Ethnomathematics
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Ethnomathematics is the study of mathematics that considers the culture in which mathematics arises. It especially focuses on the mathematics that is part of general culture, rather than formal, academic mathematics, though some ethnomathematicians study formal mathematics as an artifact of a particular culture or cultures. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, but mainly to the relationship between the two.
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[edit] Subject matter
Subjects which are studied in ethnomathematics include but are not limited to numeral systems, formation of number names, architecture, weaving, and games of skill and chance.
[edit] 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. "Hindu-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.
Ehnomathematicians 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.
[edit] Names for numbers
This means the ways in which number words are formed. (See Menninger (1934, 1969) and Zaslavsky (1973).)
[edit] 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.)
[edit] 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.
[edit] 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.)
[edit] Mesopotamia
In ancient Mesopotamia the base for constructing numbers was 60, with 10 used as an intermediate base for numbers below 60.
[edit] 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).
[edit] 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. (See Zaslavsky (1980) for some finger-counting gestures.) One continuing use for finger counting is for people who speak different languages to communicate prices in the marketplace.
[edit] 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—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, or 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. 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.
[edit] 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 (1973) for an illustrated mathematical study of African weaving patterns.
[edit] 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. An opponent of ethomathematics 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. Moreover, to compare finger-counting with Cantor's set theory is unfair, since no one claims they are equally deep.
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 courses that emphasize ethnomathematics spend too little time on teaching useful mathematics, and teach multi-culturalism and pseudoscience instead. 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 work to which they do not apply, and this body is the main part of the subject.
[edit] See also
[edit] Further reading
- Ascher, Marcia (1991). Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, Calif.: Brooks/Cole. ISBN 0-412989417
- Closs, M. P. (ed.) (1986). Native American Mathematics. Austin, TX: University of Texas Press.
- Crowe, Donald W. (1973). Geometric symmetries in African art. Section 5, Part II, in Zaslavsky (1973).
- Eglash, Ron (1999). African Fractals: Modern Computing and Indigenous Design. New Brunswick, New Jersey, and London: Rutgers University Press. ISBN 0-8135-2613-2, paperback ISBN 0-8135-2614-0
- Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics. 2nd. ed. London: Penguin Books.
- Menninger, Karl (1934), Zahlwort und Ziffer. Revised edition (1958). Göttingen: Vandenhoeck and Ruprecht.
- Menninger, Karl (1969), Number Words and Number Symbols. Cambridge, Mass.: The M.I.T. Press.
- Powell, Arthur B., and Marilyn Frankenstein (eds.) (1997). Ethnomathematics: Challenging Eurocentrism in Mathematics Education. Albany, NY: State University of New York Press. ISBN 0-7914-3351-X
- Zaslavsky, Claudia (1973). Africa Counts: Number and Pattern in African Culture. Third revised ed., 1999. Chicago: Lawrence Hill Books. ISBN 1-556523505
- Zaslavsky, Claudia (1980). Count On Your Fingers African Style. New York: Thomas Y. Crowell. Revised with new illustrations, New York: Black Butterfly Books. ISBN 0-86316-250-9
[edit] External links
- International Study Group for Ethnomathematics
- Ethnomathematics Digital Library. Pacific Resources for Education and Learning. This is a Web collection of source and resource materials
- Ethnomathematics – a rich cultural diversity. Nova: Science in the News. Australian Academy of Science
- Ethnomathematics. Nancy Casey, Mega-Mathematics
