Friday, April 16, 2010

Why study the history of Mathematics in Africa S. of the Sahara?


Why study the history of mathematics in Africa south of the Sahara?

There are many reasons which make the general study of the history of mathematics both necessary and attractive (see e.g. Struik, 1980). There exist important additional reasons which make the research on the history of mathematics in Africa south of the Sahara indispensable.
African countries face the problem of low 'levels' of attainment in mathematics education. Math anxiety is widespread. Many children (and teachers too?) experience mathematics as a rather strange and useless subject, imported from outside Africa. One of the causes thereof is that the goals, contents and methods of mathematics education are not or not sufficiently adapted to the cultures and needs of the African peoples, as stresses the first Secretary-General of the AMU Commission for Mathematical Instruction (Eshiwani, 1979, 346; cf. Eshiwani, 1983; Jacobsen, 1984). Today's existing African educational system is "unadapted and elitist" and "favours foreign consumption without generating a culture that is both compatible with the original civilization and truly promising" (Ki-Zerbo, 1990, 4; cf. El-Tom on mathematics education and the selection of élites, 1984, 3).

The delegates to the Vth Conference of Ministers of Education and those Responsible for Economic Planning in African Member States declared that educational policy should be designed to "restore to their rightful status the African cultural heritage and the traditional social and human values that hold potential for the future " (MINEDAF,1982, 41). The mathematical heritage of the peoples of Africa has to be valued and African mathematical traditions should be 'embedded' into the curriculum (Cf.e.g. Ale, 1989; Doumbia, 1984, 1989b, Gerdes, 1985a, 1986a, 1986b, 1988d, 1990c; Langdon, 1989, 1990; Mmari, 1978; Njock, 1985; Shirley, 1986a, 1986b). And as this scientific legacy of Africa south of the Sahara is little known, research in this area constitutes a challenge to which an urgent response is necessary (Njock, 1985, 4). Also African-Americans and minorities of African descent all over the world feel the need to know their cultural-mathematical heritage (Campbell, 1977; Frankenstein & Powell, 1989; Zaslavsky, 1973, etc.; Ratteray, 1991). More generally, both in highly industrialised and in Third World countries it is becoming more and more recognised that it is necessary to multi-culturalise the mathematics curriculum in order to improve its quality, to augment the cultural confidence of all pupils and to combat racial and cultural prejudice (cf. e.g. D'Ambrosio, 1985a; Ascher, 1984; Bishop, 1988a, b; Joseph, 1987; Mellin-Olsen, 1986; NCTM, 1984; Nebres, 1983; Zaslavsky, 1989a, 1991).

Broad conception of 'history' and 'mathematics'

Most histories of mathematics devote only a few pages to Ancient Egypt and to northern Africa during the 'Middle Ages´. Generally they ignore the history of mathematics in Africa south of the Sahara and give the impression that this history either did not exist or, at least, is not knowable / traceable, or, stronger still, that there was no mathematics at all south of the Sahara (cf. Lumpkin, 1983; Njock, 1985). "Even the Africanity of Egyptian mathematics is often denied" (Shirley, 1986b, 2). Prejudice and narrow conceptions of both 'history' (cf. Ki-Zerbo, 1980, General Introduction) and of 'mathematics' form the basis of such (eurocentric) views (cf. Joseph, 1987, 1991).

At the 17th International Congress of Historical Sciences, Humphrey (1990, 4) stressed that "Any narrow definition of science in modern terms would make it difficult for us to understand its origins and the variable forms it has taken in different cultures". In the case of mathematics, authors like Ale, D'Ambrosio, Ascher & Ascher, Bishop, Doumbia, Gerdes, Njock, Shirley and Zaslavsky consider 'mathematics' as a pan-cultural phenomenon and propose a broad conception, including counting, locating, measuring, designing, playing, explaining, classifying, sorting...

Pioneer study

Zaslavsky's ' Africa Counts ' is a pioneer work in the area of the history of mathematics south of the Sahara. She offers her book as "a preliminary survey of a vast field awaiting investigation" (1973a, vi). Her task was not an easy one: in face of "the inadequacy of easily accessible material... ", she had to search "the literature of many disciplines - history, economics, ethnology, anthropology, archaeology, linguistics, art and oral tradition - ..." (1973a, vi).
She used a broad perspective on mathematics; her study deals with, what she calls, the 'sociomathematics' of Africa: she considers "the applications of mathematics in the lives of African people, and, conversely, the influence that African institutions had upon the evolution of mathematics" (1973a, 7). The concept of sociomathematics may be considered a forerunner of the concept of ethnomathematics. It is ethnomathematics as a discipline that studies mathematics (and mathematical education) as embedded in their cultural context - the (development of) different forms of mathematical thinking which are proper to cultural groups, like ethnic, professional, and age groups.

For the (possible) relationships between ethnomathematics and the history of mathematics, see (in general) D'Ambrosio (1985b) and (in the case of Africa) Shirley (1986b) and Gerdes (1990e).
The application of historical and ethnomathematical research methods has contributed, as will be shown, to the knowledge and understanding of the history of mathematics in Africa, or, at least, of some further mathematical elements in African traditions, in addition to the information gathered in ' Africa Counts '.

The beginnings

Zaslavsky presented as early evidence for (proto-)mathematical activity in Africa a bone dated at 9000-6500 B.C., dug up at Ishango (Zaire). The bone has what appear to be tallying marks on it, notches carved in groups. The bone's discoverer, De Heinzelin, interpreted the patterns of notches as an "arithmetical game of some sort, devised by a people who had a number system based on 10 as well as a knowledge of duplication and of prime numbers". Marshack, on the contrary, explains the bone as early lunar phase count. Their views, summarized in (Zaslavsky, 1973a, 17-19), are reproduced recently in (Fauvel & Gray, 1987, 5-7). Later, the dating of the Ishango bone has been reevaluated, from about 8000 B.C. to 20,000 B.C. (Marshack, 1991). Zaslavsky (1991b) raises the question "who but a woman keeping track of her cycles would need a lunar calendar?" and concludes that "women were undoubtedly the first mathematicians!".

Bogoshi, Naidoo & Webb report in 1987 on a still much older "mathematical artefact": "A small piece of the fibula of a baboon, marked with 29 clearly defined notches, may rank as the oldest mathematical artefact known. Discovered in the early seventies during an excavation of Border Cave in the Lebombo Mountains between South Africa and Swaziland, the bone has been dated to approximately 35000 B.C.". They note that the bone "resembles calendar sticks still in use today by Bushmen clans in Namibia" (1987, 294).
A research project looking for numerical representations in San (Bushmen) rock art has recently been started by Martinson (University of the Witwatersrand, South Africa). From the surviving San hunters in Botswana - "the oldest pattern of life found in the world today..." - , Lea and her students at the University of Botswana have collected information. Her papers describe counting, measurement, time reckoning, classification, tracking and some mathematical ideas in San technology and craft. The San developed very good visual discrimination and visual memory as needed for survival in the harsh environment of the Kalahari desert (Lea, 1987, 1989, 1990a, 1990b).


The middle and upper reaches of the Niger River have played an important role in West African history. This area in Western Sudan was a base for the camel caravan routes crossing the Sahara to the Mediterranean, while a black nation is reputed to have existed there from around the 3rd or 4th centuries A.C.
The Mali Empire flourished in the 13th century, with the city of Timbuktu on the banks of the Niger River as an intellectual, artistic and religious center. The Songhai Empire reigned in the 15th century, followed notably by the Bambara Kingdom in the 17th and 18th centuries. From the latter half of the 16th century, Mali experienced a period under Moroccan control in its north party.
In the 19th century the French army advanced into the region, making Mali a part of French West Africa from 1898 to 1960. Mali became an autonomous republic within the French Community in 1958, formed the Mali Federation with Senegal in April 1959, and gained independence in its own right on September 22, 1960.

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