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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).
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224:"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 are compounded from tens and units words in one way, and the non-multiples of ten from twenty-one to ninety-nine are compounded from tens and units words in a different way. Larger numbers are also formed on a base of ten and its powers ("
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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
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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 is its racial implications. Ethnomathematicians purport that the prefix "ethno" should not be taken as
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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. This description
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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.
575:
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
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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
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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
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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
131:"N.C. Ghosh included Ethnomathematics in the list of Folk Mathematics" Vide : Lokdarpan- a Journal of the Department of Folklore, Kalyani University and Rabindra Bharati Patrika- a Journal of Rabindra Bharati University, Kolkata, India. Lokashruti - a Journal of Govt. of West Bengal, India.
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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
632:
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
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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.
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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
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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)
127:
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
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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
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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
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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
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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. Following the analysis of Indonesian folk weaving patterns and
123:"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 and others)? The essential issue is the relation between intentionality and
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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
268:
language counts similarly to
English, but the unit is placed before the tens 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
801:
Various mathematics 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
599:
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.
697:; 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.
1100:(Eglash et al. 2006) 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.
450:
For example, what is in
English written out as "twelve thousand three hundred forty five" is "一万二千三百四十五" (simplified) / "一萬二千三百四十五" (traditional) whose characters translate to "one ten-thousand two thousand three hundred four ten five".
35:. 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
199:
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.
962:(Powell and Frankenstein, 1997 quoting D'Ambrosio) Powell, Arthur B., and Marilyn Frankenstein (eds.) (1997). Ethnomathematics: Challenging Eurocentrism in Mathematics Education, p.7. Albany, NY: State University of New York Press.
308:(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.
637:
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.
1831:
The focus of submitted papers should be on the aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, and sociological aspects of doing, learning, and teaching mathematics.
1733:
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).
166:, 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,
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More recently, curriculum changes proposed by the Seattle school district drew criticism to ethnomathematics. Some people judged the proposed changes, which involved a framework for blending math and
681:(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
46:. 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.
344:
Counting in Welsh combines the vigesimal system (counting in twenties) with some other features. The following system is optional for cardinal numbers nowadays, but mandatory for ordinal numbers.
700:
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
69:"The mathematics which is practiced among identifiable cultural groups such as national-tribe societies, labour groups, children of certain age brackets and professional classes".
59:
252:" was used between American English and British English. These have since been reconciled, and modern English speakers universally refer to 1,000,000,000 as 'one billion'.
624:
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).
1706:
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.
1574:
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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.
248:" are not part of the original English system; they are scholarly creations based ultimately on Latin.). There were historical inconsistencies in the way the term "
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91:"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
154:
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
995:(D'Ambrosio, 1985) D'Ambrosio. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5, 44-8.
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Many West African languages generally base their number words on a combination of 5 and 20, derived from thinking of a complete hand or a complete set of
3491:
1688:
Eglash, R., Bennett, A., O'Donnell, C., Jennings, S., and Cintorino, M. "Culturally Situated Design Tools: Ethnocomputing from Field Site to Classroom."
944:(D'Ambrosio, 1999, 146) D'Ambrosio. (1999). Literacy, Matheracy, and Technoracy: A Trivium for Today. Mathematical Thinking and Learning 1(2), 131-153.
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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.
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The majority of research in this area has been about the intuitive mathematical thinking of small-scale, traditional, indigenous cultures, including
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1176:
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has been oversimplified", and seeks to explore the emergence of mathematics from various ages and civilizations throughout human history.
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textbook criticized in an op-ed by Marianne M. Jennings) and among other shortcomings found it guilty of repeating debunked claims about
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1856:
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D'Ambrosio. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5, 44–8.
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743:. Woven designs in cloth or carpets (to name two) commonly have some kind of symmetrical arrangement. A rectangular carpet often has
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by countering the common belief that most worthwhile mathematics known and used today was developed in the Western world.
220:, there are four different systems. The units words (one to nine) and ten are special. The next two are reduced forms of
1716:
1408:
Cellular-Automata and Innovation within Indonesian Traditional Weaving Crafts: A Discourse of Human-Computer Interaction
873:
1695:
Goetzfridt, Nicholas J. (2008) Pacific Ethnomathematics: A Bibliographic Study. Honolulu: University of Hawai'i Press.
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and games, Malloy (1997) about African American students and Flores (1997), who developed instructional strategies for
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1983:
920:
1567:
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1113:. Revised edition (1958). Göttingen: Vandenhoeck and Ruprecht. (See Menninger (1934, 1969).) Menninger, Karl (1969),
510:) keep count by using the four spaces between their fingers rather than the fingers themselves. This is known as an
3653:
2908:
2326:
2298:
853:, for incorporating questions like "How important is it to be right?" and "Who gets to say if an answer is right?"
708:). Children use sticks to draw these in the dirt or sand, and of course the game can be played with pen and paper.
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mathematics. More specific examples of the political impact of mathematics are seen in Knijik's 1993 study of how
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Number words in Chinese are assembled from the words for "one" through "nine" and words for powers of ten.
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The following is a sampling of some of the definitions of ethnomathematics proposed between 1985 and 2006:
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1044:(1991). Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, Calif.: Brooks/Cole.
744:
654:
646:
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101:"The investigation of the traditions, practices and mathematical concepts of a subordinated social group".
55:
1812:
Pacific Resources for Education and Learning. This is a Web collection of source and resource materials.
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Crowe, Donald W. (1973). Geometric symmetries in African art. Section 5, Part II, in Zaslavsky (1973).
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748:
43:
20:
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85:"Mathematics…is conceived as a cultural product which has developed as a result of various activities".
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1911:
1904:
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Berczi, Sz. (2000): Katachi U Symmetry in the Ornamental Art of the Last Thousands Years of Eurasia.
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888:
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36:
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1899:
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relating to race, but rather, the cultural traditions of groups of people. However, in places like
507:
1318:(D'Ambrosio. (1985). "Ethnomathematics and its place in the history and pedagogy of mathematics".
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95:, but not necessarily by the group of origin, as mathematical knowledge or mathematical activity".
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3189:
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2921:
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2413:
2098:
1993:
1946:
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163:
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New York: Thomas Y. Crowell. Revised with new illustrations, New York: Black Butterfly Books.
1215:. New York: Thomas Y. Crowell. Revised with new illustrations, New York: Black Butterfly Books.
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279:
Sing a song of sixpence, / a pocket full of rye. / Four and twenty blackbirds, / baked in a pie.
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150:
Some of the systems for representing numbers in previous and present cultures are well known.
62:. Since D'Ambrosio put forth the term, there has been debate as to its precise definition.).
112:) of explanation, of understanding, and of coping with the natural and cultural environment (
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2026:
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while spending too little time on core mathematical content, and that this often results in
823:
658:
217:
82:"The codification which allows a cultural group to describe, manage and understand reality".
40:
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Less well known systems include some that are written and can be read today, such as the
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1988:
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182:
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92:
1211:(See Zaslavsky (1980) for some finger-counting gestures.) Zaslavsky, Claudia (1980).
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traditional architectural ornaments, the geometry of Indonesian traditional motifs of
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1968:
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705:
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309:
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171:
124:
54:
The term "ethnomathematics" was introduced by Brazilian educator and mathematician
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978:(D'Ambrosio 1997) D'Ambrosio. (1997). "Foreword", Ethnomathematics, p.xv and xx.
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2423:
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1963:
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D'Ambrosio. (1999). Literacy, Matheracy, and Technoracy: A Trivium for Today.
1147:"Bien écrire et parler juste, guide pratique d'expression et de communication".
803:
747:
in the overall pattern. A woven cloth may exhibit one of the seventeen kinds of
678:
577:
551:
503:
464:
460:
76:
28:
1515:"Seattle Schools Lead Controversial Push to 'Rehumanize' Math - Education Week"
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3262:
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2800:
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2643:
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1978:
616:
1493:
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2008:
1865:
568:
167:
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88:"The study and presentation of mathematical ideas of traditional peoples".
3564:
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229:
186:
155:
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249:
245:
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32:
615:' writings about how mathematics can be used in the school systems of
1958:
662:
621:
751:; see Crowe (2004) for an illustrated mathematical study of African
1738:
Ethnomathematics: Challenging Eurocentrism in Mathematics Education
1440:
1416:
2493:
1998:
761:
757:
720:
716:
564:
547:
511:
317:
241:
193:
1670:
New Brunswick, New Jersey, and London: Rutgers University Press.
1879:
1723:. Revised edition (1958). Göttingen: Vandenhoeck and Ruprecht.
3450:
2836:
1838:
1538:
677:
A mathematical game that is well known in European culture is
236:. Residues of ancient counting by 20s and 12s are the words "
232:"). One may suspect this is based on an ancient tradition of
189:, in order, for digits 1–9, tens 10–90, and hundreds 100–900.
1740:, p. 7. Albany, NY: State University of New York Press.
1482:"'Rain Forest' Algebra Course Teaches Everything but Algebra"
467:, with 10 used as an intermediate base for numbers below 60.
162:
are a family of systems, originating in India and passing to
1834:
1809:
1711:
The Crest of the Peacock: Non-European Roots of Mathematics.
1612:
Ethnomathematics: A Multicultural View of Mathematical Ideas
1234:
Ethnomathematics: A Multicultural View of Mathematical Ideas
1736:
Powell, Arthur B., and Marilyn Frankenstein (eds.) (1997).
522:
This area of ethnomathematics mainly focuses on addressing
1322:, 5, 44-8., 1987; Borba, 1990; Skovsmose and Vithal, 1997)
480:
of hundreds of African languages is badly oversimplified.
1668:
African Fractals: Modern Computing and Indigenous Design.
1380:. Third revised ed., 1999. Chicago: Lawrence Hill Books.
1195:. Third revised ed., 1999. Chicago: Lawrence Hill Books.
1130:. Third revised ed., 1999. Chicago: Lawrence Hill Books.
661:, Brazilian construction foremen, and various tribes in
1778:
Third revised ed., 1999. Chicago: Lawrence Hill Books.
822:
unduly emphasizes ethnomathematics in order to promote
1825:
This is NASGEm's refereed journal on ethnomathematics.
1109:(See Menninger (1934, 1969).) Menninger, Karl (1934),
776:; see Situngkir and Surya (2007) for implementations.
208:
This means the ways in which number words are formed.
1776:
Africa Counts: Number and Pattern in African Culture.
731:
may also be viewed as subjects for ethnomathematics.
1378:
Africa Counts: Number and Pattern in African Culture
1193:
Africa Counts: Number and Pattern in African Culture
1128:
Africa Counts: Number and Pattern in African Culture
3517:
3484:
3336:
3293:
3203:
3165:
3132:
3084:
3056:
3003:
2950:
2907:
2814:
2697:
2569:
2372:
2206:
2007:
1872:
60:
American Association for the Advancement of Science
685:, to name only one country where they are found.
558:The philosophy and cultural nature of mathematics
316:do not use these forms, preferring more standard
1566:but its sources remain unclear because it lacks
953:(D'Ambrosio, 1997, may paraphrases Ascher 1986)
1730:. Cambridge, Massachusetts: The M.I.T. Press.
1432:Deconstructing Bataknese Gorga Computationally
739:One way mathematics appears in art is through
304:as used in France, one sees some differences.
281:It persists in some children's songs such as "
3462:
2848:
1850:
1376:(Zaslavsky, 1973) Zaslavsky, Claudia (1973).
196:, which recorded numbers on knotted strings.
192:A completely different system is that of the
8:
1692:, Vol. 108, No. 2. (2006), pp. 347–362.
1468:Contemporary Issues in Mathematics Education
1126:Zaslavsky (1973) Zaslavsky, Claudia (1973).
72:"The mathematics implicit in each practice".
244:", and "gross". (Larger number words like "
3469:
3455:
3447:
2855:
2841:
2833:
1857:
1843:
1835:
1456:The Third Mathematics Education Revolution
1117:. Cambridge, Massachusetts: The MIT Press.
789:conventional mathematics operations, like
1597:Learn how and when to remove this message
1439:
1415:
554:civilizations, have also been presented.
27:is the study of the relationship between
463:, the base for constructing numbers was
346:
937:
98:"The mathematics of cultural practice".
1660:Austin, TX: University of Texas Press.
1470:(Press Syndicate, Cambridge, UK, 1999)
75:"The study of mathematical ideas of a
58:in 1977 during a presentation for the
1829:The Journal of Humanistic Mathematics
628:The mathematics of different cultures
7:
3608:
3417:
1790:Count On Your Fingers African Style.
1644:Mathematical Thinking and Learning 1
1614:Pacific Grove, Calif.: Brooks/Cole.
768:that eventually made a new genre of
641:Some examples and major contributors
608:Some examples and major contributors
537:Some examples and major contributors
502:In contrast to finger counting, the
332:) for 80 (only in Swiss French) and
3429:
1759:Fisika Batik (The Physics of Batik)
1709:Joseph, George Gheverghese (2000).
1213:Count On Your Fingers African Style
571:, an internalist. On the one hand,
185:method of using the letters of the
1816:Journal of Mathematics and Culture
108:as modes, styles, and techniques (
14:
1810:Ethnomathematics Digital Library.
1513:Gewertz, Catherine (2019-10-23).
1171:(in French), Fides, p. 253,
3607:
3598:
3597:
3428:
3416:
3405:
3404:
3392:
1627:D'Ambrosio. (1997). "Foreword",
1543:
1169:Dictionnaire du français acadien
425:two on five-ten and three-score
116:) in distinct cultural systems (
16:Study of mathematics and culture
3313:Computational complexity theory
1728:Number Words and Number Symbols
1713:2nd. ed. London: Penguin Books.
1430:Situngkir, Hokky (2015-08-24),
1406:Situngkir, Hokky (2015-08-12),
1320:For the Learning of Mathematics
1291:(Langermann and Simonson, 2002)
1115:Number Words and Number Symbols
1349:(Pixten,1987 and Ascher, 1991)
550:, and of Islamic, Hebrew, and
1:
348:Examples of numbers in Welsh
164:medieval Islamic civilization
1788:Zaslavsky, Claudia (1980).
1658:Native American Mathematics.
1150:Sélection du Reader's Digest
874:Culturally relevant teaching
544:mathematics of ancient Japan
529:The area stresses that "the
104:"I have been using the word
1656:Closs, M. P. (ed.) (1986).
1191:Zaslavsky, Claudia (1973).
921:Teaching for social justice
649:, the indigenous people of
506:(indigenous Americans from
141:Numerals and naming systems
3672:
3363:Films about mathematicians
2226:Countries by ethnic groups
2221:Contemporary ethnic groups
1761:. Gramedia Pustaka Utama.
587:
518:The history of mathematics
514:(base-8) counting system.
487:
3593:
3386:
2932:Philosophy of mathematics
2872:
1486:Christian Science Monitor
900:Pedagogy of the Oppressed
422:dau ar bymtheg a thrigain
403:two on five-ten on score
3368:Recreational mathematics
1726:Menninger, Karl (1969),
1552:This section includes a
1153:(in French). p. 21.
818:Some critics claim that
400:dau ar bymtheg ar hugain
3634:Mathematics and culture
3253:Mathematical statistics
3243:Mathematical psychology
3213:Engineering mathematics
3147:Algebraic number theory
2338:Torres Strait Islanders
2183:Ethnopsychopharmacology
1927:In-group and out-groups
1690:American Anthropologist
1610:Ascher, Marcia (1991).
1581:more precise citations.
1232:Ascher, Marcia (1994),
1091:(D'Ambrosio, 1999, 146)
864:Anti-racist mathematics
735:Mathematics in folk art
590:Anti-racist mathematics
414:half hundred and seven
275:Sing a Song of Sixpence
50:Development and meaning
3399:Mathematics portal
3248:Mathematical sociology
3228:Mathematical economics
3223:Mathematical chemistry
3152:Analytic number theory
3033:Differential equations
2584:Cultural appropriation
2484:Lineage-bonded society
1480:Jennings, Marianne M.
1236:, Chapman & Hall,
647:Aboriginal Australians
567:, an externalist, and
531:history of mathematics
3649:Traditional knowledge
3639:Mathematics education
3478:Mathematics education
3378:Mathematics education
3308:Theory of computation
3028:Hypercomplex analysis
2609:Ethnic interest group
2444:Ethnicity in censuses
2394:Cultural assimilation
1895:Ethnolinguistic group
1167:(2009), Fides (ed.),
820:mathematics education
780:Mathematics education
749:plane symmetry groups
588:Further information:
44:mathematics education
21:mathematics education
3358:Informal mathematics
3238:Mathematical physics
3233:Mathematical finance
3218:Mathematical biology
3157:Diophantine geometry
2822:Minzu (anthropology)
2791:Separatist movements
2654:Ethnographic village
2479:Legendary progenitor
2114:Transidioethnography
1917:Hyphenated ethnicity
1912:Ethnographic realism
1905:Ethnoreligious group
1666:Eglash, Ron (1999).
1653:, 15/1. 11–28. Tokyo
1340:(Gay and Cole, 1967)
926:Mathematics and arts
916:Social progressivism
889:Informal mathematics
869:Cultural imperialism
745:rectangular symmetry
37:mathematical systems
3373:Mathematics and art
3283:Operations research
3038:Functional analysis
2689:Multinational state
2684:Model minority myth
2571:Multiethnic society
2489:Linguistic homeland
1900:Ethnonational group
1721:Zahlwort und Ziffer
1111:Zahlwort und Ziffer
508:Northern California
411:hanner cant a saith
349:
56:Ubiratan D'Ambrosio
3575:Cognitively guided
3318:Numerical analysis
2927:Mathematical logic
2922:Information theory
2669:Middleman minority
2629:Ethnic pornography
2624:Ethnic nationalism
2529:Pantribal sodality
2469:Imagined community
1994:Symbolic ethnicity
1922:Indigenous peoples
1890:Ethnographic group
1821:2009-08-17 at the
1772:Zaslavsky, Claudia
1554:list of references
1462:2016-08-06 at the
1022:(D.Ambrosio, 1987)
657:in North America,
347:
288:2019-03-23 at the
3654:Critical pedagogy
3621:
3620:
3560:Modern elementary
3535:Three-part lesson
3444:
3443:
3043:Harmonic analysis
2830:
2829:
2751:Ethnic stereotype
2664:Indigenous rights
2649:Ethnographic film
2634:Ethnic theme park
2594:Dominant minority
2589:Diaspora politics
2579:Consociationalism
2514:National language
2399:Cultural identity
2389:Cross-race effect
2333:Aboriginal groups
1701:978-0-8248-3170-7
1607:
1606:
1599:
1178:978-2-7621-3010-2
879:Critical pedagogy
772:batik designs as
695:nine men's morris
659:Pacific Islanders
440:
439:
436:hundred less one
216:For instance, in
204:Names for numbers
3661:
3611:
3610:
3601:
3600:
3471:
3464:
3457:
3448:
3432:
3431:
3420:
3419:
3408:
3407:
3397:
3396:
3328:Computer algebra
3303:Computer science
3023:Complex analysis
2857:
2850:
2843:
2834:
2721:Ethnic cleansing
2716:Ethnic bioweapon
2599:Ethnic democracy
2208:Groups by region
2158:Ethnomethodology
2141:Ethnomathematics
2131:Ethnolinguistics
2027:Ethnoarchaeology
1859:
1852:
1845:
1836:
1629:Ethnomathematics
1602:
1595:
1591:
1588:
1582:
1577:this section by
1568:inline citations
1547:
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894:Multiculturalism
836:Focus on Algebra
824:multiculturalism
793:(Andrew, 2005).
688:Another kind of
655:Native Americans
494:Many systems of
381:one on five-ten
350:
106:ethnomathematics
25:ethnomathematics
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3268:Systems science
3199:
3195:Homotopy theory
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2999:
2946:
2917:Category theory
2903:
2868:
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2810:
2756:Ethnic violence
2704:
2702:ethnic conflict
2700:
2693:
2674:Minority rights
2614:Ethnic majority
2565:
2549:Detribalization
2504:Nation-building
2439:Ethnic religion
2382:
2378:
2368:
2275:Central America
2202:
2173:Ethnophilosophy
2168:Ethnomusicology
2146:Ethnostatistics
2104:Person-centered
2074:Autoethnography
2003:
1868:
1863:
1823:Wayback Machine
1806:
1717:Menninger, Karl
1631:, p.xv and xx.
1603:
1592:
1586:
1583:
1572:
1558:related reading
1548:
1544:
1537:
1535:Further reading
1532:
1523:
1521:
1512:
1511:
1507:
1498:
1496:
1479:
1478:
1474:
1466:, published in
1464:Wayback Machine
1453:Richard Askey:
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1358:(Kyselka, 1981)
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1317:
1313:
1309:(Barton, 1996).
1308:
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1282:(Sesiano, 2002)
1281:
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1082:(Knijnik, 1998)
1081:
1077:
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1040:(Ascher, 1991)
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844:Dogon astronomy
816:
799:
782:
766:Hokky Situngkir
764:is analyzed by
737:
671:
643:
630:
610:
592:
586:
560:
539:
520:
496:finger counting
492:
490:Finger counting
486:
484:Finger counting
473:
457:
445:
342:
302:French language
298:
290:Wayback Machine
262:German language
258:
234:finger counting
214:
206:
160:Arabic numerals
148:
143:
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125:epistemological
93:anthropologists
52:
17:
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3555:Computer-based
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3346:Mathematicians
3342:
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3338:Related topics
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3273:Control theory
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2798:
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2783:
2781:Indigenization
2778:
2773:
2771:Ethnosymbolism
2768:
2763:
2758:
2753:
2748:
2743:
2741:Ethnic penalty
2738:
2733:
2728:
2723:
2718:
2713:
2707:
2705:
2698:
2695:
2694:
2692:
2691:
2686:
2681:
2679:Model minority
2676:
2671:
2666:
2661:
2659:Ethnopluralism
2656:
2651:
2646:
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2604:Ethnic enclave
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3135:
3134:Number theory
3131:
3125:
3122:
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3115:
3112:
3110:
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3066:Combinatorics
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3018:Real analysis
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2761:Ethnocentrism
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2726:Ethnic hatred
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2038:
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2034:
2033:
2030:
2028:
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2023:
2020:
2018:
2015:
2014:
2012:
2010:
2006:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1975:
1974:Polyethnicity
1972:
1970:
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1954:Monoethnicity
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271:nursery rhyme
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41:multicultural
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3409:
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3323:Optimization
3185:Differential
3109:Differential
3076:Order theory
3071:Graph theory
2975:Group theory
2796:Xenocentrism
2746:Ethnic slurs
2736:Ethnic party
2699:Ideology and
2619:Ethnic media
2561:White ethnic
2554:Neotribalism
2509:Nation state
2449:Ethnofiction
2380:ethnogenesis
2351:
2344:
2325:
2294:Central Asia
2286:
2239:
2232:
2215:
2188:Ethnoscience
2178:Ethnopoetics
2140:
2126:Ethnohistory
2064:Ethnogeology
2052:Ethnozoology
2042:Ethnoecology
2032:Ethnobiology
2017:Anthropology
1969:Panethnicity
1885:Ethnic group
1789:
1775:
1758:
1737:
1727:
1720:
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1689:
1678:, paperback
1667:
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1573:Please help
1565:
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1518:
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1485:
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597:South Africa
593:
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573:Internalists
561:
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524:Eurocentrism
521:
501:
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474:
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310:Swiss French
306:Soixante-dix
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198:
191:
176:
172:real numbers
149:
117:
113:
109:
105:
77:non-literate
64:
53:
24:
18:
3525:Traditional
3435:WikiProject
3278:Game theory
3258:Probability
2995:Homological
2985:Multilinear
2965:Commutative
2942:Type theory
2909:Foundations
2865:mathematics
2731:Ethnic joke
2539:Tribal name
2524:Origin myth
2499:Mythomoteur
2424:Ethnic flag
2409:Development
2094:Netnography
2069:Ethnography
2059:Ethnocinema
2037:Ethnobotany
1964:Nationality
1579:introducing
804:African art
723:(and other
690:geometrical
679:tic-tac-toe
504:Yuki people
471:West Africa
461:Mesopotamia
459:In ancient
455:Mesopotamia
222:Anglo-Saxon
29:mathematics
3628:Categories
3263:Statistics
3142:Arithmetic
3104:Arithmetic
2970:Elementary
2937:Set theory
2806:Xenophobia
2801:Xenophilia
2776:Indigenism
2711:Allophilia
2644:Ethnocracy
2534:Statistext
2464:Historical
2357:Indigenous
2309:South Asia
2245:Indigenous
1979:Population
1524:2020-07-31
1499:2024-03-27
1441:1510.01140
1417:1508.02868
933:References
810:students.
741:symmetries
617:Mozambique
328:(formerly
3644:Ethnology
3540:Singapore
3509:Australia
3485:Geography
3190:Geometric
3180:Algebraic
3119:Euclidean
3094:Algebraic
2990:Universal
2766:Ethnocide
2639:Ethnoburb
2544:Tribalism
2327:Australia
2319:West Asia
2299:East Asia
2270:Caribbean
2255:Greenland
2136:Ethnology
2009:Ethnology
1947:Influence
1866:Ethnicity
1494:0882-7729
834:examined
814:Criticism
622:Brazilian
569:Aristotle
370:five-ten
168:fractions
128:qualify".
79:culture".
3603:Category
3585:Critical
3570:Informal
3530:Exercise
3518:Approach
3497:New York
3411:Category
3167:Topology
3114:Discrete
3099:Analytic
3086:Geometry
3058:Discrete
3013:Calculus
3005:Analysis
2960:Abstract
2899:Glossary
2882:Timeline
2786:Nativism
2454:Ethnonym
2419:Ethnarch
2375:Identity
2362:European
2240:Americas
2084:Critical
2079:Clinical
1873:Concepts
1819:Archived
1774:(1973).
1757:(2007).
1755:Surya Y.
1719:(1934),
1587:May 2012
1460:Archived
857:See also
808:Hispanic
797:Examples
713:checkers
546:, Iraq,
336:for 90.
326:huitante
324:for 70,
322:septante
318:Latinate
286:Archived
230:thousand
187:alphabet
156:integers
146:Numerals
3613:Commons
3423:Commons
3205:Applied
3175:General
2952:Algebra
2877:History
2815:Related
2474:Kinship
2414:Endonym
2404:Demonym
2352:Oceania
2109:Salvage
1575:improve
1073:, 1996)
1071:Presmeg
770:fractal
753:weaving
725:mancala
683:England
651:Liberia
443:Chinese
367:pymtheg
334:nonante
330:octante
320:forms:
300:In the
250:billion
246:million
228:" and "
226:hundred
218:English
212:English
114:mathema
33:culture
3550:Reform
3124:Finite
2980:Linear
2887:Future
2863:Major
2346:Europe
2260:Mexico
2250:Canada
2234:Africa
2099:Online
1959:Nation
1796:
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1765:
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966:
663:Africa
477:digits
392:score
296:French
256:German
179:Hebrew
170:, and
118:ethnos
3580:Ethno
3545:Saxon
3351:lists
2894:Lists
2867:areas
2494:Mores
2216:World
2119:Video
1999:Tribe
1651:FORMA
1560:, or
1436:arXiv
1412:arXiv
762:batik
758:Batak
721:oware
717:chess
706:graph
704:on a
565:Plato
552:Incan
548:Egypt
512:octal
389:ugain
340:Welsh
266:Dutch
242:dozen
238:score
194:quipu
183:Greek
136:Areas
2288:Asia
1984:Race
1880:Clan
1794:ISBN
1780:ISBN
1763:ISBN
1742:ISBN
1697:ISBN
1680:ISBN
1672:ISBN
1633:ISBN
1616:ISBN
1490:ISSN
1382:ISBN
1238:ISBN
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1197:ISBN
1173:ISBN
1132:ISBN
1046:ISBN
980:ISBN
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1571:.
1527:.
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