Historical Fragments of The Development of the Ethnomathematics Program

 

Milton Rosa

Encina Preparatory High School – San Juan Unified School District

Sacramento, California

1400 Bell Street

Sacramento CA, 95825.

(916) 971-7538

milrosa@hotmail.com

 

Daniel Clark Orey

California State University, Sacramento

6000 J Street

Sacramento CA, 95819-6079

(916) 278 5531

orey@csus.edu

 

Abstract

This article is designed to serve as an introduction to as well as a historical background to ethnomathematics as a program of study.  The authors seek to show that ethnomathematics includes the mathematical ideas, perspectives and practices of individuals in different cultures as manifested, diffused, and transmitted in diverse modes.  Since this particular area of research comes from Brasil, our work focuses primarily from the Brazilian perspective that teaches that ethnomathematics forms the intersection set between cultural anthropology and academic mathematics.  Any study of the history of ethnomathematics and its proponents helps us to clarify the importance of this perspective in terms of mathematics education as a cultural artifact and interlinks to an emerging global perspective.  The development of ethnomathematics is documented here, principally as part of the study of the development of scientific, anthropological, and mathematical ideas and practices made by distinct cultural groups.

[Keywords: history, ethnomathematics, etymological, culture, knowledge]

 
Introduction
According to D’Ambrosio, (1) “We are now living a moment that is similar to an intellectual effervescence found at the end of the Middle Ages.  There is a justification for this fact, therefore, to conceptualize a new Renaissance.  Ethnomathematics is one of the manifestations of this new Renaissance (2001:29)”.  Seen in this perspective, he offers a very interesting example related to the beginning of an ethnomathematics perspective.  He also states that “When Australopithecus first chose and began to chip a piece of rock with the objective of picking the flesh off a bone, his mathematical mind was disclosed.  To select a rock, it is necessary to evaluate its dimensions, and, to chip it, it is necessary to evaluate and compare dimensions (D’Ambrosio 2001:33).”  The comparison and evaluation of dimensions is one of the most elementary manifestations of mathematical thinking.  This mathematical idea was developed by Australopithecus may be considered as one of the first example of Ethnomathematics.

Since the beginning of humankind, every culture has developed its own mathematical ideas and different practices.  In some cases, these ideas and practices have been transmitted and diffused from one culture to another.  Some of what we call “modern academic mathematics” has roots in ancient Egypt and Mesopotamia and then grew rapidly when it spread to ancient Greece.  Many mathematical ideas and practices that were written in ancient Greek were translated into Arabic.  At about at the same time, mathematical ideas and practices from India were translated into Arabic.  As commerce and conquest continued across Asia, new ideas were translated into Latin and became part of the mathematics of Western Europe.

However, there were other regions, which have also developed significant mathematical ideas and practices.  The mathematical ideas from regions such as China, southern India, Japan, Mesoamerica, regions in Africa and South America were useful for individuals belonging to cultural groups in those regions.  Therefore, because of colonial globalization (Orey and Rosa, in press), this mathematical knowledge did not acquire as much influence on current international or academic and scientific mathematics.  Academic and scientific mathematics is widely known as Western mathematics.  Teresi (2002) stated that for some historians, mathematicians, and philosophers, the failure to acknowledge the success of non-Western cultures in regards to the development of mathematics derives not just from ignorance but also from conspiracy because the roots of European civilization originated in Africa and Asia.

In this context, an ethnomathematics program is a research field that can be described as the study of the history of mathematical ideas and practices that are found in diverse and specific cultural contexts (Rosa 2000).  This program arose to confront current scientific taboos that mathematics is universal and not rooted in culture and traditions.  Historically, the evolution of this confrontation, through systemized methods of study, has manifested itself only recently.

 

The Etymological Root of Ethnomathematics

D’Ambrosio (1993) used the resource of etymology to name this program.  He used three modified Greek roots, ethno, mathema, and tics to explain what he understands to be ethnomathematics.  In this perspective, according to D’Ambrosio (1985) ethnomathematics is the mathematics practiced by distinct cultural groups identified as indigenous societies, groups of workers, professional classes, and groups of children of a certain age group, etc.  This (2) Dambrosian perspective of ethnomathematics is the motive by which specific cultures (ethno vs. ethnic) developed over history, the techniques and the ideas (tics = techne) to learn how to work with measures, calculations, inferences, comparisons, classifications, and the ability to model the natural and social environment in which we use to explain and understand phenomena (mathema).  Seen in this context, D’Ambrosio (1990) proposes that this program of study represents a methodology for the discovering and analysis of the processes that transmit, diffuse, and institutionalize mathematical knowledge (ideas and practices) originating from diverse cultural groups through history.  Hence, the history of ethnomathematics is identified with the history and culture of a specific group.  Ethnomathematics as a program and its connections with history, philosophy, and pedagogy of mathematics is the recognition of this fact.  In this perspective, mathematics is a cultural endeavor rooted in traditions, which is identified with the history and development of specific civilizations.  Therefore, mathematics is not exclusively associated with the history, philosophy and pedagogy of the specific mathematics called Western Mathematics that was originated in ancient Mediterranean civilizations.  This form of mathematics was imposed on the entire planet after the great navigations of the 15th century by the ongoing processes of colonization and globalization.

 

Ethnomathematics, Culture, and Anthropology

One of the most important concepts of ethnomathematics is the association of the mathematics found in diverse cultural contexts.  Ethnomathematics as a research paradigm is much wider than traditional concepts of mathematics and ethnicity or any current sense of “multiculturalism”.  D’Ambrosio (1990) refers to “ethno” as that related to distinct cultural groups identified by cultural traditions, codes, symbols, myths, and specific ways of reasoning and inferring.  In so doing, ethnomathematics may be considered as the way that various cultural groups mathematize because it examines how both mathematical ideas and mathematical practices are processed and used in the daily activities.  It can be also described as the arts or techniques developed by diverse peoples to explain, to understand, and to cope with their own environment (D'Ambrosio 1992). 

In accordance to Barton (1996) ethnomathematics embraces the mathematical ideas thoughts and practices as developed by all cultures.  From his perspective, a body of anthropological research has come to focus on both the intuitive mathematical thinking and the cognitive process that are largely developed in minority cultural groups in Australia, Africa, the Pacific Islands, and the Americas.  Ethnomathematics is a program that seeks to study how distinct peoples have come to understand, comprehend, articulate, process, and ultimately use mathematical ideas, concepts, and practices that may solve problems related to their daily activity.  According to Barton (1996) ethnomathematics is not only the study of mathematical ideas because it is also the study of anthropology and history. In this perspective, the study of the cultural history of mathematics and mathematics attempts to identify the historical mathematical contributions of different cultures across the world.  Seen in this context, the focus of ethnomathematics consists essentially of a serious and critical analysis of the generation and production of the mathematical knowledge and intellectual processes, the social mechanisms in the institutionalization of knowledge; and the diffusion of this knowledge (Rosa and Orey in press).  In this much more holistic context of mathematics that uses an anthropological perspective to include diverse perspectives, patterns of thought, and histories, the study of the systems comes to reflect, understand, and comprehend extant relations among all of the components Rosa (2000) and has allowed us to define ethnomathematics as the intersection of cultural anthropology, mathematics, and mathematical modeling which is used to translate diverse mathematical practices.

Figure 1: Ethnomathematics as an Intersection of Three Research Fields

All individuals possess both anthropological and mathematical concepts.  These concepts are rooted in the universal human endowments of curiosity, ability, transcendence, life, and death.  They characterize our very humanness.  Awareness and appreciation of cultural diversity that can be seen in our clothing, methods of discourse, our religious views, our morals, and our own unique world view allow us to understand each aspect of the daily life of humans (Rosa and Orey in press).

The unique cultural background of each person represents a set of values and the unique way of seeing the world as it is transmitted from one generation to another.  The principals of anthropology that are relevant to the work of ethnomathematics includes the essential elements of culture such as language, economy, politics, religion, art, and the daily mathematical practices of diverse groups of people.  Since, cultural anthropology gives us tools that increase our understanding of the internal logic of a given society; detailed anthropological studies of the mathematics of distinct cultural groups most certainly allows us to further our understanding of the internal logic and beliefs of different peoples.

The Development of the Ethnomathematics Program

We do not really know when an interest in the mathematical practices of other cultures was first expressed.  However, the interest in different mathematical ideas was manifested from early history through isolated and not systemized situations.  The earliest observations of distinct mathematical practices probably occurred at the same time as the first travels to different regions of the world.  Travelers who came in contact with local cultures observed different customs that no doubt included different mathematically related ideas and practices such as counting, measuring, estimating, classification, ordering, selecting, classifying, and modeling.

As initial processes of cultural interaction occurred, individuals observed and learned about different cultures, and registered their observations, often through story telling and myth.  In these early reports people recognized that different cultural practices existed and they also started to write about diverse mathematical ideas and practices.  Even though an absence of early records has delayed true understanding of these processes, and many of these stories are hindered by magical tales, these initial observations of practices used by early scientists, philosophers, and mathematicians influenced many mathematical concepts when they were brought back from early adventures and travels.  The development of writing allowed mathematics historians to piece together knowledge accumulated by early civilizations.

This context allows us to point out that in a world of a diversity of perspectives, academic formation and cultural backgrounds, based on both the Brazilian and United States cultures, the following historical fragments were considered in assisting the writers to describe the development of the ethnomathematics program.

 

Around 484-425 BC, The Greek historian Herodotus wrote one of the earliest accounts of diverse cultures.  In 440 BC, he wrote a book called The Histories, in which he shared his observations of the different cultures, practices, customs, habits, and mathematical practices of the peoples encountered during his travels around his known world.  He dispassionately described and made anthropological observations, which valued and described the cultures he interacted.  Herodotus also recorded the geometry that he learned from his time amongst the Egyptians.  The Egyptian geometry he recorded was related to the system of valuation of productive areas of the time.  This aspect of the Egyptian mathematical knowledge evidences a production system that is related to the social-political-economic structures of the Egyptian culture that interacted with their environment, and which demanded the measurement of land and arithmetic to deal with their economy and with the calculation and counting of time.  According to D’Ambrosio (2001), as this system of knowledge was developing in civilizations around the Mediterranean Sea, at the same time, the indigenous people of the Amazon and other civilizations from the Andes, China, and Sub Sahara in Africa were also developing their own unique and diverse ways to know, understand, comprehend, and interact with their own unique environments.  They developed a sophisticated production and social system that equally used measurement of space and time.

 

For many traditional philosophers, mathematicians, and historians the intervening “Dark Ages” in Europe, were depicted as a period of scientific-technological-mathematical inactivity (Joseph, 1991).  During this period, Europe lost whatever light had been cast on its art, sciences, mathematics, and philosophy by the ancient Mediterranean civilizations.  However, at that time, there were great scientific and mathematical developments elsewhere in the world.  For example, there were the complex interactions between the many diverse Asian civilizations that generated an intense cultural dynamic.  Seen in this context, mainly through trade and commerce, China interacted with India and then India influenced the Arabs.  At the same time, Arabs and Hindus simultaneously maintained a relationship with the Hellenic world.  Thus, even though with the supposed stagnation of the mathematical development during the Dark Age, Joseph (1993) affirms that, at that time, the Arabs translated, refined, and synthesized the scientific knowledge that originated in India, China, Egypt, and Greece.  For example:

In this perspective, Teresi stated that “our modern numerals – 0 through – 9 were developed in India…, during the so-called dark age of mathematics” (2002:32).  In accordance to Joseph (1991) the importance of the Arabs and other civilizations such as China and India as transmitters and creators of the mathematical knowledge has been ignored in the study of the development of mathematical concepts during the Dark Age.

 

When, in the 7th century, the Arabs invaded Europe, they brought with them their traditions and mathematical knowledge as acquired earlier in India.  They also influenced Medieval Europe by exchanging food, customs, culture, sciences and new forms of technology.  In turn, when the Europeans went on to conquer and colonize the peoples across the world, they introduced this system into the “New World”.

 

In the 11th century, the internationalization of mathematical knowledge was not solely influenced by Western cultures.  The agents of creation of knowledge were located in other regions of both the known and unknown world (Sen, 2002).  At that time, the invention of zero and the notion of place value had been attributed to the Hindus around the 9th century.  However, according to Cajori (1993), Diaz (1995), and Jr. Merick (1969), the earliest systematic use of a symbol for zero in a place value system must be attributed to the Mayan because they utilized these representations in (3) stelas, (4) lintels, murals, monuments, and other objects, which are found in various Mayan archeological sites, centuries before the Hindus.  Teresi stated “Pre-Columbian achievements in the New World have long eluded traditionalists.  The Maya invented zero about the same time as the Indians, and practiced math and astronomy far beyond that of medieval Europe.  Native Americans built pyramids and other structures in the American Midwest larger than anything then in Europe (2002:13).”

By the end of the 11th century, the growing diffusion of mathematical knowledge brought accelerated technological progress to various parts of the world.  For example, place value notation was transmitted to Arab peoples through religious and commercial activities, war, and conquest.

 

In the 14th century, the Arab historian and philosopher Ibn Khaldun (1332-1406) examined social, psychological, economic, and environmental factors that affected the development, ascension and fall of different civilizations.  In his study, Khaldun analyzed several economic policies and demonstrated the consequences for both local and distant communities (Atiyeh and Oweiss 1988).  These facts accompanied by a mathematical knowledge that strongly contributed to the defense of communities against the injustice and oppression of the ruling class.

 

In the 15th century, the number system used by the Greeks and Romans was cumbersome and not very practical for many uses.  This system did not satisfy the demand and necessities of the new capitalist societies that were emerging and developing in the European continent.  The adoption of the decimal number system used by the Hindus and brought to Europe by the Arabs made perfect sense and has allowed for the explosive growth in what we currently call “the sciences”.  The Hindus also took advantage of this same cultural interchange by learning important concepts of Greek mathematicians by way of the Arab culture.  They assimilated habits and customs of the Arab culture and had learned important concepts Greek mathematical concepts because Arab and Persian translations of scientific Greek and Egyptian texts became quickly available in India.  Another important factor in this interchange was the influence of Islamic architecture that added new architectural elements architectural into the Hindu architecture such as floral motives, decorative tiles, arched roof, vaults, and domes. 

In light of the above, Rashed (1989) offers an excellent example of the cultural dynamism shared by these cultures.  He comments about the contributions of the Greek-Arab-Hindu for the development of the mathematical concept of the chord of the circle. Greeks and Hindus also explored the existing relationship between the ray and the chord of the circle.  However, Hindu mathematicians also studied the half-chord concepts.  In this context, both the Greeks and Hindus influenced the studies of al-Khwarizmi in relation to the chord because they used the concept of half-chord of the Hindus in conjunction with the comparisons of Ptolemy’s ratio-angle (theorem on chords) and constructed a meticulous table for the values of the sine, which is very similar to the one that is currently in use (Elert 1994).

 

At the end of 15th and the beginning of the 16th centuries, explorers provided descriptions of different aspects of “exotic” cultures that they encountered in Asia, Africa, and the Americas.  Early chroniclers of the Americas reported observations and registered data they collected about the cultures they encountered in their explorations.  Using a process that can be considered ethnomathematical in nature, Juan Diaz Freyle published in 1556, in Mexico City, the first book of arithmetic of the new world entitled El Sumario compendioso de las quentas de plata y oro que en los reinos del Pirú son necessarias a los mercaderes y todo genero de tratantes: Con algunas reglas tocantes al arithmética (A Compendium Summary of the Accounts of Silver and Gold that in the Kingdoms of Peru are Necessary to Merchants and All Kinds of Dealers: With Some Rules Concerning Arithmetic).  In his book, Freyle described the arithmetic practiced by the indigenous people.  It is important to observe that this book described the process of the indigenous people’s assimilation of the conquering people’s mathematical knowledge.  This can be perceived as a transformation of the native mathematical system through a cultural dynamic perspective.  However, in accordance to D'Ambrosio (1999), this book was removed from circulation and the Aztec Arithmetic was replaced by the Spanish arithmetical system. The book also contains tables used in the conversion of exchange and the taxes used in transactions with gold and silver.  It also explains how to use the “rule of three” to convert of the amount of raw gold that was necessary to coin different types of European currencies.  By sharing this perspective, Grattan-Guinness affirms that when Europeans invaded and conquered the northern part of the new world during the early 16th century, they “began to apply commercial arithmetic to the purchase of citizens in North America from local chiefs and kings, and the later sale of those still alive, to entrepreneurs and landowners across to the Americas” (1997:112).  He also stated that the Europeans “made little effort to conserve the culture of either their slaves or of the indigenous tribes” (1997:113).

 

D’Ambrosio (2001) states that the book The History of Brazil (A História do Brasil) was written in 1627 by Frei Vicente do Salvador, and only published in 1888 by Capistrano de Abreu.  In the narratives of this masterpiece, Frei Vicente claimed that Brazilian native populations did not possess a system of numbering for more than five and that they utilized their toes and fingers to count bigger quantities.  He also referenced indigenous mathematics when narrating the exchange system, in which native Brazilians exchanged one product for another, a process of one-to-one correspondence (biunivocal correspondence) without the utilization of a standard system of measures.

 

·        HISTORICAL FRAGMENT 9

The ascension of the Portuguese, Spanish, French, Dutch, English, and Belgian Empires in 18th and 19th centuries contributed to increasing contact between cultures (both the colonized and the colonizer).  This context allowed for an increased development of global commerce, a greater spread of the growing capitalist economy, and the eventual industrialization of Europe. 

Newly industrialized countries continued their search for new lands as sources of supply, cheap human power and for raw materials to be manufactured at low costs.  At the same time, millions of Europeans from the lower classes were encouraged to immigrate to newly established colonies in promise of better lives.  These further cultural exchanges allowed for an accumulation of data and information of distinct cultural groups that were “found” in the colonies.  In the late 19th century, the first forms of what would become modern anthropology began to be systematized.  As different cultures were studied during the ongoing processes of colonization, the customs of diverse cultural groups also became objects of study by many early European anthropological societies.  These aspects led to present day social, cultural, and economical transformations of all societal and cultural groups on the planet.

 

·        HISTORICAL FRAGMENT 10

In the early decades of the 20th Century, the mathematical ideas and practices by select cultural groups awakened the interest of anthropological and historical societies and educational institutions.  For example, German historian and philosopher Oswald Spengler related in his book The Decline of the West, in 1918, that the history of two cultures can be shown across similar standards and that all cultural aspects, as for example, art, politics, mathematics and sciences, possess differences from one culture for another.  In this book, Spengler tried to understand the nature of mathematical thinking and tried to understand mathematics as a vivid cultural manifestation (D’Ambrosio, 2001).

 

·        HISTORICAL FRAGMENT 11

This context allowed Cassius Jackson Keyser, an American mathematician to write books in the area of mathematics and philosophy by examining the foundations and structure of mathematics and science attempting to apply these structures to human interaction.  In 1922, he wrote Mathematical Philosophy: A Study of Fate and Freedom where he described mathematics as the science of exact thought or rigorous thinking.  Keyser stated that one of its distinctive characteristics is a precision, sharpness, and completeness of definitions.  In the meantime, in his point of view, mathematical philosophy has very little to do with “mere calculations” or with numbers as such or with formulas because it is a philosophy where precise, sharp, and rigorous thinking is essential.  From his perspective, those who deliberately refuse to think mathematically deliberately violate the supreme law of intellectual rectitude.  For many years Keyser meditated upon the nature of mathematics and its connections with different spheres of the human life.

 

·        HISTORICAL FRAGMENT 12

Various attempts were made by some mathematicians and philosophers to consider mathematics as an integral part of culture.  Austrian philosopher Ludwig Wittgenstein (1931) wrote Culture and Value, providing insights into the relation between the real world and mathematics, religion, language, culture, and philosophy.

 

·        HISTORICAL FRAGMENT 13

In the 3rd decade of the 20th Century, a form of pre-ethnomathematics emerged, where researchers in parts of Africa collaborated in the development of early ethnomathematical ideas.  For example, according to Gerdes (2001), in 1938, Otto Raum wrote Arithmetic in Africa. In this book, he affirmed that arithmetic problems should be secluded practices and that there were living mathematical experiences used by students in their own cultural context.

 

·        HISTORICAL FRAGMENT 14

The further development of both the philosophical and anthropological ideas pertaining to interactions of mathematics and culture spread even faster during the 1940’s.  This was in part due to the massive movements of populations, solders, and the explosive growth in all the cognitive sciences during World War II.  In 1947, an American anthropologist Leslie White published the article The Locus of Mathematical Reality: an Anthropological Footnote, in which he explained that to understand mathematics as a cultural product signifies the recognition of the human influence on mathematics.  For White, formulae as well as other mathematical tools are to some extent cultural artifacts, which depend on the unique interaction between individuals and groups, with races and with nations.  Algorithms and other forms of mental calculations do have certain cultural connections (Orey 1999).

 

·        HISTORICAL FRAGMENT 15

In 1948, historian and Dutch mathematician, Dirk Jan Struik published, The Concise History of Mathematics, Volumes I & II, in which the author documents his understanding of the institutional and social forces that influenced research in mathematics at the time.  Struik found that social context interacts with the output of mathematical knowledge.  At this time “Some other mathematicians and philosophers ... also realized that mathematics has a cultural context but stopped short of probing other cultures” (Ascher and Ascher 1997:44).  In this context, people with a certain background of mathematical knowledge were looking for ways to understand, comprehend, and acquire knowledge related to the significance of mathematics in human culture.

 

·        HISTORICAL FRAGMENT 16

The various aspects of mathematics such as the usefulness and applications of mathematics in solving problems posed in other fields of the human knowledge was a concern of Morris Kline.  In his book Mathematics in the Western Culture (1953) he describes for us a remarkably fine account of the influences mathematics exerts on the development of philosophy, the physical sciences, religion, and the arts in Western life.  However he stated that “the assertion that mathematics has been a major force in the molding of a modern culture, as well as a vital element of that culture, appears too incredible or, at best, a rank of exaggeration (Kline 1953:3). In our opinion, this disbelief continued to this day amongst many “western” academic scientists, mathematicians and historians.  According to Teresi in Kline’s classic work (5) Mathematics: A Cultural Approach, Kline acknowledges that the Babylonians and Egyptians pioneered mathematics long before the Greeks, but he dismisses them as pragmatics” (2002:29).  We believe that Kline’s passion for Western mathematics blinded him from appreciating important non-Western contributions to the development of the mathematical knowledge.

 

·        HISTORICAL FRAGMENT 17

Interest in the relationship between mathematics and culture began a vigorous emergence in the 1950’s.  An American topologist (6) Raymond Louis Wilder was perhaps the first educator who clearly relates mathematics and culture, with a talk titled The Cultural Basis of Mathematics, at the International Congress of Mathematics in 1950, in Cambridge, Massachusetts.  In 1981, he wrote Mathematics as a Cultural System where he described the nature of mathematics and its relation to society from the standpoint of cultural anthropology.  In his point of view mathematics is regarded as a subculture of the general culture, and its current state and development is subject to cultural influences.  According to Ascher and Ascher, “The late Raymond L. Wilder, the primary mathematical spokesman for the importance of relating mathematics and culture, used his understanding to describe the processes of mathematical development in the West” (1997: 44).  For Wilder, mathematics develops between two cultural kinds of influence.  He wrote that the first cultural influence related to mathematics arising out of the cultural environment in which a determined group is inserted.  The environmental cultural influence is an answer to the needs of social interactions.  The second cultural influence is related to the cultural inheritance transmitted by elements of a group, and is an answer used to solve internal mathematical problems that are owned by the group.

 

·        HISTORICAL FRAGMENT 18

In the 1960’s, Japanese conceptual algebraist Yazoo Akizuki proposed an emphasis on a reflexive form of mathematics, and proposed that the history of science and mathematics should be taught at all school levels.  However, the most interesting point from the argumentation of Akizuki was the recognition of mathematics as a cultural product and that peculiar ways for the resolution of mathematical problems exists (D’Ambrosio 2003, Orey 2000).  In his opinion, oriental philosophies and religions are very different from those of the West.  This perspective allowed him to believe that there also exist different modes of thinking and processing even in mathematics.

Beginning in the 1970’s, anthropological research began to demonstrate a definite interest in the peculiar ways of (7) “mathematization” in order to understand the development of mathematical ideas, processes, and practices.  Akizuki’s proposal was marked by a growing conscience of researchers and mathematics educators regarding socio-cultural aspects of mathematics. 

 

·        HISTORICAL FRAGMENT 19

At the same time six other important facts were fundamental in the development of an ethnomathematics program:

1)  In 1973, Claudia Zaslavsky published Africa Counts: Number and Patterns in African Culture, considered by many as a pioneering work in English that coherently organized the knowledge of African people from an educational-pedagogical perspective, through the development of the history and practice of mathematical activities of people from the African Sub-Sahara.  Zaslavsky showed how mathematics was prominent in African routine life and that it has helped in the present mathematical development of concepts.

2)  In 1976, Brazilian mathematician Ubiratan D’Ambrosio organized and presided the section: called Why Teach Mathematics? within the Topic Group: Objectives and Goals of Mathematics Education during the Third International Congress of Mathematics Education, in Karlsruhe, Germany.  D’Ambrosio argued that it was important to distinguish cultural roots in mathematics from mathematics education (Ferreira, 2004).

3)  In 1977, D’Ambrosio first utilized the term ethnomathematics in a lecture given at the Annual Meeting of the American Association for the Advancement of Science, in Denver, Colorado.

4)  The consolidation of the term culminated with the lecture of Opening Socio-cultural Bases of Mathematics Education given by D’Ambrosio at ICME 5, in Australia, in 1984.  This meeting officially instituted ethnomathematics as a program and field of research (D’Ambrosio, 2002).

5)  In 1985, Ubiratan D’Ambrosio wrote his masterpiece Ethnomathematics and its Place in the History and Pedagogy of Mathematics.  This fundamental article “represents the first comprehensive, theoretical treatment in English of ethnomathematics.  These ideas have stimulated the development of the field” (Powell and Frankenstein1997b:13).  In 2003, this article was selected to be part of the NCTM book Classics in Mathematics Education Research, because this article has had a major influence on recent research in mathematics education.

6) In 1985, the International Study Group on Ethnomathematics (ISGEm) was founded, which has enabled ethnomathematics as a program to grow internationally. 

 

·        HISTORICAL FRAGMENT 20

It is very important to highlight the importance of Ubiratan D’Ambrosio, a Brazilian mathematician and philosopher, in relation to the development of the ethnomathematics program.  He is the most important theoretician in this field.  He has also the role of leadership and dissemination of the ideas involving ethnomathematics around the world and is applications in mathematics education. Powell and Frankenstein stated “D’Ambrosio’s broader view of ethnomathematics accounts for the dialectical transformation of knowledge within and among societies.  Moreover, his epistemology is consistent with Freire’s (1970, 1973) in that D’Ambrosio views mathematical knowledge as dynamic and the result of human activity, not static and ordained (1997a:8)”.  D’Ambrosio studies in the area of sociopolitical issues established a strong relationship between mathematics, anthropology, culture, and society.  Gerdes, Powel and Frankenstein and Orey consider D’Ambrosio “the intellectual father of the ethnomathematics program” (1997b:13).  D’Ambrosio was also selected as one of the most important mathematicians of the twentieth century in the area of sociopolitical issues and ethnomathematics (Shirley 2000).  In 2005, D’Ambrosio was awarded with the second Felix Medal of the International Commission on Mathematical Instruction (ICMI) that acknowledges his role in the development of mathematics education as a field of research.

 

ETHNOMATHEMATICS IN THE PRESENT TIME

In subsequent years, papers, investigations, and researches in the field of ethnomathematics was presented locally (Multicultural Conference and SAME Conference in Sacramento, California), regionally (First Northern California Conference on Ethnomathematics, Asilomar Conference on Mathematics (CMC - Northern California), nationally (National Council of Teachers of Mathematics, First Bolivian Congress on Ethnomathematics, First and Second Brazilian Congresses on Ethnomathematics, Math and Social Justice Conference), and internationally (International Congress of History of Sciences and International Congress of Mathematics Education).  Studies and discussions concerned about issues in ethnomathematics also happened in a succession of meetings, working groups, newsletters, theses, dissertations, conferences, and congresses.

The International Study Group on Ethnomathematics organized in September 1998 in Granada, Spain, the First International Congress on Ethnomathematics.  The Second International Congress on Ethnomathematics was held in Ouro Preto, Brazil, in August 2002.  The Third International Congress on Ethnomathematics took place in Auckland, New Zealand, in February 2006.  The Third Brazilian Congress on Ethnomathematics will be held in Niterói, state of Rio de Janeiro, in 2008 and the Fourth International Congress on Ethnomathematics will be held in Baltimore, Maryland in 2010.  These events collaborated and will continue to construct further collaborations with the ongoing evolution of research, inquiry, investigation and study in ethnomathematics.  The growing number of books, chapters, and articles published in journals, magazines, and newspapers of diverse languages, the diversity of studies, theories, theses and dissertations submitted in universities in many countries are indicators of the vitality of this new area of research (D’Ambrosio, 2004).

At the beginning of the 21st century, greater and more sensitive understandings of mathematical ideas, processes, and practices and ideas from diverse cultural groups have become increasingly available through the growth of the fields of ethnology, culture, history, anthropology, linguistics, and ethnomathematics.  The insight from many ongoing theoretical investigations and research studies in many countries demonstrate the possibility of the sensitive internationalization of mathematical practices and ideas expressed in different cultural contexts.  The main objective of this perspective is to raise self-confidence, to enhance creativity, and to promote cultural identity of diverse cultural groups.

 

Final Considerations

The objective of this article was to present historical fragments in relationship to the development of ethnomathematics as a program.  In so doing, the authors used our own different particular cultural backgrounds to highlight the importance of these historical fragments presented here.  These facts are limited by our own personal views and experiences and are based on our ongoing work and research in both Brazil and the United States.

We believe that the acknowledgment and recognition of contributions that individuals from diverse cultures have made to mathematical understanding, along with the recognition and identification of diverse practices of a mathematical nature through history can help to develop a sense of value for diverse cultural forms of knowledge by raising the confidence and self-esteem of individuals who belong to these groups.  Essentially, the historical aspects of ethnomathematics lead us to a critical analysis of the generation and production of mathematical knowledge and the intellectual processes of this production, the social-cultural mechanisms of institutionalization of knowledge (academics), and its diffusion and transmission (education).  This aspect helps to increase understanding of the universality of mathematics, while revealing mathematical ideas, cognition processes, and practices of groups from different “ethnos”.  It is possible to begin this process by introducing an account of the historical evolution, and the recognition of the natural, anthropological, social, and cultural factors that shape the human development of mathematical thought.

NOTES

(1) Estamos vivendo agora um momento que se assemelha à efervescência intelectual da Idade Média.  Justifica-se, portanto, falar em um novo renascimento.  Etnomatemática é uma das manifestações desse novo renascimento (D’Ambrosio 2001:29).

(2) According to the authors of this article, the Dambrosian pedagogical approach to the ethnomathematics program emphasizes the influences of anthropological-socio-cultural factors on the teaching and learning of mathematics.  D’Ambrosio states that much of the mathematics used in daily practice, as affected by distinctive modes of cognition, may be quite different from that which is taught in school.  He indicates that many cultural differentiated groups "know" mathematics in ways that differs from academic mathematics in the school curricula.

(3) Maya rulers regularly erected a stela, called a stone tree, a monument generally sculpted in only one block of stone that contains codes and inscriptions.  On stone stela the Maya showed themselves in costumes with symbols that were associated with their own World

(4) Lintel is the designation given by modern archaeologists to an ancient Maya limestone or wood carvings that contains text of Maya hieroglyphics.

(5) Teresi (2002) states that this book “Worth reading for the author’s opinion of non-Western mathematics” (p.423).

(6) Raymond Louis Wilder (1896-1982) led the development in topology in the United States.  He was also a pioneer in the study of mathematical history from the point of view of anthropology. 

(7) Mathematization is the development of a given problem, that is, the transformation of a problem into mathematical language.  This can be accomplished by the formulation of the hypotheses; classification of the data and information as important and non-important to the hypotheses, selection of important variables, selection of symbols adapted for these variables and description of those relationships in mathematical terms.

 

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