Pacific Resources for Education and Learning logo Originally published in the International Study Group on Ethnomathematics (ISGEm) Newsletter, Volume 6, Number 2, July 1991. Located at: http://web.nmsu.edu/~pscott/isgem62.htm.
Article reproduced 2003 with permission of the ISGEm Newsletter editor for use in the Ethnomathematics Digital Library (www.ethnomath.org) developed by Pacific Resources for Education and Learning (www.prel.org).

Recent Ethnomathematical Research in Mozambique

Most "mathematical" traditions that survived colonization and most "mathematical" activities in the daily life of the Mozambican people are not explicitly mathematical. The mathematics is "hidden." The first aim of the project "Ethnomathematics in Mozambique" is to "uncover" this "hidden" mathematics. As some traditions are nowadays rather obsolete, the "uncovering" often means also a tentative reconstruction of past knowledge.

In our study On the Awakening of Geometrical Thinking (1985) and our book Ethnogeometry: Cultural-Anthropological Contributions to the Genesis and Didactics of Geometry (concluded 1986, published 1990) some anthropological research methods were developed in order to "uncover" and reconstruct "hidden" mathematical thinking (cf. also [1]). The basic method then proposed for recognizing implicit mathematics may be characterized as follows: When analyzing 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?

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, quite a lot of times, the only solution of a production problem. Applying this method in the period 1986-1990, new results have been obtained. Abdulcarimo Ismael (Department of Mathematics, Higher Pedagogical Institute, Maputo) did in 1989 fieldwork in the northern Mozambican province of Nampula. In his provisional report, he reveals interesting aspects of the (implicit) mathematical knowledge displayed by basketweavers. During our stay as a visiting professor at the State University of Sao Paulo (UNESP, Rio Claro, April-May 1988)--lecturing a postgraduate course on ethnomathematical research methods--we collected a series of Amer-Indian baskets and initiated their analysis.

It came out that to guarantee the beautiful, symmetrical wall ornamentation, the artisans had to use (and develop) arithmetical tools like multiplication and to know some of their properties like commutivity (see Chapter 3 in [2]. In two research papers 0n Ethnomathematical Research and Symnmetry (Chapter 2 in [2]) and Fivefold Symmetrv and (basket) Weaving in Various Cultures, we explain why basketweavers "prefer" certain symmetries.

As this method for recognizing "hidden" mathematics had been developed in the context of analyzing material production, like that of baskets, mats, pots, houses and fishtraps, the question of the possibility of extending the method to other spheres of production--such as artistic and/or symbolic production--had to be posed (objective 1), in view of the success of the method in the first field.
On analyzing, by the same method, spiral ornaments on the walls of old Egyptian tombs, it came out that ancient Egyptian artisans probably might have known how to construct a square equal in area to the sum of the areas of two given squares, which could have led to the discovery of the so-called Theorem of Pythagoras (see Chapter 4 in [2]).

Then we tried to apply the method to the analysis of traditional African and Asian designs, in particular the Tchokwe sand drawings [Angola, with relationship to the Luchazi (Zambia) and Makonde (Mozambique) graphic traditions] and the--from a technical point of view- related Tamil (South India) threshold designs. It came out that the aforementioned method for recognizing "hidden" mathematical thinking, as such, was not immediately applicable. The method had to be adapted and "refined." Instead of starting by posing the question why the (material) products possess the form they have, the researcher had first of all to ask "which are the cultural values that lay at the basis of the drawing tradition?" and only then, in view of these underlying cultural standards, to pose the question "why do these drawings possess the 'form' they have?".

Both Tchokwe and Tamil traditions are similar in the sense that the drawers use the same mnemonic device for the memorization of their standardized pictograms. After cleaning and smoothing the ground they first set out an orthogonal net of (equidistant) points. Then the curves are drawn in such a way that they surround the dots without touching them. Many such traditional Tamil threshold designs are "monolinear," i.e. made out of one closed, smooth line. In Reconstruction and Extension of Lost Svmmetries: Examples From the Tamil of South India (Chapter 6 in [2]), there is an investigation of a series of Tamil patterns which do not conform to their cultural standard, as they are composed of two, three or more superimposed closed paths. An analysis of possible construction errors shows that these "polylinear" designs are probably "degraded" versions of originally monolinear patterns.

Furthermore, it became possible to reconstruct these original patterns and to make explicit some of the geometrical knowledge of their inventors (transformation roles, geometrical algorithms, extension and generalization). The success obtained in developing the adapted and "refined" method (objective 2) for recognizing "hidden" mathematics and in applying it to the Tamil designs, stimulated its application in other contexts such as the Tchokwe sand drawings.

With the colonial penetration and occupation, the Tchokwe sand drawing tradition has been disappearing. Our analysis of the sand drawings that have been reported by missionaries and ethnographers, shows how symmetry and monolinearity played an important role as cultural values in this tradition. We succeeded (objective 1) in reconstructing classes of Tchokwe sand drawings that had been lost over time and in showing that the Tchokwe drawing experts had invested general construction rules and had discovered "theorems" about transformation rules, algorithms, dimensions and rules for the chaining of monolinear patterns to bigger monolinear patterns. The first results have been included in [2, p.120-189] and have been extended in [3] and [4].

It had been suggested by us that the origin of the mnemonic technique used in the Tchokwe and Tamil drawing tradition lies probably in weaving and as some of their designs may be characterized as plaited-strip-patterns ([3], p.7), we looked for such patterns in other cultural contexts. In Chapter 8 of our Ethnomathematical Studies [2], p. 190-209, (in German) we present the first results of this excursion:

In On Culture, Geometrical Thinking and Mathematics Education (Chapter 9 in [2]) we summarized our experimentation (until 1987) with the incorporation of traditional African cultural elements into mathematics education (objective 3). The paper confronts a widespread prejudice about mathematical knowledge, that mathematics is "culture- free," by demonstrating alternative constructions of euclidean geometrical ideas developed from the traditional culture of Mozambique. As well as establishing the educational power of these constructions, the paper illustrates the methodology of "cultural conscientialization" in the context of teacher training.

In A Widespead Decorative Motif and the Pythagorean Theorem (Chapter 10 in [2]), we gave concrete examples of multi-culturalizing the mathematics curriculum, using a well-known African and also Scandinavian ornament motif as a starting point for doing and elaborating mathematics in the classroom. At the same time it is shown that there exists an infinity of (new) proofs for this theorem (cf. objective 4.1 and 4.4; see also our paper How Many Proofs of the Pythagorean Proposition do There Exist?, published in Sweden). In Chapter 11 of [2] we relate our first reflections on the possibilities of using the Tchokwe sand drawings in the mathematics classroom. The examples given in this paper range from the study of arithmetical relationships, symmetry, similarity, and Euler graphs to the determination of the greatest common divisor of two natural numbers.

Later on, a reflection on the results obtained in the historical reconstruction of the above-mentioned Tamil and Tchokwe designs and on the geometrical algorithms involved led to the formulation of a first series of geometric problems of the type Find the Missing Figures (Published also in the Swedish journal Namnaren).

In 1988 and 1989 we conducted further didactical experiments and concluded in early 1990 a book with problems of this type, entitled Lusona: Geometrical Recreations From Africa (English version [5] and Portuguese version [6]).

Many--both reported and reconstructed--Tchokwe drawings are aesthetically appealing and the analysis of the geometric algorithms involved stimulated their generalization and the invention of new patterns. In Examples of Algorithms and Monolinear Motifs Inspired by the Tchokwe Sona (Chapter 4 in [3], in [5] and in Pickover's The Pattern Book: Recipes of Beauty) we present some beautiful designs we found in this context.

The study of the mathematical potential (cf. objective 4.4) of the traditional Tchokwe designs and of their generalizations constitutes a new and attractive area of mathematical research. Already in 1987 we were stimulated by an analysis of a class of Tchokwe drawings to discover A Physical Model for the Determination of Prime Numbers (Chapter 14 in [2]). In 1988 we found that a whole class of Tchokwe ideograms satisfy a common construction principle. An analysis of all possible curves that satisfy the same construction principle, led to the discovery of some theorems, proved in 1989. The proofs are included in Chapter 15 of [2] and explained to a broader public in [7] and in Chapter 5 of [3]. Early 1990 we summarized our historical, educational and mathematical results in a manuscript for a book, entitled Geometry of the African Sona: History. Education. Recreation. Art Design ([3]). At the end of the introduction to this book, we summanze:
"The study of the Tchokwe drawing tradition, threatened with extinction during the colonial period, is not only interesting for historical reasons. The incorporation of this sona tradition in the curriculum, both in Africa and in other parts of the world, will contribute to the revival and valuing of the old practice of the sona experts, and will reinforce the comprehension of the value of the artistic and scientific heritage of Africa. It may contribute to the development of a more productive, more creative and multicultural mathematics education. Furthermore, an analysis of the Tchokwe patterns stimulates the development of new mathematical research areas."

References

[1] Paulus Gerdes, How to recognize hidden geometrical thinking? A contribution to the development of anthropological mathematics: For the Learning of Mathematics, Montreal, 1986, Vol.6, No.2, p.10-12, 17
[2] Paulus Gerdes, Ethnomathematische Studien Dr.Sc.nat.thesis, Leipzig, 1989, 360 p.
[3] Paulus Gerdes, Geometric of the African "sona": History, Education, Recreation, Art Design, 1990, 130p.
[4] Paulus Gerdes, Geometria dos "sona" africanos : historia, educacao, recreacao, desenho artistico, 1990, 130.
[5] Paulus Gerdes, Lusona : Geometrical Recreations from Africa,
[6] Paulus Gerdes, Lusona : Recreacoes geometricas de Africa, 1990, 120 p.
[7] Paulus Gerdes, On mathematical elements in the Tchokwe "sona" tradition to be published in :For the Learning of Mathematics, Montreal, 1990 11 p.