Introduction

Many of us equate modern mathematics with Europe, North America and ancient Greece. The growth of a word-wide or internationalized culture (globalization), has made for numerous discoveries from outside this traditional sphere. Most notably: ethnomathematics.

The field of ethnomathematics - has important Brazilian roots. In recognizing this, I have chosen the Portuguese word "jeito" - the unique Brazilian way of getting things done, as the name for this paper for a number of reasons. Most importantly, the use of this Brazilian word for this paper honors the two founders of the concept of ethnomathematics, the late Paulo Freire and my dear friend, mentor and colleague, Ubiratan D'Ambrosio. The word "jeito" seems apropos for the title of this work, as so much of ethnomathematics has come to be connected to the culture, people, and history of Brazil through the philosophy and work of Paulo Freire.
 
 

Overview, Description and Background Information

Ethnomathematics forms the intersection set between mathematics and cultural anthropology. The term, coined in 1968 by Brazilian educator Ubiratan D'Ambrosio, has its roots in the ideas and philosophy of the late Paulo Freire. Many ethnomathematicans seek to document the kind of mathematics used by indigenous peoples in numerous locations world-wide. As well, ethnomathematics possesses a strong link to multicultural mathematics. Many ethnomathematicans are involved in active research - the documentation and empowerment of people through the mathematics that particularly underrepresented peoples have used for centuries. D'Ambrosio has described ethnomathematics by looking at its etymology,

Ethno is today accepted as something quite broad, referring to "cultural context", and therefore including considerations such as language, jargon, codes of behavior, myths, and symbols; mathema is a difficult root which approaches a meaning of "to explain, to know, to understand"; and tics comes without a doubt from techne, which is the same root as art and techniques. Thus we can say that ethnomathematics is the art or techniques of explaining, knowing, and understanding diverse cultural contexts.

Powell & Frankenstein (1997) stated that ethnomathematics has come to include the documentation and study of cultural-related learning styles; historical developments in mathematics and technology; prominent people in various cultural contexts who have made contributions to the field of mathematics; cultural applications of "nontraditional" mathematics; and various forms of mathematics that may draw upon the interests, abilities, and talents of all students.

The book produced by Powell and Frankenstein, is likely to become the seminal work in the field and has raised the dialogue of ethnomathematics to an unprecedented level. This dialogue has created a need for practical and hands-on knowledge about the "doing of" ethnomathematics. Questions arising from the reading of Powell and Frankenstein might be, "How can we adapt this philosophical approach to the day to day practical aspects of teaching? How might we look for, or find ethnomathematics to bring into our own classroom?, and finally, What makes an activity ethnomathematical, how does it, or does it differ from "regular" mathematics?"

The most important philosophical difference between a traditional and an ethnomathematical perspective is that ethnomathematics recognizes, encourages, and honors the belief that all people do mathematics with in their own unique and personal context, and that this ability may take many forms. Indeed it emerges from within each individual through their individual interaction with their cultural and physical environment. It also recognizes that everyone does mathematics, therefore there is no such thing as a non-math person - ethnomathematics is closely tied to issues of access and equity.
 
 

Further Definitions

As stated earlier, the field of ethnomathematics forms the intersection set between mathematics and cultural anthropology. Many ethnomathematicans are involved in active field research which focuses on the documentation and empowerment of people through the mathematics that they have traditionally used and the evolving use of mathematics in their own unique cultural environment. As stated earlier, ethnomathematics includes the documentation and study of culturally-related learning styles; historical developments in mathematics and technology; prominent people in various cultural contexts who have made contributions to the field of mathematics; applications of "nontraditional" mathematics; multicultural aspects of mathematics education, and various forms of mathematics that may draw out the diverse interests, abilities, and talents of all students.

Borba, states that mathematics is considered a form of expression, a language of communication. He says,

A language is a code understandable only to people who have participated in common past experiences. Each language expresses a way of knowing developed by a group of human beings. One way of knowing is mathematics. Mathematical knowledge expressed in the language code of a given sociocultural group is called "ethnomathematics." In this context "ethno" and "mathematics" should be taken in the broad sense. "Ethno" should be understood as referring to cultural groups, and not as the anachronistic concept of race; "mathematics" should be seen as a set of activities such as ciphering, measuring, classifying, ordering, inferring, and modeling.

D'Ambrosio (1985) has further described ethnomathematics "as the mathematics practiced among 'identifiable cultural groups'." These diverse groups include people found in national-tribal societies; labor groups; children of a certain age bracket, abilities and interests; and diverse professional classes in any number of social settings. Academic mathematics produced by professional mathematicians can thus be seen as ethnomathematics (Borba, 1990), because it was produced by an identifiable cultural group and because it is not the only mathematics that has been produced.
 
 

Feasibility

An important question about an ethnomathematics approach is its feasibility. It is said that it can be achieved easily as a somewhat spontaneous practice by inspired teachers, but is seldom taught to prospective teachers. In other words, can ethnomathematics be implemented as a pedagogical practice?

The work of Geraldo Pompeu Júnior sought to "to investigate how an 'ethno' approach to mathematics can be incorporated into the school curriculum, and what consequences, it has for the teaching and learning of mathematics " (Pompeu, 1992). Pompeu found that an ethnomathematically based pedagogy carries with it the concept of mathematics as a debatable, interactive and non-passive subject. Mathematical knowledge is to be shared between student and teacher, and not just passively transmitted from the teacher to the student. Teachers using an ethnomathematical pedagogy allow students to develop a personal role in active learning strategies and in their own assessment procedures.

Many ethnomathematicans, look for ways in which to integrate the outside-school or folk mathematics experiences to in-school or academic mathematics experiences. The traditional mathematics found in many schools offers learners an efficient way of instilling in them a sense of co-dependency and failure and outright aversion to mathematics. The mathematical competencies, which are often lost in the first years of schooling (NCES, 1997, 1996) are essential at this stage for future access to everyday life and labor opportunities. D'Ambrosio (1985) wrote,

The former, let us say spontaneous, abilities have been downgraded, repressed, and forgotten while the learned ones have not (yet) been assimilated either as a consequence of a learning blockage, or of an early drop-out, or even as a consequence of failure or many other reasons. Mathematics in schools shall be such that it facilitates knowledge, understanding, incorporation and compatibilization if known and current popular practices into the curriculum. In other words, recognition and incorporation of ethnomathematics into the curriculum. It is necessary to identify within ethnomathematics a structured body of knowledge.
 
 

Walking the Mystical Way with Practical Feet

The purpose of this paper was to briefly outline and define for the reader what ethnomathematics as a field has come to mean. The following discussion outlines in more practical terms what it is that the practitioner needs to know in order to create, use and develop practical ethnomathematical experiences for learners. Earlier this writer asked three questions:

1. How can we adapt this philosophical approach to the day to day practical aspects of teaching?

One adaption is for teachers, parents and learners to learn about and share with thier students the day to day uses of mathematics. Ethnomathematical perspectives encourage teachers to connect very concrete and practical uses of mathematics to ongoing classroom activity. For many children, sports statistics, bus routes and schedules, and other practical and concrete connections assist learners to become empowered mathematically. The second question asks:

2. How might we look for, or find ethnomathematics to bring into our own classroom?

In the bibliographic resources below there are countless examples of teachers discovering ethnomath. The most powerful one for this writer has been here in California where we have large numbers of immigrant children. My student teachers are encouraged to teach their children to become "jr. anthropologists" as it were. The children are given a list of questions, and trained to interview parents,m grandparents or other "elders" in their community. Some of the questions the children seeks answers to are: "How did you learn and memorize the basic facts (+, -, x, etc) when you were a child?", How do you do certain algorithms?". This becomes a rich data source for the class as they learn that many people learn to add, subtract, multiply, and divide in different ways (i.e. the French - Southeast Asian form or algorithm is different than the American way of doing it). Children then learn a variety of different strategies for problem solving. The class then creates a book, and resource that is then placed in the library. And finally,

3. What makes an activity ethnomathematical, how does it, or does it differ from "regular" mathematics?"

An ethnomathematical activity is different from "regular" math in one major way - it is student centered. Traditional mathematics is teacher driven and assessed. Where as ethnomathematics encourages the teacher to link history, culture, community, and Howard Gardener's "multiple intelligences" into an integrated whole. It also requires that the teacher move away from the textbook as the curriculum to the needs of the child and community as the driving force of the curriculum.
 
 

BIO

Daniel C. Orey, earned a doctorate in curriculum and instruction in multicultural education from the University of New Mexico in 1988. His field research took him to Highland Maya Guatemala and the city of Puebla, Mexico. He has been a member of the faculty of the School of Education at California State University, Sacramento since that time. He is a founding board member and has served as Vice President for North America (1996 - present); General Secretary (1995) of the Sociedade Internacional para Estudas da Criança . He currently is a J. William Fulbright Foreign Scholar at the Pontifícia Universidade Católica de Campinas, Brazil. He can be reached at: Daniel C. Orey, Ph.D.; c/o School of Education; California State University, Sacramento; 6000 J Street; Sacramento, CA 95819-6079. tel. (916) 278 5531; FAX: (916) 278 5904; email: orey@csus.edu; http://edweb.csus.edu/courses/orey.

Bibliography

Araujo, M. A.. (1997). "Matemática: Acerto de contas." Educação (197): 16-19. São Paulo: Editora Segmento.

Benatti, L. (1997, August 4). "Disciplina liga matemática ao cotidiano." São Paulo: Folha de São Paulo.

Borba, M. (1990). "Ethnomathematics and education." For The Learning of Mathematics (10)1: 39-43.

CSDE. (1992). Mathematics Framework for California Public Schools. Sacramento: California State Department of Education.

Davis, B. (1996). Teaching mathematics: toward a sound alternative. New York: Garland Publishing, Inc.

D'Ambrosio, U. (1996). Educação matemática: da teoria à prática. São Paulo: Papirus Editora.

_____________. (1985a). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics. 5(1).:44-48.

_____________. (1985b). Socio-cultural bases for mathematics education. Campinas: UNICAMP.

_____________. (1984). Intercultural transmission of mathematical knowledge: Effects on mathematical education. Campinas: UNICAMP.

_____________. (1983). Successes and failures of mathematics curricula in the past two decades In: Proceedings of the Fourth International Congress of Mathematics. (pp. 362-364). Boston: Berkhäuser

Draften, A., Strauss, R., & Swanney, D. (1992). Brazil: A travel survival kit (second edition). Hawthorn, VIC., Australia: Lonely Planet Publications.

Economist, The. (1997, October 18). Brazil: Schooling the multitudes. The Economist. p. 38.

Freire, P. (1970). Pedagogy of the oppressed. New York: Publisher.

_______. (1973). Education for critical consciousness. New York: Publisher.

_______. (1997). Pedagogia da autonomia: Saberes necessários à prática educativa. São Paulo: Editora Paz e Terra.

_______. (1987). Na escola que fazemos: Uma refexão interdisciplinar em educação popular. Petrópolis: Editora Vozes

NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

______. (1991). Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.

______. (1995). Assessment standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Orey, D. (1988). "Mathematics for the 21st Century." In: Teaching Children Mathematics, Reston, VA.: National Council of Teachers of Mathematics.

______. (1996). Introduction in Krebs, R. J., Vieira, L. F., Vieira, J. L. L., and Beltrame, T. S. Desenvolvimento Humano: Uma área emergente da ciência do movimento humano. Universidade Federal de Santa Maria, Laboratório do Desenvolvimento Humano, Centro de Educação Física e Desportos - CEFD.

______. (1996). "Linking pre-service teachers and mathematics reform." CMC Communicator 20, (4).

______. (1989). "Ethnomathematical perspectives on the NCTM Standards." Int'l Study Group on Ethnomathematics Newsletter. (5)1. pp 5-7.

______. (1987). Logo programming language in a Mexican primary school: relationships to cognition, geometric skills, and computer attitudes. In L. Takeuchi, Proceedings - University Association of Research Scholars, Sacramento.

______. (1986). Introducing computers in the curriculum. In J. I. Bustamonte (Ed. ), Proceedings from the III International Symposium: La computación y la educación. Puebla, México.

______. (1986). With P. Norton, The Española Schools Computer Curriculum. Española Public Schools, and the Center for Education and Technology, University of New Mexico.

______. (1986). "Romancing the turtle: Using Logo in multicultural environments." International Logo Exchange 1(3).

______. (1985). El Logo como puente transcultural. In J. I. Bustamonte (Ed. ). Proceedings from the II International Symposium: La computación y la educación. México, D. F.

______. (1984). "Logo goes Guatemalan: an ethnographic study." The Computing Teacher 12(1): 46-47.

______. (1984). ISHF Volunteer Sourcebook. Alamogordo, NM: International Space Hall of Fame.

______. (1982). "Mayan Math." The Oregon Mathematics Teacher. Portland, OR.

Page, J. A. (1995). The Brazilians. New York: Addison-Wesley Publishing Company.

Pinker, S. (1995). The language instinct: How the mind creates language. New York: Harper.

Powell, A. B. & Frankenstein, M. (eds.) (1997). Ethnomathematics: challenging eurocentrism in mathematics education. Albany, NY: State University of New York Press.

TIMSS. (1997). Third International Mathematics and Science Survey. See http://nces.ed.gov/timss.