Explorations in Ethnomathematics and Ethnoscience in Mozambique

2 Paulus Gerdes (Editor)

EXPLORATIONS IN ETHNOMATHEMATICS AND ETHNOSCIENCE IN MOZAMBIQUE

2014

3 Title: Explorations in Ethnomathematics and Ethnoscience in Mozambique

Editor: Paulus Gerdes C. P. 915, Maputo, Mozambique [email protected]

First edition (1994): Ethnomathematics Research Project Instituto Superior Pedagógico (since 1995: Universidade Pedagógica) Maputo, Mozambique

Authors: 1 Mário Baloi: M.Ed. (Physics, Gustrow, Germany), Department of Physics. ISP, P. O. Box 3276, Maputo José Barros: M.Ed. (Chemistry, Gustrow, Germany), Department of Chemistry, ISP, P. O. Box 3276, Maputo Marcos Cherinda: M.Ed. (Mathematics, Gustrow, Germany), Department of Mathematics, ISP, P. O. Box 2923, Maputo Jan Draisma: M.Sc. (Mathematics, Amsterdam, Netherlands), Department of Mathematics, ISP, P. O. Box 2025, Beira Abdulcarimo Ismael: M.Ed. (Mathematics, Dresden, Germany), Department of Mathematics, ISP, P. O. Box 2923, Maputo Felisberto Lobo: M.Ed. (Biology, Gustrow, Germany), Department of Biology, lSP, P. O. Box 3276, Maputo Abílio Mapapá: M.Ed. (Mathematics & Physics, Maputo), Department of Mathematics, lSP, P. O. Box 2923, Maputo Adão Matonse: M,Ed. (Physics, Dresden, Germany), Department of Physics, ISP, P. O. Box 3276, Maputo

1 See “Where are the authors in 2014” on pages 93-96. 4 Cristiano Pires: M.Ed. (Biology, Gustrow, Germany), Department of Biology, ISP, P. O. Box 3276, Maputo Luís Ramos: M.Ed. (Chemistry, Gustrow, Germany), Department of Chemistry, ISP, P. O. Box 3276, Maputo Horácio Simão: M.Ed. (Mathematics & Physics, Maputo), Department of Physics, ISP, P. O. Box 2923, Maputo Daniel Soares: M.Ed. (Mathematics, Gustrow, Germany), Department of Mathematics, ISP, P. O. Box 2025, Beira

Linguistic revision: Jill Gerrish & Joanna Smith: Department of English, ISP, Maputo

Second edition (2014): Faculty of Natural Sciences and Mathematics, Pedagogical University, Maputo, Mozambique

International distribution (Printed version and eBook): www.lulu.com/spotlight/pgerdes

Map of Mozambique

6 Contents

page Map of Mozambique 6 0 Preface (Paulus Gerdes) 9 1 On the Origin of the Concepts of “Even” and 13 “Odd” in Makhuwa Culture (Abdulcarimo Ismael) 2 Mathematical-educational exploration of 21 traditional basket weaving techniques in a children’s “Circle of Interest” (Marcos Cherinda) 3 Popular counting methods in Mozambique 31 (Daniel Soares & Abdulcarimo Ismael) 4 How to handle the theorem 8+5=13 in (teacher) 39 education? (Jan Draisma) 5 Symmetries and metal grates in Maputo – 63 Didactic experimentation (Abílio Mapapá) 6 Symmetric ornamentation on wooden spoons 71 from Sofala Province – a study in progress (Daniel Soares) 7 Strip patterns on wooden spoons from 75 Inhambane Province – a study in progress (Marcos Cherinda) 8 Ideas about Nature: Traditional Interpretations 79 of Thunder and Lightning in Catembe and Physics Teaching (Mário Baloi) 9 Perspectives in Ethnophysics (Mário Baloi, 87 Adão Matonse & Horácio Simão) 10 Perspectives in Ethnobiology (Cristiano Pires 91 & Felisberto Lobo) 11 Perspectives in Ethnochemistry (José A. Barros 95 & Luís Ramos) Where are the authors in 2014? 99

7 Cover of the 1994 edition

8 and Ethnoscience in Mozambique

Paulus Gerdes

Prefácio

Three important publications on the challenges to the South in general and to education in Africa in particular, appeared in 1990: * “The challenge to the South”, The Report of the South Commission, led by the former President of Tanzania, Julius Nyerere (Nyerere, 1990); * “African Thoughts on the Prospects of Education for All”, selection from papers commissioned for the Regional Consultation on Education for All, Dakar, 27-30 November 1989 (UNESCO, 1990); * “Educate or Perish: Africa’s Impasse and Prospects”, study directed by Joseph Ki-Zerbo. These studies delineate the societal and educational background that must be taken into account in any reflection on Mathematics and Science Education in Africa for the 21st Century. “The challenge to the South” criticizes development strategies that minimize cultural factors. Such strategies only provoke indifference, alienation and social discord. The development strategies followed until now “have often failed to utilize the enormous reserves of traditional wisdom and of creativity and enterprise in the countries of the Third World”. Instead, the cultural wellsprings of the South should feed the process of development (Nyerere, p. 46). An important feature of “African thoughts” is the fact that two themes keep recurring in all contributions: the focus on the crisis in African contemporary culture and the theme of African languages (as vehicles of culture and media of education). The crux of the crisis of African cultures is the issue of African cultural identity (UNESCO, p. 9). A people’s cultural identity (including their awareness of such an identity) is seen as the springboard of their development effort (UNESCO, p. 10).

9 Explorations in Ethnomathematics

Africa needs culture–oriented education that would ensure the survival of African cultures, if it emphasized originality of thought and encouraged the virtue of creativity (UNESCO, p. 15). Scientific appreciation of African cultural elements and experience is considered to be “one sure way of getting Africans to see science as a means of understanding their cultures and as a tool to serve and advance their cultures” (UNESCO, p. 23). “Educate or Perish...” shows that today’s African educational systems favour foreign consumption without generating a culture that is both compatible with the original civilization and truly promising. Unadapted and elitist, the existing educational systems feed the crisis by producing economically and socially unadapted people, and by being heedless of entire sections of the active population. Education for all, as discussed by Ki-Zerbo, should be an attempt to encourage the development of initiative, curiosity, critical awareness, individual responsibility, respect for collective rules, and a taste for manual work. Africa needs a “new educational system, properly rooted in both society and environment, and therefore apt to generate the self- confidence from which imagination springs “ (Ki-Zerbo, p. 104). Reminding us of the apt African proverb “When lost, it’s better to return to a familiar point before rushing on”, Ki-Zerbo stresses “Africa is in serious trouble, not because its people have no foundations to stand on, but because ever since the colonial period, they have had their foundations removed from under them” (Ki-Zerbo, p. 82). This is probably particularly true in the case of mathematics and science. Here lies one of the principal challenges to African mathematics and science educators. Many African children (and teachers too!) experience mathematics and natural science as rather strange subjects, imported from outside Africa. In order to overcome this psychological and cultural blockage to the learning and development of mathematics and science; the African scientific heritage, traditions and practices have to be embedded’ or ‘incorporated’ into the curriculum. How should we respond to this challenge? Ki-Zerbo (1990, p. 87) stresses that all educational renovation in Africa has to be based on research. This appeal is indeed necessary, as, according to Hagan in “African thoughts”, “In Africa, there is

10 and Ethnoscience in Mozambique

generally a surprising lack of research to back up proposals for educational reforms” (p. 24). It is in the context of trying to contribute to a response to the aforementioned challenge, that research begun in ethnomathematics and ethnoscience in Mozambique. 1 Ethnomathematical and ethnoscientific studies analyse * scientific traditions that have survived colonization, and activities in peoples daily life with scientific components, and look for ways to incorporate them into the curriculum; * cultural elements that may serve as a starting point for doing and elaborating mathematics and science, both in and outside school. A type of mathematics and science education is intended that succeeds in valuing the scientific knowledge inherent in the culture by using this knowledge to lay the foundations of providing quicker and better access to the scientific heritage of the whole of humanity. At Mozambique’s Institute Superior Pedagógico, ethnomathematical research started at the end of the 1980, continuing earlier research done at the Eduardo Mondlane University from the end of the seventies. 2 Ethnoscientific research began more recently. “Explorations in Ethnomathematics and Ethnoscience” presents a collection of papers written by various lecturers at the Instituto Superior Pedagógico, based both in Maputo, situated in the South of Mozambique, and in Beira, in the central Sofala Province. The ethnomathematical papers reflect on some mathematical ideas involved in basket and mat making in the North of the country; on languages and mental arithmetic, on popular counting practices all over Mozambique; on symmetries and metal grates; and on the decoration of spoons in Sofala and the South-eastern lnhambane Province. A paper on the traditional interpretation of lightning and thunder in the southern Catembe region and physics teaching is a first

1 Cf. e.g. the book by the U. D’Ambrosio, “father” of ethnomathematics and ethnoscience in the context of education (D’Ambrosio, 1990). 2 Cf. the list of books published by the “Ethnomathematics Research Project” on the last page of this volume and (Gerdes, 1990a, 1990b, 1992, 1994). 11 Explorations in Ethnomathematics

publication in the field of ethnophysics. Short articles on perspectives in ethnoscience related to physics, biology and chemistry conclude this collection. With the expansion of this research line it is hoped to contribute to the preparation of a curriculum reform that guarantees that mathematics and science education in the 21st century in Mozambique is indeed “in tune with African traditions and socio-cultural environment” (UNESCO, 1990, p. 14).

The Editor May 25, 1994

References

D’Ambrosio, Ubiratan (1990): Etnomatemática: arte ou técnica de expIicar, Editora Ática, São Paulo, 88 pp. Gerdes, Paulus (1990a): Ethnogeometrie. Kulturanthropologische Beiträge zur Genese und Didaktik der Geometrie, Verlag Franzbecker, Hildesheim / Bad Salzdetfurth, 360 pp. [preface by Peter Damerow] ___ (1990b): Vivendo a matemática: desenhos de África, Editora Scipione, São Paulo, 68 pp. ___ (1992): Sobre o despertar do pensamento geométrico, Federal University of Parana, Curitiba, 1992, 105 pp. [preface by Ubiratan D’Ambrósio] ___ (1994): Vivendo a matemática: Geometria dos trançados, Editora Scipione, São Paulo, 48 pp. (in press). Ki-Zerbo, Joseph (1990): Educate or Perish: Africa’s Impasse and Prospects, UNESCO-UNICEF, Dakar / Abidjan, 109 pp. Nyerere, Julius (ed.) (1990): The challenge to the South: The Report of the South Commission, Oxford University Press, Oxford, 325 pp. UNESCO (1990): African Thoughts on the Prospects of Education for All, UNESCO-UNICEF, Dakar / Abidjan, 193 pp.

12 and Ethnoscience in Mozambique

Chapter 1

Abdulcarimo Ismael

On the Origin of the Concepts of “Even” and “Odd” in Makhuwa Culture 1

This paper presents an ethnomathematical approach on the origin of the concepts of “even” and “odd” on the basis of an analysis of basketry among the Makhuwa (Northern Mozambique). Other contexts such as traditional hunting, social relationships, the human body form and counting methods, culturally linked either with the concepts of “even” and “odd” or to the importance of the number “two” will also be discussed.

Introduction

The Makhuwa, a Bantu people, also referred to as Emakhuwa or Macua 2 (consisting of about 3,231,000 [1980], corresponding to 25%

1 This paper was presented at the Symposium on Ethnomathematics, Ethnoscience and the Recovery of the History of Science at the XIXth International Congress of History of Science, Zaragoza (Spain), August 22-29, 1993. However, preliminary results of this work were presented at the section of History of Mathematics and Mathematics Education during the 3rd Pan-African Congress of Mathematicians, held in Nairobi (Kenya), August 20-28, 1991. 2 In Cameroon (Central Africa) there lives the Maka-Makua people (see Diop, p. 378), related to the Makhuwa in Mozambique. 13 Explorations in Ethnomathematics

of the country’s population) inhabit northern Mozambique, predominantly the province of Nampula 3 and some regions of the provinces of Niassa, Cabo Delgado and Zambézia (see map). Traditionally and at present, the Makhuwa are mostly peasants. They grow essentially maize, rice, cassava and cashew nuts. Besides working on the land for subsistence, they also make different types of baskets and, for a number of purposes, sculptures, which reflect the reality of their life and work, which is complemented by fishing. The latter is in the case of populations living on the coast. Their skilful craftwork has made them famous and their works of art are highly admired, especially by tourists. Concerning the origin of the concepts of “even” and “odd” in the history of mathematics, little research seems to have been done. Lefèvre (1977) attempts to establish the origin of these concepts in the Pythagorean Theory. Becker (1934) made also a contribution through a tentative reconstruction of these concepts in the theorems 21-34 of the IXth book of Euclid’s “Elements.” Gerdes was the first to suggest that the origin of these concepts and their importance might be related to their role in craftwork, particularly in the making of baskets, and in pictographic art. He presented examples of the importance of these concepts in the productive process (see Gerdes, 1985, 1990, 1991, 1992). During my fieldwork in Nampula (1989, 1993) I came across to situations, which support Gerdes’ general hypothesis. This paper presents a reflection about the origin of the concepts of “even” and “odd” in the context of basket and mat weaving by Makhuwa craftsmen.

3 The author is a Makhuwa from ‘Ilha de Moçambique’ (Mozambique Island, Nampula Province). 14 and Ethnoscience in Mozambique

Echava handbag

The handbag echava 4 (see Figure 1.1) is a type of basket, which is very much appreciated by women and used to carry small objects for personal use, such as rings, bracelets, documents and even money.

Figure 1.1

The knowledge and techniques of making echava are passed on from father-basket-maker to son. It must be stressed that this technique is known only by some men and for this reason, and as time goes by, relevant technical details have gradually been lost. In fact, at present, fewer people possess the traditional knowledge of making echava than before. Usually to make the echava you use carefully chosen leaf strips of the mikhutha plant, of the same length. These strips are put together pili-pili, meaning ‘two by two’, in knots making a right angle. After a

4 In the southern part of Mozambique, in Inhambane, a kind of handbag exists called gipatsi, which is similar to the echava of the province of Nampula. See (Gerdes & Bulafo, 1994). 15 Explorations in Ethnomathematics

knot has been made each strip is divided into two others smaller strips (see Figure 1.2). The knots are also counted pili-pili. In other words, the total number of knots has to be even.

Figure 1.2

After this, the two parts of the strip on one side of the knot are entwined to the two strips on the other side of the next knot. Then you keep on putting together and entwining from both sides of the knot up to the last knot, which is also entwined to the first (see Figure 1.3). In this a cylinder form is obtained. The finishing touches consist of sewing the bottom and hemming the opening part and fixing the handles.

Figure 1.3

If you bend the object, the number of knots on one side must, in the opinion of the basket maker, necessarily be equal to number of knots on the other side. Otherwise the handbag is not considered oligana, what means ‘beautiful,’ ‘perfect,’ or ‘complete’. On the contrary, if we have pili-pili ni mosa (two by two and one) knots, the 16 and Ethnoscience in Mozambique

handbag would be considered ovirigana, i.e. ‘ugly’, imperfect’, or ‘incomplete’. In other words, only if the number of knots is even, will the handbag be symmetrical and beautiful. The concept of ‘even’ which is developed in this productive process corresponds to ‘enumerable in pairs’ and is related to the notion of beauty.

Ntthato mat of straw

The ntthatto mat is an object used to sit and eat meals on as well as to sleep. In urban areas, the mat is also used as a rug. Ntthatto is also made of leaves of the mikhuta plant. The knowledge of how to make it is passed on from father to son.

Figure 1.4

To make the mat, carefully selected strips of straw are used, so that they have more or less the same length and width. These are carefully counted in pairs: pili-pili. The strips are then interlaced “two down – two up” and folded to make a 45o angle to the side of the stroke (see Figure 1.4). A series of four to ten of these rectangular strokes are afterwards knitted together and the end product is the ntthatto. According to the weaver the straw mats are beautiful and perfect, only if the entwinement is always made in pili-pili. However, if the artisan during the entwinement, goes beyond, one rather than two

17 Explorations in Ethnomathematics

strips, the mat will become, still in his opinion, ugly, imperfect or as it is said in the local language, ntthatto n’ovirigana, which means an ‘ugly straw mat’.

Other cultural contexts

Traditionally, in the northern region of Mozambique, it is thought that people who go hunting must be in even number. In this way they will be luckier in the hunt. Hunting involving an odd number of people is always avoided. It is believed that each hunter should have a partner. The concepts of ‘even’ and ‘odd’ developed in this context, which are related to the notions of luck and bad luck, have probably been transferred from the concepts of ‘even’ and ‘odd’ developed in the referred process of making of the echava and the ntthatto. The concepts of ‘even’ and ‘odd’ as present in the basketry and also in traditional hunting are related to the counting process, ‘two by two (and one)’. Other cultural aspects exist in the Makhuwa tradition where the importance of the number ‘two’ is once again stressed, as the following examples will show. According to the Makhuwa tradition it is more ‘beautiful’, both for men and women, to be married than to be single. Two people together cause fewer problems. A couple, a man and a woman, living together is better than a man or a woman alone. On the other side, if there is a third person in a couples relationship, that will not look good. The Makhuwa have a proverb esiiri enkhanla ya tthénli (cf. Matos, p. 178) which means that a secret can only be kept for a long time, if it is shared by only two people. If somebody else knows the secret too, then it is not a secret any more. In other words, a secret is secret only if only two people share it. A complete human being has two arms, two hands, two eyes, etc. When (s)he has only one eye, one arm or only one hand, etc., (s)he is incomplete.

18 and Ethnoscience in Mozambique

Concluding remarks

The most important result of this work is that the ‘two by two’ in the analysed cultural contexts is associated with perfection, beauty, completeness and luck. On the contrary ‘two by two and one’ is associated with something ugly, bad, imperfect, incomplete and unlucky. Through other examples the importance of the number ‘two’ as such was shown. As among the Makhuwa, also in other cultures the awareness of “two” might have been reinforced and the awareness of “two by two” (even) and of “two by two plus one” (odd) might have been awoken in basket and mat weaving or in other production processes. This growing awareness could find its reﬂection in several beliefs and proverbs. Perhaps the origin of binary counting systems known in various parts of the world (cf. e.g. Struik) may be explained in the same way.

References

Baptista, Abel dos Santos (1951): Monografia etnográfica sobre os Macuas, Agência Geral do Ultramar, Lisbon, 59 pp. Becker, Oskar (1934): Die Lehre der Geraden und Ungeraden im Neunten Buch der Euklidischen Elemente, in: Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien. Band 3, Berlin, pp. 533-553. Diop, Cheikh Anta (1979): Nations nègres et culture, Presence Africaine, Paris, Vol. 2. Gerdes, Paulus (1985): Zum erwachenden geometrischen Denken (mimeo), Eduardo Mondlane University, Maputo, 260 pp. ___ (1990): Ethnogeometrie, Kulturanthropologische Beiträge zur Genese und Didaktik der Geometrie, Verlag Barbara Franzbecker, Hildesheim, 360 pp. ___ (1991): Cultura e o despertar do pensamento geométrico, Instituto Superior Pedagógico, Maputo, 147 pp. ___ (1992): Sobre o despertar do pensamento geométrico, Federal University of Paraná, Curitiba, 103 pp.

19 Explorations in Ethnomathematics

Gerdes, Paulus & Bulafo, Gildo (1994): Sipatsi: Technology, Art and Geometry in Inhambane, Instituto Superior Pedagógico, Maputo, 103 pp. Ismael, Abdulcarimo ed. (1989): Relatório sobre o trabalho de campo realizado na Província de Nampula (Norte de Moçambique): Cestaria e Matemática, Instituto Superior Pedagógico, Maputo, 28 pp. (unpublished manuscript). Lefèvre, Wolfgang (1977): Die Lehre der Geraden und Ungeraden – Zur Rekonstruktion des Ursprungs der Wissenschaftlichen Mathematik in der griechichen Antike, in: Materialien zur Analyse Berufspraxis der Mathematiker, Universität Bielefeld, Bielefeld, pp. 89-111. Matos, Alexandre V. (1982): Provérbios Macuas, Instituto de Investigação Científica do Ultramar, Lisbon, 376 pp. Struik, Dirk (1948 [1987]): A concise history of mathematics, Dover, New York, 228 pp.

20 and Ethnoscience in Mozambique

Chapter 2

Marcos Cherinda

Mathematical-educational exploration of traditional basket weaving techniques in a children’s “Circle of Interest”

Introduction

“Círculo de interesse” (circle of interest) is a term used in Mozambique to refer to extra-curricular activities that are organized around the interests of pupils, in which the pupils voluntarily participate during their leisure time. Gerdes conducted a series of “circles of interest” among lecturers and students of mathematics teacher educator programmes. Their general focus was ethnomathematics. In several of these “circles”, traditional basket and mat weaving techniques were explored (see Gerdes, 1986, 1988; 1993). In February 1992 we invited some of the students of the M.Ed. Programme at the Instituto Superior Pedagógico, who had taken part in the “circles of interest” organized by Gerdes, to join me in conducting an experimental “circle of interest” among pupils of the Lhanguene Junior Secondary School in Maputo. Six students volunteered. The children’s “circle of interest” took place from March to July 1992 with weekly sessions of three to four hours on Saturday mornings. Thirty pupils – 16 girls and 14 boys – from grade 8 took part from the beginning. Motivated by the enthusiasm of the participants, almost every Saturday more children turned up. On the last day we invited the teachers from their school and the children’s parents to see what had been produced by the children: various objects

21 Explorations in Ethnomathematics

made out of cardboard strips woven according to traditional Mozambican basket weaving techniques. The enthusiastic parents wanted to learn how to do what their children were already able to do...

Children’s activities

During the “circle of interest”, the children constructed geometrical models inspired by traditional basket weaving. Using cardboard strips, the participants were able to rediscover mathematical considerations involved in the weaving. The scheme for producing models was the following: We took a woven object such as a hat, a basket, or a mat as our source of inspiration. Using the object, we examined the weaving pattern. The children learned first the basic weaving technique and then they were asked to use it to create their own objects. For example, inspired by the hexagonal zigzag weaving of a hat, the children produced a “snake”, a “crown”, a letter of the alphabet (M), etc. (see Figure 2.1). With the help of the student-tutors, the children also made spatial shapes constructed with the weaving technique used for the fabrication of a fish trap. They created geometrical shapes such as cylinders, spheres, half-spheres, etc. and other forms too (see Figure 2.2). By the process of model construction itself, there are geometrical relationships that must be taken into account by the children, in order to produce the desired object. For instance, to obtain the interlacement shown in Figure 2.1b, the wall strip has to be folded exactly as in Figure 2.3a. The question that came from the children was why this wall strip had to be folded exactly like in that way. What happens if it is folded as shown, for example, in Figure 2.3b? What is the measure of the angle concerned?

22 and Ethnoscience in Mozambique

Source or inspiration: Weaving pattern woven hat

Models Figure 2.1

23 Explorations in Ethnomathematics

Source or inspiration: Weaving technique woven fish trap

Models Figure 2.2

24 and Ethnoscience in Mozambique

a b Figure 2.3

As a mathematics educator, when I was working with the children my goal was not only to have them imitate traditional artisan techniques but also to stress that they may create new models, inspired by the weaving pattern or algorithm of a given object. In this process, an active interplay between children’s accumulated knowledge and imagination from their life experience takes place. For instance, a child who wanted to make an “L” (see Figure 2.4), found out that it is not possible to use the “hat-weave” and turn through a right angle. But alternative L’s turned out to be possible (see Figure 2.5). 1

Figure 2.4

1 Many such examples are given by Gerdes in “On the awakening of geometrical thinking” (1985b), recently published in a short popular edition, “Culture and the awakening of geometrical thinking” (1992).

25 Explorations in Ethnomathematics

Figure 2.5

Concluding remarks

At the end of our “circle of interest”, the pupils were able to create their own models, inspired by traditional basket weaving techniques. For instance, all children constructed letters of alphabet to compose their names, using the hexagonal zigzag hat weaving technique. This meant that everyone had to think about how to obtain each letter of his or her name. Through this kind of activity, we used elements of the children’s traditional culture to heighten their self- confidence. They were able to create, invent and explore, and so they began to feel that the mathematics they learnt at school also related to their life and their society / culture. When pupils become self- confident in relation to the informal knowledge of their environment and classroom mathematics, they can develop their mathematical thinking more naturally and creatively. We are referring not only to individual self-confidence. It is closely related to and reinforced by social and cultural self-confidence as stressed by Gerdes (1982, 1985a): when the African peasants’, and other workers’, sons and daughters can appreciate that their ancestors, collectors, hunters, shepherds, peasants, artisans, etc. were able to develop their own mathematics, then today they – the descendants –

26 and Ethnoscience in Mozambique

may feel themselves more confident that they can appropriate mathematics and develop them creatively. This self-confidence becomes more salient when the children realize the relationship between the application of the creative mathematical power of their people to modern technology. For instance, let us look at a handle of a basket, a shoelace, and the covering of a wire cable. In Figure 2.6 we can see that in each case the weaving technique is the same.

Figure 2.6

27 Explorations in Ethnomathematics

With examples of this nature, we begin to expose the myth that the mathematical power, which supports today’s industrial technology, has exclusively European roots. The Mozambican people, and many other formerly colonized peoples, are owners of a cultural power that contributed and contributes to the development of mathematics, science and technology. By including the M. Ed students in the children’s “circle of interest”, we intended to provide pedagogical training in the organization and running of “circles of interests”. This training is necessary for them as future teachers. It is relevant to show them that it is possible to explore mathematical ideas and relationships using artefacts of local cultural traditions. In this way, we intend to stimulate future teachers to popularize the exploration of African mathematical and technological knowledge in schools across the country.

References

D’Ambrosio, Ubiratan (1990): Etnomatemática, Arte ou técnica de explicar e conhecer, Editora Ática, São Paulo, 88 pp. Gerdes, Paulus (1982): Mathematics for the Benefit of the People, CARIMATHS, Paramaribo, 13 pp. ___ (1985a): Conditions and strategies for emancipatory mathematics education in underdeveloped countries, For the Learning of Mathematics, Montreal, Vol. 5, No. 1, 15-20. ___ (1985b): Zum erwachenden geometrischen Denken, Universidade Eduardo Mondlane, Maputo, 260 pp. ___ (1986): How to recognize hidden geometrical thinking? A contribution to the development of anthropological mathematics, For the Learning of Mathematics, Montreal, Vol. 6, No. 2, 10- 12, 17. ___ (1988): On culture, geometrical thinking and mathematics education, Educational Studies in Mathematics, Dordrecht / Boston, 1988, Vol. 19, No. 3, 137-162; and in: Bishop, A. (ed.), Mathematics Education and Culture, Kluwer Academic Publishers, Dordrecht / Boston, 1988, 137-162.

28 and Ethnoscience in Mozambique

___ (1991a): Etnomatemática: Cultura, Matemática, Educação, Mozambique’s Higher Pedagogical Institute, Maputo, 115 pp. ____ (1991b): Ethnogeometrie, Kulturanthropologische Beiträge zur Geschichte und Didaktik der Geometrie, Franzbecker Verlag, Hildesheim, 360 pp. ___ (1992): Cultura e o despertar do pensamento geométrico, Mozambique’s Higher Pedagogical Institute, Maputo, 147 pp. ___ (1994): Geometria dos trançados, Editora Scipione, São Paulo, 64 pp. (in press)

29 Explorations in Ethnomathematics

30 and Ethnoscience in Mozambique

Chapter 3

Daniel Soares and Abdulcarimo Ismael

Popular Counting Methods in Mozambique

Within the context of the course ‘Mathematics in History’ 1 we taught on the M.Ed. Programme at the Instituto Superior Pedagógico in Maputo and Beira, and as a part of the particular unit “Systems of numeration and counting practices”, we have carried out surveys among our students on ‘popular counting systems’, since 1989. Gerdes and Nhancolo also carried out similar surveys in 1987-89 among computer science students at the Eduardo Mondlane University in Maputo, as did Ismael in 1992 among student teachers at the Instituto Médio Pedagógico in Nampula. This paper presents a description of the various counting practices that are used in several parts of Mozambique.

Knots

In the province of Cabo Delgado, malundu (singular: lilundo, also vilundu) which means ‘knots’, is a counting method used for recording the months of the year, the age of the children, the duration of long hunts or long trips and for counting of the months of pregnancy.

1 Paulus Gerdes initially taught this course with the assistance of Daniel Soares in Beira, and of Abdulcarimo Ismael, Marcos Cherinda and the late Mauricio Nhancolo in Maputo. 31 Explorations in Ethnomathematics

On long hunts, a piece of material or a rope of approximately one meter is used. On the first day of hunting a lilundo is made. On each subsequent day, when the sun rises, a new knot is made. For better control the man most trusted in the group and respected in the village keeps the rope. Sometimes a second thread is used for counting the number of attacks made each day in the forest. In this latter case, a knot is made during each rest period. On the last day of hunting, the knots are counted and in this way the duration of the hunt is determined. Normally after hunting the malundu are carefully preserved in order to compare with the duration and number of attacks from previous or future hunts. To record the months of the year, the 12th moon, also known as ‘Ramadan’ by the Moslems, the month when they fast is taken as a reference point. When the new moon appears, immediately after ‘Ramadan’, a lilundo is made in a piece of material or cord. In the following months, more malundu are made successively up to the number 11 that is up to the 11th moon. This is the time when everybody must fast. The appearance of the expected moon means, at the same time, the end of a year. To record the age of children, the parents use the same system, which is used for measuring the months of the year. For example, if the child was born during the 4th moon that is during the 4th Moslem month, a lilundo is made in the piece of material or cord, which is used to control the months of the year. After 12 months the first knot is made in the rope to measure the age. There is no particular difference between the piece of cloth or rope for measuring Moslem months or years, and the piece of cloth for recording children’s ages. The person keeping the pieces must be able to distinguish between them. In some cases they are kept in different places; they are hung on one of the walls of the parents’ bedroom or kept in a traditional trunk made of bamboo. For the counting of the months of pregnancy, a thread (usually from sisal or pineapple tree leaves) is tied to the pregnant woman’s waist and a knot is made at each new moon. By counting the knots the proximity of the ninth month of pregnancy can be determined. Apart from the aforementioned cases, this way of counting can still be found in other contexts. For example, the duration of the

32 and Ethnoscience in Mozambique

construction of a house or the duration of the development phases of a child. For instance, the time a child takes from birth up to beginning to sit can be measured, from creeping to walking etc., by doubling the knots in the main phases. In this last case, the mother or the child’s grandmother keeps the rope. Nluthó or nlutché which also means ‘knot’, is a method used in northern province of Nampula for counting animals and in some cases for recording the time. For counting animals, knots are made in a thread according to the number of animals. At night, when the animals are taken back to the stable, they are counted using the nlutché for control. In the southern province of Maputo, fundzu (plural: mafundzu) also means ‘knot’. This fundzu counting is used for recording the quantity of cattle (cows or goats) that someone has. When a cow or a goat has calves or kids, a fundzu is made on the sisal thread reserved for the corresponding species. In the event of selling, slaughter or the death of an animal, a knot is untied.

Stones

In the province of Gaza xiguama xa xibala i.e. a bag, which contains small stones (sometimes it contains wild fruits seeds but it keeps the same name) is used as a counting device. It is applied to record the number of animals in a kraal. Every day the owner of the herd goes to the kraal with his xiguama xa xibala in order to check his cattle. He takes the animals out one by one, while at the same time taking the stones out of the sack and putting them one by one into another bag, or more commonly, onto the ground. If a stone remains in the xiguama xa xibala he knows that an animal is missing, which may have been stolen or have gone to a different kraal. Very often the owner cannot tell in words how many cattle he has got in the kraal, but through this method he is able to know if his animals are complete or if there are some missing. In general, a man can keep a hundred or more animals in a kraal. This operation is done daily, in the morning before the cattle set off for the pasture. At the time the cattle are brought back to the kraal, it is not possible to do the check because the animals return to the kraal when it is relatively dark. When there is a 33 Explorations in Ethnomathematics

slaughter, death or loss of any animals, the owner takes the number of stones corresponding to the number of missing animals away from the sack. In the case of reproduction, the operation is the opposite: a number of stones corresponding to the number of calves born is put into the xiguama. In the North-western province of Tete mwala, (plural: myala) which means stone, is a way of counting usually used to record the number of domestic animals such as chickens, cattle, goats and pigs. Sometimes this method is also used to measure the quantities of crops harvested in the fields. Our informer learned this counting method from his parents and neighbours when he helped them to check the animals. In the morning, the checker, who may be a boy or a girl, goes to the stable or kraal, which is usually built at a distance of approximately a hundred metres, and stands at the gate with a certain quantity of myala, small stones, in his hands. When the animals go out of the stable to the pasture (s)he puts one mwala on the ground for each animal, or passes one mwala from one hand to the other. At the end (s)he keeps the stones corresponding to the number of animals, which went out of the stable. (S)he keeps the stones for different kinds of animals in different places. When the animals are taken back to the stable, the checker stands at the gate with the stones s/he had kept that morning. (S)he repeats the process. If one or more stones remain, the controller understands the disappearance of part of the cattle, and sets out to look for them. Sometimes it happens that the number of cattle has increased. In such cases the checker tries to find out if any neighbour has lost an animal from his / her herd. In general, a breeder has between 20 and 50 cows and between 100 and 200 goats. The use of this counting method to measure crops harvested from the fields (sacks of maize, other kinds of cereals or even bunches of maize and other cereals, which are kept in the storehouse) is not very important because these kinds of cereals are kept inside the house and are therefore less likely to disappear.

34 and Ethnoscience in Mozambique

Folds

In the central province of Zambézia, mulobuó 2 refers to a method used mainly for the counting of the number of goals scored by a team at the end of a football match. The boys take a coconut tree leaf and they extract the leafs’ main vein, also known as a “toothpick”. Thus, it is divided into two parts and each team takes one, which from that moment on is called mulobuó (plural: milobuó or milebuò), and will be kept by each captain, usually at his waist. As soon as a goal is scored, the successful team makes a fold in its mulobuó, under the careful watch of the opposing team in order to avoid irregularities. At the end of the match the teams have a meeting where they count the folds of each mulobuó. The team, which has more folds in its mulobuó is the winner. When, for some reason, there is not enough time for the counting, the teams do not actually count the folds or goals, but simply compare the lengths of the two milobuó from the first fold to the last. The team with the longer mulobuó will celebrate the victory. In such cases it is not important to know by how much the team has won; the main thing is to know whether you have won or lost. Figure 3.1 shows a mulobuó.

Figure 3.l

2 In Chuabo language, the word mulobuó may mean a trace, mark or, in games, it may mean a point. 35 Explorations in Ethnomathematics

Marks

In the province of Gaza swivate or timarka (singular: marka) are used as a way of recording the amount of money or animals paid as lobolo (bride price) for later checking or comparison. Our informer learnt this method from his maternal grandfather. For each animal (goat, sheep, cow, etc. ) or 100 escudos note (which was worth a lot at that time) they used to make a mark – marka – with an axe on the nearest tree to the house, preferably a big or long-living tree such as a hemp tree. The marks were grouped and well organized in order to make it possible to record more lobolos, with timarka, on the same tree. There were no special marks identifying money or animals. They were distinguished from memory. For example, a person had to remember that for a certain lobolo sheep or goats had been paid, and that the timarka would give the quantities. In the province of Inhambane sticks and tree trunks are used, for example, to record people’s ages, the number of animals in a herd, and animals killed during a hunt for later counting. For example, after the birth of a baby, people go to a big old tree, which has many more years to live, and they make a cut on its trunk, which represents the birth. This cut is different from those which will be made later. Annually more cuts are made on the trunk, next to the previous one, until the child grows and knows his/her own age. Generally the cuts are made until the person reaches the age of 20 or 21. That means, they stop at an age when the person is able to look after him / herself (e.g. working in the case of boys, or marrying in the case of girls). To count the animals in a herd, people cut a stick from a tree and make marks by making small cuts, corresponding to the number of the existing animals in the kraal. The marking process is done in the following way: First all the animals of the same species are put in the kraal. Then they are taken out one by one, and simultaneously cuts are made on a stick. Each stick can bear a maximum of a 100 cuts. After this marking process the stick is given to the head of the family for him to check the animals at any time he likes. In the case of reproduction, the stick is taken and marks are made according to the number of new born animals. During hunts a stick is also taken and each time an animal is killed a corresponding cut is made.

36 and Ethnoscience in Mozambique

Mbara is that part of a bunch of coconuts which looks like a bean pod. In the province of Inhambane it is used as method of recording the number of coconut trees already climbed by a worker during a certain period of time. Before starting the climbing job, the worker prepares a mbara. After climbing down from the first coconut tree, from where he dropped some coconuts, the climber takes his mbara and makes a cut on it with a knife. The same process is repeated after climbing down from the second, third or fourth coconut tree and so on. At the end of the job he counts the number of cuts made which will correspond to the total number of trees climbed. In general a worker climbs about twenty coconut trees in a morning.

Figures or traces

In the province of Nampula okwenhenha is a way ‘of counting used by basket makers by making traces on the ground to record the number of strips they need for a basket.

Figure 3.2

In the same province ualaca or ualaquela, which means counting, is a way used to find out who is the winner in card games (nttejo). Card games are one of the most popular entertainments in the Memba district. Before the game starts, some drawings are made on a piece of paper for each team (see Figure 3.2a). After a game the valuable cards are counted – ualaca – and a small circle (dot) is drawn at the end of one of the straight lines of the winning team (see Figure 3.2b). The game continues while the partial counting, ualaca, are been done and more circles are drawn on the winner’s side. When a team 37 Explorations in Ethnomathematics

has won six games, it means the six ends are filled with circles, then a big circle is drawn around the whole drawing and that means a victory for the respective team (see Figure 3.2c). The process of drawing the big circle is called othuca and it means to close or to win well. The final counting or ualaquela is done at the end of many games. The winning team is the one, which has drawn the highest number of big circles.

38 and Ethnoscience in Mozambique

Chapter 4

Jan Draisma

How to handle the theorem 8 + 5 = 13 in (teacher) education? 1

During the last few weeks, in Beira (Mozambique) and in Northern Transvaal (South Africa), I have been thinking a lot about what my contribution to the PDME-2 Conference could be. The pre- conference workshops in Kwandebele, Lebowa, Venda and Gazankulu, in which my colleague Evaristo Uaila and I participated, particularly complemented my ideas. Eventually I decided to organize my thoughts and present my personal experience according to the various occasions that I have come across the theorem 8 + 5 = 13. The theorem is well known to many of you, but for young children, during some stage in their lives, the problem 8 + 5 = ? is a serious one (cf. e.g. Hatano, 1982, p. 216; Treffers & De Moor, 1990, p. 40-49), and for many primary school teachers in Mozambique, the problem of how to help children find the answer is serious too. I first met some difficulty related to the theorem when I was teaching at the Frelimo Secondary School, 2 Bagamoyo (Tanzania) in

1 Paper presented at the Second International Conference Political Dimensions of Mathematics Education (PDME-2), Broederstroom, South Africa, April 2-5, 1993 2 Frelimo = Frente de Libertação de Moçambique (Mozambique Liberation Front), founded in Dar es Salaam, Tanzania, 1962. On the history of Frelimo and its educational policies during the armed struggle for Independence: Mondlane (1969), Machel (1970, 1974). 39 Explorations in Ethnomathematics

1970, where some of my students in Grade 5 had not yet memorized multiplication tables. I also discovered that some of the students had memorized these tables only in a specific, written form: they would know the answer to, say, “How much is eight times seven?” if the problem were presented in the following, written form: 8 7

But if the problem were presented orally or in the following written form: 8 x 7 = ? then these students usually found it difficult to give the answer. I also got the impression that for some students the difficulties in memorizing multiplication tables originated in not having memorized all the basic facts of addition, especially those with sums greater than ten. So, in the 5th Grade, I had to pay attention to ways of finding the results of problems of the type 8 + 5 = ? In 1971 Frelimo’s second teacher training course for primary school teachers of the liberated areas in Mozambique, and from the Tunduru Educational Centre (in Southern Tanzania, run by Frelimo’s Department of Education and Culture), was held in Bagamoyo. My task was to teach Mathematics and its Didactics. We had at our disposal the first Mathematics schoolbooks written for the 1st Grade (Kindler & Minter, 1971). These Mathematics books and the corresponding teacher’s guide (Kindler, 1971) suggested that teachers should teach the following method: 8 + 5 = 8 + (2+3) = (8+2) + 3 = 10 + 3 = 13

Machel’s text of 1974 consists of an analysis of a crisis that the Frelimo Secondary School in Bagamoyo went through. 40 and Ethnoscience in Mozambique

I shall refer to this method as a two-step one: instead of adding 5, you add first 2 and then 3. On the teacher training course I tried to convince the teachers that this was a nice method. On the six-month course we devoted about half the time to the didactical problems of Grade 1. In Lusaka on 7th September 1974, the Portuguese Government and Frelimo signed the historical agreements that brought Portuguese colonial rule in Mozambique to an end. A Transitional Government was empowered on 20th September. In January 1975 I had the pleasure of flying to Beira, Mozambique, in order to participate in the First National Conference on Education. A very heterogeneous group of more than 400 teachers, representing all provinces and all types of education – official, private, missionary and Frelimo, including educational officers from the former colonial government and of the liberation movement – decided to do away with the Portuguese syllabi used in Mozambican schools and write new ones for all subjects, from Kindergarten to Grade 11 – at that time the end of pre-university schooling (Ministério da Educação, 1975). At that conference I met, among others, a well-known Portuguese didactician for primary education – a very experienced primary school teacher working at the Primary Teacher Training College (Escola de Magistério Primário) in Lourenço Marques (now Maputo). After the Conference, I moved to Lourenco Marques, and one day I happened to visit the Primary Teacher Training College, where I met the abovementioned didactician again. During our conversation he mentioned the Grade 1 Mathematics books of the Frelimo schools, and made the following remark on the two-step method for the 8 + 5 = ? problem: Why does the book suggest a two-step procedure? Why not have the children calculate directly 8 plus 5 gives 13? I tried in vain to convince him that the method of doing addition by counting on, that he had in mind, was not so direct as he thought, especially when numbers get bigger. In 1975, the Ministry of Education and Culture decided to reprint the Frelimo Mathematics books for Grades 1 and 2, in order to use them as material to be studied in teacher education. The books were 41 Explorations in Ethnomathematics

reprinted with a few minor changes. I was involved in preparing the books for reprinting and we took the liberty of introducing a minor simplification in the presentation of the 8 + 5 = 13 calculation:

8 + 5 = 8 + 2 + 3 instead of 8 + 5 = 8 + (2+3) = 10 + 3 = (8+2)+3 = 13 = 10 + 3 = 13 Later the Ministry named a commission to analyse the feasibility of introducing the same books for use in primary schools. The commission included the aforementioned didactician and Portuguese mathematicians working at the Eduardo Mondlane University (EMU), Maputo. The Commission gave a negative recommendation to the Ministry and the books were shelved. As the import of Portuguese schoolbooks stopped after Independence, and for several years no books were produced in Mozambique, 3 primary teachers and pupils had to work practically without schoolbooks. Writing schoolbooks and teacher’s guides proved to be a much harder task than many had imagined. The first Mathematics books for use in schools were produced more than 5 years after Independence: — Grade 1 Pupil’s Book (National Institute for the Development of Education (INDE), 1981); — Grade 1 Pupil’s Exercise Book (INDE, 1981); — Grade 1 Teachers Guide (INDE, 1981); — Grade 5 Pupil’s Book (2 volumes, INDE, 1981); — Grade 6 Pupil’s Book (INDE, 1982); — Grade 10 Pupils Book (4 volumes, EMU, 1980); — Grade 11 Pupil’s Book (2 volumes, EMU, 1981).

3 The only exception being the Mathematics books of the “Colecção Moçambique”, for grades 1 to 4, published by Livraria Minerva Central (Lourenço Marques) a few years before Independence. 42 and Ethnoscience in Mozambique

The first systematic production of syllabi, pupils’ books and teacher’s guides started with the introduction of the National System of Education (NSE) in 1982. The curriculum materials were written by subject teams of the Department of Curriculum Planning, in the National Institute for the Development of Education, Maputo. I had the pleasure of working in the Mathematics Section of INDE, from 1981 to 1990. When we started thinking about the syllabi for Grades 1 and 2, we had to discuss how to handle the basic number facts of addition and subtraction. The rather heterogeneous team decided to pay more attention to mental arithmetic than was traditionally done in primary schools. For the basic number facts of addition, of the type 8 + 5 = 13, we suggested the use of the two-step method, represented in the pupil’s exercise books in the following way (Draisma et al., 1983): 8 + 5 = ? 8 + 2 = 10 10 + 3 = 13 8 + 5 = 13 Our idea was that this presentation follows the different steps of reasoning more closely than the presentation used by Kindler & Minter, 1971. In refresher courses for primary teachers, teacher educators and subject advisers of the Ministry of Education and Provincial Directorates of Education, we discussed the new syllabi (integrated into the teacher’s guides) and the new pupils’ books, paying much attention to mental arithmetic. At the same time, in the new 3-year course for primary teacher training (entrance level: 6 years of primary education), 4 the same methods were emphasized, or, rather, in the 1- year course, organized for the up-grading of teacher educators who had only had the regular 1-year course after Grade 6, these methods were studied. However, recent studies show that many pupils in

4 After Independence, teachers for Lower Primary School were trained in a 1-year course after 6 years of primary education. See Annex 1 for an overview of types and levels of teacher training for Lower Primary Schools that existed in Mozambique before and after Independence. 43 Explorations in Ethnomathematics

Grades 1 - 4 continue to use elementary counting strategies in order to solve, not only 8 + 5 = ?, but also 48 + 25 = ? These pupils use their fingers and sometimes also their toes, or a non-structured series of strokes (Kilborn, 1990; Draisma, 1992). At least in some cases, the difficulties of children abandoning counting strategies in favour of more efficient calculation strategies, start with the teacher, as is shown by the following explanations given by a primary teacher during an interview by a group of students of the Higher Pedagogical Institute in Beira: Question: How would you teach the addition eight plus seven to a Grade 1 child, who does not yet know the written procedure? Answer: In Grade 1 we teach the numbers from one to twenty, using concrete materials, like sticks and pebbles. In order to teach seven plus five or eight plus seven, we count groups of sticks or pebbles, put them together and finally count them all Later during the same interview: Question: And in Grade 2, how do the pupils solve fifteen minus eight” Answer: They don’t learn that in Grade 2, because of the “transport” (= “borrowing”). They can only do it with sticks. The teacher shows the method, making fifteen strokes and counting them one by one. (S)he then counts eight of them, from right to left, dividing them and says: “Then they count the rest; there are seven left”. And during a second session of the interview with the same teacher: Question: In Grade 2, the limit is one hundred. Can the pupils solve problems like ninety-two minus fifty-five? Answer: Yes, they can. It is easy to count ninety-two sticks, take away fifty-five and count the rest. Question: But is that a practical way of doing it?

44 and Ethnoscience in Mozambique

Answer: Yes, it is practical, because every child has his / her own sticks and keeps them on his / her desk. In 1986 this teacher finished a 3-year teacher training course after Grade 6, and in 1991 completed Grade 9 of general secondary education (Draisma, 1993, p. 33-39). According to the perspectives of the National System of Education (See Assembleia Popular, 1981, p. 93), all primary school teachers (for lower and upper primary) should have at least 10 years of general education and a 2 or 3-year course of professional training. For upper primary teachers (who are subject teachers in Grades 6 and 7) this level of training was introduced in the eighties. For lower primary teachers (who teach all subjects in Grades 1 - 5), this level of training will be introduced in the nineties. The idea of improving the level of pre-service training of lower primary teachers, from 7 + 3 to 10 + 2 years, seems to be a good one. But where are the teacher educators to be found, able to give a better professional training at a higher level? At the moment, a number of the teacher educators who are working in the (6 + 3)-year course are themselves primary teachers who have the same level of education, plus a one year, subject-oriented course of specialization. Another group of primary teacher educators was trained as subject teachers (Mathematics and Physics) for general secondary education, in a (9 + 2)-year course. In the medium level Pedagogical Institutes, where the subject teachers for upper primary level get their training, the majority of the teacher educators themselves were trained as subject teachers for secondary schools. So, thinking about the planned introduction of medium level training (10 years of general education plus 2 years of professional training) of lower primary teachers, a group of members of the Mathematics Section of INDE proposed the creation of a full university-level programme of Mathematics Education for Primary Schools, with the main aim of training specialists who may be responsible for the subjects of Mathematics and Didactics of Mathematics at the Primary Teacher Training Colleges. The proposal was approved by Mozambique’s Instituto Superior Pedagógico in April 1990 and in November of the same year a specific entrance examination was held in every provincial capital of the

45 Explorations in Ethnomathematics

country. Out of a total of 170 candidates – all of them experienced primary teachers and teacher educators – 45 students were selected, representing the eleven provinces. The programme opened officially in August 1991, at the Beira branch of the Instituto Superior Pedagógico. During the first 4 years, corresponding approximately to a level of a B.Ed. degree, the main subjects are: Mathematics and its Didactics, Psychology of learning Mathematics, History of Mathematics, Thought and Language, Mathematics, Portuguese, English, Natural Science, History and Anthropology. The second stage of the course, with a duration of 2 years and leading to a M.Ed. degree, will be organized as distance education, requiring the presentation of papers and participation in training seminars. Silva & Draisma (1991) give a more detailed description of the planned curriculum. One of the main subjects of the course is Mathematics and its Didactics. It is globally divided into Arithmetic and its Didactics, and Geometry and its Didactics. I became responsible for Arithmetic and its Didactics. So in Beira again I had to face the question: how to handle the problem of finding the sum of two digit numbers, for the case where the sum is greater than ten, in teacher education. We decided to conduct a series of interviews in the “real world”. At the end of 1991 our students interviewed the following people in Beira: – 15 illiterate women who had just started a literacy course in the local Sena language; – 14 children of school age (7 - 12 years), who did not go to school; – 13 pupils of Grades 1 and 2. The main results of these interviews are presented in Draisma (1992). We found that the illiterate women calculated with great ease, using the number-words in the Sena language. One of the easier tasks of the interview was: how much is eight plus five? I was present at one of the first interviews, where a lady answered immediately: Thirteen. The student-interviewer asked: How did you find the answer? The lady again answered immediately: Eight plus two ten; ten plus three, thirteen. You understand how grateful I

46 and Ethnoscience in Mozambique

was to the lady: she showed our students in a very natural way how to calculate with number-words, using a method that many primary teachers consider artificial, difficult and unnecessary because “there are more direct ways of finding the result”. The main result of these interviews was, that our students discovered that: a) it is possible to calculate just with number-words; you don’t need the written symbols; b) calculation with number-words has its own rules, that make use of the way in which the number-words are formed (some students had thought that arithmetic always requires the written symbols); c) all methods of mental arithmetic, suggested by the 1983 syllabus and corresponding teachers guides and pupils’ books, were used by the women we interviewed; we found also some interesting alternatives to the methods of the syllabus; d) calculation with number-words may be supported by gestures. A different kind of result was that there were no significant differences between the methods used by schoolchildren and the methods used by children of the same age that don’t go to school: in general the children used counting strategies supported by their fingers (and sometimes their toes too). We discussed these results with our students. I asked them: How do you explain that the illiterate women calculate so well, without ever having been to school, and without knowing how to write numbers? The students said: “Adults face many problems in their life that cannot remain unsolved; so they are forced to find solutions” (Draisma, 1991, p. 1-13). This seems to be a very straightforward explanation. Then I posed a second question: How do you explain that there are no significant differences between the arithmetical abilities of schoolchildren and street-children of the same age? What is the use of going to school, if children who don’t attend school are doing as well as the school pupils of the same age? An embarrassing question for students who are experienced primary school teachers themselves,

47 Explorations in Ethnomathematics

because the apparent answer and subsequent conclusion is: there is no difference, so it is better to close the schools. And when these children grow up, they will learn how to solve arithmetical problems, because life will force them to. This is when we started analysing in greater detail the interviews done in the Sena language and the methods of mental arithmetic used by the illiterate women. I also asked my students to solve the same problems using the system of spoken number-words of their own languages. In our group of students nearly twenty different Mozambican languages are represented. 5 We found that the methods of calculation depend on the peculiarities of each system of spoken number-words. In the Sena language, the system of number-words uses ten as a base, just like English or Portuguese. But there are other Mozambican languages that use a combination of base five and base ten. In the table on the next page we present the number-words for Sena and Changane (cf. Gerdes, 1992; Veloso & Draisma, 1992). In Mozambique, the main languages south of the Save river and north of the Zambeze river use the auxiliary base five, in addition to the general base ten structure. The main languages of the central provinces of the country use a base ten numeration system. The most important new discovery of the students, on the relation between calculation and language, was: e) Languages that use a combination of base five and base ten offer specific possibilities for calculation that do not exist in languages with a simple base ten number system.

5 Linguists will call some of these languages variants or dialects of the same main language. Using the provisional classification of Firmino (1989), which is based on Guthrie (1948), our 43 students represent at least 16 different dialects / variants belonging to 8 main languages. 48 and Ethnoscience in Mozambique

n English Portuguese Sena Changane (Tsonga) 1 one um cibodzi xin’we 2 two dois piwiri swimbirhi 3 three três pitatu swinharhu 4 four quatro pinai mune 5 five cinco pixanu ntlhanu 6 six seis pitanthatu ntlhanu ni xin’we 7 seven sete pinomwe ntlhanu ni swimbirhi 8 eight oito pisere ntlhanu ni swinharhu 9 nine nove pipfemba ntlhanu ni mune 10 ten dez khumi khume 11 eleven onze khumi na khume ni xin’we cibodzi 12 twelve doze khumi na khume ni swimbirhi piwiri 13 thirteen treze khumi na khume ni pitatu swinharhu 14 fourteen catorze khumi na khume ni mune pinai 15 fifteen quinze khumi na khume ni ntlhanu pixanu 16 sixteen dezasseis khumi na khume ni ntlhanu ni pitanthatu xin’we 17 seventeen dezassete khumi na khume ni ntlhanu ni pinomwe swimbirhi 18 eighteen dezoito khumi na khume ni ntlhanu ni pisere swinharhu 19 nineteen dezanove khumi na khume ni ntlhanu ni pipfemba mune 20 twenty vinte makumawiri makume mambirhi

49 Explorations in Ethnomathematics

Let us take the problem 8 + 5 = ? as an example. In Portuguese you say: oito mais cinco é igual a treze Translated into English: eight plus five equals thirteen One uses three different number-words: oito (eight), cinco (five) and treze (thirteen). If you calculate in English, one also needs three different number-words: eight, five and thirteen, and there is no relation between these words. But if one uses a language with a combined base five / base ten number system, the problem takes on a different form. For instance, in Changane, 6 the main language spoken in the Mozambican province of Gaza, people will say ntlhanu ni swinharhu (= eight) kupatsa (= add, join) ntlhanu (= five). Children could use the method of “counting all” or “counting further”, as in any other language, in order to find the sum. But the expression ntlhanu ni swinharhu gives an alternative: add ntlhanu and ntlhanu; the sum is khume. Then add khume and swinharhu; the result is khume ni swinharhu. So the problem 8 + 5 = ?, which is difficult for children during a certain phase, is solved in a natural way, using the very simple number fact 5 + 5 = 10 (ntlhanu ni ntlhanu, khume); and what seems to be the second step (ten plus three, thirteen), is no step at all in Changane: khume ni swinharhu swihamba (makes) khume ni swinharhu, because the name for thirteen, in Changane (and in most other Mozambican languages) is just ten-and-three. In Mozambique, the language of instruction in all levels of education is Portuguese, the official language of Mozambique. 7 But for the majority of children who start primary school, Portuguese is a

6 The Mozambican Changane language is the same as the South- African Tsonga language. In Mozambique, the most common designation for the language is Changane (see Ribeiro 1965, p. vii), although some authors use also the name Tsonga (Firmino 1989, Sitoe 1986, Ribeiro 1965, Conselho Coordenador do Recenseamento 1983). 7 According to the 1980 census, only 1,2% of the Mozambican population had Portuguese as its mother tongue and only 24,4% of the adult population speaks the Portuguese language (Conselho Coordenador do Recenseamento, 1983; Coelho & Barca 1986, p. 46). 50 and Ethnoscience in Mozambique

new language they have had little contact with and which they have to learn and use at school. Recently, the Ministry of Education, through the National Institute for the Development of Education (INDE) started some pilot projects of bilingual literacy courses and bilingual schooling, i.e., types of education in which both the mother tongue (or local language) and the official language (Portuguese) are studied and used. 8 Our Mathematics department at the Beira branch of the Instituto Superior Pedagógico has some involvement in these projects because of the question of the language(s) of instruction to be used in the Mathematics lessons. We have stressed to our colleagues of INDE that the experience of other African countries should be studied. In southern Africa there are examples of the use of local languages in primary education: either as the language of instruction or as a subject. But in general, Mathematics is taught from the beginning in the English language, with very few exceptions: in Tanzania, Mathematics is taught in Kiswahili, at least in primary school (Mmari, 1978, p. 322-327); in South Africa, within the framework of “Bantu Education”, the local languages were used, during a certain phase, as language of instruction for all subjects, including Mathematics. But in Zimbabwe, where the local languages Ndebele and Shona are used in many primary schools, Mathematics is taught exclusively in English. In Zambia, where the local languages are taught as a subject, Mathematics is taught from the beginning in the English language. The separation between Mathematics and the local language is such that, for instance, the Short English-Nyanja Vocabulary used in

8 In 1991 a bilingual literacy project for women was started in the central Sofala province. The languages involved are Sena and Portuguese. In 1993 a project of bilingual primary education started in some schools of the southern Gaza province (Changane and Portuguese) and some schools in the North-western province of Tete (Nyanja and Portuguese). Projects for other languages are being prepared. An overview of types of bilingual education and possible implications for Mathematics teaching and learning are given by Zepp 1989, chapter 8. 51 Explorations in Ethnomathematics

primary schools does not even contain the Nyanja number-words for two, three, ten, etc. (Price, 1983). So, when we were preparing our visit to South Africa, we asked the organizers of PDME-2 to give us the opportunity of coming into contact with teachers who have experience in the teaching of Mathematics in the local language. And, days before the conference, we participated in workshops organized by NOTMO (the Northern Transvaal Mathematics Organization) in the regions of Kwandebele, Lebowa, Venda and Gazankulu. In the last three places we had the opportunity of discussing the specific questions: a) What should be the language(s) of instruction in our primary schools: an official language like English or Portuguese, or a local language? b) What should be the language of instruction in Mathematics lessons? On 1st April, we had an interesting discussion with some 150 or 200 teachers in Gazankulu. We already knew some of the Mathematics books in Tsonga that were used in Gazankulu a few years ago (Gleimius et al.; Fletcher). What opinions did we collect from the Gazankulu teachers? a) Some of them said that it is better to use English as the language of instruction in the Mathematics lessons from the beginning, because, in general, the children already know the English number-words when they come to primary school. So it is not necessary to use the Tsonga number-words. This was also the opinion of some of the teachers who had experience in using Tsonga as the language of instruction in the Mathematics classes. b) Others said that in the first Grades of primary school the mother tongue should be used for all subjects, because it is the only means of communication possible. It is a mistake to think that you can communicate effectively in the classroom, using a new language. c) Another participant said that all instruction, including Mathematics, should be done in the mother tongue, because the children should be able to communicate with their grandparents. In particular, they should be able to tell, 52 and Ethnoscience in Mozambique

in their mother tongue, about the mathematical activities of the classroom. d) Finally, one of the participants told us that he had been teaching arithmetic in Tsonga, using the official, “simplified” Tsonga base ten numeration system (cf. Sitoe, 1986), introduced “to suit educational purposes” (Ouwehand, 1965, p. 62), but that he continued to use, at home, the original base five / base ten number-words (see the table on the next page). These were opinions that were presented directly to us. But we had an interesting complementary experience: during the workshop led by Arthur Powell, the participants, organized in small groups, had to do some calculations on a so-called Conway sequence. These were elementary calculations, mainly with natural numbers, whole numbers and fractions. We noted that in some groups the discussions were conducted in Tsonga, but for the number-words the English names were used. In other groups, the whole discussion was conducted in Tsonga, except for the calculations with fractions, for which the English names were used. Our students in Beira would not be able to conduct this kind of mathematical discussion in their mother tongues: nearly all their mathematical knowledge and experience depends on the use of the Portuguese language.

53 Explorations in Ethnomathematics

n Tsonga (Changane) Tsonga (Bantu Education, Gazakulu) 1 xin’we xin’we 2 swimbirhi swimbirhi 3 swinharhu swinharhu 4 mune mune 5 ntlhanu ntlhanu 6 ntlhanu ni xin’we tsevu 7 ntlhanu ni swimbirhi nkombo 8 ntlhanu ni swinharhu nhungu 9 ntlhanu ni mune kaye 10 khume khume 11 khume ni xin’we khume ni xin’we 12 khume ni swimbirhi khumembirhi 13 khume ni swinharhu khumenharhu 14 khume ni mune khumemune 15 khume ni ntlhanu khumentlhanu 16 khume ni ntlhanu ni khumetsevu xin’we 17 khume ni ntlhanu ni khumenkombo swimbirhi 18 khume ni ntlhanu ni khumenhungu swinharhu 19 khume ni ntlhanu ni mune khumekaye 20 makume mambirhi khumbirhi or makumembirhi 23 makume mambirhi ni khumbirhinharhu swinharhu

Conclusions

54 and Ethnoscience in Mozambique

1. Let children use their hands, not only for counting but also for calculating: the gestures used by many people correspond directly to the base five number-words of languages like Tsonga. But even when children use a base ten spoken numeration system, as in English, Portuguese, Zulu or Sena, the use of gestures with both hands gives access to some of the advantages of the base five numeration systems and may help children to find easily, by gesture calculation, the sums of the type eight plus five equals thirteen. 9 2. Let children use the knowledge contained in the languages they know, independently of what the language of instruction is. 3. In particular, even when the language of instruction is an official language (like English or Portuguese), use should be made of children’s knowledge in other languages. In Mozambique, and probably in other African countries too, children have contact with several languages, especially in the urban areas. 4. The idea of simplifying the spoken numeration systems existing in the local Bantu languages is understandable for greater numbers (three or more digits), because of the length of the expressions, especially in languages that use the auxiliary base five. However, it seems to me that the use of the base five numerals may constitute an important resource for children during the earlier stage of mastering the basic number facts of addition and subtraction and of the discovery of the first methods of oral and mental arithmetic. In the case of greater numbers, where pupils will have the written symbols at their disposal, any eventual simplification of an existing numeration system should take into account the possibilities of early Mathematics learning. Throwing away the base five number-words for six, seven, eight and nine may be a mistake.

9 Several authors living in countries with base ten spoken numeration systems recognize the importance of an auxiliary base five in the transition from counting strategies to calculation strategies. Cf. e.g. Hatano (1982, p. 213, 216: Japan), Treffers & De Moor (1990: The Netherlands). 55 Explorations in Ethnomathematics

References (in brackets: translation of some of the titles)

Assembleia Popular (1981): Sistema Nacional de Educação. Linhas Gerais (Resolução nº 11/81 of 17 December 1981) [National System of Education. General outline], Maputo Coelho, M. & Barca, A. (1986): Atlas geográfico – Volume 1, Ministério da Educação, Maputo & Esselte Map Service, Stockholm Conselho Coordenador do Recenseamento (1983): Os distritos em números [The districts in numbers], volumes 1 – 10, Direcção Nacional de Estatística, Maputo Draisma, J.; Soares, M. C.; Burre, G. & Marques, H. (1983): Eu vou à escola – Matemática, 1ª classe [I go to school – Mathematics, Grade 1], Vol. 2, INDE, Maputo Draisma, J., Kuijper, J., Neeleman, W. & Tembe, A. (1986): Mathematics education in Mozambique, in Proceedings of the Fourth Symposium of the Southern Africa Mathematical Sciences Association (SAMSA), Department of Mathematics, University of Swaziland, Kwaluseni, 56-96 Draisma, J. (1991): Trabalho Escrito Individual de 13.02.1992 – Respostas Dadas e Comentários [Written individual test of February 13, 1992 – Students’ answers and lecturer’s comment], Higher Pedagogical Institute – Beira Branch, Beira (internal paper, 40 pp.) ___ (1992): Arithmetic and its didactics. Report on activities (lectures and practical activities) realized during the 1st and 2nd semester (August 1991 – June 1992), Instituto Superior Pedagógico, Beira, 42 pp. ___ Ed. (1993): Entrevistas sobre o cálculo mental em algumas escolas primárias da Ponta-Gêa (Beira) –Volume 1: Entrevistas a professores; Aulas observadas [Interviews on mental arithmetic in some primary schools of Ponta-Gêa (Beira) – Volume 1: Interviews with teachers; Class observations], Instituto Superior Pedagógico, Beira (internal paper, 81 pp.) Firmino, G., Afido, P.; Heins, J.; Mbuub, S. & Trinta, M. (1989): Relatório do primeiro seminário sobre a padronização da

56 and Ethnoscience in Mozambique

ortografia de línguas moçambicanas [Report of the first workshop on the standardization of the orthography of Mozambican languages], INDE-EMU/NELIMO, Maputo Fletcher, C.: Tinhlayo le’tintshwa ta ka Juta - Ntangha 3, 4, Juta & Company, Cape Town (translated into Tsonga by R. Mabale) Gerdes, P. (1981): Changing mathematics education in Mozambique, in Educational Studies in Mathematics, Reidel, Dordrecht/Boston, Vol. 12, 455-477. ___ Ed. (1992): A numeração em Moçambique [Numeration in Mozambique], Instituto Superior Pedagógico, Maputo, 159 pp. Gleimius, E.; Stals, T. N. & Marivate, C. T. D.: Tinhlayo ta tsakisa – Ntangha 2, Varia Books, Alberton Guthrie, M. (1967): Comparative Bantu, London Hatano, G. (1982): Learning to add and subtract: a Japanese perspective, in Carpenter, T. P.; Moser, J. M.; Romberg, T. A. (Eds), Addition and subtraction: a cognitive perspective, Lawrence Erlbaum, Hillsdale, New jersey Kilborn, W. (1990): Evaluation of textbooks in Mozambique. Mathematics grade 1 – 3, University of Gothenburg, Gothenburg Kindler, J. & Minter. W. (1971): Matemática – 1ª Classe, published by the Conferência das Organizações Nacionalistas das Colónias Portuguesas (CONCP), for the use in the primary schools of MPLA (Angola), PAIGC (Guiné Bissau) and FRELIMO (Mozambique). Kindler, J. (1971): Manual do professor – Matematica 1ª classe [Teacher’s guide –Mathematics, Grade 1], Departamento de Educação e Cultura, Frelimo Machel, S. M. (1970): Educate man to win the war, create a new society and develop our country, in Machel, Mozambique: sowing the seeds of revolution, Committee for Freedom in Mozambique, Angola & Guiné Bissau, London Machel, S. M. (1974): Faire de l’école une base pour que le peuple prenne le pouvoir, in Machel, S.: Le processus de la revolution démocratique populaire au Mozambique, Editions L’Harmattan, Paris

57 Explorations in Ethnomathematics

Ministério da Educação e Cultura, 1973: Ensino primário. Organização política e administrativa. Programas e directrizes pedagógicas [Primary education. Political and administrative organization. Syllabi and pedagogical guidelines], Imprensa Nacional de Moçambique, Lourenço Marques Mmari, G., 1978: The United Republic of Tanzania: Mathematics for social transformation, in Swetz, Frank (Ed.), Socialist Mathematics Education, Burgundy Press, Southampton PA, 301- 350 Mondlane, E. (1969): The struggle for Mozambique, Penguin Books, Harmondsworth, Middlesex (Reprinted with a new Foreword and Biographical Sketch by Zed Press, London, in 1983) Ouwehand, M. (1962): Everyday Tsonga, Sasavona Publishers, Braamfontein Price, T. (1983): A short English – Nyanja vocabulary, NECZAM – National Educational Company of Zambia, Lusaka (First published in 1959. Revised in accordance with standardized Zambian Orthography in 1983) Ribeiro, A. (1965): Gramática Changana (Tsonga), Editorial “Evangelizar”, Caniçado Silva, J. & Draisma, J. (1991): Master’s Degree in Mathematics Education for Primary Schools, paper presented to the 8th Symposium of the Southern Africa Mathematical Sciences Association (SAMSA), held in Maputo, 16-19 December 1991. Sitoe, B. (1986): Byi xile! Curso experimental de Tsonga [Hello! Experimental course of Tsonga], volumes 1 and 2, EMU, Maputo Treffers, A. & De Moor, E. (1990): Proeve van een national programma voor het reken—wz’skunde onderwijs op de basisschool. Deel 2: Basisvaardigheden en cijferen [Proposal of a national syllabus for the teaching of arithmetic-mathematics at the basic school. Part 2: Basic abilities and written computation], Zwijsen, Tilburg Veloso, M. & Draisma, J. (1992): Bukhu ya kupfundzisa makonta. Malongero a cisena [Mathematics book in Sena language], Instituto Nacional do Desenvolvimento da Educação, Maputo

58 and Ethnoscience in Mozambique

Zepp, R., 1990: Language and Mathematics Education, API Press, Hong Kong

List of abbreviations

CFPP Centro de Formação de Professores Primários [Training Centre for (lower) Primary Teachers – run by the State, after Independence] CONCP Conferência das Organizações Nacionalistas das Colónias Portuguesas [Conference of Nationalist Organizations of the Portuguese Colonies] EHPP Escola de Habilitação de Professores de Posto Escolar [School for the Training of (African lower primary) Teachers – run by the Roman Catholic Church, before Independence] EMU Eduardo Mondlane University (Maputo) FRELIMO Frente de Libertação de Moçambique [Mozambique Liberation Front] IMP lnstituto Médio Pedagogico [Medium level Pedagogical Institute – created by the State in 1983, for the training of Primary Teachers in a 2 or 3-year course after 10 years of general education] INDE National Institute for the Development of Education (Maputo) ISP Instituto Superior Pedagógico [Higher Pedagogical Institute] (since 1995 Universidade Pedagógica – Pedagogical University) LEMEP Licenciatura em Educação Matemática do Ensino Primário [Master’s Degree Programme in Mathematics Education for Primary Schools] NELIMO Núcleo de Estudo de Línguas Moçambicanas [Nucleus for the Study of Mozambican Languages – based at the Eduardo Mondlane University] NSE National System of Education (Mozambique, since 1983)

59 Explorations in Ethnomathematics

SAMSA Southern Africa Mathematical Sciences Association

Annex 1:

Types and levels of Teacher Training for Lower Primary Schools in Mozambique (Grades 1 - 5)

Before Independence (1975)

Location Level of teacher training Training Entrance level Duration of Institution Professional Training Schools in the Grade 4 no training rural areas (for “blacks”) Schools in the Grade 4 1 month Inspectorate rural areas Schools in the Grade 4 4 years Escola de rural areas (including 2 Habilitação de years of general Professores de education) Posto (EHPP) Schools in the Grade 9 2 years Escola de urban areas (for Magistério “whites” and Primário “assimilados”) 10 After Independence (1975)

All primary Level of teacher training Training schools Entrance level Duration of Institution Professional

10 On the Portuguese colonial policy of assimilation, see Mondlane (1983, 48-50) 60 and Ethnoscience in Mozambique

Training 1976 – 1986 Grade 6 1 year Training Centre for Primary Teachers (CFPP) 1983 – 1990 Grade 6 3 years Training Centre for Primary Teachers (CFPP) 1990 – 199? Grade 7 of 3 years Training Centre NSE for Primary Teachers (CFPP) 199? – 20?? Grade 10 of 2 years Instituto Médio NSE Pedagógico (IMP)

61 Explorations in Ethnomathematics

62 and Ethnoscience in Mozambique

Chapter 5

Abílio Mapapá

Symmetries and metal grates in Maputo – Didactic experimentation 1

Introduction

Whoever visits Mozambique’s capital, Maputo, can appreciate an interesting feature: many buildings have metal grates on their windows and doors for protection. The owners of the buildings desire a beautiful shape for their grates. For many of them this means that the grates must be symmetrical. During my fieldwork in 1991 I collected over a hundred of different grate patterns. I interviewed the craftsmen who make them. In this paper some patterns will be presented together with the craftsmen’s nomenclature. Their classification will be compared with the mathematical classification of two-dimensional symmetry patterns that has its origin in crystallography. With a group of students of Maputo’s Upper Primary Teacher Training College I explored the possibilities of using the metallic grates – belonging to the cultural environment of teacher and students

1 Paper prepared for the Boleswa Regional Conference Science and Mathematics Education, University of Botswana, October 1993. It is based on my M.Ed. thesis “Simetrias e gradeamentos metálicos em Maputo”, Instituto Superior Pedagógico, Maputo, 1992 (Supervisor: Paulus Gerdes). 63 Explorations in Ethnomathematics

– as a starting point for the study of the aforementioned mathematical classification of symmetries. My experiences with this didactic experimentation will be brieﬂy described.

S-type Snake-type

Rhombus-type Ladder-type Figure 5.1

The artisans’ classification

It is interesting to see that the Maputo grates makers give names to their products. They choose the names of the grate designs through comparison with objects used frequently, like a letter of the alphabet or features that appear in the Nature. When the letter “S” appears in the grate pattern they call it a grate of the “S-type”. If a pattern appears which is similar to a snake they call it a grate of the “snake-type”. 64 and Ethnoscience in Mozambique

When they produce grates that have rhombus shaped holes, they call them grates of the “rhombus-type”. If shapes can be identified in the grates like ladders, the makers call them “1adder- type” grates. Figure 5.1 presents examples of these types. Do the grates makers use mathematics, when they are making these products? Mathematical aspects appear clearly in the making of grates: the measurement for correct cut and correct bend of the steel bars. Symmetries occur. Is it only occasional?

A classification of two-dimensional grate patterns by symmetries

Most of the grate patterns display symmetries. If a plane figure admits translations in two or more directions it is called a two-dimensional pattern. Eighty five of the grating patterns I collected are two-dimensional; the rest are one-dimensional, i.e. they are strip patterns. Classifying two-dimensional patterns by symmetry leads to 17 distinct classes (cf. e.g. Wasburn & Crowe, 1988, p. 58, 128). Of these 17 classes I found examples of 12 of them among the grating patterns I had collected. Table 5.1 presents the distribution of the grating patterns by class; the classes are indicated by the so-called short international notation, which had its origin in crystallography. Figure 5.2 displays illustrative grating patterns for each of these 12 classes. The flowchart reproduced from Wasburn & Crowe (1988, p. 128) explains both how to classify a given pattern and the notation.

cm pm p1 pmm cmm pmg pgg p2 p4m p4g p4 p6m 4 2 1 28 22 3 1 8 12 1 1 2

Table 5.1

Patterns belong to the classes pmm, cmm and p4m, appeared more than ten times. On the other side patterns belonging to the classes pg, p3, p3m1, p31m and p6 did not appear. Why did this happen?

65 Explorations in Ethnomathematics

Flow chart for the seventeen two-dimensional patterns

66 and Ethnoscience in Mozambique

p1 pm

cm p2

pmm cmm Figure 5.2 (First part)

67 Explorations in Ethnomathematics

pmg pgg

p4 p4m

p4g p6m Figure 5.2 (Second part)

68 and Ethnoscience in Mozambique

Didactic experimentation

When I had studied the classification of two-dimensional patterns by their symmetry groups and had applied this theory to the gratings in Maputo, I posed myself the following question: As gratings belong to the cultural environment of students of the teacher education institutions in Maputo, is it possible to introduce them in a geometry course for these students and have them discover the different classes? Might it be a way to consolidate their knowledge about symmetries in the Euclidean plane? With a group of volunteers who were students of the third year of Maputo’s Upper Primary Teacher Training College I explored in a workshop in 1992 the possibilities of using the metal grates as a starting point for the study of the aforementioned mathematical classification of plane pattern by symmetries. One objective of this didactic experimentation was to apply the geometry and algebra already studied by students to the analysis of grating patterns. The didactic experimentation took place in three phases. Without my help, the students collected first some of the grate patterns in Maputo. Thereafter the students compared their collections and tried to analyse them in function of the symmetries they display. My role was that of questioning their results and so to stimulate their reflection. Once it was found that different symmetry classes of patterns exist, the aforementioned flowchart was introduced and the students were invited to invent their own patterns, including ones belonging to classes not represented by the grates’ patterns. ‘ The students found the theme interesting. One problem was that the workshop was near to the end of the semester and the students were worried about upcoming exams, so that time was short for this workshop with voluntary participation. I hope to be able to conduct more workshops and to pursue my didactic experimentations.

69 Explorations in Ethnomathematics

References

Washburn, D. & Crowe, D. (1988): Symmetries of culture: Theory and practice of plane pattern analysis, University of Washington Press, Seattle.

70 and Ethnoscience in Mozambique

Chapter 6

Daniel Soares

Symmetric ornamentation on wooden spoons from the Sofala Province – a study in progress

In 1991 I arrived in Beira, the capital of Mozambique’s central Province of Sofala, and started to work as a mathematics lecturer at the Beira branch of the Institute Superior Pedagógico. At local markets, interestingly decorated wooden spoons drew my attention. Initially it seemed that there were only a few decorative motifs. Gradually it came out that there are in fact many, so I began to make a collection of these spoons.

Photograph 6.1 Photograph 6.1 shows three spoons; their motifs display a double axial symmetry. We are dealing in this case with woodcarvings on the haft of the spoon. Before carving, the haft has been blackened by fire.

71 Explorations in Ethnomathematics

Photographs 6.2 and 6.3 show details of two other spoons produced in the same manner. This time insects are represented.

Photograph 6.2

Photograph 6.3

Another technique of decorating wooden spoons consists of carving the figures with hot knives. Photographs 6.4, 6.5 and 6.6 display three examples with double, axial and half turn symmetry respectively. I intend to expand the collection and to interview the artisans who produce the spoons in order to understand better their skill, knowledge and art. Further, it is intended to analyse possibilities to explore mathematically and educationally the art of wood carving in Sofala.

72 and Ethnoscience in Mozambique

Photograph 6.4

Photograph 6.5

73 Explorations in Ethnomathematics

Photograph 6.6

74 and Ethnoscience in Mozambique

Chapter 7

Marcos Cherinda

Strip patterns on wooden spoons from Inhambane Province – A study in progress

My colleague Daniel Soares showed me his collection of wooden spoons from the Sofala Province in September 1993. I found the collection very interesting as an example by which we can establish a relationship between mathematics and Mozambican culture. In January 1994 I visited the Children and Adults Special Centre of Education in the Jangamo District in Inhambane Province. During that time I met the artisan Marcelino Nhanala, who told me about his work on wooden spoon ornamentation. As in Sofala Province, two different decorating techniques are used. The old technique consists of carving the decorative figure with a hot metallic object. Most of the motifs done by this technique are circles and segments of straight lines. These are done because of the instruments used, such as a stamper. The artisan uses a piece of iron pipe to make circles and a straight iron piece (e.g. a knife) to make a straight line segments (see Photograph 7.1).

75 Explorations in Ethnomathematics

Photograph 7.1

The newer technique consists of blackening the surface of spoon on which the decorative figures will be made with fire. After that, the decoration is made by carving the blackened area, using a knife (see Photo 7.2). At the local markets in Maxixe and Inhambane City I took photographs of several decorative motifs. According to the classification of strip patterns based on symmetry groups, five different classes of the seven possible ones are represented in my collection so far. Figure 7.1 shows us one example of each class. We intend to collect more decorative motifs on spoons from Inhambane Province, and will compare them with motifs from other parts of the Country.

76 and Ethnoscience in Mozambique

Photograph 7.2

77 Explorations in Ethnomathematics

Figure 7.1

78 and Ethnoscience in Mozambique

Chapter 8

Mário Baloi

Ideas about Nature: Traditional Interpretations of Thunder and Lightning in Catembe and Physics Teaching

Introduction

Interviews conducted by the author about lightning and thunder among future physics teachers in Mozambique showed that more than 90% of them are heavily influenced by traditional-popular interpretations of these natural phenomena. This led the author to analyse in more depth these interpretations through field work in Southern Mozambique. This paper describes the traditional interpretations by tinyanga in the Matutuíne District, of thunder and lightning discharge to the ground. Their interpretations will be analysed in view of the explanations by “physical science” in history. The paper closes with a reflection about physics teaching in Mozambique.

Traditional interpretation of thunder and lightning

We carried out fieldwork from September to November 1992 in Catembe in the Matutuíne District, south of the Bay of Maputo (formerly Delagoa Bay). Two of our students Elidja Machavane and Patrício José Manuel, who were born and lived for many years in this region, helped me to identify the informants and to carry out the

79 Explorations in Ethnomathematics

interviews. To communicate with the informants we spoke Ronga, the language spoken in the Matutuíne District, which is rather similar to Changana, my own mother tongue. Our principal informants were Buhlafa Siyinda, Hlunguza Nwamba Tembe, Pedro Matlaba and Tchali Ndawana, who were born in Matutuíne and are living in Catembe. They are locally known generally as tinyanga (singular: nyanga), 1 and some of them have been trained as such in Natal-Kwazulu (South Africa). 2 But they belong to a special category of tinyanga: the tingedla (singular: ngedla). The tingedla are the traditional agents who can mediate between the people and the heavens, avoiding further thunder and lightning discharge in living areas or using their wisdom to bring bad luck to certain persons. They are also involved in other activities such as prediction, producing rains, modifying people’s luck, and in communicating with ancestors’ spirits. The population believes that the tinyanga and tingedla can negotiate the heavens’ provocations – thunder and lightning – with the ancestors’ spirits, aiming to avoid further thunder and lightning discharge on their houses. It is also believed that their personal enemies can cause thunder and lightning. Therefore it is thought if those who suffered from lightning discharge should communicate their accident to the tingedla to avoid further trouble with future thunder and lightning. If one does not look for a nyanga or ngedla one must leave the place or the home otherwise other lightning discharges will probably happen. According to the tingedla two kinds of thunder and lightning exist, one caused by Man, one caused by Nature. In order to explain them, I have to introduce some expressions in the Ronga language. Generally the word tilo means “heaven”. The expression tilo dza kuhambiwa means literally “heaven made”; it refers to “thunder and /

1 I limit myself to the role of the tinyanga in relationship to thunder and lightning. Other roles have been analysed by Junod (1934, p. 415) and (Polanah, 1987, p. 39). 2 Cf. the study of Kriege & Martzburg (1936, pp. 310-320) about the role played by the so-called “Heaven-Herds” in the interpretation of lightning among the Zulu. 80 and Ethnoscience in Mozambique

or lightning made”, i.e. caused by human beings. 3 The expression tilo dza tumbuluku means literally “heaven of Nature”; it refers to “thunder and/or lightning of the nature”, i.e. caused by Nature. Tilo dza duma means “heaven makes noise”, i.e. “heaven thunders”. The expression kurumela tilo, literally “to send heaven”, means “to send thunder and / or lightning”. After a “thunder or lightning” discharge, the ngedla must go to that place to extract “lightly stuff” from the underground that is caused by thunder and / or lightning discharge to the ground. 4 This material will be then be kept in an animal horn called mondzo or in another object that has the shape of a chicken (locally known as ngwama). Thereafter a ritual ceremony is organized to prevent further thunder and lightning discharge at the same place. The ngedla uses several roots of trees and plants to communicate with “spirits in heaven”. The “lightly stuff” in the horn is kept away from enemy spirits, to avoid a further lightning discharge. A rod is constructed at the place of lightning discharge (functioning as a future “lightning rod” for protection). 5

Fragments of the history of the physical conception of lightning

In the history of physics there are a lot of assumptions about lightning that have been refuted or adapted by today’s science. The following examples collected a book by Park (1895, pp. 570-580) show that the interpretation, analysis and concept of lightning have been the subject of controversy: “Lightning was due to a mixture of nitrous sulphurous vapours in the air”; “sulphurous odour following a lightning stroke”; “flickering lights in the globe were compared to the

3 The word lihati meaning only “lightning” is also used in southern Mozambique. 4 In fact, in the case of lightning discharge on sand, the heat melts it and produces the so-called fulgurite. I suppose that some tinyanga or tingedla might occasionally have had observed such fulgurite. 5 In further research I will try to find evidence of the existence of a traditional lightning conductor. 81 Explorations in Ethnomathematics

lightning flash”; “crackling and sparks of rubbed amber are a resemblance of thunder and lightning”; “electric fire and lightning are of the same nature”; “the electrical matter agrees with fire and are composed by similar particles. If they are forced together, then the fire becomes more lightning is produced by a great quantity of fire driven together and is of the same nature of electricity celestial fire amasses, envelopes, and discharges itself with explosion called thunder”; “lightning-stroke consist of immense number of sparks and are more powerful than an electric spark. The combined exploration of sparks cause the thunder”; “lightning-stroke is due to ignited gas”; “lightning is due to sulphur fired in the air and earthquakes are of the same substance ignited underground”. Park quotes the physicist D. Lardner who wrote 1844 that “it would be a curious and interesting result of scientific investigation to demonstrate that the thunder of heaven elaborates in the clouds the chief ingredients of the counterfeit thunders which man has invented for the destruction of his fellows” (Park, p.570). “Magic lightning” in Mozambique is based on similar assumptions, as it admits the existence of a counterfeit thunder or lightning produced by Man. Benjamin Franklin wrote statement about “electrical fluid” and was first to design an experiment to prove that lightning was electrical. This experiment was successfully performed and sparks were observed to jump from an iron rod during a thunderstorm. In his time the problem was to explain the nature of the lightning, i.e. lightning as an electric phenomenon (cf. Park, 1895). In the textbooks and scientific dictionaries lightning is today treated, for instance, as follows: high- energy luminous electric discharge between a charged cloud and a point on or connected to Earth’s surface: this is know as forked lightning. Alternatively the discharge may occur between two charged clouds or between oppositely charged layers of the same cloud: this is known as sheet lightning. The potential difference required to initiate a ﬂash is about 108 Volt. Generally there is a downward leader stoke, i.e. partial discharge, followed by an upward return stoke, the later being much more luminous. The average current is about 10 000 amperes but maximum values of around 20 000 amperes, associated with a temperature of about 30 000K, have been obtained. A typical lightning flash consists of four or five stokes at about 40 milliseconds apart “ (Lord, 165-166).

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Perspectives for teaching “lightning” in Mozambican schools

The teaching of the subject of “lightning” in Mozambican schools has, in my opinion, to take into account not only the “Euro- American” history of the development of the attempts made to explain what lightning is, but also traditional interpretations of and practices related to “lightning”. Our fieldwork and interviews conducted among our students give us some reasons for redefining strategies of “lightning” teaching in Mozambican schools. I interviewed 68 physics teacher students to find out what they think about thunder and lightning. Only 10% of these students do not present any traditional explanation about thunder and lightning. Seventy % of the students had acquired a traditional interpretation of thunder and lightning before going to secondary school. Only 28% think that from what they learned in secondary school, they could explain the phenomenon of lightning. Pupils in secondary schools but also students of higher education are significantly influenced by some of the cultural meaning of lightning. This means that children have developed their ideas about lightning under cultural influence. How can be these ideas become meaningful in the context of physics teaching? During classroom activities my students were requested to “defend” their own viewpoints based on traditional education. Generally I verified through dialogue that most of my teacher students have strong beliefs about lightning. In the traditional context people constructed their own concepts that enabled them to impose some sort of meaning on the world; through them the reality is given sense, order and coherence. How we perceive the world is determined, in the present case, by the “traditional” concepts developed over the centuries. Now with my students we challenged and discussed those concepts, interpretations and practices, using some scientific ideas existing at that present moment in classroom. I observed among my students a certain shortage of scientific information from secondary school. At the moment I am looking for ways to improve the integration of ideas from the traditional culture in the classroom. 83 Explorations in Ethnomathematics

Internationally quite a lot of research in physics learning and teaching is being undertaken to find out new approaches. Scott, Asoko & Driver (1991, 310-327) analyse the so-called “teaching strategies based upon cognitive conﬂict and its resolution”. According to those scientists, cognitive conflict has been used as the bases for developing a number of approaches to teaching for conceptual change. These approaches involve promoting situations where the student’s existing ideas about some phenomenon are made explicit and are then directly challenged in order to create a stage of cognitive conflict. Attempts to resolve this conﬂict provide the first steps to any subsequent learning. 6 In further research I will experiment with these new approaches in my lecturing of “Didactics of Physics” in order to find those better ways for incorporation of traditional conceptions, interpretations and practices, like those about thunder and lightning, in formal education.

References

Junod, H. (1934 [1974]): Usos e costumes dos Bantos, Volume 2, Imprensa Nacional de Moçambique, Lourenço Marques. Kriege, E. & Martzburg, P. (1936): The social system of the Zulu, Shuter & Shooter, Pietermaritzburg. Lord, M. (1986): Dictionary of Physics, Macmillan, London. Park, B. (1895): The Intellectual Rise In Electricity, A History, D. Appleton and Company, New York.

6 One of them is called the “Generative Learning Model of Teaching” proposed by Cosgrove and Osborne (1985), which is organised into four phases: Preliminary phase (teacher needs to understand the scientists view, the children’s view, his/her own view); Focus phase (opportunity for pupils to explore the context of the concept, preferably within a “real” everyday situation. Learners to engage in clarification of own views); Challenge phase (learners debate the pros and cons of their current views with each other and the teacher introduces the science view); Application phase (opportunities for application of new ideas across a range of contexts). 84 and Ethnoscience in Mozambique

Polanah, L. (1987): O NHAMUSSORO e as outras funções mágico- religiosas, Coimbra University, Coimbra. Scott, P.; Asoko, H. & Driver, R. (1991): Teaching for Conceptual Change: A review of Strategies, in: R. Duit, F. Goldberg & H. Niedderer (Eds.), Research in Physics Learning: Theoretical Issues and Empirical Studies, IPN, Kiel, pp. 310-329. Uman, M. (1971 [1986]): All About Lightning, Dover, New York.

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Chapter 9

Mário Baloi, Adão Matonse & Horácio Simão

Perspectives in Ethnophysics

The difficulties faced by the Faculty of Natural Sciences and Mathematics at the Instituto Superior Pedagógico, in the process of teaching different subjects in the field of physics – related to the behaviour of the students and their attitudes towards the importance and the role of physics in society, as well as their understanding of several physical concepts and phenomena – indicate that the way the students learned physics at secondary school must be changed. Pupils feel physics to be something alien to their lives, and at the same time they find it difficult to apply their knowledge to the resolution of practical everyday problems. Staff and students appreciate that the aforementioned situation is not only a result of current poor teaching conditions, but also of the necessity to bring this science closer to the cultural environment of the pupils. The research recently begun in ethnophysics must be seen in this context. For the beginning it has been important for staff members to know the answers to these questions:

1. How do people in Mozambique understand natural phenomena and how do they explain them? What concepts do they use? 2. How do people traditionally solve practical everyday problems, what technology do they use, and how do they explain the techniques (methods) they use?

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The following topics have been selected for a first study. During the last two years students have prepared some papers on these topics and presented them at “Scientific Journeys” for students in our Faculty.

A small timbila seen from below and from above Figure 9.1

Musicians’ assumptions about traditional music and instruments

In the Province of Inhambane, lecturer Mário Baloi has started fieldwork among Chope musicians. The aim is to find out what descriptions of traditional musical instruments, like the famous timbila (a type of xylophone), are given by the musicians (See Figure 9.1). They are asked to describe their concepts and ideas about the principal parts of the instruments, their function, sound production and the way of playing them. For example, how does the musician tune his

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instrument? It is also intended to compare the processes of music teaching in traditional culture in Mozambique, and their relationship to physics and Euro-music teaching.

Brandy-making

The student Filipe Viola analyses the know-how involved the distillation process and the calorimetry used in the traditional brandy- making process. He analyses the technology applied in the western- central province of Manica. “How is the technology explained?”, “How is the distillation process interpreted” and “How is the quality of the brandy evaluated by its producers?” are some of the questions Viola posed. He also proposes some ways to link relevant aspects from the traditional distillation process with topics in the physics programme at secondary school.

Traps

Many people from the District of Boane in Maputo Province regularly make traps. More than a hundred traps are constructed, mostly during the weekend, and located in the uncultivated region of Gumi or Matsekela in that district. Some of the constructors are professional hunters; others are poachers with many years of practice. These traps – the small mechanical devices producing great force – are made to catch small animals. There are different kinds of these traps, which are locally known as “lithaka”, “lithamu”, “nteve” and “ligubu”. The lecturers Horácio Simão and Mário Baloi, together with the student Felisberto Cupane, interviewed several hunters and paid special attention to the mechanism of these traps and the process of production. They have tried to understand how the producers analyse the process of hunting, how they make judgments about the performance of traps, which concepts (in the local language) they use and how they evaluate their experience of getting small animals to fall into the traps. The researchers expect to use these mechanical devices to challenge the students in the classroom about the physical

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interpretation, with reference to the forces involved and to take measurements of strengths and force.

These topics constitute examples of early research. As seen in the previous chapter, Mário Baloi advanced with a study of lightning and thunder interpretation and the didactics of physics. We intend to establish a larger research programme at our institution. In the context of this research programme the following focuses will be considered: knowledge developed among different ethnic groups in different regions of the country will be compared. We will try to reconstruct the (historical) development of physical- technological know-how. The incorporation of the research results in school education and teacher education will be explored. We believe that the involvement of students in these research activities will give them an opportunity for better understanding of their environment and for developing new ideas about physics teaching and the “incorporation” of the environment. We also believe that by applying the results of this research in schools we can contribute to the reduction of the gap between school curricula and the social and cultural environment of the pupils.

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Chapter 10

Cristiano Pires & Felisberto Lobo

Perspectives in Ethnobiology

In a context like Mozambique where there exists, especially in the rural areas, accumulated wisdom of appropriated (indigenous) technologies, the development of ethnoscience in general and of ethnobiology in particular, acquires vital importance for the people. In this way ethnobiology is becoming a new and attractive research field at the Department of Biology of Mozambique’s Instituto Superior Pedagógico (ISP). Ethnobiological activities are aimed at developing environmental awareness so as to provide students with a better understanding of their surrounding environment. Another important objective is to investigate ways and models of interaction between biological indigenous practices and “modern” biological science in order to provide the students with a better understanding of science. The investigation also aims to introduce new topics in the school biology curricula as well as in out-of-school education. The integration of cultural knowledge in school programmes will enable science teaching to proceed from the known to the unknown. In this way local thinking and understanding of Nature can create a basis for science teaching. We also have in view the dissemination, at a national level, of several indigenous practices from different regions of the country, as there are significant cultural differences among them. In this paper examples both of research already done and of on- going investigations will be presented, as well as some perspectives.

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Mulala

“Mulala” is a widely propagated plant (Euclea natalensis), whose roots are used for cleaning teeth. One of our students, Rafael Bernardo, 1 wrote his M.Ed. thesis on the “mulala”, analysing its inhibitor action on mouth micro flora and comparing it with the toothpastes sold in Maputo. His results show that there are significant differences between filtered and non-filtered extracts, in their effect on bacteria (Candida albicans, Lactobacillus spp, Streptococcus mutans) and on fungi (Saccharomyces cerevisiae). The antibiotics compared with the effects of “mulala” were Nystatin and Tetracycline. Different kinds of “mulala” show different results concerning their action on the mentioned micro-organisms. The student also described other traditional uses of the “mulala”. It is applied for the treatment of different diseases. Its leaves are used in combination with the leaves of another species (Tabernaemontana elegans) for the treatment of malaria. The leaves are first boiled and the patient bathes in the infusion. The roots of “mulala” serve as a means to combat lepra, diarrhoea and skin diseases.

Indigenous contraceptive methods

Ana Wamir is writing a M.Ed. thesis on the use of some indigenous contraceptive methods during the lactation period. She is trying to find out the methods traditionally used to stop fertilization. She also intends to learn about how these methods act on the hormonal system of the wife. So far, she found “nembenembe” (Cassia petersiana) and “n’kuhlo” or mafureira in Portuguese (Trichillia emetica) as examples of plants with contraceptive effects. It seems important to disseminate the traditional methods at a national level so that more people can use them for family planning. In this way such methods may be integrated in the biology school curricula.

1 Rafael Lambo Bernardo (1993): O efeito microbicida da “mulala” e a sua contribuição para a hygiene bucal, 70 pp. 92 and Ethnoscience in Mozambique

Traditional methods of treating asthma

Silva Mujovo is writing his M.Ed. thesis on traditional methods of treating asthma. Asthma is one of the more widespread diseases in Mozambique. Traditional healers possess various methods (plants, animals, etc.) of prevention and treatment of asthma. For example, garlic (Allium sativum) and chameleon are used. Mujovo proposes to investigate the different ways used by them in treatment of the different kinds of this disease.

Pests and parasites

We also have in view an investigation of indigenous methods of combating both plant pests and plant parasites. For example, rural people use “nembenembe” (Trichillia emetica) as a repellent against animal parasites of some kind of agriflora. The use of such traditional methods may avoid or, at least, reduce the use of chemical pesticides with their negative effects on the environment. The integration of this type of indigenous knowledge into the biology curriculum may also contribute to a better awareness of factors that affect the quality of the environment.

Language and biology teaching

So far (almost) all teaching in Mozambican schools takes place in the official language of the country, i.e. in Portuguese. We propose to consider linguistic aspects of biology teaching. In this respect we will try to analyse the possibility of using African languages in teaching some biological concepts and contents in schools. This investigation aims to evaluate under which pedagogical situations, and up to which level it would be necessary and possible to introduce African languages in biology teaching.

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Chapter 11

José A. Barros & Luís Ramos

Perspectives in Ethnochemistry

Indigenous knowledge in Chemistry may play a basic but also a very important role in Mozambican society. If we see Chemistry as a science that contributes to a deeper understanding of the changes of substances, which in turn teaches how to benefit from Nature, we can say that Chemistry emerged all over the world long before it became “scientific” in 17th century Europe. In Mozambique, we can surely find a lot of chemical - technological experiences in the daily life of the population, in particular in the countryside, that can be used and developed and may be introduced in the school Chemistry curricula, in order to improve the quality of the teaching of chemistry and to facilitate the popularisation of this science. Using our own examples of traditional technology and chemical knowledge, for example, of substances like soap, salt, paint, ceramics, oil, alcohol, or methods like distillation, extraction and painting, a bridge can be laid between traditional and formal education. In this way the pupils’ interest in Chemistry may be enhanced. This seems to us extremely important, as presently Chemistry is one of the less popular subjects in school and only a few students continue Chemistry at a higher level. In the two Chemistry departments at Mozambique’s Instituto Superior Pedagógico (in Maputo and Beira) research in Ethnochemistry has just started. The following topics constitute research themes currently under study.

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Traditional extraction and use of metals

It is intended to collect information about the knowledge involved in working with iron, gold and other metals and to reconstruct the local history of the use of metals as well as the methods for their extraction. Some results can be used to enrich the reflection on the history of chemistry. Both scientific articles and didactic- methodological recommendations are expected to be elaborated.

Traditional methods of oil production

The oil based on the “mafurra” seed is largely used by the Mozambican population in their food and health system. The “mafureira” (Trinchilia emetica) is a plant, which can be found all over Mozambique, particularly south of the Save river with high incidence in Zavala and Inharrime in the Inhambane Province and also in the Gaza Province. From this plant the “mafurra” oil is extracted. This oil is commercialized locally and was used in soap industry during the colonial period. Artisans extract the oil and secondary products, using water as solvent, heat and salt for the sedimentation of secondary products and ceramic materials, like “ndzeca”, “tchigalangwana” and “tchibembe” as boilers. They also use “ngoto”, “inpahata” and “ngureti” as recipients for the collection and conservation of products, as well as traditional filter materials (like “tchirema” and “inseke” made of straw or palm leaves, which are also involved in the process). Oleaginous products are basic in the food and health system for the Mozambican population. We think that a theme like oil can and should be developed and introduced in the school curricula, in an interdisciplinary way. The study is already in progress, and now laboratory analysis on the constitution of the oil and plant linked to their use as medicine have to be done.

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Soap production

Soap production through the use of “n’hlehlua” (Dicerocarium eriocarpum zangebarium) is a wide-spread traditional technique. Dicerocarium eriocarpum zangebarium is a creeping plant, which is common and spontaneous in many regions of the country. People use its leaves for washing the hair and clothes. The plant contains a chemical substance whose action and feature can be compared with saponina. In this way we try to find out which chemical nature this substance has.

Mulala

The roots of the “mulala” (Euclea natalensis) are not only used directly for cleaning teeth (cf. “Perspectives in Ethnobiology”), but also for the treatment of several diseases like leprosy and diarrhoea after extracting the biologically active drugs from the roots. Various other traditionally used plants can also be analysed in order to find the biological active drugs and the relationship between those and the microorganisms that cause disease as well as the secondary effects in the patient. Examples like this are very important for students in the Chemistry and Biology Education Masters Program, because they involve a whole range of concepts of Biology and Chemistry, showing the importance of interdisciplinarity very effectively.

These are only some examples of indigenous knowledge from the “traditional Chemistry world” that may be used to improve Chemistry teaching and teacher education in a cultural historical and multidisciplinary perspective.

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Where are the authors in 2014?

Mário Suarte Baloi Ph.D. 2002 Möglichkeiten zur Verbesserung des Physikunterrichts durch stärkere Ausprägung der Schülertätigkeit, des projekt-orientierten und des fächerübergreifenden Aspekts am Beispiel der Behandlung des Magnetischen Feldes und der Elektromagnetischen Induktion in Klasse 9, Technische Universität Dresden, Germany. Director, Centre for Educational Technology, Pedagogical University, Maputo. Email: [email protected]

José António Barros (1958 - …) Ph.D. 1997 Entwicklung und Erprobung eines Konzepts zur laborpraktischen Ausbildung von Chemielehrern in anorganischer Chemie mit einem Minilabor unter besonderer Berücksichtigung mosambikanischer Bedingungen, Pädagogische Hochschule Heidelberg, Germany. Deputy Dean, Faculty of Natural Sciences and Mathematics, Pedagogical University, Maputo. Email: [email protected]

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Marcos Cherinda (1963 - …) Ph.D. 2002 The use of a cultural activity in the teaching and learning of mathematics: The exploration of twill weaving in Mozambican classrooms, University of the Witwatersrand, Johannesburg, South Africa. Dean, Faculty of Natural Sciences and Mathematics, Pedagogical University, Maputo. Email: [email protected]

Jan Draisma (1944 - …) Ph.D. 2006 Teaching gesture and oral computation in Mozambique: four case studies, Monash University, Clayton, Australia. Retired. Email: [email protected]

Abdulcarimo Ismael (1962 - …) Ph.D. 2001 An ethnomathematical study of Tchadji – about a Mancala type board game played in Mozambique and possibilities for its use in Mathematics Education, University of the Witwatersrand, Johannesburg, South Africa. Academic Director, Lúrio University, Nampula. Email: [email protected]

Felisberto Lobo M.Ed. in Biology. Lecturer, Department of Biology, Faculty of Natural Sciences and Mathematics, Pedagogical University, Maputo. Email: [email protected]

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Abílio Mapapá Standard Bank, Maputo, Mozambique. Email: [email protected]

Adão Henrique Matonse Ph.D. 2009 Development and evaluation of hill slope discharge models at a watershed scale, State University of New York, USA City Research Scientist, New York City Environmental Protection, Bureau of Water Supply, Water Systems Operations. Email: [email protected]

Cristiano Vicente Pires (1960 - …) Ph.D. 2000 Kosten-Nutzen-Bilanzen in einer Räuber-Beute- Beziehung: Transport-, Zeit- und Energieparameter im Trichterbau- und Nahrungsverhalten von Eoroleon nostras Fourcov (Insecta, Planinennia, Myrmeleontidae), Universität Rostock, Germany. Associate Professor, Department of Biology, Faculty of Natural Sciences and Mathematics, Pedagogical University, Maputo. Email: [email protected]

Luís Ramos (deceased in 2013)

Horácio Simão M.Sc. in Physics. General Director of the Office for Planning, Studies and International Cooperation, Mozambican Tax Revenue, Ministry of Finance. Email: [email protected] or [email protected]

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Daniel Bernardo Soares (1961 - 2011) Ph.D. 2010 The incorporation of geometry involved in the traditional house building in mathematics education in Mozambique: The cases of the Zambézia and Sofala provinces, University of the Western Cape, South Africa.

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