(Mostly excerpts from Eglash, R. "When math worlds collide: intention and invention in ethnomathematics." Science, Technology and Human Values , vol 22, no 1, pp. 79-97, Winter 1997.)

0) Introduction

 Ethnomathematics is typically defined as the study of mathematical concepts in cohesive social groups, with an emphasis on small-scale or indigenous cultures. Working in many different areas of the world, Ascher (1990), Closs (1986), Crump (1990), D'Ambrosio (1990), Gerdes (1991), Njock (1979), Washburn and Crowe (1988), Zaslavsky (1973), and many others (see Fisher 1992, Shirley 1995 for reviews), have provided mathematical analyses of a variety of indigenous patterns and abstractions, while drawing attention to the role of conscious intent in these designs.

1) Five Subfields in ethnomathematics

a. Non-western mathematics consists primarily of historical studies (e.g. Cajori 1896), with a cultural focus (which has continued in contemporary works, such as Joseph 1991) on state empires such as the ancient Chinese, Hindu and Muslim civilizations. It is epistemologically based on the idea of direct, literal translations of nonwestern mathematics to the western tradition. For example, Needham (1959; 137) shows how the Chinese Chu Shih-chieh triangle can be mapped onto Pascal's triangle by a rotation of ninety degrees.

b. Mathematical anthropology uses mathematical modelling in ethnographic and archaeological studies to describe material and cognitive patterns, generally without attributing conscious intent to the population under study. The patterns are instead seen as the structural basis of underlying social forces, or as epiphenomena resulting unintentionally from the nature of the activity itself. Classificatory systems for kinship (e.g. Morgan 1871) were the first of these models. Later refinements of mathematical anthropology (e.g. Kay 1971) expanded this analysis to a variety of social phenomena, and increasingly complex mathematical tools.

c. sociology of mathematics. Under this rubric the methodologies most closely associated with STS, that of the social construction of science, are applied to the work and community of professional mathematicians (Restivo 1993). This is not to suggest that sociology is merely a sub set of anthropology; the term simply derives from its common usage, and from the sociologists' emphasis on urban settings in the west. It is important to note that such theories vary along the weak/strong axis. For some there is simply social "influence" such as differing areas of inquiry; a far cry from the "thoroughly social" portrait of strong constructivism.

d. vernacular mathematics. Borrowing from architecture, we can use this term to specifically focus on those who, while distinctly outside any mathematical professionalism (of either west or non west), would not qualify under the old-fashioned anthropological category (now primarily used in Discovery Channel narrations) of an "ancient cultural tradition."
Examples include the "Street Mathematics" of Nunes et al. (e.g. calculation by peasant pushcart venders), the "Situated Cognition" of Lave (e.g. European women's knitting as algebra), and similar designations (Gerdes (1994) lists titles such as "folk mathematics," "informal mathematics," and "non-standard mathematics").

e. indigenous mathematics. Cultural locations of this research emphasize small-scale (indigenous, traditional) societies (this is how Ascher (1990) defines her use of the term "ethnomathematics"). The epistemological basis is not restricted to methods of direct translation, as in (a), but also includes the types of pattern analysis seen in the modelling approach of (b). Unlike mathematical anthropology, however, this research generally strives to include conscious intent as an important component of the analysis.

2) The problem of primitivist romanticism and orientalism

The cultural categories for these five sub-fields are by no means arbitrary; they reflect both traditional anthropological concepts and their recent challenges. From the traditional point of view, those societies with complex social organization (e.g. labor specialization and political hierarchy) will tend to have greater technological complexity, i.e. "higher mathematics." Since this has often been used to justify ethnocentric or racist stereotypes, ethnomathematics can be directly applied to opposing such mythologies.

 This opposition is not as simple as it might seem, however. First, primitivist ethnocentric discourse is often only considered in terms of the the mean-spirited talk of "savages." But the well intentioned romanticism of "children of the forest" can be just as damaging, for it portrays indigenous peoples as unconscious, animal-like extensions of the ecosystem. Ethnomathematics requires attention to the intentional, conscious aspects of knowledge systems, which means that it will have to discuss indigenous errors as well as success (for example there are many documented cases of indigenous ecological disasters; cf. Diamond 1988).

Second, as Said (1978) pointed out, there is another stereotype category in which the subjects are not too close to nature, but rather too far from it. The "arabesque mind" of the Muslim, the Hindu who thinks only of karma, the Jew who thinks only of money, and the Buddhist who is divorced from emotion are all examples. Thus the British, who could not justify colonizing India for a primitive lack of mathematics (Adas 1989), could criticize Indian culture for not concretizing its mathematics to produce engineering: they were too abstract, just as primitives were too concrete, and only whiteness held the proper balance. Given this formulation, it does not necessarily combat racial prejudice to extol the virtues of mathematical achievements in Chinese, Indian, and Islamic empires; for the same reason that Charles Murry's Bell Curve text can promote pro-white racism by proposing genetically higher IQ in "Jews and Asians." Unfortunately this tends to be ignored by some texts, such as "Multicultural Mathematics" (Nelson 1993), which emphasized only Chinese, Hindu, and Muslim examples. It is true that the mathematics of these cultures are easier to translate into the standard educational curriculum, but in doing so the emphasis merely reinforces orientalism and primitivism. A similar problem occurs in the focus on ancient Egypt from Afrocentric curricula (c.f. discussion of the "Portland Baseline Essays" in Oritz de Montellano 1993, Martel 1994).

3) The problem of multicultural mathematics

 Nelson's text is, however, a rigorous and scholarly account of the mathematics in these cultures. Certain other attempts to write under the "multicultural mathematics" rubric have been far less attentive to the educational requirements, sacrificing mathematical content for a third-world cultural gloss. What is missing from these failed attempts at "inclusiveness" is in part the insistance on intentional indigenous mathematics. In my experience, most students and teachers are delighted to find real examples of African geometric algorithms or Native American applications of probability. But these examples take a great deal of skill to discover, analyze, and combine with standard mathematics curricula. What goes under the name of multicultural mathematics is too often a cheap short-cut that merely replaces Dick and Jane counting marbles with Tatuk and Esteban counting coconuts. Of the few texts that do use indigenous math, almost all examples are restricted to primary school level. Again, this restriction might unintentionally imply primitivism (e.g. that mathematical concepts from African culture are only child-like).

While there are many worth-while projects that have used the "multicultural mathematics" label, far too many have shown a lack of reflection on these pedagogical issues. Rather than working with the cultural concepts held by the students themselves -- either by researching the students' own cultural self-identity or attempting to better educate the students about the hetergeneous complexities of their local heritage -- these poorly applied versions assume a static, broadly essentialist ethnic identity. Under this essentialist assumption, a child from Puerto Rico may find herself confronted with Incan llamas, as if she should automatically be familar with any artifact from the universe of Latin American societies simply because she is "Latina" (Zolkower 1996).

In addition, this essentialist approach leans too heavily on the crutch of "self-esteem," as if all cultural barriors could be reduced to a self-imposed shame (ignoring, for example, the dangers of primitivist romanticism, as noted previously, or the systematically misdirected pride recently exposed in John Hoberman's text, Darwin's Athletes). In contrast, ethnomathematics directly addresses the over-emphasis on biological determinism that creates a learning deterrent for students of all ethnic groups, including whites. Geary (1994) reviews cross cultural studies which indicate that while children, teachers and parents in China and Japan tend to view difficulty with mathematics as a problem of time and effort, their American counterparts attribute differences in mathematics performance to innate ability (which thus becomes a self fulfilling prophecy).

 Finally, we should note that the focus of multicultural mathematics has been in changes to the mathematics curriculum. Why should math teachers bear the brunt of curricular reform? African art, for example, is increasingly used in secondary schools across the nation. Yet students often learn of the geometric basis for Greek architecture or Renaissance painting, while commentary on African works are often restricted to discussion of "naturalness" or "emotional expression." Including ethnomathmatics in the arts would allow math teachers who would like to include ethnomathematics components in their teaching to refer to examples that the students are already engaged with, and would provide art teachers with new tools for design and analysis. Similar advantages could be obtained in other disciplines.

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