of Multicultural Education
Multicultural Mathematics Curriculum
The Mathematics-Literature Connection
Counting with Fingers and Words
Numerals: Symbols for Numbers
Recording and Calculating: Tallies, Knots and Beads
How People Use Numbers
Geometry and Measurement in Architecture
Geometry, Measurement and Symmetry in Art
Data Analysis and the Culture of the Community
Games of Many Cultures
Multicultural Mathematics in Practice

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The Mathematics of Nature: The Canamayté Quadrivertex

Rogelio Pitracho Díaz
Mathematics Museum, University of Querétaro



When we speak of Geometry, usually we think of Euclidean Geometry. Rarely do we think of the Geometry that we inherited from our American ancestors, especially the Mayas and the Aztecs. This work will attempt to take us back in time to our Mathematical Roots, to the Geometry of Nature. Thus we hope to motivate a more profound investigation of that geometry, because currently we are ignorant of it.

The Canamayté Quadrivertex is the central square in the row of squares on the back of the Yucatecan rattlesnake (Crotalus Durissus Tzabcan Yucateco).





Figure 1: The Ajau Cán with thegeometric pattern on its skin.



Figure 2: A vertical square with a cross inserted symmetrically in the square with its axis centered.



The four sides of the square represent the number 4, which is an attribute of the Sun and of this snake, and also symbolize the four cardinal points and the quadrature of the Sun and the Moon (Figure 2).



The Dawn of Mayan culture

The dawn of Mayan culture, what is called the formative period, dates from 500 BC. Later came the so-called classical period between 300 and 800 AD. The height of the Mayan civilization occurred around the year 700 AD. The decline of the empire has been situated between 800 and 925 AD. In its full splendor the Mayan culture raised great ceremonial centers in southeast of what is now Mexico (Yucatán, Campeche, Quintana Roo, much of Tabasco and the eastern half of Chiapas), as well as almost all of Guatemala, and parts of Honduras and El Salvador.



Figure 3: Map of the Mayan region



Figure 4: One version of a Mayan calendar



Mayan Arithmetic

The concern of the Mayas to measure time led them to make calendric and astronomical calculations as precise as those that are made today by modern astronomers. In referring to the Mayas, Thompson (1966, p. 178) said "Careful and patient observation over hundred of years, transmission of data from one generation to another, and flexible minds willing to discard inaccurate calculations were the essentials of success".

To make precise calendric and astronomical calculations a measuring and counting tool was necessary. There emerged the vigesimal (base 20) system, and the most important invention of the Zero. It is through their writing with pictographic images that we have come to know the customs and decipher the knowledge of the Mayas, including their positional numeration system.

The Mayan vigesimal system uses only three elements (point, line and a variety of representations of zero). It should be pointed out that there exists another way to represent Mayan numerals: with the use of various representations of a human head, which were considered to be the gods of each of the numbers in the vigesimal system.

It would appear that the only reason that the pre-hispanic cultures chose the vigesimal system is the obvious one: not only can we count with are fingers, but also with our toes. The Mayas prefer-red to use all the "digits", those associating the vigesimal system with my own self (el Propio yo).

The Mayas represented zero in various ways. In the top row of Figure 6 below are variations of a snail and of a bivalve shell. These representations appeared in the Mayan codices studied by Ernest Förstemann. The bottom row are forms of zero that have been found in stelae, tablets, monuments and other archeological objects.



Figure 6: Various Mayan forms of zero.



The most common Mayan representation, outside of the codices, is a flower or incomplete flower. Nevertheless, a variant of the head did exist, just like there were for other numbers. In it appears the profile of a Mayan face with a hand "making horns" down by the jaw. There is also a spiral sign on the forehead, which is believed to have some relation to the snail forms that appear in the codices.

The Mayas used the zero to refer to dates and time periods in diverse monuments and texts. Nevertheless, on occasions it neither represented the absence of units nor the empty set, but instead denoted the end of a period of time or date and the beginning of the next one. Thompson (1966, p. 182) suggested with respect to zero that

This was a discovery of fundamental importance. That it was not an obvious one is shown by the failure of any people of our Western world to make it. Even the great philosophers and mathematicians never found this simple way of lightening their laborious calculations. Indeed, it was not known in Europe until it was passed on to our ancestors, after the Maya Classic period had ended, by the Arabs who had learned of it from India.



Figure 7: Stela 18 at Uaxactún which has the oldest sculpted zeros in the world.



The cosmogonic idea of the Canamayté can be understood from the following passage from the Popol Vuj, sacred book of the Maya-Quiché:

It is with great detail that the description and narration of how everything, Heaven and Earth, was formed is given, how the four corners and four sides were made (that is, governed by the square of Crótalus Durissus), how the four stakes were measured and placed, how the cord was folded and extended.

In order that the extent of the Sky and the Earth might be measured they were made of four sides by Tz'akol Bitol. That is to say, they were made by the mother and father of life and of creation; she who gives light and he who looks out for the well-being of the true race of true daughters and sons; thinkers who had wisdom about everything wherever there is Sky and Earth, lakes and seas.

According to the Mayas, creation was carried out in accordance with the geometric principle of the rattlesnake. The Mayan creator is a mathematician. The root of Tz'akol is Tsa or Tza, that is Tzamná or Itzamná, which comes from Tzab, rattlesnake, which is onomatopoeic with the sound of the rattle. Tz'akol is then the same god-man-bird-serpent of a rattlesnake known by the name Cuculcán or Quetzal-cóatl, the mathematician and geometer par excellance (for being the creator of all things).

The plumed serpent, that is indigenous, signifies time, chronology, calendar; which is shown by the fact that not only Zamná, but Cuculcán and Quetzalcóatl, were reputed to be the inventor of the science of measuring time. They had the rattlesnake as their emblem and were called Can y Coatl.

The number 4 is Cuculcán's, as well as Quetzalcóatl's, corresponding to the four sides of the square. The 4, considered as the sides of the square, is in the designs on the skin of the Crotalus. The Crotalus Durissus (Crotalus Durissus Durissus of Central American, Crotalus Durissus Tzabcán or Yucateco, and Crotalus Durissus Totonacus) are the rattlesnakes that on their bodies have the pattern of proportion Ad Quadratum.

The Crotalus Durissus provides the base for the astronomic arithmetic of the Mayas, as that arithmetic has the numbers 4 and 13 that are repeated in various aspects of the geometric model of the rattlesnake. Among its merits we have the mathematical basis of Mayan art, the model of the so-called false arch, 23 parallelisms of the Sun, no less that 2 axes of symmetry and the symbol of the four solstice blades (aspas solsticiales) called Can or Nahui olin.

Some particular Crótalus Durissus have 13 scales in each of four rows for a total of 52, the number of years in the Mayan and Toltec astronomical cycles known as canturía, or, in other words, the main period that combine the movements of the Sun, the Moon and Venus.

In all the Mayan region one finds stone reliefs of priests holding in both hands a ceremonial rod that has, in the center, the row of vertical squares of the Ajau Can (Figure 8). Generally, these rods have a snake head on top. They were the mathematical insignias of the wise priests that ordered the construction of the Mayan temples.



Figure 8: Relief of a Mayan Priest



Mayan Geometry

The Canamayté Quadrivertex (Figure 9) is a geometrical model that antedates all archeological and historical cultures and offered its mathematical foundations to all the precolumbian cultures. As it moves, the rattlesnake produces a dynamic geometry. The squares are transformed into rhombuses that immediately return to being what they were, thus revealing Geometry, Arithmetic, Cosmology and Architecture. Geometry was the soul of the terrestrial and celestial thought of the Mayas in the same way that mathematics was the soul of Greek culture.



Figure 9: Canamayté Quadrivertex from the skin of the Crótalus Durissus Tzabcán Yucateco



Below in Figures 10-12 is an indication of how the Canamayté Quadrivertex might be used in geometric constructions:



Figure 10: The pentagon and the star drawn with only the mathematical help of the Canamayté



Figure 11: The Canamayté inserted into another square; the cross of the octants of the Moon and its phases



Figure 12: The Canamayté of Uxmal drawn only with the pattern of the Canamayté, in the center is the flower of the lunar phases.



Figures 13-20 demonstrate how the Canamayté Quadrivertex might indicate proportions found in nature and in construction:

Figure 13: Proportion of a flower

Figure 14: Proportion of the Mayan profile

Figure 15: Proportion of a face

Figure 16: Proportion of the human body exactly as it was in the well-known drawing of Leonardo Da Vinci, illustrating the Pythagorean theory of the golden ration, the proportion Ad Quadratum

Figure 17: Proportion of a straw hut

Figure 18: The Canamayté and the first Mayan temples

Figure 19: Aerial view of a pyramid

Figure 20: Model of the Mayan arch, known as false; but authentic for them. The position of projecting stones in the arch is exactly the same as scales of the rattlesnake.

Location of the Cardinal Points

The four cardinal directions can be fixed using the Canamayté of the rattlesnake. To do so, we fix two points, one for each solstice, as in Figure 21.



Figure 21

Then, just as is stated in the Popol Vuj, we have a cord, between the two solstitial points; and, also as indicated in that book, we fold back that cord. Now we have the equinoctial point, which is the mid-point of the path between the high point of the sun in the summer and the other high point in winter (Figure 22).



Figure 22

Having done that, we look for the angle of 4654', which is the solstitial angle, by drawing a point below the equinoctial line (Figure 23).



Figure 23

Having obtained the solstitial angle, we adjust the Canamayté with the lower point and upper equinoctial pint (Figure 24).



Figure 24

Thus placed, the Canamayté indicates the four cardinal directions with its four vertices: East, South, West and North. Keep in mind that East is the principal point of Orientation, as is so well said by the word Orientation. That is to say, the true orientation is to the East, given that it is there that the sun rises, as it drifts from one solstice to the other. An indication that the orientation was obtained with the Canamayté we see in the belief that there were four huge rattlesnakes, one for each "corner" of Heaven, which were in fact the angles of the Canamayté.

In counting here the number four is fundamental, given that it is both solar and Crotalic. Counting from the east vertex of the Canamayté four scales to the left and then four to the right to be at the central scale of the Canamayté, we have the angle of the solstitial opening.

If we draw a line between the east-west vertices, we have the equinoctial point and the exact establishment of the Canamayté in its cosmographic function, given that:

a) The Canamayté with its geometric Solar-Lunar proportion and with its propitiation of the circle whose drawing it facilitates, permits the division of the circle no only by means of 45 angles, evident in said Canamayté, but also in the counting of the scales that regulate like an abacus the arithmetic division of the circle, by "degrees" according to Mayan science.

b) An indication that the Canamayté served to establish the cardinal points we find in that the Stone of the Sun (Aztec Calendar) is drawn exactly according to that pattern. That stone should lie in a horizontal position inasmuch as it shows the solstitial angle and is divided according to the movements of the Sun and Moon. It was oriented with its vertices located on the equinoctial line and cardinal points. The division of the Stone of the Sun are, without the least discrepancy, the divisions of the Canamayté.



Conclusions

From a scientific point of view it is somewhat remarkable that the figures of Euclidean Geometry are implicit in rattlesnakes; that is in Nature. Incredibly, the Crótalus Durissus also expresses the scientific bases of other disciplines such as Architecture, Arithmetic and Cosmology. That shows, yet again, that the human mind, or science, discover that which is in Nature.

Below is a comparison between Euclidean-type propositions and Mayan geometry:



E1) Two points determine a line.

M1) A point is used for unity.
A line is used to indicate five units.
The Canamayté is formed by line segments.

E2) A line can be extended from each end.

M2) The lines that calculate the solstice angle can be extended.

E3) Given a point and a segment it is possible to draw a circle.

M3) For the Mayas the Earth was the center of a circle. The center of the Universe was the Sun.
The radius of said circle was the segment given by the distance from the Earth to the Moon.
The circle was observed in the phases of the Moon indicated on the Canamayté.

E4) All right angles are congruent.

M4) In the cross inserted on the Canamayté all the angles are right angles.
The four interior squares and all the interior angles are congruent.

E5) Two lines are parallel if the do not intersect as they approach infinity.

M5) The Canamayté is a quadrivertex with opposite sides parallel, such that if they are extended toward infinity, they will never intersect.

Therefore we can ask: "If the Mayas had axiomatized their observations, what knowledge would we have today?"



References

Thompson, J. Eric S. The Rise and Fall of the Maya Civilization, University of Oklahoma Press, 1966.

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