Africa Counts: Number and Pattern in African Culturesby Claudia Zaslavsky
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This fascinating study of mathematical thinking among sub-Saharan African peoples covers counting in words and in gestures; measuring time, distance, weight, and other quantities; manipulating money and keeping accounts; number systems; patterns in music, poetry, art, and architecture; and number magic and taboos. African games such as mankala and elaborate versions of tic-tac-toe show how complex this thinking can be. An invaluable resource for students, teachers, and others interested in African cultures and multiculturalism, this third edition is updated with an introduction covering two decades of new research in the ethnomathematics of Africa.
“Anyone who reads this fine book will never again view the African peoples in quite the same light.” —The New Yorker
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Number and Pattern in African Culture
By Claudia Zaslavsky
Chicago Review Press IncorporatedCopyright © 1973 Claudia Zaslavsky
All rights reserved.
Before the time of Newton, mathematics and science had little impact on the daily lives of people. Since then mathematics and technology have worked hand in hand to bring about radical innovations in our life style. In an "advanced" country like the United States, the greatest portion of the federal budget is now spent on products whose existence depends on higher mathematics — military hardware and equipment for space travel. People are constantly being urged to purchase more cars, more electronic devices, more material goods manufactured by precisely designed machinery. The computer has taken over many jobs of office workers, technicians, factory operators and teachers. Technology, made possible by mathematics, has become the new god and the possession of its products the goal of life. Yet, the ways of this new god remain as mysterious to the uninitiated citizen of the modern world as were those of the ancient deities.
To the average person, mathematics is a subject that deals with numbers in counting and the computations of elementary arithmetic. I have had my students tell me, "I could do the proof all right, but I made a mistake in the math. I added two and three, and got six!" So, when I tell people that I am doing research in African mathematics, they often ask: "How do Africans count — like us?" I posed this question to two young colleagues, one from Kenya and the other from Tanzania. "How do you count in your own language?" They looked at me in surprise, and replied, "Just as you do here in the United States!" Nowadays school children in most of the world write Hindu numerals (popularly called "Arabic") — truly universal symbols.
The Sociomathematics of Africa
For many mathematicians, mathematics as a discipline originated in ancient Greece with the formulation of logical systems based on definitions, postulates and formally proved theorems. A more flexible definition — the British science writer, Lancelot Hogben calls it a "provisional formula" — is offered in his Mathematics in the Making (page 9): "Mathematics is the technique of discovering and conveying in the most economical possible way useful rules of reliable reasoning about calculation, measurement, and shape."
The last part of Hogben's provisional formula describes the subject matter of this book; it deals with "calculation, measurement, and shape." It would be difficult to discuss "discovery", since few records exist. "Conveying" was accomplished chiefly by word of mouth and by example, from one generation to the next. "Useful" is the key word; I shall deal with mathematics entirely from the point of view of its applications, and not as an abstract body of thought.
It is true that Africans, to our present knowledge, have only recently begun to participate in the development of pure mathematics; possible reasons will be suggested in the concluding chapter. But mathematics is in evidence in many aspects of African life. This book is concerned with the "sociomathematics" of Africa — the applications of mathematics in the lives of African people, and, conversely, the influence that African institutions had upon the evolution of their mathematics.
Geographically, this study is confined to that part of Africa which lies south of the Sahara, an area comparatively isolated from the Mediterranean, Europe, and Asia. We are concerned with a vast region extending nearly four thousand miles from north to south and over four thousand miles from east to west. Within these boundaries exist a variety of cultures and a corresponding diversity of mathematical developments.
The development of a number system depends upon need. In a small, self-contained economy in which all or most of the necessities of life are produced within the community — typical of large sections of Africa — there is little need for an extensive reckoning system. The names of numbers are frequently connected with the objects to be counted, just as we have special names for certain sets — flock, herd, brace, etc., dating back from a pastoral or agricultural society. Gesture-counting is especially necessary in the market place, where people speaking various languages gather to exchange goods. It may be customary to use beads, shells, nuts, or pebbles (the word calculate is derived from calculus, Latin for pebble) as media of exchange or counting materials, and to arrange them in sets, thus giving rise to special words.
Systems of numeration range from the few number words of some San peoples, who have been pushed into the least hospitable areas of the continent by advancing black and white populations, to the extensive numerical vocabulary of nations having a history of centuries of commerce. A characteristic of African counting is a standardized system of gestures to accompany, or even replace, the number words. The gesture languages show as much variety as the spoken ones.
Mystical beliefs about numbers are many and varied. Certain numbers are deemed to have special significance, favorable or unfavorable. In Chinua Achebe's novel Arrow of God the medicine-man counted the cowrie shells "carefully on the ground as a woman would before she bought or sold in the market in groups of six. There were four groups and he nodded his head" (p. 147). The Igbo were unique in counting cowries in units of six, and the number four has particular significance for them.
In many societies it was the use of cowrie shell currency which created both the need for number names in the higher denominations and special categories of numbers. The demands of commerce also dictated the degree of standardization of weights and measures. The villager with the longest arm sometimes set the standard for measuring cloth! Record keeping varied from knotted strings and notched sticks to the ritual of the annual census, carried out by indirect methods that circumvented the taboo on counting living creatures.
The ability to observe and reproduce patterns, both numerical and geometrical, is of great importance in Africa, where most societies have been nonliterate until recently. The Bushman in the Kalahari Desert walks miles to dig up a watery root whose location he had noted several months previously — and with no man-made markers to guide him! Cattle-herding folk have in their vocabularies dozens of words to describe their livestock on the basis of hide markings and dozens more to differentiate cattle by the shape of the horns. Each pattern in weaving, in wood carving, in cloth dying, has a special meaning. Numerical patterns are evident in games, from versions of tic-tac-toe to the universal African board game in which seeds or pebbles are transferred and captured. The scholars of Muslim West Africa associated astrology and numerology with arrays of numbers called magic squares.
African art is varied and complex; geometric patterns and symmetries appear in the beautifully carved and painted masks associated with religious practices and in the decoration of such common household objects as gourds and baskets. Patterned mats and carved wooden doors embellish the home in some cultures; in others a round thatched cottage provides temporary shelter until the herds of cattle move on to greener pastures. Geometry points to a social distinction in the Kikuyu village of James Ngugi's The River Between, where the rectangular house of the one converted Christian is conspicious among the round homes of his neighbors.
The general discussion of African numerical and geometric concepts in this book will be followed by studies in detail of southwest Nigeria and East Africa. These will enable the reader to observe closely the development of mathematical concepts in two areas having quite different historical and cultural backgrounds. The Yoruba people and the related people of Benin, in Nigeria, have been urbanized farmers and traders for centuries and have a complex numeration system noted by many investigators. East African society is varied — pastoral and agricultural, stateless and highly centralized — with cattle the mainstay of the economy.
The book concludes with a discussion of the disruptions caused by the slave trade and colonialism during the past five centuries, and their disastrous effect upon the potential development of science and technology in Africa.
Africa's Place in Writings on the History of Mathematics
Africans have made significant contributions to the development of counting and numbers and deserve a place in studies dealing with this subject. But what do we find? Even recently published books suffer from inadequacy of material, exhibit an outdated point of view, and repeat incorrect information.
Until the nineteenth century, Europeans had few contacts with Africa other than commerical dealings and the slave trade. In the late nineteenth century, stimulated in part by missionaries and commerical adventurers, the European countries engaged in a scramble for African territory. The continent was divided among the powers, who sent in administrators to set up colonial governments. They were soon joined by anthropologists. In a short time numerous studies became available of the many ethnic groups, explaining their customs, religion, dress, language, art — every facet of life. It was necessary to describe their numeration systems; how else could the colonial administrations employ Africans for wages and levy taxes upon them? Much valuable descriptive material became available to European and American readers.
In Great Britain there arose a school of anthropologists, led by E. B. Tylor, having a point of view based on their interpretation of the new doctrine of evolution. Their thesis was this: man evolved from a primitive to an advanced state during the course of many millennia. The white man had arrived at the highest level, in contrast to the "primitive savages" of "Darkest Africa," who were still in the very early stages of evolution. Tylor's Primitive Culture became the leading reference work for anthropologists, ethnologists, and even for writers of the history of mathematics. Using Tylor as his source, the mathematical historian Florian Cajori wrote in 1896 (page 3):
Of the notations based on human anatomy, the quinary and vigesimal systems are frequent among the lower races, while the higher nations have usually avoided the one as too scanty and the other as too cumbrous, preferring the intermediate decimal system.
The quinary and vigesimal systems are common in Africa!
L. L. Conant's The Number Concept, published in 1896, is one of the few books in English which discuss the numeration systems of many African peoples. However, Conant's point of view is completely colored by the prevailing attitude toward Africans as "primitive savages"; they were deemed hardly human. He dismisses the amazingly complex numeration system of the Yoruba people of southwest Nigeria with these words (page 32):
Nor on the other hand, is the development of a numeral system an infallible index of mental power, or of any real approach toward civilization. A continued use of the trading and bargaining faculties must and does result in a familiarity with numbers sufficient to enable savages to perform unexpected feats in reckoning. Among some of the West African tribes this has actually been found to be the case; and among the Yorubas of Abeokuta the extraordinary saying, "You may seem very clever, but you can't tell nine times nine," shows how surprisingly this faculty has been developed, considering the general level of savagery in which the tribe lived.
Conant sees the occurrence of numbers up to a million among South African people as "remarkable exceptions" to the "law" that "the growth of the number sense keeps pace with the growth of the intelligence in other respects" (p. 33). Such was the extent of the prejudice against dark-skinned peoples that he turned all the principles of logic upside down in contradiction of the scientific method. One would expect that, when confronted with evidence that refutes his "law," the scientist should begin to doubt its validity or else apply his "law" to draw the logical conclusion about the high level of intelligence of his subjects.
Lévy-Bruhl's How Natives Think, originally published in French in 1910, contains excellent criticism of the Tylor school of anthropology, and of the data-gathering techniques upon which they relied (page 20):
They observed what seem to them most noteworthy and singular, the things that piqued their curiosity; they described these more or less happily. ... Moreover, they did not hesitate to interpret phenomena at the time they described them; the very idea of hesitation would have seemed quite unnecessary. How could they suspect that most of their interpretations were simply misapprehensions, and the "primitives" and "savages" nearly always conceal with jealous care all that is most important and most sacred in their institutions and beliefs?
These false interpretations were copied from one book to another, and translated from one European language to another. Lévy-Bruhl's own work was marred by his distinction between the "pre-logical" or "mystical" mentality of the "lower societies" (les sociétés inférieures), and the "logical" mental activity of civilized peoples.
This "scientific" attitude is analyzed in an unpublished manuscript, Science and Africa, by Frank E. Chapman, Jr., a brilliant young black man now serving life imprisonment. (See the Appendix for his autobiography.)
There is so much talk about a "primitive type of mind" and an "advanced type of mind." These conceptualists rarely pause to consider what it is in fact that they are talking about; they never stop to consider that this whole business of "mind types" is merely a collocation of convenient verbalizations; indeed, the scientist's behavior is very similar to the "primitive" he is talking about when he verbalizes about "mind types." This stifling "mind" concept cripples scientific analysis, and only when greater emphasis is put on psychological behavior patterns (in a given social context) will it be understood once and for all ... that the difference in psychological make-up is due, more or less, to the differences in social conditions, which have nothing whatsoever to do with "mind types."
Karl Menninger's Number Words and Number Symbols, published in 1969 in an English translation of the 1958 revised German edition, deals extensively with the development of numeration systems throughout the world. On African systems there is very little: a few number words of two unspecified languages, Francis Galton's story of the difficulty of purchasing sheep from the "primitive" Damara people, the repetitive counting of the Bushmen, a "Masai girl wearing annual rings, which show her to be 23 years old" (page 33). This caption accompanies a photograph of a young woman wearing a spiral metal collar. The text states: "Young unmarried girls of the Masai, a warlike tribe of herdsmen living on the slopes of Mt. Kilimanjaro, each year add one heavy brass ring around their necks, so that their precise age can be known from this extraordinary necklace of annual rings." Hollis, in his book The Masai, published in 1905, has the identical photo, as well as a picture of a married woman wearing a similar necklace with only three coils. He makes a clear distinction between the style of ornament worn by uncircumcised girls and those worn by women. However, I was told by Mr. Ole Kantai, a Maasai instructor at the University of Nairobi, that a coiled necklace is worn only by a married woman as a symbol of her husband's affection. I also described the Menninger interpretation to Mr. Onesmo ole Moiyoi, a young Maasai now studying at an American medical college. He wrote:
The number of rings in a Maasai girl's necklace has no bearing whatever on her age. Such a statement is, I think, a reflection of the lack of understanding regarding the mechanics of making iron necklaces. The caption implicitly points out the necessity of making a necklace once every year for each girl that wears one. This is not done.
Excerpted from Africa Counts by Claudia Zaslavsky. Copyright © 1973 Claudia Zaslavsky. Excerpted by permission of Chicago Review Press Incorporated.
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Meet the Author
Claudia Zaslavsky is a mathematics teacher and the author of Multicultural Mathematics, The Multicultural Math Classroom, Math Games and Activities from Around the World, and other books. She lives in New York City.
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