Ethnomathematics

In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiable cultural groups". It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, and mainly to lead to an appreciation of the connections between the two.

The development and meaning of "ethnomathematics"
The term "ethnomathematics" was introduced by the Brazilian educator and mathematician Ubiratan D'Ambrosio in 1977 during a presentation for the American Association for the Advancement of Science. Since D'Ambrosio put forth the term, people - D'Ambrosio included - have struggled with its meaning ("An etymological abuse leads me to use the words, respectively, ethno and mathema for their categories of analysis and tics from (from techne)". ).

The following is a sampling of some of the definitions of ethnomathematics proposed between 1985 and 2006:


 * "The mathematics which is practiced among identifiable cultural groups such as national-tribe societies, labour groups, children of certain age brackets and professional classes".
 * "The mathematics implicit in each practice".
 * "The study of mathematical ideas of a non-literate culture".
 * "The codification which allows a cultural group to describe, manage and understand reality".
 * "Mathematics…is conceived as a cultural product which has developed as a result of various activities".
 * "The study and presentation of mathematical ideas of traditional peoples".
 * "Any form of cultural knowledge or social activity characteristic of a social group and/or cultural group that can be recognized by other groups such as Western anthropologists, but not necessarily by the group of origin, as mathematical knowledge or mathematical activity".
 * "The mathematics of cultural practice".
 * "The investigation of the traditions, practices and mathematical concepts of a subordinated social group".
 * "I have been using the word ethnomathematics as modes, styles, and techniques (tics) of explanation, of understanding, and of coping with the natural and cultural environment (mathema) in distinct cultural systems (ethnos)".
 * "What is the difference between ethnomathematics and the general practice of creating a mathematical model of a cultural phenomenon (e.g., the "mathematical anthropology" of Paul Kay [1971] and others)? The essential issue is the relation between intentionality and epistemological status. A single drop of water issuing from a watering can, for example, can be modeled mathematically, but we would not attribute knowledge of that mathematics to the average gardener. Estimating the increase in seeds required for an increased garden plot, on the other hand, would qualify".

Numerals
Some of the systems for representing numbers in previous and present cultures are well known. Roman numerals use a few letters of the alphabet to represent numbers up to the thousands, but are not intended for arbitrarily large numbers and can only represent positive integers. Arabic numerals are a family of systems, originating in India and passing to medieval Islamic civilization, then to Europe, and now standard in global culture—and having undergone many curious changes with time and geography—can represent arbitrarily large numbers and have been adapted to negative numbers, fractions, and real numbers.

Less well known systems include some that are written and can be read today, such as the Hebrew and Greek method of using the letters of the alphabet, in order, for digits 1–9, tens 10–90, and hundreds 100–900.

A completely different system is that of the quipu, which recorded numbers on knotted strings.

Ethnomathematicians are interested in the ways in which numeration systems grew up, as well as their similarities and differences and the reasons for them. The great variety in ways of representing numbers is especially intriguing.

Names for numbers
This means the ways in which number words are formed.

English
For instance, in English, there are four different systems. The units words (one to nine) and ten are special. The next two are reduced forms of Anglo-Saxon "one left over" and "two left over" (i.e., after counting to ten). Multiples of ten from "twenty" to "ninety" are formed from the units words, one through nine, by a single pattern. Thirteen to nineteen, and in a slightly different way twenty-one through ninety-nine (excluding the tens words), are compounded from tens and units words. Larger numbers are also formed on a base of ten and its powers ("hundred" and "thousand"). One may suspect this is based on an ancient tradition of finger counting. Residues of ancient counting by 20s and 12s are the words "score", "dozen", and "gross". (Larger number words like "million" are not part of the original English system; they are scholarly creations based ultimately on Latin.)

German
The German language counts similarly to English, but the unit is placed first in numbers over 20. For example, "26" is "sechsundzwanzig", literally "six and twenty". This system was formerly common in English, as seen in an artifact from the English nursery rhyme "Sing a Song of Sixpence": ''Sing a song of sixpence, / a pocket full of rye. / Four and twenty blackbirds, / baked in a pie.''

French
In the French language as used in France, one sees some differences. Soixante-dix (literally, "sixty-ten") is used for "seventy". The words "quatre-vingt" (literally, "four-twenty", or 80) and "quatre-vingt-dix" (literally, "four-twenty ten" 90) are based on 20 ("vingt") instead of 10. Swiss French and Belgian French do not use these forms, preferring more standard Latinate forms: octante for 80 and nonante for 90.

Mesopotamia
In ancient Mesopotamia, the base for constructing numbers was 60, with 10 used as an intermediate base for numbers below 60.

West Africa
Many West African languages base their number words on a combination of 5 and 20, derived from thinking of a complete hand or a complete set of digits comprising both fingers and toes. In fact, in some languages, the words for 5 and 20 refer to these body parts (e.g., a word for 20 that means "man complete"). The words for numbers below 20 are based on 5 and higher numbers combine the lower numbers with multiples and powers of 20. Of course, this description of hundreds of languages is badly oversimplified; better information and references can be found in Zaslavsky (1973).

Finger counting
Many systems of finger counting have been, and still are, used in various parts of the world. Most are not as obvious as holding up a number of fingers. The position of fingers may be most important. One continuing use for finger counting is for people who speak different languages to communicate prices in the marketplace.

In contrast to finger counting, the Yuki people (indigenous Americans from Northern California) keep count by using the four spaces between their fingers rather than the fingers themselves. This is known as an octal (base-8) counting system.

The history of mathematics
This area of ethnomathematics mainly focuses on addressing Eurocentrism by countering the common belief that most worthwhile mathematics known and used today was developed in the Western world. The area stresses that the history of mathematics has been oversimplified, and seeks to explore the emergence and mathematics from various ages and civilizations throughout human history.

Some examples and major contributors
D'Ambrosio's 1980 review of the evolution of mathematics, his 1985 appeal to include ethnomathematics in the history of mathematics and his 2002 paper about the historiographical approaches to non-Western mathematics are excellent examples. Additionally, Frankenstein and Powell's 1989 attempt to redefine mathematics from a non-eurocentric viewpoint and Anderson's 1990 concepts of world mathematics are strong contributions to this area. Detailed examinations of the history of the mathematical developments of non-European civilizations, such as the mathematics of ancient Japan, Iraq, Egypt, and of Islamic, Hebrew, and Incan civilizations, have also been presented.

The philosophy and cultural nature of mathematics
The core of any debate about the cultural nature of mathematics will ultimately lead to an examination of the nature of mathematics itself. One of the oldest and most controversial topics in this area is whether mathematics is internal or external, tracing back to the arguments of Plato, an externalist, and Aristotle, an internalist. On the one hand, Internalists such as Bishop, Stigler and Baranes, believe mathematics to be a cultural product. On the other hand, externalists, like Barrow, Chevallard and Penrose, see mathematics as culture-free, and tend to be major critics of ethnomathematics. With disputes about the nature of mathematics, come questions about the nature of ethnomathematics, and the question of whether ethnomathematics is part of mathematics or not. Barton, who has offered the core of research about ethnomathematics and philosophy, asks whether "ethnomathematics is a precursor, parallel body of knowledge or precolonized body of knowledge" to mathematics and if it is even possible for us to identify all types of mathematics based on a Western-epistemological foundation.

Political math
The contributions in this area try to illuminate how mathematics has affected the nonacademic areas of society. One of the most controversial and provocative political components of ethnomathematics are its racial implications. Ethnomathematicians purport that the prefix "ethno" should not be taken as relating to race, but rather, the cultural traditions of groups of people. However, in places like South Africa concepts of culture, ethnicity and race are not only intertwined but carry strong, divisive negative connotations. So, although it may be made explicit that ethnomathematics is not a "racist doctrine" it is vulnerable to association with racism.

Another major facet of this area addresses the relationship between gender and mathematics. This looks at topics such as discrepancies between male and female math performance in educations and career-orientation, societal causes, women's contributions to mathematics research and development, etc.

Some examples and major contributors
Gerdes' writings about how mathematics can be used in the school systems of Mozambique and South Africa, and D'Ambrosio's 1990 discussion of the role mathematics plays in building a democratic and just society are examples of the impact mathematics can have on developing the identity of a society. In 1990, Bishop also writes about the powerful and dominating influence of Western mathematics. More specific examples of the political impact of mathematics are seen in Knijik's 1993 study of how Brazilian sugar cane farmers could be politically and economically armed with mathematics knowledge, and Osmond's analysis of an employer's perceived value of mathematics (2000).

The mathematics of different cultures
The focus of this area is to introduce the mathematical ideas of people who have generally been excluded from discussions of formal, academic mathematics. The research of the mathematics of these cultures indicates two, slightly contradictory viewpoints. The first supports the objectivity of mathematics and that it is something discovered not constructed. The studies reveal that all cultures have basic counting, sorting and deciphering methods, and that these have arisen independently in different places around the world. This can be used to argue that these mathematical concepts are being discovered rather than created. However, others emphasize that the usefulness of mathematics is what tends to conceal its cultural constructs. Naturally, it is not surprising that extremely practical concepts such as numbers and counting have arisen in all cultures. The universality of these concepts, however, seems harder to sustain as more and more research reveals practices which are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways (see Section 2.1: Numerals and Naming Systems).

One of the challenges faced by researchers in this area is the fact that they are limited by their own mathematical and cultural frameworks. The discussions of the mathematical ideas of other cultures recast these into a Western framework in order to identify and understand them. This raises the questions of how many mathematical ideas evade notice simply because they lack similar Western mathematical counterparts, and of how to draw the line classifying mathematical from non-mathematical ideas.

Some examples and major contributors
The majority of research in this area has been about the intuitive mathematical thinking of small-scale, traditional, indigenous cultures including: Aboriginal Australians, ) the indigenous people of Liberia, Native Americans in North America, Pacific Islanders, Brazilian construction foremen, and tribes in Africa.

Games of skill
An enormous variety of games that can be analyzed mathematically have been played around the world and through history. The interest of the ethnomathematician usually centers on the ways in which the game represents informal mathematical thought as part of ordinary society, but sometimes has extended to mathematical analyses of games. It does not include the careful analysis of good play&mdash;but it may include the social or mathematical aspects of such analysis.

A mathematical game that is well known in European culture is tic-tac-toe (noughts-and-crosses). This is a geometrical game played on a 3-by-3 square; the goal is to form a straight line of three of the same symbol. There are many broadly similar games from all parts of England, to name only one country where they are found.

Another kind of geometrical game involves objects that move or jump over each other within a specific shape (a "board"). There may be captures. The goal may be to eliminate the opponent's pieces, or simply to form a certain configuration, e.g., to arrange the objects according to a rule. One such game is Nine Men's Morris; it has innumerable relatives where the board or setup or moves may vary, sometimes drastically. This kind of game is well suited to play out of doors with stones on the dirt, though now it may use plastic pieces on a paper or wooden board.

A mathematical game found in West Africa is to draw a certain figure by a line that never ends until it closes the figure by reaching the starting point (in mathematical terminology, this is a Eulerian path on a graph). Children use sticks to draw these in the dirt or sand, and of course the game can be played with pen and paper.

The games of checkers, chess, oware (and other mancala games), and Go may also be viewed as subjects for ethnomathematics.

Mathematics in folk art
One way mathematics appears in art is through symmetries. Woven designs in cloth or carpets (to name two) commonly have some kind of symmetrical arrangement. A rectangular carpet often has rectangular symmetry in the overall pattern. A woven cloth may exhibit one of the seventeen kinds of plane symmetry groups; see Crowe (2004) for an illustrated mathematical study of African weaving patterns. Several types of patterns discovered by ethnomathematical communities are related to technologies; see Berczi (2002) about illustrated mathematical study of patterns and symmetry in Eurasia.

The mathematics of different cultural groups
Rather than looking at the mathematics of different cultures, this area focuses on the mathematics of different social groups based on activities, occupation, age, gender, etc. This is another area in which examining the connections between gender and mathematics arises. One can explore the mathematics involved in things like traditional women's work, like knitting, embroidery and sewing, how monitoring menstruation may have been one of the first examples of mathematical use.

There are mathematical components to a whole range of areas not generally connected to mathematics. For example, study the geometrical basis of graffiti, the transformational geometry and iteration in cornrow hairstyles, taxation and the creativity of children with decorative arts can all encompass and rely on mathematical concepts.

Mathematics education
Ethnomathematics and mathematics education addresses first, how cultural values can affect teaching, learning and curriculum, and second, how mathematics education can then affect the political and social dynamics of a culture. One of the stances taken by many educators is that it is crucial to acknowledge the cultural context of mathematics students by teaching culturally based mathematics that students can relate to. Can teaching math through cultural relevance and personal experiences help the learners know more about reality, culture, society and themselves? Robert (2006)

Another approach suggested by mathematics educators is exposing students to the mathematics of a variety of different cultural contexts, often referred to as multicultural math. This can be used both to increase the social awareness of students and offer alternative methods of approaching conventional mathematics operations, like multiplication.(Andrew, 2005)

Examples
Various educators have explored ways of bringing together culture and mathematics in the classroom, such as: Barber and Estrin (1995) and Bradley (1984) on Native American education, Gerdes (1988b and 2001) with suggestions for using African art and games, Malloy (1997) about African American students and Flores (1997), who developed instructional strategies for Hispanic students.

Criticism
There has been criticism of ethnomathematics. Criticism comes in three forms.

First, some have objected to applying the name "mathematics" to subject matter that is not developed abstractly and logically, with proofs, as in the academic tradition descended from Hellenistic Greeks like Pythagoras, Euclid, and Archimedes and comparable traditions in China, Japan, and India. They posit that mathematics consists of objective truths that are not subject to cultural differences. An opponent of ethnomathematics may claim that pointing out that many cultures have arrived at different ways of counting on their fingers is not as insightful, on objective terms, as Cantor's work on infinity, for example. Moreover, some academic mathematicians feel that ethnomathematics is more properly a branch of anthropology than mathematics. An ethnomathematician might reply that ethnomathematics is not meant to be a branch of mathematics nor of anthropology, but combines elements of both in something different from either.

Second, some critics of ethnomathematics claim that most books on the subject emphasize the differences between cultures rather than the similarities. These critics would like to see emphasis on the fact that, for example, negative numbers have been discovered on three independent occasions, in China, in India, and in Germany, and in all three cultures, mathematicians discovered the same rule for multiplying negative numbers. Pascal's triangle was discovered in China, India and Persia long before it was discovered in Europe, and all found exactly the same properties as did the European discoverers. These critics would like to see ethnomathematics emphasize the unifying aspects of mathematics. An ethnomathematician may reply that these critics overlook the central role in ethnomathematics of how mathematics arises in ordinary life.

Third, some critics claim that mathematics education in some countries, including the United States, unduly emphasizes ethnomathematics in order to promote multiculturalism while spending too little time on core mathematical content, and that this often results in pseudoscience being taught. An example of this criticism is an article by Marianne M. Jennings in The Christian Science Monitor, April 2, 1996, titled "'Rain Forest' Algebra Course Teaches Everything But Algebra". Another example is the article "The Third Mathematics Education Revolution" by Richard Askey, published in Contemporary Issues in Mathematics Education (Press Syndicate, Cambridge, UK, 1999), in which he accuses Focus on Algebra, the same Addison-Wesley textbook criticized by the Christian Science Monitor, of teaching pseudoscience, claiming for South Sea islanders mystic knowledge of astronomy more advanced than scientific knowledge. The student of ethnomathematics can answer such criticisms by saying that there is a large body of good ethnomathematical research to which they do not apply, and this body is the main part of the subject.