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Chapter 24: Ethnomathematics and Mathematics Education!
PAULUS GERDES Department of Mathematics, Universidade Pedagogica, Maputo, Mozambique
This chapter analyses the emergence of ethnomathematics as a field of re-search. It starts with the work of some isolated forerunners like Wilder and Raum, and moves to D' Ambrosio's ethnomathematical research program, and the simultaneous gestation of other concepts, like indigenous, socio-, in-formal, spontaneous, oral, hidden, implicit, and people's mathematics. It compares various conceptualisations and paradigms of ethnomathematics. The influence of Freire's ideas on a series of scholars working in the field of ethnomathematics is stressed. The second part of the chapter presents an overview of ethnomathematical literature, continent by continent. The third and final part discusses some of the basic assumptions associated with the use of ideas from ethnomathematics in education. Some complementary and par-tially overlapping trends in educational experimentation are considered from an ethnomathematical perspective.
1. HISTORY OF ETHNOMATHEMATICS: AN OVERVIEW
This chapter analyses the emergence of ethnomathematics as a field of re-search, and then presents an overview of ethnomathematicalliterature conti-nent by continent. It concludes with some illustrations of educational experimentation and research within an ethnomathematical perspective.
1.1 Early Advocates of Ethnomathematics
Ethnomathematics, which may be defined as the cultural anthropology of mathematics and mathematical education, is a relatively new field of interest, one that lies at the confluence of mathematics and cultural anthropology. Tra-ditionally, the dominant view saw mathematics as a 'culture-free', 'universal' phenomenon, and ethnomathematics emerged later than other ethnosciences. Among mathematicians, ethnographers, psychologists and educationalists, Wilder, White, Fettweis, Luquet and Raum may be registered as the principal forerunners of ethnomathematics.
909 A.J. Bishop et al. (eds.), International Handbook of Mathematics Education, 909 - 943 1996 Kluwer Academic Publishers,
In an address entitled 'The cultural basis of mathematics', delivered in 1950 to an international congress of mathematicians, Raymond L. Wilder stated that it was not new to look at mathematics from a cultural perspective: 'Anthropologists have done so, but as their knowledge of mathematics is gen-erally very limited, their reactions have ordinarily consisted of scattered re-marks concerning the types of arithmetic found in primitive cultures' (Wilder, 1950, p.260). However, Wilder (1950) said, there were noteworthy exceptions -like, for example, the arguments put forward in an article enti-tled 'The Locus of Mathematical Reality: An Anthropological Footnote' written by the anthropologist L. White (1947). Wilder (1950) summarised his ideas in the following way:
In man's various cultures are found certain elements which are called mathematical. In the earlier days of civilization, they varied greatly from one culture to another so much that what was called 'mathematics' in one culture would hardly be recognized as such in certain others. With the increase in diffusion due, first, to exploration and invention, and, secondly, to the increase in the use of suitable symbols and their subsequent standardization and dissemination in journals, the mathematical elements of the most advanced cultures gradually merged until... there has resulted essentially one element, common to all civilized cultures, known as mathematics. This is not a fixed entity, however, but is subject to constant change. Not all of the change represents accretion of new material; some of it is a shedding of material no longer, due to influential cultural variations, considered mathematics (pp.269-270).
Wilder pointed out that there are some 'borderline' practices and concepts which are difficult to place either in mathematics or outside mathematics. Later Wilder elaborated his ideas in two books, Evolution of Mathematical Concepts (1968) and Mathematics as a Cultural system (1981).
White started his study by asking the question 'Do mathematical truths re-side in the external world, there to be discovered by man, or are they man-made inventions?' (White, 1947/1956, p.2349). In seeking an answer, he as-serted that 'mathematics in its entirety, its 'truths' and its 'realities', is part of human culture' (p.235l), and concluded with the statement that mathematical truths 'are discovered but they are also man-made'. He went on to assert that although mathematical truths are 'the product of the mind of the human spe-cies', they 'are encountered or discovered by each individual in the mathe-matical culture in which he grows up' (p.2357). For White, mathematics did not originate with Euclid and Pythagoras - or even in ancient Egypt or Mes-opotamia - but is 'a development of thought that had its beginning with the origin of man and culture a million years or so ago' (p.236l).
Wilder and White did not seem to be aware of the studies by the German mathematician, ethnologist and pedagogue, Fettweis (1881-1967i on early mathematical thinking and culture, or of the reflections of the French psy-chologist Luquet on the cultural origin of mathematical notions (Luquet, 1929). Likewise, Raum's (1938) book, Arithmetic in Africa, was not known among the mathematicians and anthropologists of his time. It contained the substance of a course given in the Colonial Department of the University of London Institute of Education. The foreword stated that education 'cannot be truly effective unless it is intelligently based on indigenous culture and living interests' (Raum, 1938, p.4). One of the principles of good teaching 'lays down the importance of understanding the cultural background of the pupil and relating the teaching in school to it' (Raum, 1938, p.5).
Other mathematicians, anthropologists and educators were slow to take up these early reflections of Wilder, White, Fettweis, Luquet and Raum. The prevailing idea in the first half of the century was that of mathematics as uni-versal, basically aprioristic form of knowledge. A reductionist tendency tend-ed to dominate mathematics education, with culture-free models of cognition in the ascendency. 3
1.2 Ubiratan D'Ambrosio: Intellectual Father of the Ethnomathematical Program.
In the 1970's several pervasive factors combined to stimulate reflection not only on the place of mathematics in school curricula but also, more generally, within society. Simultaneously, questions about the role of mathematics edu-cation research, and any implications of such research for curriculum devel-opment and for teaching and learning, began to be asked. Among these pervasive factors were:
a) the failures of the hasty transplantations of 'New Mathematics' curricula from the North to the South in the 1960's;
b) the importance attributed in the newly politically independent states of the Third World to the concept of 'education for all', including mathematics education, in the quest for economic independence; and
c) public unrest about the involvement of mathematicians and mathematical research in the Vietnam war.
At the end of the 1970's and the beginning of the 1980's, a growing aware-ness ofthe societal and cultural aspects of mathematics and mathematical ed-ucation began to emerge within the ranks of mathematicians.4 Evidence to support this statement can be found in the summaries of sessions at various international meetings of mathematicians, mathematics educators, and edu-cation policy makers in which the societal objectives of mathematical educa-tion were earnestly considered - for example, at the 1976 International Congress on Mathematical Education (ICME3, Karlsruhe, Germany), the
1978 Conference on Developing Mathematics in Third World Countries (Khartoum, Sudan) [see El Tom, 1979], the 1978 Workshop on Mathematics and the Real World (Roskilde, Denmark) [see Booss and Niss, 1979], the ses-sion on Mathematics and Society at the 1978 International Congress of Math-ematicians (Helsinki, Finland), the 1981 Symposium on Mathematics-in-the-Community (Huaraz, Peru), and the 1982 Caribbean Conference on Mathe-matics for the Benefit of the Peoples (Paramaribo, Surinam).
Ubiratan D' Ambrosio, a Brazilian mathematician and mathematics educa-tor, played a dynamic role in all these initiatives. It was during that period that he launched his 'ethnomathematical program', and, at the Fourth Internation-al Congress of Mathematics Education in 1984 (ICME4, held in Adelaide, Australia), he presented in the opening plenary lecture his reflections on the 'Socio-cultural Bases for Mathematics Education' (D'Ambrosio, 1985a).
D'Ambrosio (1990) proposed an ethnomathematical program as a 'meth-odology to track and analyse the processes of generation, transmission, diffu-sion and institutionalization of (mathematical) knowledge' in diverse cultural systems (p.78). D'Ambrosio (1985b) contrasted 'academic mathematics', that is to say the mathematics which is taught and learned in the schools, with 'ethnomathematics', which he described as the mathematics 'which is prac-tised among identifiable cultural groups, such as national-tribal societies, la-bour groups, children ofa certain age bracket, professional classes, and so on' (p.45).
According to D' Ambrosio (1985b), the 'mechanism of schooling replaces these practices by other equivalent practices which have acquired the status of mathematics, which have been expropriated in their original forms and re-turned in a codified version' (p.47). Before and outside school almost all chil-dren in the world become 'matherate' - that is to say, they develop the 'capacity to use numbers, quant