Ethnomathematics
Encyclopedia
In mathematics education
, ethnomathematics is the study of the relationship between mathematics
and culture
(D'Ambrosio, 1999, 146). Often associated with "cultures without written expression" (D'Ambrosio, 1997, may paraphrases Ascher 1986), it may also be defined as "'the mathematics which is practised among identifiable cultural groups'" (Powell and Frankenstein, 1997 quoting D'Ambrosio). 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 mathematics which is practiced among identifiable cultural groups such
as national-tribe societies, labour groups, children of certain age brackets and
professional classes" (D‘Ambrosio, 1985).
"The mathematics implicit in each practice" (Gerdes, 1986).
"The study of mathematical ideas of a non-literate culture" (Ascher, 1986).
"The codification which allows a cultural group to describe, manage and
understand reality" (D‘Ambrosio, 1987).
"Mathematics…is conceived as a cultural product which has developed as a
result of various activities" (Bishop, 1988).
"The study and presentation of mathematical ideas of traditional peoples"
(Ascher, 1991).
"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" (Pompeu, 1994).
"The mathematics of cultural practice" (Presmeg, 1996).
"The investigation of the traditions, practices and mathematical concepts of a subordinated social group" (Knijnik, 1998).
"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)" (D'Ambrosio, 1999, 146).
"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." (Eglash et al. 2006).
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.
, 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).
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.
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 Latin
ate forms: octante for 80 and nonante for 90.
the base for constructing numbers was 60, with 10 used as an intermediate base for numbers below 60.
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).
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.
(Shigeru, 2002), Iraq (Robson, 2002), Egypt (Ritter, 2002) and of Islamic (Sesiano, 2002), Hebrew (Langermann and Simonson, 2002) and Incan (Gilsdorf, 2002) civilizations, have also been presented.
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. (Barton, 1996).
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.
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.
foremen (Carraher, 1986) and tribes in Africa (Zaslavsky, 1973 and Gerdes, 1991).
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 (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.
. 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. Several types of patterns discovered by ethnomathematical communities are related to technologies; see Berczi (2000) about illustrated mathematical study of patterns and symmetry in Eurasia.
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.
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.
African American students and Flores (1997), who developed instructional strategies for
Hispanic students.
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.
Mathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
, ethnomathematics is the study of the relationship between mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and culture
Culture
Culture is a term that has many different inter-related meanings. For example, in 1952, Alfred Kroeber and Clyde Kluckhohn compiled a list of 164 definitions of "culture" in Culture: A Critical Review of Concepts and Definitions...
(D'Ambrosio, 1999, 146). Often associated with "cultures without written expression" (D'Ambrosio, 1997, may paraphrases Ascher 1986), it may also be defined as "'the mathematics which is practised among identifiable cultural groups'" (Powell and Frankenstein, 1997 quoting D'Ambrosio). 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)." (D'Ambrosio 1997)). Below 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" (D‘Ambrosio, 1985).
"The mathematics implicit in each practice" (Gerdes, 1986).
"The study of mathematical ideas of a non-literate culture" (Ascher, 1986).
"The codification which allows a cultural group to describe, manage and
understand reality" (D‘Ambrosio, 1987).
"Mathematics…is conceived as a cultural product which has developed as a
result of various activities" (Bishop, 1988).
"The study and presentation of mathematical ideas of traditional peoples"
(Ascher, 1991).
"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" (Pompeu, 1994).
"The mathematics of cultural practice" (Presmeg, 1996).
"The investigation of the traditions, practices and mathematical concepts of a subordinated social group" (Knijnik, 1998).
"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)" (D'Ambrosio, 1999, 146).
"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." (Eglash et al. 2006).
Numerals
Some of the systems for representing numbers in previous and present cultures are well known. Roman numeralsRoman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
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
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...
are a family of systems, originating in India and passing to medieval Islamic civilization
Islamic Golden Age
During the Islamic Golden Age philosophers, scientists and engineers of the Islamic world contributed enormously to technology and culture, both by preserving earlier traditions and by adding their own inventions and innovations...
, then to Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
, 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
Alphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...
, in order, for digits 1–9, tens 10–90, and hundreds 100–900.
A completely different system is that of the quipu
Quipu
Quipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...
, 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. (See Menninger (1934, 1969) and Zaslavsky (1973).)English
For instance, in EnglishEnglish language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
, there are four different systems. The units words (one to nine) and ten are special. The next two are reduced forms of Anglo-Saxon
Old English language
Old English or Anglo-Saxon is an early form of the English language that was spoken and written by the Anglo-Saxons and their descendants in parts of what are now England and southeastern Scotland between at least the mid-5th century and the mid-12th century...
"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 languageGerman language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
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
Nursery rhyme
The term nursery rhyme is used for "traditional" poems for young children in Britain and many other countries, but usage only dates from the 19th century and in North America the older ‘Mother Goose Rhymes’ is still often used.-Lullabies:...
"Sing a Song of Sixpence
Sing a Song of Sixpence
Sing a Song of Sixpence is a well-known English nursery rhyme, perhaps originating in the 18th century. It is also listed in the Roud folk song index as number 13191.-Lyrics:...
": Sing a song of sixpence, / a pocket full of rye. / Four and twenty blackbirds, / baked in a pie.
French
In the French languageFrench language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
as used in France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan 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
Swiss French
Swiss French is the name used for the variety of French spoken in the French-speaking area of Switzerland known as Romandy. Swiss French is not to be confused with Franco-Provençal/Arpitan or Romansh, two other individual Romance languages spoken in areas not far from Romandy.The differences...
and Belgian French
Belgian French
Belgian French is the variety of French spoken mainly in the French Community of Belgium, alongside related minority regional languages such as Walloon, Picard, Champenois and Gaumais. The French spoken in the Democratic Republic of the Congo, Rwanda, and Burundi, which were formerly Belgian...
do not use these forms, preferring more standard Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
ate forms: octante for 80 and nonante for 90.
Mesopotamia
In ancient MesopotamiaMesopotamia
Mesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...
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 digitsDigit (anatomy)
A digit is one of several most distal parts of a limb, such as fingers or toes, present in many vertebrates.- Names:Some languages have different names for hand and foot 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 countingFinger counting
Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in...
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.
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 JapanJapanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...
(Shigeru, 2002), Iraq (Robson, 2002), Egypt (Ritter, 2002) and of Islamic (Sesiano, 2002), Hebrew (Langermann and Simonson, 2002) and Incan (Gilsdorf, 2002) 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, comequestions 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. (Barton, 1996).
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 (D'Ambrosio, 1985, 1987; Borba, 1990; Skovsmose and Vithal, 1997). 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 havegenerally 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 (Harris,1991), the indigenous people of Liberia (Gay and Cole, 1967), Native Americans in North America (Pixten,1987 and Ascher, 1991), Pacific Islanders, (Kyselka, 1981), Brazilian constructionforemen (Carraher, 1986) and tribes in Africa (Zaslavsky, 1973 and Gerdes, 1991).
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
Tic-tac-toe
Tic-tac-toe, also called wick wack woe and noughts and crosses , is a pencil-and-paper game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The X player usually goes first...
, 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
Nine Men's Morris
Nine Men's Morris is an abstract strategy board game for two players that emerged from the Roman Empire. The game is also known as Nine Man Morris, Mill, Mills, Merels, Merelles, and Merrills in English....
; 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
Eulerian path
In graph theory, an Eulerian trail is a trail in a graph which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is a Eulerian trail which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of...
on a graph
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
). 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
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
, oware
Oware
Oware is an abstract strategy game of Akan origin. Part of the mancala family, it is played throughout West Africa and the Caribbean. Among its many names are Ayò , Awalé , Wari , Ouri, Ouril or Uril , Warri , Adji , and Awélé...
and other mancala
Mancala
Mancala is a family of board games played around the world, sometimes called "sowing" games, or "count-and-capture" games, which describes the game-play. Mancala games play a role in many African and some Asian societies comparable to that of chess in the West, or the game of Go in Eastern Asia...
games, and Go
Go (board game)
Go , is an ancient board game for two players that originated in China more than 2,000 years ago...
may also be viewed as subjects for ethnomathematics.
Mathematics in folk art
One way mathematics appears in art is through symmetriesSymmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. 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. Several types of patterns discovered by ethnomathematical communities are related to technologies; see Berczi (2000) 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 themathematics 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?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.
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) aboutAfrican 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
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
like Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
, Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, and Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
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
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
's work on infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
, for example. Moreover, some academic mathematicians feel that ethnomathematics is more properly a branch of anthropology
Anthropology
Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...
than mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
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
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, in India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, and in Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, and in all three cultures, mathematicians discovered the same rule for multiplying negative numbers. Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
was discovered in China, India and Persia
Iran
Iran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as 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
Mathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
in some countries, including the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, unduly emphasizes ethnomathematics in order to promote multiculturalism
Multiculturalism
Multiculturalism is the appreciation, acceptance or promotion of multiple cultures, applied to the demographic make-up of a specific place, usually at the organizational level, e.g...
while spending too little time on core mathematical content, and that this often results in pseudoscience
Pseudoscience
Pseudoscience is a claim, belief, or practice which is presented as scientific, but which does not adhere to a valid scientific method, lacks supporting evidence or plausibility, cannot be reliably tested, or otherwise lacks scientific status...
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
Richard Askey
Richard Allen Askey is an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials are an important schematic in organising the theory of special polynomials...
, published in Contemporary Issues in Mathematics Education (Press Syndicate, Cambridge, UK, 1999), in which he accuses Focus on Algebra, the same Addison-Wesley
Addison-Wesley
Addison-Wesley was a book publisher in Boston, Massachusetts, best known for its textbooks and computer literature. As well as publishing books, Addison-Wesley also distributed its technical titles through the Safari Books Online e-reference service...
textbook criticized by the Christian Science Monitor, of teaching pseudoscience, claiming for South Sea Islanders mystic
Mysticism
Mysticism is the knowledge of, and especially the personal experience of, states of consciousness, i.e. levels of being, beyond normal human perception, including experience and even communion with a supreme being.-Classical origins:...
knowledge of astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
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.
See also
- Anti-racist mathematicsAnti-racist mathematicsAnti-racist mathematics is a branch of education reform theory that sees a need to form a curriculum to counter a perceived bias in mathematics...
- Cultural imperialismCultural imperialismCultural imperialism is the domination of one culture over another. Cultural imperialism can take the form of a general attitude or an active, formal and deliberate policy, including military action. Economic or technological factors may also play a role...
- Culturally relevant teachingCulturally relevant teachingCulturally Relevant Teaching is a pedagogy that recognizes the diverse cultural characteristics of students from different ethnic backgrounds and adjusts teaching methods to account for this diversity Culturally relevant teachers display cultural competence: skill at teaching in a cross-cultural...
- Critical pedagogyCritical pedagogyCritical pedagogy is a philosophy of education described by Henry Giroux as an "educational movement, guided by passion and principle, to help students develop consciousness of freedom, recognize authoritarian tendencies, and connect knowledge to power and the ability to take constructive...
- EthnocomputingEthnocomputingEthnocomputing is the study of the interactions between computing and culture. It is carried out through theoretical analysis, empirical investigation, and design implementation...
- Informal mathematics
- MulticulturalismMulticulturalismMulticulturalism is the appreciation, acceptance or promotion of multiple cultures, applied to the demographic make-up of a specific place, usually at the organizational level, e.g...
- Pedagogy of the oppressedPedagogy of the OppressedPedagogy of the Oppressed is the most widely known of educator Paulo Freire's works. It proposes a pedagogy with a new relationship between teacher, student, and society...
- PostmodernityPostmodernityPostmodernity is generally used to describe the economic or cultural state or condition of society which is said to exist after modernity...
- Social progressivism
- Teaching for social justiceTeaching for social justiceTeaching for social justice is an educational philosophy designed to promote socioeconomic equality in the learning environment and instill these values in students. Educators may employ social justice instruction to promote unity on campus, as well as mitigate boundaries to the general curriculum...
Further reading and references
- Ascher, Marcia (1991). Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, Calif.: Brooks/Cole. ISBN 0-412-98941-7
- D'Ambrosio. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5, 44-8.
- D'Ambrosio. (1997). "Foreward", Ethnomathematics, p.xv and xx. ISBN 0-7914-3352-8.
- D'Ambrosio. (1999). Literacy, Matheracy, and Technoracy: A Trivium for Today. Mathematical Thinking and Learning 1(2), 131-153.
- Berczi, Sz. (2000): Katachi U Symmetry in the Ornamental Art of the Last Thousands Years of Eurasia. FORMA, 15/1. 11-28. Tokyo
- 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
- Eglash, R., Bennett, A., O'Donnell, C., Jennings, S., and Cintorino, M. "Culturally Situated Design Tools: Ethnocomputing from Field Site to Classroom." American Anthropologist, Vol. 108, No. 2. (2006), pp. 347–362.
- Goetzfridt, Nicholas J. (2008) Pacific Ethnomathematics: A Bibliographic Study. Honolulu: University of Hawai'i Press. ISBN 978-0-8248-3170-7.
- Harrison, K. David. (2007) When Languages Die: The Extinction of the World's Languages and the Erosion of Human Knowledge. New York and London: Oxford University Press.
- Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics. 2nd. ed. London: Penguin Books.
- Menninger, KarlKarl Menninger (mathematics)Karl Menninger was a German teacher of and writer about mathematics. His major work was Zahlwort und Ziffer , about non-academic mathematics in much of the world...
(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.
- Luitel, Bal Chandra and Taylor, Peter. (2007). The shanai, the pseudosphere and other imaginings: Envisioning culturally contextualised mathematics education. Cultural Studies of Science Education 2(3).
- Powell, Arthur B., and Marilyn Frankenstein (eds.) (1997). Ethnomathematics: Challenging Eurocentrism in Mathematics Education, p.7. Albany, NY: State University of New York Press. ISBN 0-7914-3351-X
- Zaslavsky, ClaudiaClaudia ZaslavskyClaudia Zaslavsky was an American educator and ethnomathematician. She advanced the study of the links between mathematics and world cultures, especially with her pioneering book Africa Counts, that extended to Africa the work of Karl Menninger about mathematics in ordinary life in other parts of...
(1973). Africa Counts: Number and Pattern in African Culture. Third revised ed., 1999. Chicago: Lawrence Hill Books. ISBN 1-55652-350-5 - 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
External links
- Ethnomathematics Digital Library. Pacific Resources for Education and Learning. This is a Web collection of source and resource materials
- Journal of Mathematics and Culture This is NASGEm's refereed journal on ethnomathematics