ON THE HISTORY OF MATHEMATICS IN AFRICA SOUTH OF THE SAHARA*

* slightly adapted version of a paper presented by Paulus Gerdes at the Third Pan-African Congress of Mathematicians, Nairobi, 20-28 August 1991.This paper forms part of the study "Recent research on the History of Mathematics in Africa: an overview" by A.Djebbar and P.Gerdes

AMUCHMA NEWSLETTER #9

Introduction

In her classical study "Africa Counts: number and pattern in African culture" (1973a; review by Wilder, 1976), C.Zaslavsky presented an overview of the available literature on the history of mathematics in Africa south of the Sahara. She discussed written, spoken and gesture counting, number mysticism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric form, and Crowe contributed a chapter on geometric symmetries in African art. Since the publication of Zaslavsky's overview many scholars, students, teachers and laymen alike - both in Africa and abroad - have become interested in the mathematical heritage of Africa south of the Sahara. The African Mathematical Union (AMU) included a History section in the 2nd Pan-African Congress of Mathematicians (Jos, Nigeria,1986) with as one of its purposes "to encourage more reports and exchanges of references and ideas on historical studies of African mathematics" (Shirley, 1986b, 3). The success of this section stimulated the formation of the AMU Commission on the History of Mathematics in Africa (AMUCHMA). In order to stimulate research on the history of mathematics in Africa in general, and to promote the divulgation of the research findings and the exchange of information in this field, AMUCHMA has published since 1987 a newsletter in English, French and Arabic. In this paper an overview of research (findings) on or related to the history of mathematics in Africa south of the Sahara is presented. Topics like counting and numeration systems, numerology, mathematical games and puzzles, geometry, graphs, Islam and mathematical development, international connections, and history of mathematics curricula will be included. Attention will also be paid to the objectives of research in the history of mathematics in Africa, to methodology, to the relationship with ethnomathematical research and to the uses of research findings in mathematics education. Some possible directions for further research will be identified.

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).

Numeration systems

Zaslavsky's discussion of written, spoken and gesture counting and numeration systems is primarily based on Almeida (1947: Guine Bissau), Armstrong (1962: Yoruba, Nigeria), Atkins (1961), Delafosse (1928), Herskovits (1939: Kru, Liberia -Ivory Coast), Lagercrantz (1968: tally-systems), Mann (1887: Yoruba), Mathews (Northern Nigeria), A.V.O. (1964: Hima, Uganda), Raum (1938), Schmidl (1915), Thomas (1920), Torrey (1963), Williamson (1943: Dabida, Kenya). In the meantime other sources came also to the fore, like Seidenberg (1959, 1963, 1976), Santos (1960: Tchokwe, Angola), Hazoume (1983: Gun, Gen and Bariba). During the last years, a whole series of research projects on spoken and written numeration systems in Africa is being carried out, e.g. On:

* counting in traditional Ibibio and Efik societies (I.O.Enukoha, University of Calabar, Calabar, Nigeria);

* numeration among the Fulbe (Fulani) (S.O.Ale, Ahmadu-Bello-University, Bauchi, Nigeria);

* pre-Islamic ways of counting (Y.Bello, Bayero University, Nigeria);

* counting in Nigerian languages (Ahmadu-Bello-University, Zaria; cf. Shirley, 1988b);

* pre-colonial numeration systems in Burundi (J.Navez, University of Burundi, Bujumbara);

* learning of counting in Côte d'Ivoire (cf. Zepp, 1983c);

* numeration systems used by the principal linguistic groups in Guinea (S.Oulare, University of Conakry);

* counting among the various ethnic groups in Kenya (J.Mutio, Kenyatta University, Nairobi);

* traditional counting in Botswana (H.Lea, University of Botswana, Gaberone);

* numeration systems and popular counting practices in Mozambique (Higher Pedagogical Institute, Maputo / Beira; cf. Soares, 1991).

A important study - from the point of view of its contents and the methodological debate it initiates - is E.Kane's doctoral dissertation (1987) on "The spoken numeration systems of west-atlantic groups and of the Mandé". Kane (Cheik-Anta-Diop-University, Dakar-Fann, Senegal) analyses numeration in about twenty languages spoken in Senegal. He realised the necessity of basing his research on ethnomathematics, trying to understand mathematical ideas in relationship to the general culture in which they are embedded. Therefore he did preparatory research in four domains: African linguistics, history of numeration systems, works of Africanists and African languages spoken in Senegal (as understood by interviewing many speakers of the same and different languages). He shows that spoken numeration systems, like the one of the Mandé, are susceptible to reform and evolution. Kane develops a methodology for the analysis of numeration systems that is adapted to the specificities of 'oral cultures'.

Number symbolism

Zaslavsky dedicated a chapter to number symbolism, superstitions and taboos on counting (1973, 52-57; cf. Williamson & Timitimi (1970, Ijo, Nigeria)). Vergani (1981) wrote a Ph.D.thesis on number symbolism among the Tchokwe of Angola (see below: networks). Ojoade (1988) published a paper on the number 3 in African lore, highlighting the sacredness, mysticism and taboos attached to it (Cf. Also Nicolas, 1968). In Page (1987) objects of African art, mostly from the Yoruba (Nigeria) are analysed in function of the involved repetitions. The twofold objects evoke the most usual dichotomies: good/bad, life/death; the threefold objects evoke sometimes a hierarchy; the fourfold objects may be associated with the directions in space. Probably by searching systematically the ethnographical literature as well as romances, biographies, etc. A lot more information on number symbolism in African cultures may be found. For instance, the anthropological study of Thornton explains the significance of the number 9 among the Iraqw of Tanzania (1980, p.96, 167, 183). Number symbolism may have a rational basis. E.g. Makua basket makers in northern Mozambique call odd numbers or odd quantities of plant strips 'ugly', and they have good reasons to do so (Ismael, 1991; cf. Earlier discussion of 'even' and 'odd' numbers in basketry in Gerdes, 1985a). Certainly, further collecting of oral data may throw new light on African numerology.

Riddles and puzzles

Zaslavsky (1973, 109-110) presents a riddle from the Kpelle (Liberia) about a man who has a leopard, a goat, and a pile of cassava leaves to be transported across a river, whereby certain conditions have to be satisfied: The boat can carry no more than one at a time, besides the man himself; the goat cannot be left alone with the leopard, and the goat will eat the cassava leaves if it is not guarded. How can he take them across the river? Ascher (1990) places this river-crossing problem in a cross-cultural perspective and analyses mathematical-logical aspects of story puzzles of this type from Algeria, Cape Verde Islands, Ethiopia, Liberia, Tanzania and Zambia. More difficult to solve is an 'arithmetical puzzle' from the Valuchazi (eastern Angola and northwestern Zambia), recorded and analysed by Kubik (1990): "This..dilemma tale is about three women and three men who want to cross a river in order to attend a dance on the other side. With the river between them there is a boat with the capacity for taking only two people at one time. However, each of the men wishes to marry all the three women himself alone. Regarding the crossing, they would like to cross in pairs, each man with his female partner, but failing that any of the other men could claim all the women for himself. How are they crossing?" (Kubik, 1990, 62). In order to solve the problem or to explain the solution, auxiliary drawings are made in the sand. Fataki (1991) describes riddles he learnt as a child in Uganda.

Art and symmetries (see also below: cross-cultural psychology)

Njock (University of Yaoundé, Cameroon) characterises the relationship between African art and mathematics as follows: "Pure mathematics is the art of creating and imaginating. In this sense black art is mathematics" (1985, 8).

Mathematicians have analysed mostly symmetries in African art. Symmetries of repeated patterns may be classified on the basis of the 24 different possible types of patterns which can be used to cover a plane surface (cf. The so-called 24 plane groups due to Federov, 1891). Of these, seven admit translations in only one direction and are called strip patterns. The remaining 17 that admit two independent translations are called plane patterns. In chapter 14 of (Zaslavsky, 1973), Crowe applies this classification to decorative patterns that appear on the raffia pile cloths of the Bakuba (Zaire) (Cf. Crowe, 1971), on Benin bronzes, and on Yoruba adire cloths (Nigeria), showing that all seven strip patterns occur and many of the plane patterns. Crowe continued this research and published a catalog of Benin patterns (Crowe, 1975) and a symmetry analysis of smoking pipes of Begho (Ghana) (Crowe, 1982a; cf. Also Crowe, 1982b). In Washburn and Crowe (1988) a number of patterns from African contexts are classified in the same way. Recently Washburn (1990, ch.5) showed how a symmetry analysis of the raffia patterns can diferentiate patterns produced by the different Bakuba groups. Although the use of the crystallographic groups in the analysis of symmetries in African art attests and underlines the creative imagination of the artists and artisans involved and their capacity for abstraction (cf. Meurant, 1987), these studies do not focus on how the artists and artisans themselves classify and analyse their symmetries. This is a field open for further research. Zaslavsky (1979) gives some examples of strip and plane patterns, and of bilateral and rotational symmetries, occurring in African art, architecture and design. Why do symmetries appear in human culture in general, and in African craftwork and art, in particular? This question is addressed by Gerdes in a series of studies. He analyses the origin of axial, double axial, and rotational symmetry of order 4 in African basketry (Gerdes, 1985a, 1987, 1989a, 1990c, 1991c). In (Gerdes, 1991b) it is shown how fivefold symmetry emerged quite "naturally" when artisans were solving some problems in (basket)weaving. The examples chosen from Mozambican cultures range from the weaving of handbags, hats, and baskets to the fabrication of brooms.

Langdon (1989, 1990) describes the symmetries of 'adinkra' cloths (Ghana) and explores possibilities for using them in the classroom. In a similar perspective, Harris (1988) describes and explores not only the printing designs on plain woven cloths from Ghana, but also symmetries on baskets from Botswana and 'buba' blouses from the Yoruba (Nigeria).

Games

Among the games with mathematical 'ingredients' referred to in (Zaslavsky, 1973, 102-136) are counting rhymes and rhythms, three-in-a-row-games, arrangements, games of chance and board games. Zaslavsky (1982) gives more information on three-in-a-row-games in Africa. Russ (1984; review by Crowe, 1987) presents the rules and a brief history of 'mancala' games (cf. Townshend, 1979), also known as Ayo, Bao, Wari, Mweso, Ntchuva, etc. Zaslavsky (1989, AMUCHMA3, p.6) suggests that it could be important in the reconstruction of the history of mathematical thinking in Africa to investigate further mathematical aspects of traditional games. As a starting point, she indicates, along with Russ (1984), the following publications: Béart (1955), Centner (1963), Driedger (1972), Klepzig (1972), and Pankhurst (1971). Béart's "Games of West Africa" has been reviewed by Doumbia (1989a). To this list, Nsimbi's "Omweso: a game people play in Uganda" (1968), Mizoni's "Strategic games in Cameroon and their mathematical aspects" (1971), Deledicq & Popova's "Wari and solo, the African calculation game" (1977), Ballou's "Rules and strategies of the awélé game" (1978) and Crane's "African games of strategy" (1982) may be added. Crane informs about some of the most common types of African games involving strategy and mathematical principles, like games of alignment (Shisima (Kenya), Achi (Ghana), Murabaraba (Lesotho)), 'struggle-for-territory' games (Sega (Egypt), Kei (Sierra Leone), and 'Mancala' games, both two-row versions (Oware (Ghana), Awélé (Ivory Coast), Ayo and Okwe (Nigeria)) and four-row versions (Omweso (Uganda), Tshisolo (Zaire)). Bell and Cornelius (1988) give some information on Achi (Ghana), Dara (Nigeria), Sega (Egypt) and on 'Mancala' games. Retschitzki (1988) and N'Guessan (1988) analyse the learning of strategies and tactics of the 'awélé' game. Important is the research of Doumbia (1989b) and her colleagues at the Mathematical Research Institute of Abidjan (Ivory Coast). Their work on traditional African games deals with classification, solution of mathematical problems posed by the games and analysis of the possibilities of using these games in the mathematics classroom. Their conclusion - as revealing as it is - that rules of some games, like Nigbé Alladian, show a traditional, at least empirical knowledge of probabilities, will certainly stimulate further research. Vergani (Open University, Lisbon) prepares a monograph on mathematical aspects of intellectual games in Angola. Mve Ondo (Omar-Bongo-University, Gabon) published recently (1990) a study on two 'calculation games', i.e. The 'Mancala' games, Owani (Congo) and Songa (Cameroon, Gabon, Equatorial Guinea). The possible relationship between visual memory and concentration as necessary for success in many African games (cf. Paul, 1971) and the development of mathematical ideas also deserves further attention.

Geometry and architecture

Chapter 13 of (Zaslavsky, 1973a, 155-171) (Cf. Zaslavsky, 1989) is dedicated to geometric form in architecture. More information on the geometric shapes and on the ornamentation of traditional African buildings may be found in (Denyer, 1978). (Anon., 1987) presents a bibliography on African architecture. Prussin calls attention to the fact that in West Africa the mathematician-scholar and the architectural design-builder might often be the same person (1986, 208). She refers to the relationship between magic squares and the structure of domes and remarks that "a number of 'adinkra' [Ashanti, Ghana] stamp patterns directly associated with Islam were also used in the architectural setting" (Prussin, 1986, 240). Rohrman (1974) and Matthews (1974) describe house decoration and mural painting in southern Africa. A publication of NTTC (1976) gives a catalog of geometric patterns used on house walls in Lesotho. These studies may serve as a starting point for further research on geometry and ornamentation of buildings. Eglash and Broadwell (1989) are interested in possible relationships between modern fractal geometry and traditional (knowledge about) settlement patterns in Africa. Gerdes (1985a) describes the geometrical know-how used in laying out the circular or rectangular house plans in Mozambique. His student Mahanjane (1989) uncovers geometrical knowledge applied in the construction of traditional granaries for maize and beans in the Gaza province in southern Mozambique.

Uncovering 'hidden' mathematical ideas: geometrical form

Many 'mathematical' ideas and activities in African cultures are not explicitly mathematical. They are often intertwined with art, craft, riddles, games, graphic systems, and other traditions. The mathematics is often 'hidden'. How may this 'hidden' knowledge be uncovered? And as some traditions are nowadays (becoming) obsolete, this 'uncovering' often also means a tentative reconstruction of knowledge as it existed in the past. Gerdes (1985a) explores the concept of 'hidden' mathematics and develops some methods in order to 'uncover' and reconstruct 'hidden' geometrical thinking (cf. Also Gerdes, 1986a, 1986b, 1987, 1990c, 1990e). One of these methods may be characterised as follows: when analysing the geometrical forms of traditional objects - like baskets, mats, pots, houses, fishtraps - the researcher poses the question: why do these material products possess the form they have? In order to answer this question, the researcher learns the usual production techniques and tries, at each stage of the production process, to vary the forms. Doing this, the researcher observes that the form generally represents many practical advantages and is, most frequently, the only possible or the optimal solution of a production problem. By applying this method, it becomes possible to bring to the fore knowledge about the properties and relations of circles, angles, rectangles, squares, regular pentagons and hexagons, cones, pyramids, cylinders, symmetry, etc., that was probably involved in the invention of the production techniques under consideration (see Gerdes, 1985a, 1990c).

Networks, graphs or 'sanddrawings'

One section of Zaslavsky (1973a, 105-109) was dedicated to networks, based on Torday's information (1925) on the Bushongo (actual Zaire) and Bastin's study (1961) of decorative art of the Tchokwe (Angola) (For educational use of Zaslavsky's analysis of the Bushongo networks, see NCTM (1984) and Whitcombe & Donaldson (1988)). She had not had access to the ethnographical information on such networks published by Baumann (1935, 222-223), Hamelberger (1952) and Dos Santos (1961). Since the publication of "Africa Counts" large ethnographical collections of networks have become available: Pearson (1977: 'sandgraphs' observed in the 1920s in the Kwandu-Kuvanga and Muxiku provinces of Angola); Fontinha (1983: 'sona' or 'sanddrawings' collected principally among the Tchokwe of northeastern Angola during the 1940s and 1950s); and Kubik (1986, 1987a, 1987b, 1988: networks observed among the (Va)luchazi in northwestern Zambia during the 1970s). In order to facilitate the memorisation of their standardised 'sona', the drawing experts used the following mnemonic device. After cleaning and smoothing the ground, they first set out with their fingertips an orthogonal net of equidistant points. Now one or more lines are drawn that 'embrace' the points of the reference frame. By applying their method the drawing experts reduce the memorisation of a whole drawing to that of mostly two numbers (the dimensions of the reference frame) and a geometric algorithm (the rule of how to draw the embracing line(s)). Most drawings belong to a long tradition (cf. Redinha, 1948). They refer to proverbs, fables, games, riddles, animals, etc. And play an important role in the transmission of knowledge and wisdom from one generation to the next. In Kubik's view the 'sona' "transmit empirical mathematical knowledge" (1987a, 450). The geometry of the 'sona' is a "non-euclidean geometry": "The forefathers of the Eastern Angolan peoples discovered higher mathematics and a non-Euclidian geometry on an empirical basis applying their insights to the invention of these unique configurations" (Kubik, 1987b, 108). He calls attention to the symmetry of many 'sona', the implicit rules for construction and rules for anchoring figures of the same type. The ethnographical publications of collections of 'sona' drew the attention of mathematicians. Ascher and Gerdes conducted research on the 'sona', independently one of another. Ascher's study (1988, 1991) deals with geometrical and topological aspects of 'sona', in particular, with symmetries, extension, enlargement through repetition, and isomorphy. Gerdes (1989a, 120-189) analyses symmetry and monolinearity (i.e. A whole figure is made up of only one line) as cultural values, classes of 'sona' and corresponding geometrical algorithms for their construction, systematic construction of monolinear groundpatterns, chain and elimination rules for the construction of monolinear 'sona'. It is suggested that the 'drawing experts' who invented these rules probably knew why they are valid, i.e. They could prove in one or another way the truth of the theorems that these rules express. He advances also with the reconstruction of lost symmetries and monolinearities by means of an analysis of possible drawing errors in reported 'sona' (for an introductory summary of his research findings, see: Gerdes, 1990d, 1991d and 1991e). Inspired by his historical research findings, Gerdes experimented with possibilities to use the 'sona' in mathematics education, in order to value and revive a rich scientific tradition that had been vanishing (Cf. Gerdes, 1988a, b; 1989a, b, c; 1990a; 1991a and g; cf. Ratteray, 1991). He also initiated a mathematical exploration of the properties of some (extended) classes of 'sona' (see Gerdes, 1989a, 288-297). In a similar way, Kubik's research stimulated a mathematical investigation by Jaritz on a particular class of 'sona' (1983).

Monolinear patterns appear also in other African contexts. For instance, Prussin (1986, 90) displays a symmetric monolinear pattern on a Fulbe warrior's tunic from Senegal. The study of the types and spread of monolinear patterns throughout the African continent deserves further research.

Example of an as yet unexplored area

One of the many areas of African culture which have not been studied as yet in view of their inherent mathematical aspects, is that of string figures. Elsewhere such an analysis has already begun: De Paula & De Paula (1988) studied the geometry of string figures among the Tapirapé Indians in Brazil; Moore (1986) analysed string figures from the Navaho and other North American Indians and explored their potential for mathematics education. In the case of Africa south of the Sahara, Wedgwood & Schapera's information (1939) on string figures of Botswana may serve as a starting point.They refer to studies on string figures in Central Africa, Liberia, Nigeria, Sierra Leone, South Africa and Tanzania.

In general, once the mathematical character or aspects of cultural elements are recognized, one may try to track the history of the mathematical thinking involved and its (possible) relationships to other cultural-mathematical 'threads' and try to explore their educational and scientific potential.