Mathematics Education in Mozambique

Paulus Gerdes

MATHEMATICS EDUCATION IN MOZAMBIQUE

Papers from the Early 1980s Paulus Gerdes

www.lulu.com http://www.lulu.com/spotlight/pgerdes Paulus Gerdes

Mathematics Education in Mozambique: Papers from the Early 1980s

2014 Paulus Gerdes

Title: Mathematics Education in Mozambique: Papers from the Early 1980s

Author:

Paulus Gerdes

C. P. 915, Maputo, Mozambique [email protected]

Proofreading: Lesira L. Gerdes

Publisher:

ISTEG-University, Boane, Mozambique

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

Contents

Page

0 Introduction 9 1 Changing Mathematics Education in 11 Mozambique 0 Introduction 11 1 History of Education and of Mathematics 12 Education in Particular 1.1 Traditional 12 1.2 Colonial 13 1.3 In the Liberated Areas 15 1.4 Since Independence 16 2 The Training of Mathematics Teachers 20 2.1 Historical Retrospective 20 2.2 The Teacher Training Courses 20 2.3 Further Training for Teachers 25 2.4 Professional Situation 28 3 National Seminar on the Teaching of 30 Mathematics 4 Open Questions 35 Notes and references 37 2 The First Mathematics Olympiads in 41 Mozambique Esselina Macome’s Autobiography 41 0 Introduction 42 1 Why Mathematics Olympiads were introduced 44 in the People’s Republic of Mozambique 1.1 Necessity to Popularise Mathematics 44 1.2 Looking for Ways to Popularise 45 Mathematical Knowledge 2 Learning from other Experiences 46

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2.1 Objectives 46 2.2 How to Achieve these Objectives? 47 3 The First Maputo Mathematics Olympiad 48 3.1 Mature Situation 48 3.2 Realization 49 3.3 Correction 49 4 Analysis of Results 49 4.1 Distribution 50 4.2 Social Background 53 4.3 Composition by Gender 53 4.4 ‘A Great Success’ 54 5 Extension 55 6 The Future 56 7 Postscript 1983 56 Appendix 1: Short Autobiographies of Some 60 Winners in the First Mathematics Olympiads 1 Abdulcarimo Ismael 60 2 Lourenço Lázaro Magaia 62 3 Fernando Victor Martins Saide 64 Appendix 2: Problems in the Maputo Mathematics 66 Olympiad (1981) at the Level of the 10th and 11th Grades Appendix 3: Problems in the Maputo Mathematics 69 Olympiad (1982) at the Level of the 7th, 8th and 9th Grades References 70 Notes 71 3 Conditions and Strategies for Emancipatory 73 Mathematics Education in Underdeveloped Countries Point of departure: mathematics education cannot 73 be neutral Mathematics education for emancipation. How? 74

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Problematizing reality in classroom situations 75 Creating confidence 79 A. Cultural strategies 79 B. Social strategies 85 C. Individual-collective strategies 85 Concluding remarks 87 References 89 4 Culture, Mathematics and Curriculum 91 Development in Mozambique 0 Confrontation 91 1 Education-policy in Mozambique 92 2 The fight against mathematical 95 underdevelopment: some lessons from other Third World countries Curriculum transplantation 95 Selective perspective 96 Necessity of adaptation to local culture and 97 needs Profound criticisms of actual mathematics 98 education 3 Towards a socialist mathematics-education- 100 policy 4 Incorporation of mathematical traditions into the 104 curriculum First example 106 Second example 110 Third example 112 References 115 Appendix 1 119

Books in English by the same author 121

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12 Mathematics Education in Mozambique

INTRODUCTION

The book Mathematics Education in Mozambique contains four papers from the early 1980s. The first paper, CHANGING MATHEMATICS EDUCATION IN MOZAMBIQUE, was published in 1981 in the international journal Educational Studies in Mathematics (Dordrecht / Boston, Vol. 12, 455-477). It is a modified version of the text Mathematics Education in the People’s Republic of Mozambique, elaborated in accordance with the questionnaire ‘Comparative study of the development of Mathematics Education as a professional discipline in different countries,’ organised by Gert Schubring (University of Bielefeld, Germany) for the Fourth International Congress on Mathematical Education, Berkeley, USA, 1980. The paper presents a brief description and analysis of mathematics education in different phases of the history of Mozambique, in pre-colonial and colonial times, during the National Liberation Struggle and after its Independence in 1975. The successes and problems that still had to be resolved in the post-independence period constitute the second part of the article, where particular attention has been given to teacher training and the first National Seminar on the Teaching of Mathematics. The second paper, THE FIRST MATHEMATICS OLYMPIADS IN MOZAMBIQUE, was published in 1984 in Educational Studies in Mathematics (Dordrecht / Boston, Vol. 15, 149-172). It explains why Mathematics Olympiads were introduced in the People’s Republic of Mozambique. It contains a description and analysis of the results — distribution of scores, types of problems, social background and composition by gender of participants and winners — of the first Mathematics Olympiads, together with short autobiographies of some winners. These autobiographies offer some insight into the social aspects of the development of their mathematical talents. The third paper, CONDITIONS AND STRATEGIES FOR EMANCIPATORY MATHEMATICS EDUCATION IN UNDERDEVELOPED COUNTRIES, was published in 1985 in the international journal For the Learning of Mathematics (Montreal, Vol. 5, No. 1, 15-20). It resulted from the revision of an invited paper 9 Paulus Gerdes

presented at the Caribbean Conference on Mathematics for the Benefit of the Caribbean Communities and its Reflection in the Curriculum, organised by the Inter-American Committee on Mathematical Education, Paramaribo, Suriname, October 18-21, 1982. The paper was reproduced in the books Is mathematics teaching neutral? (University of Cape Town, 1987, 10-15), edited by Chris Breen, and Mathematics Education: Major Themes in Education (Routledge, London, 2010, Volume 1, 179-189), edited by Alan J. Bishop. In the paper, I reflect on my experiences in Mozambique. I present examples of problematizing reality in mathematics classroom situations and suggest some conditions and strategies for mathematics education to become emancipatory in ‘developing’ countries; in particular, some cultural, social, and individual-collective strategies for creating confidence among teachers and pupils. The fourth paper, ON CULTURE, MATHEMATICS AND CURRICULUM DEVELOPMENT IN MOZAMBIQUE, is the short version of an invited paper presented at the Seminar “Mathematics and Culture”, Bergen, Norway, September 26-28, 1985. This version was published in the book Mathematics and Culture: A seminar report (Caspar Forlag, Rådal, 1986, 15-42), edited by Stieg Mellin-Olsen and Marit Johnsen Høines. The paper discusses the education policy in Mozambique and the fight against mathematical underdevelopment. It draws some lessons from other developing countries regarding curriculum transplantation, selective perspective of curricula, and the necessity of incorporating local culture and needs. Furthermore, the paper stresses the need for incorporation of ‘mathematical traditions’ into the curriculum, and presents several examples for reflection by mathematics educators. Naturally, the book, Mathematics Education in Mozambique: Papers from the Early 1980s, reflects the cultural, social and political contexts in which the papers were written. Nevertheless, I express the hope that the reproduction of these papers may stimulate among its readers an interest in some critical issues in the development of mathematics in Mozambique or elsewhere.

Paulus Gerdes August 27, 2014

10 Mathematics Education in Mozambique

Chapter 1 CHANGING MATHEMATICS EDUCATION IN MOZAMBIQUE 1 Educational Studies in Mathematics 12 (1981), 455-477

In this paper a brief description and analysis is given of mathematics education in different phases of the history of Mozambique, in the feudal and colonial times, during the National Liberation Struggle and after the Independence in 1975. The successes and problems that still have to be resolved of the post-independence period constitute the second part of the article, where particular attention has been given to teacher training and the first National Seminar on the Teaching of Mathematics.

0. INTRODUCTION

Mathematics education in Mozambique is no exception to the evident but often forgotten ‘rule’, that — in the formulation of Howson 2 — “mathematical education does not take place in a vacuum.” In our view, the history of mathematical education in Mozambique can be understood as an integral part of the processes of change of society and of education in particular. To achieve such an understanding it is necessary, according to Neander3, to investigate the history of mathematics education at four interrelated levels: (1) economic development, (2) general political development, (3) development of mathematics, and (4) development of the educational system. It is not yet possible to analyse thoroughly these levels and their interdependencies in the concrete case of Mozambique, but we hope to bring forward sufficient elements of such an analysis to make understandable the first achievements and the outstanding problems of the post-independence period. 4 We begin this paper with a brief review of the history of education and mathematics education in Mozambique, focussing on

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changes in these important parameters of mathematics education: 5 for whom, by whom, with what objectives, curriculum, teaching methods, and achievements. In the second part, we discuss one of the central problems in the present phase: teacher training. Finally, we give an account of the first National Seminar on the Teaching of Mathematics (May 1980) and draw attention to some of the most important ‘open’ questions.

1. HISTORY OF EDUCATION AND OF MATHEMATICS EDUCATION IN PARTICULAR

After ten years of people’s war, led by the Mozambique Liberation Front (FRELIMO), Portuguese colonialism had been defeated and Mozambique attained its independence. On 25 June, 1975, the People’s Republic of Mozambique was proclaimed. In 1977, FRELIMO became a vanguard party of the working classes, to guide the country on the road to Socialism. In correspondence with the different socio-political situations the country has passed through, four different kinds of education 6 have to be distinguished in the history of Mozambique: traditional-feudal, colonial-capitalist, in the liberated areas, and after independence (socialist).

1.1. Traditional

Periodization of kinds of education Figure 1 12 Mathematics Education in Mozambique

All societies use formal and informal education in different degrees. Traditional education in Mozambique was weakly institutionalised (initiation schools, ‘secret societies’). Most training was informal: emulation of older children, listening to stories, watching and helping adults go about their daily tasks, singing and dancing, and games (incl. mathematical games such as ntchuva and a version of the ‘three-in-a-row’ game muravarava). In the initiation schools, the children were trained under strict discipline by little sleep, hard labour, long walks, cold, etc., for obedience to the rules of society. 7 The aim of education was to bring the new generations to accept blindly the traditions of feudal society held up to them as dogma: the authority of the elders, tribal sentiments, contempt for women, and superstitions which reﬂect a superficial understanding of nature. In order to deal with problems related to production empiric- mathematical ideas were developed and transmitted to the young children. In Nampula province, for example, children learn to use a stick to draw a circle in the sand to place fish equidistant from a fire for drying. In Cabo Delgado province, children learn that the way to make sure that the base of a hut really is rectangular is by making sure that the two diagonals are of equal length. Measure, number, and reckoning systems are also developed. 8 Traditional education was imbued with a magic-religious view of the world, exemplified by the following taboo against the counting of men: “What? You are counting us? Whom do you want to see disappear?”

1.2. Colonial

At Independence, the illiteracy rate in Mozambique was still 93% — one of the highest in the world. Colonial education did not reach the Mozambican masses. There existed two categories of the school system. Government schooling was reserved almost exclusively for the children of the settlers, particularly at the levels of secondary and higher education. The few Mozambicans who had access to schools were taught at the Roman Catholic mission schools. 13 Paulus Gerdes

The black African children were taught the history and geography of Portugal. They were taught to despise their own culture, to submit themselves to the colonial and religious authorities, and to accept as valid the values of the colonial-capitalist society. 9 The objective was to transform some Mozambicans into an elite — into ‘assimilados’ or ‘black Portuguese’ — who could be docile servants of the interests of colonialism. Those Mozambicans were taught some mathematics to “calculate better the compulsory quota of cotton production or to be more lucrative ‘boss-boys’ in South-African mines.” 10 The teaching of mathematics played an important role in diffusing the values of the ruling class, such as private property, the exploitation of man by man, the plundering of the colonised territories, and contempt for women. Consider the ideological context of the following exercises, taken from mathematics textbooks of that period: * In Lourenco Marques, Mr. Abílio has a house rented to four tenants who pay him 4,000$, 5,000$, 4,300$ and 5,200$ per month. By applying the distributive property, calculate his annual income. * In a factory, the men earn 45$, the women 30$ and the apprentices 15$ per day ... * In a certain year Mozambique produced 90,000 tons of cotton, 85,500 tons were exported to the Metropolis. What percentage of the production stayed in the Province? School achievement was very poor. For example, in 1970 only 39% of the children passed from one grade to the next. The mathematics curricula were the ones used in Portugal, during the 1960s ‘enriched’ by a formal introduction of the vocabulary of set theory, without, however, using it to understand mathematics. Mathematics seemed to be identified and confined with written arithmetic. Very little attention was given to mental arithmetic and geometry. The achievements of colonial mathematics education seem to us rather poor, looking to fundamental errors made by those university students — blacks as well as whites — who went through primary and secondary school before Independence:

14 Mathematics Education in Mozambique

1.3. In the Liberated Areas

Since its creation in 1962, FRELIMO gave great importance to education, and schools were immediately set up in the areas liberated from colonialism. The people’s will and determination made it possible to overcome the lack of teaching materials and of trained teachers. To overcome the teacher shortage, pupils of the fourth year taught the second year; first year pupils helped with literacy classes, and so on. The creative initiative of people found ways to deal with the shortage of teaching materials. For example, they made chalk from sticks of dry cassava and then covered used sheets of paper with charcoal and wrote on them with twigs. In meetings and seminars at district and provincial level, teachers shared their experiences, increased their knowledge together, and analysed their objectives, curricula, and methods of teaching. At the Second Conference of FRELIMO’s Department of Education and Culture in 1970, the general aims of the new education were clearly defined. 12 In brief, they were: to create solidarity, remove racial and tribal discrimination; to acquire a scientific attitude, open and free of all the weight of superstition and dogmatic traditions; emancipate women; and lead all concerned to recognise the necessity of serving the people, participating in production, respecting manual 15 Paulus Gerdes

labour, freeing the capacity for initiative, and developing a sense of responsibility. Mathematics education was adapted to the concrete conditions of the armed struggle. It was necessary to acquire mathematical knowledge to handle weapons, to build air-raid shelters, for simple nursing tasks, and for the planning and management of production. In accordance with this line of orientation, we find the following typical problems in the first and second year mathematics textbooks produced by FRELIMO: * 23 peasants are working in a field. At mid-day 6 guerrilla ﬁghters arrive to help them, from a military base near to their village. How many people are working in the ﬁeld now? * A nurse at a health post vaccinated 32 children yesterday. Today he vaccinated another 29 and he plans to vaccinate the remaining 16 children tomorrow. How many children will he have vaccinated at the end of his campaign? * 25 students are practising their traditional dancing for a celebration at their school. They are practising in 5 equal groups. How many are in each group? * Yesterday there were 22 women in a literacy class. Today 5 more women came to the class. How many women are there in the class now? Arithmetic exercises such as these contributed towards the diffusion of new values such as solidarity among men, national consciousness, and the liberation of women. By the end of the war there were more than 30,000 pupils in the primary schools of the liberated areas. More than 500 pupils were attending secondary school.

1 .4. Since Independence

Immediately after independence, private and missionary schools were abolished. Education was nationalised and made free of charge, resulting in an educational explosion.

16 Mathematics Education in Mozambique

Primary and secondary school enrolment (unity 100,000) Figure 2

In 1973 there were 589,000 children in primary schools. In 1978 there were 1,419,000 out of a total population of about 11,000,000 at that time. There were 33,000 pupils in secondary schools in 1974. In 1978 there were 82,000. During the First National Literacy Campaign (1978), 130,000 were taught in the priority centres (sectors of collective production such as factories, state enterprises and agricultural cooperatives; the FRELIMO Party; the Mozambican Women’s Organisation, etc.). For the Second National Literacy Campaign (1980) it was planned to teach 300,000 workers. The nationalisation of formal education made it possible to standardise primary school education (from the first to the fourth year), and to allocate resources more equally among the different regions of the country and thus to greatly abolish the regional (town- countryside), racial, and social discrimination of the colonial education system. This nationalisation with its subsequent massification of education, was the first guarantee of the democratisation of education — of an education in the interest of the broad masses of the population. So, developing the experience gained in the liberated areas, the new revolutionary education in the People’s Republic of

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Mozambique is aimed at forming a Mozambican personality — a New Man, free from all complexes of superiority and inferiority, free from superstitious beliefs, self-reliant and ready to make his scientific knowledge the basis of the new society based on unity and equality. 13 One of the principles of the revolutionary education in Mozambique is that education must be permanent and progressive: permanent in that it provides possibilities for people to constantly raise their knowledge through specialised, intensive courses and seminars; progressive in that a pupil does not continue straight from kindergarten to university but advances in stages, and that at the end of each goes out to work, to increase his knowledge through practice, only later returning to the point where he left off. These are some of the aims and principles of the new education in Mozambique. There have been many advances in putting them into practice: Pupils and teachers are organised in workgroups where they help one another and share their experience, creating an atmosphere open to criticism and self-criticism. A process was begun to link the schools to social and economic activities. For two of the three months of the ‘long holidays’, the pupils participate in cleaning their school, working in the fields, or in industrial production. New programmes were progressively introduced, as were curricula and new teaching methods. Manual productive work was introduced into the schools (e.g., how to raise rabbits) as a way of combating the elitist scorn for manual labour, at the same time reducing the costs of running the schools. The enthusiastic educational explosion that took place in Mozambique reﬂects the eagerness of the population to acquire knowledge after the dark ages of colonial occupation. In the first four years after Independence, villagers opened already more than 7,000 new schools. However, enthusiasm alone is not enough. Organisation, planning and training were and are very much needed to overcome many difficulties, particularly lack of teachers and lack of publications. 14 There is a great shortage of teachers with scientific, pedagogical, and didactic training (e.g., in 1979 more than 90% of the mathematics teachers in secondary schools had no professional training).

18 Mathematics Education in Mozambique

The programmes were developed in a hurry in 1975, in an attempt to respond to the political and socio-economic changes in the country. The mathematics curriculum for primary school was simpliﬁed (e.g., the arithmetic programme was reduced so the pupils learned only to operate well with natural numbers and to handle linear measures and money), in order to cope with the difficulties that stem from the weak knowledge of the Portuguese language, the medium of instruction and communication. Already in 1975 the first teachers’ manuals had been produced. But the first national mathematics textbooks and exercise books for pupils of the first grade will leave the presses only in 1981 (who could write them before?). At the other end of the educational spectrum, professors of the Eduardo Mondlane University have now produced the mathematics textbooks for the tenth and eleventh grades. But much other educational material is still in short supply. Unsatisfactory teacher training and overcrowded classrooms, combined with negative inﬂuences of the traditional-feudal and colonial-capitalist societies, produce poor school achievement. Only 27% of the children who began their formal education in 1976 reached the end of primary school after the scheduled four years. The highest failure rates were in Portuguese and mathematics. In 1979, the low levels of attainment caused the Central Committee of the FRELIMO Party and the People’s Assembly to analyse thoroughly the educational situation. They considered the educational explosion as an important victory against underdevelopment. On the other hand, they considered that the training of teachers had not accompanied the educational explosion. So they concluded that, in order to eradicate illiteracy and to assure compulsory education as soon as possible, the Ministry of Education and Culture had to plan (and limit) the entrance to the schools, accelerate the training of teachers, and enhance the social status of teachers. In the next section, we deal with some of the implications of these decisions of the FRELIMO Party and the People’s Assembly for the training of mathematics teachers.

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2. THE TRAINING OF MATHEMATICS TEACHERS

2.1. Historical Retrospective

Under the Portuguese, there were two different types of training schools for primary school teachers, corresponding to the two categories of the school system (see Section 1.2). The first — of a low standard and controlled by the Roman Catholic Church — trained Mozambicans for the mission schools. Institutes of the second type, whose students were mostly of Portuguese origin, trained teachers for the Government schools, attended by the children of settlers and ‘assimilados.’ There did not exist any training institute for secondary school teachers in Mozambique before liberation. With the coming of Independence and the nationalisation of education, the few rather well qualiﬁed teachers left the country en masse. This aggravated the teacher shortage that resulted from the educational explosion (see Section 1.4). An immediate solution had to be found, just as during the Liberation War, pupils of the fourth year started to teach the second year, etc. In this way, thousands of well- intentioned persons with no professional teacher training at all were appointed as teachers. By 1978, more than 12,000 of them had already attended ‘refresher’ courses.

2.2. The Teacher Training Courses

At the same time, the training of teachers began straightaway through intensive courses. In 1975, 10 Primary School Teacher Training Centres (PSTTC) were created, one in each province. Their courses had a duration, which expanded from 6 months at the start, to 12 months in 1980. The sixth year of schooling is the level necessary for entry (see Table I). On leaving, the new teachers are authorised to teach primary school, i.e., from the first to the fourth year. (For the curriculum, see the first

20 Mathematics Education in Mozambique

column of Table III.) Up until 1980, 7,662 primary school teachers had graduated from the PSTTC.

TABLE I

In 1977 the Secondary School Teacher Training Courses (SSTTC) were opened at the Eduardo Mondlane University in Maputo. In accordance with the educational boom with its subsequent concentration of students in the lower grades on the one hand, and, the principle that revolutionary education must be permanent and progressive (see Section 1.4) on the other, it was decided to train, with respect to secondary school, first of all, teachers for the ﬁfth and sixth years. After an intensive course of one year (see Table II) (a) the students would teach for a period of two years in the fifth and sixth years, followed by a second intensive course (b). Then they could teach the fifth to ninth grades, and so on (see Table II). It was thought that in this way the training of teachers would keep pace with the growth of the number of pupils in the secondary schools. The SSTTC trained 18, 17 and 22 mathematics teachers for the fifth and sixth years in 1977, 1978, and 1979 respectively, who, after graduation, were equally distributed over the provinces.

TABLE II

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In 1978, an experimental Teachers’ Training and Education School (TTES) began to function in Maputo for the first six years, already with an eye to the future educational system in which primary schooling would have a duration of six or seven years. Comparative Table III shows the teacher training courses operating up to 1980. Under this system of PSTTC, SSTTC and TTES, it was not possible to train enough reasonably qualified teachers for the country’s needs. So, generally in accordance with the conclusions reached by the FRELIMO Party and the People’s Assembly in 1979 (see above, Section 1.4), the teacher training courses were reorganised from 1980, with the following principal changes: * Specialisation of primary school teachers to teach specific years in order to enhance their quality; * Lowering the level of entry for SSTTC and TTES in order to augment drastically the quantity of students who can be trained to become a teacher (there are still only some hundreds of students graduating from the ninth grade every year); * To abolish the polyvalent course of the TTES and to substitute them by monovalent (e.g., only mathematics) teacher training courses for the fifth and sixth year; * To start immediately in the SSTTC the training of teachers for the seventh to ninth (SSTTC 1) and tenth and eleventh grades (SSTTC 2) (see Table IV) by intensive bivalent (e.g., mathematics and physics) courses with a duration of two years.

TABLE IV

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TABLE III Comparative table of teacher training courses up to 1980

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TABLE III (Continued)

24 Mathematics Education in Mozambique

The introduction of bivalent courses, especially in the case of mathematics and physics, seems to the author rather untimely, when we take into account, for example, the poor achievement of the students in the diagnostic tests in mathematics and other subjects, when entering the teacher training courses. Table V compares the new courses. Training in mathematics teaching methods (didactics of mathematics). There is no instructor specialised in Mathematics Education in the whole country. The classes in this subject are given by experienced teachers who have shown themselves to be particularly good at teaching and are keenly interested in the improvement of the quality of mathematics teaching and in the relevance of mathematics for the development of Mozambique. In the case of the SSTTC, the instructor of didactics of mathematics is an expatriate (French), as is almost all the staff of this level. In the SSTTC mathematics/physics teacher training course, the instructors come from the German Democratic Republic, Soviet Union, Cuba, Brazil, The Netherlands, and Guinea Conakry. A course for instructors in mathematics education is currently under study. The general idea is that the future instructors for the teacher training courses, especially in the case of didactics, have to be selected from among the best students to graduate from the courses and who have shown good performances during their first years of teaching.

2.3. Further Training for Teachers Teachers may continue their training at various levels: At school level — groups of teachers are organised by years in primary schools and by subjects in secondary schools, in which the teachers prepare their lessons together, discuss their difficulties and sit in on one another’s lessons. Thus, the scientific and pedagogical weaknesses of the untrained teachers are partly compensated by the collective work of preparing lessons and sharing of experiences with more qualified colleagues. For this, the trained teachers are distributed through all the provinces, so as to be able to help the others. Moreover, in many secondary schools, the best pupils collaborate with the teacher in helping their classmates with most difficulties.

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TABLE V Comparative table of the new teachers’ training courses (1980)

26 Mathematics Education in Mozambique

TABLE V (Continued)

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At district level — ‘Zones of Educational Inﬂuence’ were created in 1976. These are groupings of various schools in the same district, in which the teachers hold periodic meetings for study, lesson planning, standardisation of methods of work, and of implementation of programmes. At provincial level — since 1977 there have been two Teaching Support Commissions in each province, one for primary schools and one for secondary schools. These are composed of the best teachers in each province, and support the work of the other teachers by visiting their schools (if there are no transport problems), writing and duplicating notes on the various subjects, organising the work of the Zones of Educational Influence and holding meetings and workshops at provincial level. At national level — The National Teaching Support Commissions, created in 1976, support and supervise the work of the Provincial Commissions. Since there are four times as many students in the ﬁfth and sixth years as in the whole of the other secondary years, the secondary level Teaching Support Commissions concentrate their help (notes on teaching methods, work books of exercises, setting of end-of-term examinations, organisation of refresher courses, etc.) on the teachers of the fifth and sixth years.

2.4. Professional Situation

The professional situation of mathematics teachers in Mozambique can only be understood in the context of the situation of teachers in general. Due to the colonial past in which the teaching profession was undervalued by the ruling class, and even despised, by comparison with other professions such as those of doctors, engineers, economists etc., in terms of social status and actual salary, it has not yet found the esteem, which it deserves. However, the Central Committee of the FRELIMO Party considered this question in 1979 (see above, Section 1.4) and, bearing in mind the important role played by the teaching profession in the People’s Republic of Mozambique, decided that the Ministry of

28 Mathematics Education in Mozambique

Education and Culture should develop a programme to enhance the teaching profession, and give prestige to the function of teachers as agents of the transformation of society. As a consequence of this decision, and as one of the steps to be taken, it is planned to establish (in 1981) a National Organisation of Teachers. At the same time, the National Commission on Prices and Salaries is reviewing the whole current system of teachers’ salaries. Mozambican Mathematics Association. Also under study is a Mozambican Mathematics Association, with the following objectives: * to contribute towards the development of mathematics and its applications for economic, social, and cultural progress in Mozambique; * to spread mathematical knowledge; * to stimulate the broad masses to take an interest and delight in mathematical creation, combating the elitist idea that mathematics is not for everyone; * to give pedagogical support to teachers and students at all levels of instruction and to stimulate the type of teaching which, linking theory with practice, places the applications of mathematics within reach of the worker and peasant masses; * to promote scientiﬁc cooperation with other countries in the ﬁeld of mathematics; * to cooperate with other organisations having similar aims.

The principal forms of its activity would be: * to promote exhibitions, talks, seminars, and holiday courses; * to produce mathematical publications; * to organise competitions, Olympiads, and mathematical study circles; * to study and publicise the traditional Mozambican games, which have mathematical aspects; * to create conditions for the study of the history of mathematics of Mozambique. 29 Paulus Gerdes

In January 1981, the first volume of the Journal of Teachers appeared. Later in 1981, a Journal on Mathematics Education will be launched.

3. NATIONAL SEMINAR ON THE TEACHING OF MATHEMATICS

The building of a socialist society in Mozambique, and the victory over under-development, which the Mozambican people aim to achieve in this decade of the 80s, will make increasing demands in all sectors, particularly education, requiring from every worker a great effort of science and technology, and especially mathematics. 15 In this context, mathematics is given a high priority. Half of the adult ‘literacy’ course is, in fact, numeracy. Mozambique’s second national seminar on teaching, following the Portuguese language teaching seminar (October 1979), was the National Seminar on the Teaching of Mathematics held in Maputo from the 5th to the 10th of May 1980, with the motto ‘Let us make mathematics a weapon in the building of socialism’. 16 It was chaired by the Minister of Education and Culture, Graça Machel, and most sessions were attended by other senior leaders of the Party and the Government, such as the Rector of the Eduardo Mondlane University, Fernando Ganhão, and the Vice- Governor of the Bank of Mozambique, Prakash Ratilal. Their presence shows clearly the importance given to the teaching of mathematics in Mozambique. This first National Seminar on the Teaching of Mathematics had the objective of making a thorough analysis of the current situation, determining to what extent the teaching of mathematics had contributed towards the improvement of the standard of living in the country, and defining new measures to improve the educational process and the training of cadres. In view of its importance, teachers of all levels, literacy workers and adult educators, workers in various departments of the Ministry of Education and Culture, as well as representatives from other ministries, were all involved in the seminar, not only as participants but also in the preparations (which took about a year). In the preparatory phase, the experiences of various provinces were studied by visits to schools, to training and refresher course centres for 30 Mathematics Education in Mozambique

teachers, and to factories and farms, and also by preparatory provincial meetings. The work of the seminar included two plenary lectures, one on the ‘Science of Mathematics’ and one on ‘Mathematics, society and class struggle.’ Furthermore, one day was reserved for each of the major topics: objectives, teaching methods and means, curriculum, and teacher training. For each of these, an introductory paper was distributed and in the afternoon and evening the delegates discussed the paper in groups of about 20 people. Their comments were combined into a report, which was available the next morning for a plenary discussion. The reflection on the ‘Science of mathematics’ contributed towards the demystiﬁcation of mathematical science, by showing how, growing out of the social and economic needs of human societies, it is in continual progress, intimately connected to the development of those societies. Also shown were some concrete applications of mathematics to agriculture and industrial production. For a substantial number of delegates it was a ‘shock’ to see that mathematics does not fall from heaven, is not eternal, and is not taught in order to have a mechanism of selection of the pupils, but rather that it has applications. Later in the week the delegates visited factories — for the first time in their lives for those from the countryside — and were very surprised to see so many uses of mathematics. A teacher without professional training, visiting a brewery, cried out “after all, negative numbers have a sense. They are not just a plague.” Relationships between the teaching of mathematics and the class struggle in Mozambique were considered in the second lecture, ‘Mathematics, society and class struggle’. This study made it possible to place the teaching of mathematics in the different phases of the history of the country, to analyse the influence of the traditional and the colonial society on the actual teaching of mathematics. That there are still so many authoritarian teachers who hardly permit a doubt or a suggestion from their students; that there are older teachers who do not accept the experience and the help of younger ones, even when these are their superiors or when they have a better training; that certain fundamental mathematical concepts are presented as absolute truths without being explained or demonstrated; and that many students receive passively the information transmitted by the teacher (and that 31 Paulus Gerdes

this passivity is most marked in girls) was explained as a negative heritage of traditional-feudal society, just as the persistent presence of superstitious ideas which prevent teachers from understanding the origin of the development of mathematics. That mathematics teaching continues to be too detached from its applications, and to be based on memorisation, was explained as one of the deep scars left by the colonial society. Under the Portuguese system, teaching was by rote. Students memorised for exams, but never learned to think mathematically. This was in the interest of the colonists who wanted to create only some low-level clerks who could carry out orders accurately but not be able to use any creative thinking. These vestiges of traditional-feudal and colonial-capitalist societies constitute serious blocks to the creation of the new, of the love of learning and of scientific curiosity, i.e., blocks to some of the most important objectives of socialist education. The need to set objectives for the teaching of mathematics appropriate for the present phase of development led the seminar to examine the discussion paper ‘the aims of mathematics education in Mozambique’. The general objectives of mathematics education were deduced from, on the one hand, the general objectives of socialist society and socialist education, as defined by the FRELIMO Party and, on the other hand, an historical and dialectical materialist view on mathematics, 17 its development and significance. They include objectives directly related to mathematics, such as stimulation to understand the significance of mathematics for raising production and productivity, transmission of mathematical knowledge necessary for the development of the country, etc. Other objectives related to general mental capacities, such as critical attitude, creativity, discipline of thought, logical reasoning, etc. Ideological objectives include stimulation of the view that mathematics reflects the objective reality. 18 It was much more difficult to deﬁne the specific objectives of mathematics education for the different levels and types (technical, agricultural, adult, etc.) of education. To define them, the debates concluded, it is also necessary to take into account the concrete needs of the society, the general cultural level of the population, and the exigencies of the logical system of mathematics (see Figure 3). It was stressed that different objectives apply to literacy and adult education on the one hand, which directly reach the workers on whom the

32 Mathematics Education in Mozambique

victory over underdevelopment in this decade will depend, and on the other hand, to the teaching of children, on whom will depend the country’s development in the future.

Figure 3 Also discussed were the best teaching methods to develop the student’s capacity for reasoning, analysis, investigation and application of his knowledge in practice, and to free his creative initiative. The importance of the use of teaching aids was stressed, and an exhibition was organised of teaching aids produced in the provinces using natural and locally available materials. Although some delegates recognised that their horizon was widened by the exhibition and the explanation of more active, less memorising teaching methods, they nevertheless emphasised that the actual conditions (shortage of textbooks, overcrowded classrooms, teaching different classes in the morning and afternoon etc.) make it rather difficult to apply. The last of the themes thoroughly discussed at the seminar was the ‘training and upgrading of teachers.’ A particular concern was to define the objectives of the training of mathematics teachers. It was considered that, in order for mathematics teachers to be able to carry out their work correctly they must: * know about the origins and evolution of mathematics; * understand the role of mathematics in the development of other sciences, and the role of those sciences in the development of mathematics; 33 Paulus Gerdes

* know the applications of mathematics in the various sectors of economic life, and the part it plays in increasing production and productivity; * have a basic general training in other sciences so as to be able to teach the applications of mathematics; * have a thorough knowledge of the methodology of mathematics teaching; * know how to produce and use teaching materials. Many guidelines are drawn up for the training and upgrading of mathematics teachers. Among others, the following may be mentioned: * In all training and refresher courses, visits must be organised to factories, producer cooperatives, state farms, ports and railway installations, etc., in accordance with the possibilities of each centre; * In each training or refresher course centre, there must be a production centre, in which the future teachers can apply their mathematical knowledge; * School pupils must be mobilised for the study of mathematics and for the task of teaching by: * Creating interest groups in the schools (production of teaching materials, mathematical games, collection of information on mathematics in traditional society, etc.); * Selecting the best pupils in mathematics in each class, to participate as monitors, under the supervision of the teacher, in helping the pupils with difficulties; * Organising competitions at school, district, provincial, and national level; * The entry requirements and/or the duration of the training courses must be raised as soon as possible, giving priority to those for literacy and adult education.

34 Mathematics Education in Mozambique

4. OPEN QUESTIONS

In this article, we have described and commented on major changes in mathematics education in Mozambique, especially those in the years immediately following the Liberation in 1975. Independence and the option for socialism have implied an overall democratisation of mathematics education: 19 the majority of children, and hundreds of thousands of adults, have already gained access to the science of mathematics; the mathematics teaching profession has been opened up for children of peasants and labourers; the general objectives of mathematics education have been drastically changed; and the discussion on how to improve the quality of mathematics education, curricula and teaching methods, is not any more reserved for an elite but is becoming the object of reflection of the mass of primary and secondary school teachers, literacy workers and adult education teachers, university professors and education officials as well, culminating in the first National Seminar on the Teaching of Mathematics. However, mathematics curricula lag behind; they are simplified versions of the colonial ones. Teacher training will have to fulfil a pivotal role in combating the negative influence of traditional and colonial education, if mathematics education is to become really socialist — everyone mastering mathematics and capable of thinking mathematically, to the benefit of society as a whole. Or in the words of Prakash Ratilal (see Section 3): “we are not teaching mathematics just so that workers produce more, but rather so they produce in a socialist way where people have control over production.” Finally, the author would like to draw attention to two challenging — minor? — questions in the context of making everyone capable of thinking mathematically, in the concrete case of Mozambique. With independence, the colonial mathematics textbooks were abolished because of their ideological content (see Section 1.3). But, in a counteroffensive of university professors (who later left the country) the textbooks of the liberated areas were also blocked. In that situation, would it not have been better to have introduced for a transitional period, a complete curriculum and series of textbooks from another

35 Paulus Gerdes

country that had passed through comparable conditions — underdeveloped, and constructing socialism — as was done when Angola adapted curricula and textbooks from Cuba? And at this moment, with only mathematics text books for the literacy course, the first, tenth and eleventh years? Many aspects of the mathematics to be taught in Mozambique, reﬂect a level of technological development that does not correspond to that of the productive forces in the country, but on the other hand, have to be taught in order to beat underdevelopment. How can effective bridges be built from pre-school education and experience to the mathematics to be introduced in the schools?20 How can this be done, also taking into account that mathematics has to be taught in a second language, Portuguese, which is now a vehicle of national unity? The author thinks that an answer has to be sought, in this phase, in linking mathematics very closely to applications in the environment of the students in the first stages of teaching the subject, e.g., production problems in the new cooperatives and communal villages.

Faculty of Mathematics/Faculty of Education Eduardo Mondlane University

36 Mathematics Education in Mozambique

NOTES AND REFERENCES

1 This paper is a modified version of ‘Mathematics Education in the People’s Republic of Mozambique’, elaborated in accordance with the questionnaire ‘Comparative study of the development of Mathematics Education as a professional discipline in different countries’ of Dr G. Schubring for the Fourth International Congress on Mathematical Education, Berkeley, 1980. 2 Howson, A. G.: 1978, ‘Change in mathematics education since the late 1950s – Great Britain’, Educational Studies in Mathematics, 9, 183-223. 3 Neander, J.: 1974, Mathematik und Ideologie. Zur politischen Ökonomie des Mathematiksunterrichts, Werner Raith Verlag, Starnberg, p. 7. 4 For more details see: Gerdes, P.: 1980 ‘Mathematik in Mozambique’, Materialien zur Analyse der Berufspraxis des Mathematikers, vol. 25, where we also discuss the training of mathematicians at the Eduardo Mondlane University in Maputo. 5 Cf. Griffiths, H. B. and Howson, A. G.: 1974, Mathematics: Society and Curricula, Cambridge University Press, and Freudentha1, H.: 1978. ‘Change in mathematics education since the late 1950s – Ideas and realisation: An ICMI report’, Educational Studies in Mathematics, 9, 143-145. 6 Cf. 1976, ‘Educational policy in the People’s Republic of Mozambique’, Journal of Modern African Studies, 14, 331-339. 7 See, e.g., Junod, H. A.: 1934, Usos e costumes dos Bantos, Imprensa Nacional de Moçambique, Lourenço Marques. 8 The history of mathematics in Mozambique is still awaiting investigations. Cf. Zaslavsky, C.: 1973, Africa Counts. Number and Pattern in African Culture, Prindle, Webber & Schmidt, Massachusetts. 9 For a thorough analysis of the relationship between education and submission, see Mondlane, E.: 1969, The Struggle for Mozambique, Penguin Books, Harmondsworth. Dr Mondlane was elected first President of FRELIMO.

37 Paulus Gerdes

10 Education policy ... , o. c., p. 332. 11 Suggestions on how to make a comparative (in time, space and culture) analysis of errors are welcome. 12 See Machel, S.: 1970, ‘Educate man to win the war, create a new society and develop our country’, in Machel, S., Mozambique: Sowing the Seeds of Revolution, Committee for Freedom in Mozambique, Angola & Guiné, London, 1974, pp. 37-46. 13 Education policy ... , o.c., p. 333. Cf. 1978, Central Committee Report to the Third Congress of FRELIMO, Mozambique, Angola and Guiné Information Centre, London, pp. 8, 14, 50. 14 These problems in themselves are not new. They occur in many Third World countries (cf., e.g., Eshiwani, G. S.: 1979, ‘The goals of mathematics teaching in Africa: a need for re-examination’, in Prospects, 9, 346-35 3) or after a destructive war (cf., e.g., Ehrenfeucht, A., 1978: ‘The reform of mathematics education in Poland’, Educational Studies in Mathematics, 9, 283-295). What seems to be new is the measure in which they occur in Mozambique, reflecting the explosion in student population and the backward character of Portuguese colonialism. 15 Very instructive examples of the necessity for mathematical knowledge were given by the Vice-Governor of the Bank of Mozambique, during the seminar. E.g., he commented that “one of the reasons for continuing queues in Mozambique is that some factory directors, who often have only nine years of schooling, are unable to calculate raw material needs far enough in advance to do the ordering and keep the factory running at full capacity.” (Cited in: Not Just a Question of Number, in A.I.M. Information Bulletin, No. 47, 1980, p. 8.) 16 The importance of the seminar was also reinforced by front page newspaper articles during the meeting. Internationally it got attention: Hanlon, J.: 1980: ‘You must be numerate as well as literate’, Gemini New Service, GG6656, and: Hanlon, J.; 1980: ‘Mozambique ponders maths problems’, Times Educational Supplement, July 1980. 17 See, E.G., Aleksandrov, A. D., 1956: ‘A general view of mathematics’, Mathematics, its Content, Methods, and Meaning, 38 Mathematics Education in Mozambique

M.I.T., Massachusetts, 1977, vol. 1, pp. 1-65, and: Booss, B., 1979: Trends in Mathematics, Universitat Bielefeld. 18 Cf. the classification of general objectives of mathematics education in Methodik Mathematikunterricht, Volkseigener Verlag, Berlin, 1977. 19 Does not this constitute an integral part of a process of mathematics education becoming socialist? Cf. Howson, A. G.: 1980, ‘Socialist mathematics education: does it exist?’, Educational Studies in Mathematics, 11, 285-299. 20 Cf. the already classic study of Gay, J. and Cole, M.: 1967, The New Mathematics and an Old Culture. A Study of Learning Among the Kpelle of Liberia, Holt, Rinehart and Winston, New York.

39 Paulus Gerdes

40 Mathematics Education in Mozambique

Chapter 2 THE FIRST MATHEMATICS OLYMPIADS IN MOZAMBIQUE Educational Studies in Mathematics 15 (1984), 149-172

ABSTRACT. In this paper we explain why Mathematics Olympiads were introduced in the People’s Republic of Mozambique. A description and analysis of the results – distribution of scores, types of problems, social background and composition by gender of participants and winners – of the first Mathematics Olympiads are given, together with short autobiographies of some winners. These offer some insight into the social aspects of the development of their mathematical talents.

ESSELINA MACOME’S AUTOBIOGRAPHY

I am the daughter of Marcelino Nhanlane Lucas and Sara Tembe. I was born on 3 May 1962 in Maputo where my parents lived. My father was a primary school teacher in a mission school. In 1968 my father was transferred to Magude where I, for the first time, went to school and where I ﬁnished primary school in 1973. As my parents lived in Magude and there was no secondary school there where I could continue, my father placed me in a boarding school in Maputo. In 1975 education was nationalised and I was transferred to Namaacha. In 1976 I did Standard 7 of the secondary school in Namaacha but due to the leaving of various teachers of the school I lost the year. In 1977 I repeated Standard 7 and I finished Standard 8 in 1978. In 1979 I came once again to Maputo where I did Standard 9. At the end of that year I was placed in the Mathematics and Physics Teacher Training Course. In the same year, my father retired from teaching for health reasons. My parents are now living in Xai-Xai where they work in the family garden and raise chickens and goats.

41 Paulus Gerdes

I liked to participate in the Mathematics Olympiad because it showed me what my level was. I came to the conclusion that I still lacked persistence and concentration in the solving of problems. I also took part in a geometry contest in my class and got the second prize. My liking for mathematics did not begin in the Teacher Training Course. At school I also liked mathematics. I do not know exactly why, but one of the reasons could be that my father has always asked a lot of me, especially in the field of mathematics. It could be that he liked the subject too. Furthermore, I think that the fact that our secondary school had teachers who liked to teach mathematics has contributed to my love of mathematics.

Note: Esselina Macome was the highest classiﬁed Mozambican girl in the first Maputo Mathematics Olympiad. She has been selected to continue her university studies in mathematics and physics in the German Democratic Republic.

Additional note 2014: Esselina Macome concluded in 2003 her Ph.D. in Computer Science with the thesis The dynamics of the adoption and use of ICT-based initiatives for development: results of a field study in Mozambique (University of Pretoria, South Africa). She became Deputy-Dean of the Faculty of Science of the Eduardo Mondlane University in Maputo and is currently Administrator of the Bank of Mozambique.

0. INTRODUCTION

In order to understand the introduction of Mathematics Olympiads in Mozambique it will be useful to place the country within its geographical, historical, political, economic, and educational contexts. Mozambique is an extensive country with an area of 799,380 km2, situated in southern Africa. The country had a population of about 12.6 million in 1981, and is surrounded by Tanzania and Malawi

42 Mathematics Education in Mozambique

to the north, Zambia and Zimbabwe (the former British colony of Southern Rhodesia) to the west, and by South Africa and Swaziland to the south (see Figure 1).

Places named in the article:

1. Maputo 2. Beiro 3. Chimoio 4. Mozambique Island 5. Quelimane 6. Makumbura 7. Magoe 8. Jeque 9. Mariri 10. Magude 11. Namaacha 12. Xai-Xai 13. Bajone 14. Pebane 15. Nampula 16. SWAZILAND

Map of Mozambique Figure 1

The People’s Republic of Mozambique is a rather new member of the international community of independent and sovereign states, having gained its independence on 25 June 1975. After ten years of the people’s war led by the liberation movement FRELIMO, the

43 Paulus Gerdes

Portuguese colonial regime had been defeated. In 1977, FRELIMO became the vanguard party of the working classes, to guide the country on the road to Socialism. Economic underdevelopment constitutes one of the principal aspects of the colonial heritage. Pre-independence per capita annual income varied between US$200 and US$300. The greater part of agricultural production was channelled towards the colonial metropolis, and to the urban centres where the settlers lived. Industry was almost exclusively devoted to finishing imported products, which were destined to be consumed by the colonial ‘happy few’. Economic reconstruction after Independence has been greatly obstructed by Rhodesian military incursions until 1980 and by the continuing sabotage and destruction caused by the counter-revolutionaries supported by South Africa. At independence, the illiteracy rate in Mozambique was still 93%. There were only two Mozambican mathematicians. Immediately after independence, education was nationalised and made free of charge, resulting in an educational explosion. In 1978 there were already three times more children attending primary and secondary schools than at the end of the colonial period. By 1982, the illiteracy rate had been reduced to about 75%. Having brieﬂy described Mozambique’s background, it will become more understandable – we hope – not only why, when, and how Mathematics Olympiads have been introduced in our country, but also which are the successes already achieved and which the outstanding problems still to be solved.

1. WHY MATHEMATICS OLYMPIADS WERE INTRODUCED IN THE PEOPLE’S REPUBLIC OF MOZAMBIQUE

1.1. Necessity to Popularise Mathematics

Political independence is only one necessary condition for a real independence, for the creation of prosperity and wellbeing for the whole people.

44 Mathematics Education in Mozambique

Mozambique is now laying the foundation for economic independence. The period 1981-90 has been declared the ‘Decade of Victory over Underdevelopment’, during which the Mozambican people aim to eliminate poverty, hunger, disease, illiteracy, etc. The 10-year plan stresses industrialisation, socialisation of the countryside, and the training of a skilled labour force. In order to train a labour force able to meet the development needs, the first National System of Education was introduced in 1982. Integration of the school into community life, manual work and production as an integral part of the educational process, and the development of a scientific, materialist and dialectic view of the world are the general principles guiding the National System of Education. The Government has given a high priority to mathematics education, e.g., the National Seminar on the Teaching of Mathematics (May 1980), Mozambique’s second national seminar on teaching, following the Portuguese language teaching seminar, was chaired by the Minister of Education and Culture. This seminar defined the broad goals of school mathematics education such as stimulation in the understanding of the significance of mathematics for raising production and productivity, and transmission of mathematical knowledge necessary for the development of the country. Other objectives relate to general mental capacities, such as critical attitude, creativity, discipline of thought, logical reasoning, etc., see Gerdes (l98la). The continuing high failure rates, maths anxiety and the related lack of popularity of mathematics among many children remain clearly in contradiction with a mathematics education that has to become really socialist: everyone liking and mastering mathematics, everyone capable of thinking mathematically, for the benefit of society as a whole. It is obvious that ways to popularise mathematical knowledge had to be looked for.

1.2. Looking for Ways to Popularise Mathematical Knowledge

Mozambique is not the first country to construct a socialist society, nor the first to eliminate underdevelopment; neither is it the ﬁrst country with a need to popularise mathematical knowledge. So it

45 Paulus Gerdes

was natural to look at experiences elsewhere, particularly in other socialist countries. Universal mathematics education in schools is generally the basis for the propagation of mathematical ideas and methods. However, experience shows that it is important to improve the climate of mathematical learning by a variety of extra-curricular activities: student clubs, public lectures, publication of booklets, mathematics contests, etc. Various writers have emphasised this aspect, including Gnedenko (1979), Roman (1966), and Hua (1980). Student clubs have existed in Mozambique since the days of the national liberation struggle. From 1977 onwards, a series of lectures on mathematics and its importance, history and applications has been delivered to different audiences. “Tlanu” 1, the Mozambican journal of mathematics education was launched in 1981, the ﬁrst year of the “Decade of Victory over Underdevelopment”. At the same time, the Ministry of Education and Culture published a booklet, Science of Mathematics, see Gerdes (l98lb). The time was ripe to start Mathematics Olympiads.

2. LEARNING FROM OTHER EXPERIENCES

2.1. Objectives

Mathematics Olympiads are annual ‘mathematics contests, open to all students of a certain educational level. They can be played in several successive rounds, corresponding to successive geographic units (locality, district, province, nation). Generally, Mathematics Olympiads fulfil the following two clusters of objectives in socialist countries:

(i) * “to stimulate student interest in solving non-typical mathematics problems”, Skvortsov (1978, p. 354); * “to encourage young people to study hard to master science” 2 * “attirer des élèves en nombre croissant vers une étude plus approfondie des mathématiques”, Roman (1974, p. 425); 46 Mathematics Education in Mozambique

* “to fight the prejudice that mathematics is a difficult and boring subject for exceptionally gifted people”, Suranyi, J., cited in Freudenthal (1969, p. 82). (ii) * “to assist in ﬁnding those students who have a superior interest and ability in mathematics so that they might be provided with opportunities to develop their ability to the fullest”, Skvortsov (1978, p. 354); * “an effective way to discover and train young talent” (see endnote 2); * “sélectionner les mieux préparés; attirer des jeunes gens doués vers l’enseignement supérieur specialise”, Roman (1974, p. 425).

The first of these two clusters reﬂects – in the spirit of De Coubertin’s sports Olympiads – the ultimate goal of Mathematics Olympiads: popularisation. However, this does not mean that the other one, selection, may be neglected; to develop a socialist society, a mere reliance on the spontaneous emergence of mathematically talented students is insufficient to satisfy labour needs, Skvortsov (1978, p. 352).

2.2. How to Achieve these Objectives?

In planning how to achieve the above-mentioned objectives, we read in the literature the following guidelines. Skvortsov suggests that to induce a large participation from the student body it is necessary to select the Olympiad problems so that their solution do not require any knowledge exceeding that provided by the general school curriculum (p. 359). According to Roman, importance has to be attached to selecting problems of varied complexity “aﬁn que la majorité puisse résoudre quelque chose et que la selection pour le tour suivant soit possible . . .” (p. 434). The Hungarian Suranyi advises caution. Failures in Mathematics Olympiads may discourage the students. For this reason, a correct psychological preparation is necessary: neither the teachers nor the

47 Paulus Gerdes

students should over-estimate the results in the Olympiads. The Olympiads should remain a complementary activity, a good sport rather than a duty. Moreover, Roman warns against a too competitive atmosphere. Discouragement should be avoided by a good mathematical and psychological preparation of the candidates, and “par le climat de parfaite camaderie dans la classe, par la transformation d’un insuccès en décision de se mieux préparer pour le concours suivant” (p. 434, my italics). With these objectives and guidelines in mind, 3 the mathematics community in Maputo, Mozambique’s capital, organised the first, experimental, local Olympiad in 1981.

3. THE FIRST MAPUTO MATHEMATICS OLYMPIAD

3.1. Mature Situation

In the first years after independence some isolated mathematics contests were organised in secondary schools and in the Industrial Institute and the Eduardo Mondlane University in Maputo. The time had come to start competitions on a wider scale. On the other hand, it was also clear3 that teachers who themselves had participated in mathematics contests could more easily motivate their own students to take part in Olympiads than their colleagues who did not have such experience. The first graduates from the Teacher Training Courses for secondary schools (7th to 11th grades) were due to leave at the end of 1981, see Gerdes (1981a, pp. 462-466). Once motivated by their own Olympic experience, they could constitute a national network for the propagation of Mathematics Olympiads. These were the two principal factors that led to the launch of the Mathematics Olympiads in 1981, at the level of the tenth and eleventh grades.

48 Mathematics Education in Mozambique

3 .2. Realization

After a period of mathematical and political stimulation of the students, see Gerdes (l981a, pp. 459, 474), the first Maputo Mathematics Olympiad at the level of the 10th and 11th grades was held on 4 July 1981, in a festive atmosphere at the Eduardo Mondlane University. Radio Mozambique transmitted a three-hour live programme of the Olympiad, interviewing students and teachers, and referring to the importance of mathematics. In total, 139 pupils took part voluntarily. They comprised 14% of the eligible youth of secondary schools and 34% of the students of the Mathematics and Physics Teacher Training Course (seventh to ninth grades). Only 17% of the participants were girls (14% were Mozambican girls, the others Hungarian and Portuguese).

3 .3. Correction

Each of the nine problems had been allotted a maximum number of points. The jury, with mathematicians from different institutions and of different nationalities, had to face some challenging difficulties: How to establish criteria for evaluating the solutions of all problems? How to compare an original solution containing an arithmetical error with a less original, but arduous solution where the persistent student gets the right result? How to evaluate a correct result without any justification in comparison with a well analysed and justified answer with a minor error? Obviously, the jury did not succeed in completely removing these difficulties; but, in exchanging their views, those mathematicians involved became more aware of some (hidden) criteria they normally use.

4. ANALYSIS OF RESULTS

For the full test, see Appendix 2.

49 Paulus Gerdes

4.1. Distribution

Table I gives the distribution of scores obtained by the contestants. The average score was 27, out of a possible 100. The maximum score was 67. Almost the whole student body was capable of solving one or another problem, but nobody succeeded in solving all of them.

TABLE I General distribution of scores

Some problems were very accessible. Problem 4 was solved by more than 80% of the participants. Similarly, nearly everyone received some points for the first problem (Table II).

50 Mathematics Education in Mozambique

TABLE II Distribution of scores for problem 1

Problems 6 and 8 were more selective, as Tables III and IV show. Problem 7 was too selective: only two participants gave more or less acceptable solutions. These distributions show that the problem selection satisfied the two original objectives: to inspire conﬁdence in the participants in their creative powers and to stimulate further study. Nevertheless, problem selection and correction could be improved in such a way that the average score attained would be considered positive, at least by the participants.

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TABLE III Distribution of scores for problem 6

TABLE IV Distribution of scores for problem 8

52 Mathematics Education in Mozambique

4.2. Social Background

The students who participated in the First Maputo Mathematics Olympiad had reached 10th or 11th grade level. This means that they started their formal education before the independence of Mozambique. For this reason, their social composition is rather different from the general picture. Since the educational explosion, see Gerdes (198121) after liberation, the great majority of children who attend school are children of peasants and labourers. Only 29% of the contestants were from a working class background. Of the forty highest qualified participants, 23% were from a working class background. It is difficult to extrapolate on the basis of this data. The possible discrepancy between 29% and 23% nevertheless constitutes a stimulus to reﬂect on the teaching methods used and their possible implications of social class selection. It is interesting to note, on the other hand, that the best results were achieved by children of peasants and labourers. We have already given a short autobiography of Esselina Macome, and Appendix 1 contains some more autobiographies of the winners. These offer some insight into the social aspects of the development of their mathematical talents.

4.3. Composition by Gender

As noted before in Section 3.2, 14% of the contestants were Mozambican girls, but only 5% of the forty highest qualiﬁed participants were Mozambican girls. This means not only that girls are still under-represented at this educational level (as a vestige of the position of women in traditional-feudal and colonial-capitalist societies), but suggests also that girls who reach the 10th grade are less likely to advance in science than boys at the same level. These results do not reinforce any prejudice concerning women’s mathematical abilities. On the contrary, the Mozambican girls may feel themselves stimulated by the fact that the second place in this first Maputo Mathematics Olympiad was taken by Eszter Gelencser, a Hungarian girl whose parents are working in 53 Paulus Gerdes

Mozambique. In the interest of the emancipation of Mozambique’s women, it is considered necessary to analyse the social reasons (inside and outside education) for the l4% – 5% discrepancy. In particular, it means that mathematics teachers must look at their own teaching methods and consider how to contribute more to the mathematical emancipation of women.

4.4. ‘A Great Success’

On 11 July 1981 the prize-giving ceremony for the First Maputo Mathematics Olympiad was held at the Eduardo Mondlane University. The Minister of Education and Culture, Graca Machel, gave prizes to the forty highest scoring contestants. Special prizes were awarded to the best qualified Mozambican girl and for the most original solution. After an explanation of the problems by some mathematicians, the Minister delivered an important speech on the significance of Mathematics Olympiads in Mozambique: This initiative awakens in our youth the interest and the profound desire to study, not only to know the minimum, but to know the maximum; to struggle to produce the maximum for the growth of our country. On the other hand, the initiative contributes to the necessary demystification of mathematics: ‘Mathematics is something very difficult; mathematics is something very complicated; mathematics is something that can only be understood by very special human beings’. Well listen. We have seen that students from the most varied social backgrounds (children of labourers, children of peasants, children of civil servants) not only have participated but also have won prizes. Thus they have demystified: it is not necessary to be an extra-ordinary human being; it is not necessary to have grown up in a very special environment; it is not necessary to have lived in extremely favourable conditions to be able to get good results in mathematics. What is necessary, is to study and not only to study, but to fight to get good results. [See Appendix 1 herein.]

54 Mathematics Education in Mozambique

With the example of the Mathematics Olympiads established, she said: ... we feel more convinced that we will defeat underdevelopment. We shall make mathematics an important instrument in the struggle against underdevelopment, in the struggle for the victory of socialism.

5. EXTENSION

The Minister of Education and Culture also noted that the participants of the first Olympiad were “the pioneers who opened the road of what will be in the future many other Olympiads, not only in Mathematics but in many different scientific disciplines.” The Department of Mathematics and Physics of the Eduardo Mondlane University then began work on other Olympiads. Later in 1981 they organized Mathematics Olympiads in Beira at the level of the 10th and 11th grades (73 participants) and in Maputo at the level of the 7th, 8th and 9th grades (first round comprising 373 contestants) and, finally, the First National Mathematics Olympiad at the level of the 10th and 11th grades (241 participants). For the university mathematicians (especially those recently arrived in the country) involved in teacher training, the Mathematics Olympiad for the 7th to 9th levels proved very useful in further understanding some of the difficulties with which the students arrive at the university. (See Appendix 3 for the full test.) For example, many pupils * do not know how to handle brackets; * only guess a solution without trying to control their hypotheses; * do not understand what it means to demonstrate a proposition; frequently they say that for a proposition to be true in general it is sufficient that it is true in a concrete case; * show a lack of experience with algebraic symbols; * are ‘closed’, in the sense that they try to use in a mechanical way the methods they have learned, even in situations where those methods are not applicable.

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Also they think frequently in terms of ‘what will be the trick to solve this problem?’. Two booklets with Olympic problems have been published – including autobiographies of the principal winners as worthy models of emulation for Mozambican youth – to promote the expansion of mathematical competitions throughout the country. In 1982 not only were Mathematics Olympiads organised, but also the first Olympiads in Physics and Chemistry.

6. THE FUTURE

Building on the ﬁrst experiences, it is hoped to improve and expand Mathematics Olympiads in the future: * to improve popularization and selection (see above); * to improve the content of the problems; e.g. by taking into account the recommendation of the mathematicians at the Agricultural Institute in Chimoio that it is necessary to include more realistic and practical problems; * to expand the competitions to the lower grades of general education 4 and to labourers and peasants attending literacy courses and adult education programmes; and * to organise contests for groups to promote a spirit of collective work among students in solving problems. Another idea is to take part in International Olympiads, as a first stage perhaps on a regional level under the auspices of the Southern African Mathematical Sciences Association (SAMSA), an organization of the nine countries of the Southern African Development Coordination Commission (SADCC), united in their struggle for economic independence from South Africa.

7. POSTSCRIPT 1983

As already expected (see §4.2), in 1982 and 1983 we have seen a shift in the social background of the contestants, clearly as a fruit of

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the revolution: the majority of children, and so the children of peasants and labourers, have gained access to the science of mathematics. On average, 80% of the participants in the Mathematics Olympiads at the level of the 7th, 8th and 9th grades are of working class background. The same tendency can be noted when we analyse performance by social class: also about 80% of the forty best qualified participants are of working class background. Table V illustrates this tendency.

TABLE V Social background of the winners of the Maputo Mathematics Olympiad at the level of the 7th grade (1982)

Name Profession of father Gender Score (out of 100) 1. Lucas Cossa Peasant Male 32 2. Augusto Melembe Peasant Male 30 3. Pedro Chirrime Truck driver Male 27 4. Helena Langa Truck driver Female 24 5. Hilário Buque Railway worker Male 22 Jorge Chicamba Telephone operator Male 22 7. Aniceto Nguenha Carpenter Male 20 8. Emílio Temusse Peasant Male 19 Sérgio Macamo Tailor Male 19

With regard to the composition and performance by gender of the contestants in the Mathematics Olympiads the situation is rather difficult. At national level we have not seen, until now, any significant change. On the contrary, only 13% of the participants in the First National Mathematics Olympiad (1983) at the level of the 7th, 8th and 9th grades were girls. The ﬁnal communiqué concludes: In order to guarantee the mathematical emancipation of Mozambique’s women, we have to mobilise in the next years many more girls to take part in the Olympiads.

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It is really a question of mobilisation, as the results of the Nampula Mathematics Contest of 1982, in which more than half the eligible youth took part, seem to show (see Table VI).

TABLE VI Composition by gender of the principal winners of the local Nampula Mathematics Contest (1982) at the level of the 7th grade

Name Gender Score (out of 100) 1. Inês Teixeira Female 63 2. Jeorgina Lázaro Female 54 3. Nelson Alberto Male 50 Rosa Juma Female 50

TABLE VII Performance by grade and by sex in the First National Mathematics Olympiad (1983) at the level of the 7th, 8th and 9th grades

7th 8th 9th grade grade grade Number of contestants 1st round 449 438 306 2nd round 109 108 124 Best score in 2nd round Male 54 60 70 (out of 100) Female 30 34 44 Average score in 2nd round 10 14 17

The mobilisation factor reﬂects itself also in the participation percentages. In the case of the National Mathematics Olympiad (1983), participation varied between 3% of the eligible youth of some schools and 95% for the FRELIMO Secondary School of Mariri in the

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northern Cabo Delgado province. As expected and hoped for (see §3.1), the young teachers, who themselves had participated in previous mathematics contests, were the best mobilisers. Table VII gives performance by grade and by sex for the First National Mathematics Olympiad (1983) at the level of the 7th, 8th and 9th grades. Because of communication problems, the exact number of contestants is not known as yet. At least 1300 pupils took a voluntary part. Diplomas of honour, bicycles, radios, watches, mathematics booklets, etc. have been awarded by the government to the best qualified provincial and national contestants. Although the Olympiads already constitute an important success from the point of view of participation, public attention and support, etc., some major problems still have to be resolved: * How to improve the selection of the tasks so that the community in general will consider the results as stimulating? Very low average scores (see Table VII) have aroused some criticisms and pessimistic comments. * How to improve the relationships between the tasks and the mathematics programme in the schools, when, because of the shortage of well-trained teachers, the official programme is not integrally taught in many schools. * How to introduce more practice-related exercises that will motivate all students and whose nature will be understood by most contestants, both from the towns and from the countryside. At the same time it has been concluded that the formulation of the tasks has to be simpliﬁed, as mathematics is taught in a second language, Portuguese, considered a vehicle of national unity. The solution of these challenging problems is dialectically related to the development of education in general.

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APPENDIX 1: SHORT AUTOBIOGRAPHIES OF SOME WINNERS IN THE FIRST MATHEMATICS OLYMPIADS

l. Abdulcarimo Ismael

I was born on 1 March 1962 on the island named Ilha de Moçambique. My father is Ismael Vieigas, an electrician. My mother is Merbano Habibo. When I was four years old I started in the Mohammedan School, where I advanced without problems. When I was six I started primary school, Luis de Camoes, on Ilha de Moçambique, at the nursery, where I failed for my first and last time. In the 2nd form, where I was first taught multiplication tables, I started to like maths, mainly because the teacher asked the tables and he who was right would give spankings to the others. As I always studied the lessons I was one of those who always gave the right answer. However, as I was the youngest in the class I did not give spankings as I was afraid to get them back out of class. In 1974 I finished the 4th form at primary school, and began immediately the first year of the commercial school, also on the island. In 1976 I passed on to the 7th form, which did not exist at the only school of the island. With the help of teachers and some adults an evening class was formed for the 7th form. We began the year with about 30 pupils and finished with only seven, because some gave up as they could not cope with studying at night and others were transferred to another district. Of these seven students, one failed and of the six who passed two were me and my brother. In the following year, 1978, we had to go to Maputo to continue our studies. Already accompanied by another brother who had failed the 8th form in Quelimane, we came to Maputo, this was paid for by one of my brothers who worked at the Bank of Mozambique. In Maputo we, four brothers, lived in a house where we, ourselves, were the servants. During the first months we had many problems, but as time went on we adapted ourselves. The school year began and we, the three brothers, attended the commercial school in Maputo. I had a mathematics teacher who enjoyed demonstrating what he knew about

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mathematics, and the group nicknamed him ‘the man of more advanced mathematics’. The pupil who succeeded in solving a problem, which all the others in the class could not do was considered the Champion Mathematician. There was a lot of competition and everyone tried to overthrow the Champion Mathematician. Nearly every week a mathematician was overthrown, except during the third term when I succeeded in keeping the title myself for quite a long time. This made me like mathematics even more. As all the students who ﬁnished form 9 were dependent upon placement in other schools to continue their studies, I expected to be placed in the Commercial Institute. However, this did not happen and I was placed in the Teacher Training Course of Mathematics and Physics for the 7th to 9th grade class at the Eduardo Mondlane University. I really did not like the idea of being a teacher, but as mathematics, which I liked so much, was involved I stayed on. As time went on I got good marks, this increased my love of mathematics. And through the campaigns to motivate students I also started to like my future profession in which I would always be working with mathematics.

Note: Abdulcarimo Ismael took the fourth place in the first Maputo Mathematics Olympiad and the first place in the first National Mathematics Olympiad. He has been selected to continue his university studies in mathematics and physics in the German Democratic Republic.

Additional note 2014: Abdulcarimo Ismael concluded in 2001 his Ph.D. in Ethnomathematics with the thesis 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). He became Head of the Mathematics Department of the Pedagogical University and is currently the Academic Director of the Lúrio University that has its principal campus in the city of Nampula in the North of Mozambique.

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2. Lourenço Lázaro Magaia

I am the son of Lazaro Magaia and Néria Mabika. I was born on 1 August 1961 in Sinoya, Zimbabwe, where my father, who used to be a farmer, had temporarily found work as a stonecutter. He died during the National Liberation Struggle on 1 May 1974. My mother, a widow, is a farmer and now lives in the district of Makumbura in the Tete province. I began my studies in 1968 at the age of seven, at a primary school in the district of Magoe, which only functioned up to the 2nd class. In only one year I completed the nursery and the first form, as my capacities exceeded that necessary for those classes. I did standard two in 1969, and proved myself a good student, especially in arithmetic. My father insisted that I should study. He did not allow me to neglect my studies for jest. He corrected my books and hit me if I left any exercise incomplete. Thus I always studied and even did my homework on the way home. When I did not study I took care of the few cattle my father possessed. In 1970 I enrolled at a Catholic School, the Holy Mary Mission of Makumbura, approximately 15km from home. I did not complete standard three because the National Liberation Struggle had begun in that zone. I left school and with my parents joined FRELIMO (Mozambique’s Liberation Front). In mid-1971, FRELIMO founded a ‘pilot centre’ in Jeque, in one of the liberated areas. I was one of the first pupils of this school, which taught only up to the 3rd class. Due to the great lack of teachers, the older students were recruited as teachers for the other classes. The enemy, understanding the importance of the centre, attempted to destroy it. We did not give up and studied during those moments that the bombing momentarily ceased. We hardly knew what a good meal meant, and ours were nearly always limited to meat and wild fruit. In August 1974, already in the 4th class at the ‘pilot centre’ of Dzembe, I received the sad notice of my father’s and sister’s deaths. I did not stop studying. The only way to ‘revenge’ myself for this cursed incident was to study hard, in this way realising the order my father had left me: ‘My son, don’t play! Always study’.

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I finished the 4th form in Chipera after the Lusaka Accords (on the Independence of Mozambique). In 1975 I did not study, as there was no FRELIMO secondary school inside the country. At the end of the year I was transferred to the new FRELIMO secondary school in Mariri (Cabo Delgado) where I finished my secondary education successfully. In the 6th and 7th forms I had a very able mathematics teacher – the unforgettable Ally Conde of Guinea Conakry – who made me enjoy this subject even more. He taught clearly and never failed to point out the importance of mathematics in society. I did the 9th form in 1980 and in March 1981 I had to follow the Portuguese Teacher Training Course at the Eduardo Mondlane University. I never liked the idea of being a teacher, but understood the necessity of agreeing to fulfil this task: why plant a tree in the desert? I was born and grew up with the people, I have to serve my people. The only thing I tried was to change from the Portuguese to the Mathematics and Physics Teacher Training Course. This was allowed. In 1981 I did the ﬁrst year of this course. I feel that my capacities have increased in terms of knowledge and application. Every time I teach I enjoy mathematics more and I know that, while teaching, I will always improve my knowledge.

Note: Lourenço Lázaro Magaia took the second place in the First National Mathematics Olympiad. Probably he will continue his studies abroad.

Additional note 2014: Lourenço Magaia concluded in 2006 his Ph.D. in Applied Mathematics with the thesis A video-based traffic monitoring system (University of Stellenbosch, South Africa). He is Director for Planning, Cooperation and Institutional Development of the Zambeze University that has its principal campus in the city of Beira in the centre of Mozambique.

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3. Fernando Victor Martins Saide

I was born in April 1963 in the village of Bajone, in the district of Maganja da Costa, in Zambézia. My parents are Martins Saide Mussa and Daulisa Victor, both still live in this village. My mother devoted her time to working in the fields. My father was a photographer, then a shop assistant, and later worked for a tobacco company. Now he is the Secretary for Political Economy of the FRELIMO Party in Bajone. My parents always wanted their children to go to school. When I was five I learned the ‘ABC’ in the mission school of my village. In this school I right away began to teach schoolmates who could not remember the alphabet and counting. As I was very young, the teacher treated me as an example and gave me a special desk. For me this was a stimulus to study more. I always knew the multiplication tables so I was never spanked. This school had only the first three forms. When I finished this level, I went to the government school, which was 5 kilometres from my house. There again I was the youngest of the class and the best in arithmetic and natural sciences. I added one year to my age because otherwise it would have been impossible to do the standard 4 examination. Sometimes our teacher organised competitions with arithmetical problems, which were rather difficult for our level. The rewards were sweets, tangerines, oranges or a snack, and whoever gave the correct answer first won. In one of these competitions I won a book of arithmetical problems. In standard four I helped the teacher of standard one to correct the arithmetic exercises. So I started to have a ‘good name’, just like my brother who was at that time considered the best pupil of my village. When I finished primary school, I did not go to school for two years, because my parents could not support two of their children in secondary school. During this time I went into the fields with my mother; I ﬁshed and herded goats. In 1974 I went to the secondary school in Quelimane, where at that time my father was working at the tobacco company. For the first time I learned French and mathematics, which were my favourite 64 Mathematics Education in Mozambique

subjects. In the school year 1974/75 I had four different mathematics teachers. During the last two terms, I went to live with my uncle and, at the end of the year, I went to Pebane, living with another uncle. There I had to do all kinds of work (cooking and house-keeping). My aunt gave me so much work that I was completely exhausted. I had to wake up very early to prepare tea or to go to the beach to buy fish and, as I went to school at night, I had to prepare the lessons the same night. In spite of these difficulties I continued being a good pupil, especially in French and mathematics. After that year, I went to Nampula with my eldest brother and there I ﬁnished my secondary school in 1979 (9th grade). I had a very good mathematics teacher in standards seven and eight. I liked him because of his explanations and the different ways he showed us to solve problems. My interest in this subject increased and there had been created in me a great desire to know the depths of mathematics. At that time I joined with two other schoolmates to discuss the various problems every day. We gained a lot of experience in solving mathematical problems and we advanced more than our other colleagues. So we did not have to take part in the examinations of standards seven and nine. In standard nine I volunteered to teach mathematics to standard five. It was good for me to review several basic notions to test my knowledge and my capacities. Pupils and teachers agreed that I would become a good teacher. I always liked mathematics. It is easy for me to know when I am right or wrong whilst solving a problem. In natural sciences and history, for instance, I am sometimes not sure and I am not convinced that my answers are correct and I do not see a way to verify them. In 1980 I got the opportunity to become a mathematician, I was placed in the Mathematics and Physics Teacher Training Course at the Eduardo Mondlane University in Maputo.

Note: Fernando Saide took the first place in the First Maputo Mathematics Olympiad and the fourth place in the first National Mathematics Olympiad. He has also been selected to continue his university studies in mathematics and physics in the German Democratic Republic.

65 Paulus Gerdes

Additional note 2014: Fernando Victor Saide concluded in 2000 his M.Sc. in Physical Oceanography, entitled Tides, circulation and water masses in Maputo bay (Göteborg University, Sweden). He is a lecturer at the Department of Physics at the Eduardo Mondlane University in Maputo.

APPENDIX 2: PROBLEMS IN THE MAPUTO MATHEMATICS OLYMPIAD (1981) AT THE LEVEL OF THE 10TH AND 11TH GRADES

Maximum Problems score 1. Find the missing numbers: 1 (a) 2 4 6 6 8 l0 l0 12 14 … … 1 (b) -1 1 5 11 … … 1 (c) 24 12 6 … … 3 (d) 0 1 2 3 6 l1 20 37 … … 1 (e) 2 … 8 16 … 64 2 (f) 0.5 1 2.5 5 10 … … 100 … … 9 2. Determine: (402 + 392 + 382 + … + 232 + 222 + 212) – – (202+192+ …+ 22+12) 3. 222 and 222 are examples of numbers that can be constituted by three 2s without using any other symbol. 5 (a) Determine all the numbers that can be constituted by three 2s. 9 (b) Which is the greatest of these numbers and why? 9 4. How is it possible to divide the following sketch of the moon into six parts by drawing only two straight lines?

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Fig. 2. 9 5. Two birds are sitting on the tops of two palm trees, 20m and 30m high. The trees are on opposite sides of a river 50m wide. Suddenly, the birds discover at the same time a fish at the water surface between the two palm trees. The birds dive for the fish and reach it at the same moment. Find the distance between the ﬁsh and the highest tree. 10 6. Determine the value of p such that the quadratic equation x2 + px + 12 = 0 has two solutions whose difference is equal to l. 15 7. Solve the following system of equations: x + y + z = l x2 +y2 + z2 = 1 x3 +y3 + z3 = l 10 8. Given an isosceles triangle PQR with a right angle at P, let A, B, and C be the midpoints of the squares constructed on the sides of triangle PQR as in Figure 3. Let PQ = b. Determine the area of triangle ABC.

67 Paulus Gerdes

Fig. 3. 15 9. An engineer has to construct a road between the cities A and B (see Figure 4 below). The cities are on opposite sides of a canal. The construction of the bridge is very expensive. For that reason, the bridge has to be perpendicular to the canal. Make a drawing of the cheapest road.

Fig. 4.

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APPENDIX 3: PROBLEMS IN THE MAPUTO MATHEMATICS OLYMPIAD (1982) AT THE LEVEL OF THE 7TH, 8TH AND 9TH GRADES

Maximum Problems score 10 1. Alfredo, Humberto, Pedro and Roberto are waiting in a queue. When Alfredo leaves, Humberto will remain in second position, but when Pedro leaves, Alfredo will be the first. What is the sequence of the four boys in the queue?

Fig. 5. 10 2. None of these watches marks the right time. One of them shows a 30 minutes lag, another is 40 minutes fast and one of the watches has stopped. What is the right time?

Fig. 6. 10 3. Four cubes have the same weight as six balls. Knowing that two cubes and one ball weigh together 8 kg, determine the weight of a cube and a ball. 69 Paulus Gerdes

15 4. The angles of a triangle measure 15°, 105° and 60°. How can you divide the triangle into isosceles triangles? 5. The sum of some numbers is equal to 1. Will it be possible that the sum of their squares is 10 (a) smaller than 0.01? 10 (b) greater than 100? If it will be possible, give examples. 6. Determine how many numbers with two digits exist with 7 (a) at least one 5; 7 (b) the ﬁrst digit smaller than the second; 7 (c) the ﬁrst digit greater than the second. 14 7. What will be the maximum number of Sundays in a year?

REFERENCES

Freudenthal,H.: 1969, ‘ICMI report on mathematical contests in secondary education (Olympiads)’, Educational Studies in Mathematics 2, 80-114. Gerdes, P.: 1981a, ‘Changing Mathematics Education in Mozambique’, Educational Studies in Mathematics 12, 455-477. Gerdes, P.: 1981b, A ciência matemática, INDE, Maputo. Gnedenko, B.: 1979, ‘Popularisation of mathematics, mathematical ideas and results in the USSR’, in B. Boos and M. Niss Mathematics and the Real World, Birkhäuser Verlag, Basel. Hua, L. K.: 1980, ‘Some experiences in popularising mathematical methods in the People’s Republic of China’, lecture at ICME IV, Berkeley, USA. Reprinted in M. Zweng (ed.): 1983. The Proceedings of the Fourth International Congress on Mathematical Education, Birkhäuser, Boston, pp. 16-26.

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Roman, T.: 1974, ‘Les Olympiades Mathématiques en Romanie’, Educational Studies in Mathematics 5, 425-440. Skvortsov, V.: 1978, ‘Mathematical Olympiads’, in F. Swetz (ed.), Socialist Mathematics Education, Burgundy Press, Southampton, USA, pp. 351-370.

Department of Mathematics and Physics, Faculty of Education, Eduardo Mondlane University, C.P. 257, Maputo, Mozambique

NOTES

1 The journal Thlanu, Revista de Educação Matemática is edited by the Department of Mathematics and Physics (Faculty of Education) of the Eduardo Mondlane University in Maputo. 2 From ‘Middle School Math Contest’ in China Reconstructs, Vol. 9, pp. 30-32. 3 Taking into account the personal Olympics experiences of the mathematicians from the Soviet Union, The Netherlands, and Guinea Conakry, teaching in Maputo. 4 As, for example, in the German Democratic Republic. See: 1974, ‘Olympiaden Junger Mathematiker’, in: Methodik Mathematik- unterricht, Volk und Wissen, Berlin, pp. 405-407.

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Chapter 3 CONDITIONS AND STRATEGIES FOR EMANCIPATORY MATHEMATICS EDUCATION IN UNDERDEVELOPED COUNTRIES *

Reflecting on the author’s experiences in Mozambique, he suggests some conditions and strategies for mathematics education to become emancipatory in underdeveloped countries.

Point of departure: mathematics education cannot be neutral

By its physical and intellectual labour, mankind is able to create an ever more human society. Reflecting on its realisations, discovering the laws of nature and society, mankind creates its material and intellectual tools to transform reality, both nature and society. Mathematics constitutes an integrated body of those means to understand and transform reality. An ever more human society is the most rational direction. However, mankind has at its disposal, nowadays, the means to destroy itself. Neither mathematics, nor

* This is a revised version of an invited paper presented at the Caribbean Conference on Mathematics for the Benefit of the Caribbean Communities and its Reflection in the Curriculum, organised by the Inter-American Committee on Mathematical Education, Paramaribo, Suriname, October 18-21, 1982. Published in: For the Learning of Mathematics, Montreal, Vol. 5, No. 1, 15-20 (February 1985); Reproduced in: Is mathematics teaching neutral?, University of Cape Town, Cape Town, 1987, 10-15; Reproduced in: Alan J. Bishop (Ed.), Mathematics Education: Major Themes in Education, Routledge, London, 2010, Volume 1, 179-189; Reproduced in the expanded edition of: Paulus Gerdes, Ethnomathematics and Education in Africa, ISTEG, Boane & Lulu, Morrisville NC, 2014, 233-250. 73 Paulus Gerdes

mathematics education nor mathematicians can be indifferent towards these diametrically opposed possibilities:

more human more inhuman peaceful war-like liberating  oppressing creating destructive emancipating exploiting

The history of Mozambique shows very clearly that mathematics education cannot be neutral [see Gerdes 1980, 1981 b]. During the Portuguese domination, mathematics was taught, in the interest of colonial capitalism, only to a small minority of African children [see Mondlane 1969]. And those Mozambicans were taught mathematics to be able to calculate better the hut tax to be paid and the compulsory quota of cotton every family had to produce. They were taught mathematics to be more lucrative “boss-boys” in South African mines. After ten years of the people’s war, led by the liberation movement FRELIMO, Portuguese colonialism was defeated and Mozambique achieved its Independence in 1975. Post-independence objectives of mathematics education are in the service of the construction of a socialist society [cf. Machel 1977, Ganhão 1978]. Mathematics is taught to “serve the liberation and peaceful progress of the people”. Mathematics is taught to place its applications within the reach of the worker and peasant masses. Mathematics is taught to stimulate the broad masses to take an interest and delight in mathematical creation. These are general objectives. But how are they to be achieved?

Mathematics education for emancipation. How?

Independence and the option for socialism have implied an overall democratisation of mathematics education in Mozambique: the majority of children, and hundreds of thousands of adults, have already gained access to the science of mathematics; the mathematics teaching

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profession has been opened up for children of peasants and labourers; and the discussion on how to improve the quality of mathematics education is not any more reserved for an elite but is becoming the object of reflection of the mass the teachers, from adult education and primary school teachers to university professors [cf. Gerdes 1981 b]. Overall democratisation is a necessary but not sufficient condition for mathematics education to become really emancipatory – everyone mastering mathematics and capable of thinking mathematically, to the benefit of society as a whole. On the basis of the author’s experience in the training of mathematics teachers, some other necessary conditions will be presented.

Problematizing reality in classroom situations

Let us briefly listen to some dialogues occurring in our lessons:

* A student enters the classroom, late and noisy. Confrontation. Could your parents study in colonial times? No ... Who pays for your study? (With Independence, education became free of charge in Mozambique) The peasants and labourers ... How many days does a peasant have to work to pay for your being late? What do you mean? How much does it cost? What do we have to know to calculate it? (In a concrete case it was calculated, by the class, that one student coming one hour late corresponds to throwing away one day of a peasants’ work.) * Newspaper reading, a photograph: a truck crossing a bridge over the Changana River, the bridge collapsed. Why? Was the truck too heavy? The bridge ill-built? * In a report by the Centre of African Studies at our University it is told that a tractor driver drove his machine at full speed for two hours to fetch only three loaves of bread (from a state farm in Moamba). Why did he do so? Was it justifiable? Is it reasonable? Why? Why not? How can his behaviour be explained? How can the cost be evaluated? * For a certain period, sugar production was going down. Why? Changes in the way of paying the labourers had been introduced: 75 Paulus Gerdes

from payment in terms of the number of rows of sugarcane cut down to the number of kilograms of sugarcane. Why? How can we explain the economic consequences? How can production be raised? Is mathematics involved? * Our country suffered floods in 1977 (Limpopo River) and in 1978 (Zambeze River). Why didn’t we have any floods in the early eighties? Is mathematics involved? How? * To combat speculation a food distribution system has been introduced in the city of Maputo (1981). How much rice for each person this month? How much rice a family of five persons can eat each day?

These dialogues are examples of problematizing reality (Freire’s terminology). “Reality is thrown before our feet” (in the deep sense of the original Greek “-”); one cannot fly from reality, one has to reflect. Experience shows that problematizing reality leads to consciousness, to awareness of the relevance of mathematics as a tool to understand and transform reality. It leads to political awareness (as in the case of the student who came late); to physical awareness (as in the case of the bridge that collapsed); to economical awareness (as in the example of the tractor driver), etc. Let us give some more examples of problematizing reality before drawing a second conclusion – examples that give a meaningful, rich context to mathematics education. * In Cabo Delgado province some students were told that the area of a cotton field is so many metres. Area measured in m and not in m2? Does it make sense? Why? * Mozambique’s soap production was 16300 t in 1980 and 23700 t in 1981. What will the national needs be in 1990? How can they be calculated? What will the soap production be in 1990? Regular growth of production capacities or not? Regular in what sense? Why? * In order to win the battle against underdevelopment during the decade of the eighties, the People’s Assembly approved a National State Plan for the decade. We need x school buildings by 1990. In 1981 we already had y school buildings and will 76 Mathematics Education in Mozambique

construct z more school buildings. How do we have to extend production capacities for the construction of schools? Regular growth? How regular? Linear? Parabolic? Exponential? (A possible entrance to the study of geometric or exponential progressions.) * Women fetch water in their tins, cylindrical cans, on their heads. Are they exploited? Development will lead to piped water in each house. And in the meantime? Could the tin cans be less heavy? Could more cans be produced out of the same quantity of raw materials? How? What changes are implied? (A possible entrance to the study of differential calculus.)

Here it should be emphasized that a problematizing reality approach leads to a real understanding of reality, it leads to an understanding of mathematics as a tool to transform reality. Let us give one more example to reinforce this second conclusion [see Gerdes 1982]. * Rickettsiosis is a disease that kills a lot of cattle in our country. Veterinarians discovered a medicine to cure sick animals (terramycine). How much medicine is needed for each animal? This quantity depends on what? On the colour of the animal? Its height? How does this quantity depend on the weight of the cow? Linear dependence? Exponential? Why? How do we weigh a cow? Only on well-equipped farms will you find scales able to do so. What can be done in the less developed countryside? The weight of an animal is related to ...? Why? How? Is it possible to determine the weight of a cow in an indirect way? A cow knows how to swim? What does this imply? Relation weight-volume? How can the volume of a cow be determined? Let us take a very close look at the cow: approximation to a cylinder! (See Figure 1). And so on. Reality can be changed. More animals will survive. People will have more meat and milk. Improved calculations lead to a reduction in the quantity of medicine. And, as these medicines still have to be imported, well-applied arithmetic will save foreign currency.

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Approximate cylinder Figure 1

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Creating confidence

For mathematics education to become emancipatory, it is necessary to stimulate confidence in the creative powers of every person and of every people, confidence in their capacities to understand, develop and use mathematics. At least three types of strategies to produce such confidence have to be considered.

A. Cultural strategies

Colonization has implied the underdevelopment of Mozambique as of most Third World countries. Underdevelopment is not only an economic process. Foreign domination also caused mathematical underdevelopment: African and American-Indian mathematics became ignored or despised; the mathematical capacities of certain peoples were negated or reduced to rote memorization; mathematics was presented as an exclusively white men’s creation and ability. Of the struggle against the mathematical underdevelopment and the combat against racial and colonial prejudice, a cultural reaffirmation makes a part. It is necessary to encourage an understanding that:

1) The people have been capable of and will be capable of developing mathematics

Examples:

a) In coastal zones of Mozambique fish is dried to be sold in the interior. How should the fish be dried? What if you place the fish at different distances from the fire? Some fish will be grilled, while others are kept wet. Through experience, the fishermen discovered that it is necessary to place the fish equidistant from the fire. They discovered a circle-concept, constructing a circle in sand using two sticks and a rope (see Figures 2 and 3).

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Figure 2

Figure 3 (Drawings by Marcos Cherinda) b) How can the rectangular base of a house be constructed? In some parts of Mozambique, peasants use the following technique. One starts by laying down on the floor two long sticks of equal length (see Figure 4). Then the first two sticks are combined with two other sticks, also of equal length (Figure 5).

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Figure 4 Figure 5

Now one moves the sticks until the closure of the quadrangle (one obtains a parallelogram: Figure 6). One further adjusts the figure until the diagonals are of equal length (so a rectangle is obtained: Figure 7). Now where the sticks are lying on the floor lines are drawn and the building of the house can start.

Figure 6 Figure 7

This empirical knowledge of the peasants has been propagated on a national scale, during the rabbit-raising campaign, for the construction of hutches.

2) The people’s mathematics can and will enrich the understanding of mathematics, its education and its history

Examples:

a) In the north of Mozambique, traditional villages are structured in a “circular” way (Figure 8). This village structure and e.g. the structure of a bird’s nest led to the formation of one circle-

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concept “to belong to” (already a high-level abstraction). The concept is extended by comparison of form or content (“belonging to”) to e.g. the border of a basket (Figure 9).

Street

Houses Figure 8

Circular winnowing basket Figure 9

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The fabrication of a sisal mat leads to the formation of another circle- concept (“spiral-circle”: Figure 10). We have still a third circle- concept, that of the fishermen. In the official language of Mozambique, Portuguese, there are two “circle” concepts, one referring to the area and the other to the circumference of a circle. By interference between the mother tongue and Portuguese, the medium of instruction, we have the following matrix (Figure 11).

Rolling up a string of sisal Figure 10

Mother tongue

M1 M2 M3

Portuguese P1 P1 = M1 P1 = M2 P1 = M3

P2 P2 = M1 P2 = M2 P2 = M3

Matrix of language interference: circle-concepts Figure 11

So six different, rather complicated, situations may be encountered: now I have to use the third word (= M3) in my

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mother language but the second word ( = P2) in Portuguese ... , etc. How can this be avoided? Comparing the Portuguese “circle” concepts with the square concept, we can see that there is no need to have more than one “circle” concept: it is possible to speak about the area of a square and about the circumference of a square using the same work “square”. Paradoxically, by abolishing one of the two words in Portuguese, one enriches the scientific language: only one “circle” concept is retained and so one facilitates the learning processes of the students in this multilingual context.

b) In some regions of Mozambique, there exists a “cylinder” concept that can be described as a rolled-up rectangular mat (see Figure 12). This concept can be used as a way to discover a formula for the volume of a cylinder. It can be used not only in education, but also in formulating hypotheses on how long ago the first (approximating) formulas for the volume of cylinders could have been discovered [cf. Gerdes, 1985 b].

Figure 12

3) Every people is capable of developing mathematics

This can be shown by the cultural history of mathematics. By public lectures and by the publication of booklets on African, Indian, Chinese and Arabic mathematics, attention is drawn to the fact that many people have contributed to the development of mathematics. In this manner an attempt is made to combat a Eurocentric and distorted

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vision of the history of mathematics. The booklets try to analyse, in a way accessible to a broad public, not only how, but also why and for whom, mathematics was developed in different societies at distinct times [see Gerdes 1981a, 1984 a].

B. Social strategies

Within colonial society, as in every class society, prejudices about the mathematical talents of discriminated and exploited social strata and of women are widespread. For mathematics education to become emancipatory, it is necessary to encourage an understanding that: Children of all social classes and of both sexes have been capable and will be capable of mastering, developing and using mathematics. By means of counterexamples, ranging from Hypathia of Alexandria and Gauss to the winners of our National Mathematics Olympiads, sexist prejudices as well as those against children of peasants and factory workers are demystified. Special awards and scholarships are attributed to the best-qualified girls in the Mathematics Olympiads [see Gerdes 1984 b].

C. Individual-collective strategies

All the aforementioned strategies are already individual in the sense that they enhance personal self-confidence by stimulating the individual’s relatedness to mathematics through the problematizing reality approach, by supporting the comprehension of the relevance of mathematics, and by cultural and social self-confidence. More specific – say individual-collective strategies – are presented here by examples drawn from teaching experiences.

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a) Reflection on errors

Usually “solutions” of problems are collected and presented to the students, e.g., the following two solutions of the same problem:

5 + 1 5 + 1. 5 ‒ 2 5 + 1. 5 ‒ 2 2 1) 5 ‒ 2 = 5 ‒ 2. 5 ‒ 2 = ( 5 ‒ 2) = 5 + 1. 5 ‒ 2 (1 ‒ 2)√5 ‒ √5 9 + 4√5 = 9 + 4√5 = 9 + 4√5 2) 5 + 1 ( 5 + 1).( 5 ‒ 2) 3 ‒ √5 2 5 ‒ 2 = ( 5 ‒ 2) = 9 ‒ 4√5 = 1 ‒ √5 1 ‒ 1 0 3 ‒ 4√5 = 3 ‒ 4 = ‒ 1 = 0.

The students have to analyse these trial solutions, individually. Which transitions are right, which are wrong and why? In a second stage, they are asked to compare their analyses at group level and to try to achieve a common improved examination. The groups report their findings in a classroom session for a final analysis. It is interesting to see that the students like this type of “exercise”, because it urges them to reflect, argue and rethink

b) Reflection on concept building

The students know the following definition:

+ 푛 When b ∈ 푅 0 and n ∈ 푁, we define 푏 as the positive solution 푛 in 푅 of the equation 푥 = b. We ask them to debate questions such as the following: 0 Is it possible to define 푏 ? For which numbers? Does it make any sense? 1 Is it possible to define 푏 ? Is it necessary to do so? 86 Mathematics Education in Mozambique

‒ 3 ‒ 3 Is it possible to define 푏 ? If so, calculate 8. The questions are open-ended. E.g., in the last case, one finds answers ‒ 3 ‒ 3 by analogy: 푏 is the positive solution of the equation 푥 = b or by 1 ‒ 3 3 a supposed property: 푏 = 푏 . This reflection stimulates further debate and so provides a more profound understanding of the original concept.

c) Learning to discover by discovering together

+ What will be the derivative function of f: 푅 →푅, defined by f(x) = 푥 ? The students know: Δ푦 푓(푥 + ∆푥) ‒ 푓(푥) 푥 + ∆푥 ‒ 푥 Δ푥 = ∆푥 = ∆푥 lim Δ푦 ? How do we calculate ∆푥→0 Δ푥 How do we divide Δ푦 by Δ푥 ? We have 푥 + ∆푥 = ... ? 푥 + ∆푥 = 푥 + ∆푥, Pedro suggests. Is it true? Another suggestion by Lázaro: Square all the terms: 2 ( 푥 + ∆푥 ‒ 푥) = ... Not that way? Why? Ferdinand: Use another notation 1 1 2 2 푥 + ∆푥 ‒ 푥 = (푥 + ∆푥) – 푥 How do we continue? 1 1 2 2 2 ‒ 1 2 ‒ 1 2 2 ‒ 1 (푥 + ∆푥) – 푥 = (푥 + ∆푥) – 푥 = ( (푥 + ∆푥) ‒ 푥 ) 87 Paulus Gerdes

Why do you want to work with squares? 2 2 ( 푥 + ∆푥) – ( 푥) = (푥 + ∆푥) ‒ 푥 = ∆푥 How do we arrive there? 2 2 In general 푎 ‒ 푏 = ... ? 2 2 푎 ‒ 푏 = (a ‒ b). ... ? 2 2 푎 ‒ 푏 = (a ‒ b).(a + b). And now every student advances: Δ푦 푥 + ∆푥 ‒ 푥 ( 푥 + ∆푥 ‒ 푥)( 푥 + ∆푥 + 푥) Δ푥 = ∆푥 = ∆푥 ( 푥 + ∆푥 + 푥) = ( 푥 + ∆푥)2 ‒ ( 푥)2 (푥 + ∆푥) ‒ 푥 = ∆푥 ( 푥 + ∆푥 + 푥) = ∆푥 ( 푥 + ∆푥 + 푥) = ∆푥 ∆푥 ( 푥 + ∆푥 + 푥) = 1 = 푥 + ∆푥 + 푥 and obtains the end result Δ푦 1 1 1 lim lim 푓 (x) = ∆푥→0 Δ푥 = ∆푥→0 푥 + ∆푥 + 푥 = 2 푥 By discovering together and reflecting on their process of discovery, the students enlarge their creative powers and gain self- confidence. They learn to understand the non-tautological nature of mathematical knowledge [for a more profound analysis, see Gerdes, 1985 a].

Concluding remarks

A problematizing reality approach as starting point is in itself already a confidence-creating activity. Problematizing reality, reinforced by cultural, social and individual-collective confidence- stimulating activities will contribute substantially to an emancipatory mathematics education, to enable everyone and every people to

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understand, develop and use mathematics as an important tool in the process of understanding reality, the reality of nature and of society, an important tool to transform reality in the service of an ever more human world.

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References

Cherinda, M. (1981) Secando peixe, descobrir a circunferência, Tlanu, Revista de Educação Matemática, Maputo, 1, 13-15. Freire, P. (1970) Pedagogy of the oppressed, Continuum, New York. Ganhão, F. (1978) The struggle continues: Mozambique s revolutionary experience in education. Development Dialogue, 2, 25-36. Gerdes, P. (1980) Mathematik in Mozambique, Materialien zur Analyse der Berufspraxis des Mathematikers, 25, 143-275. Gerdes, P. (1981 a) A ciência matemática. INDE, Maputo. Gerdes, P. (1981 b) Changing mathematics education in Mozambique, Educational Studies in Mathematics, 12, 455-477. Gerdes, P. (1982) Exemplos de aplicações da matemática na agricultura e na veterinária. TLANU-booklet 3, Maputo [New edition: Lulu, Morrisville NC, 2008; English language editions: Examples of applied mathematics in agriculture and veterinary science, NECC Mathematics Commission, Cape Town, 1991; ISTEG, Boane & Lulu, Morrisville NC, 2014]. Gerdes, P. (1984 a) Para que? Para quem? A matemática nos países islâmicos. TLANU-booklet 11, Maputo. Gerdes, P. (1984 b) The first Mathematics Olympiads in Mozambique, Educational Studies in Mathematics, 15, 149-172. Gerdes, P. (1985 a) Marx demystifie calculus. MEP, Minneapolis MN. Gerdes, P. (1985 b) Three alternate methods of obtaining the ancient Egyptian formula for the area of a circle, Historia Mathematica, 12, 261-268. Machel, S. (1977) Le processus de la Révolution Démocratique Populaire au Mozambique. Editions L’Harmattan, Paris. Macome, E. (1983) Hipatia de Alexandria, Tlanu, Revista de Educação Matemática, Maputo, 2, 28-33. Mondlane, E. (1969) The struggle for Mozambique. Zed Press, London.

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Chapter 4 ON CULTURE, MATHEMATICS AND CURRICULUM DEVELOPMENT IN MOZAMBIQUE 1

0. Confrontation

You are mathematics educators, are you not? So let us see, if you are good at mathematics. * Do you know how to construct a circle, given its circumference?

1 Short version of an invited paper presented at the Seminar “Mathematics and Culture”, Bergen, Norway, 26-28 September 1985. This version was published in: Mellin-Olsen, Stieg & Johnsen Høines, Marit (Eds.), Mathematics and Culture: A seminar report, Caspar Forlag, Rådal, 1986, 15-42. 91 Paulus Gerdes

* Do you know how to construct angles that measure 90°, 60° or 45°, only using the strips of paper I distributed to you? * What is the minimum number of strips of paper you need in order to be able to plait a broader strip? * Can you fold an equilateral triangle out of a square of paper? * Do you know how to construct a regular hexagon out of these paper strips? I gave you five minutes. Who solved all the problems? Nobody? How is that possible? Who solved but 4 problems? Nobody? Three of them?... Did you fail? Do not you have the necessary mathematical abilities? No, that is not the reason; you need more time, don’t you? But you are mathematicians, are you not? You need more time to analyse these non-standard problems. All right. But let me only say to you, that many of our (illiterate) Mozambican artisans know how to solve these problems... (obviously “formulated” in another way).

1. Education-policy in Mozambique

After ten years of liberation-war, led by the FRELIMO- movement, Portuguese colonialism was defeated in 1974. Mozambique achieved its (political) Independence on 25 June 1975. With the development of the liberation struggle a decisive question was posed: what type of society to build in the liberated areas? FRELIMO’s President Samora Machel summarizes the debate of those days as follows: “New elements appeared within Mozambican society who proposed to substitute themselves for the fleeing exploiters, attempting to re-establish the capitalist exploitation practised by the Portuguese, in new forms. And we asked, was this really the objective of our fight? Was this really the objective of our

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sacrifice? The reply of the masses was clear: to reject any restoration of capitalist exploitation. They asserted that they were fighting for total liberation, not to substitute one exploiter for another whatever his colour” (Samora Machel, 1978, 402). During the ten years of liberation-war, the principles, which now guide Mozambican society, had been forged: the construction of a socialist society became the aim. The experience of the armed struggle taught us that education is not neutral; education has always a particular objective: one, which expresses the interests of the dominant class. The aim of traditional-feudal education was to bring the new generations to accept blindly the traditions of the feudal society held up to them as dogma: the authority of the elders, tribal sentiments, contempt for women, and superstitions. Superimposed upon this was the colonial-capitalist education that appears “as a process of denying the national character, alienating the Mozambican from his country and his origin and, in exacerbating his dependence on abroad, forcing him to be ashamed of his people and his culture” (Samora Machel, 1978, 401). It was necessary to break these chains’ of traditional and colonial education. In the liberated areas a new type of school was born, that was “much more than a place for the transmitting of knowledge... it was a centre for the formation of the New Man, the person with a national culture, with a Mozambican personality” (Ganhão, 1978, 28). The general aims of the new education were clearly defined at the Second Conference of FRELIMO’s Department of Education and Culture in 1970. In brief, they were: to create solidarity, to remove racial and tribal discrimination; to acquire a scientific attitude, open and free of all the weight of superstition and dogmatic traditions; to emancipate women; and to lead all concerned to recognise the necessity of serving the people, participating in production, respecting manual labour, freeing the capacity for initiative, and developing a sense of responsibility. With Independence rose the possibility to try to implement these general objectives on a national scale. In 1975, the illiteracy rate in 93 Paulus Gerdes

Mozambique was still 93%; by 1982 this rate had already been reduced to about 75%. Immediately after Independence, both private and missionary schools were nationalized and made free of charge. Thousands of new (provisional) schools were built. This led to an education-explosion: already three times more children are attending primary and secondary schools than at the end of the colonial period. Peasant- and labourer- children gained massive access to schools: “Nationalization in the education sector created the base for a democratization of education. We were able to start combating the obscurantist and elitist methods of bourgeois colonial education” (Ganhão, 1978, 33; italics p.g.). In 1983, the time had become ripe to introduce the (first) National System of Education, whereby the principle of seven year compulsory schooling is put into practice. The Minister of Education and Culture defined, among others, the following guidelines: * “Seven classes of compulsory schooling permit the acquisition of a basic education, suited to the needs of society and suited to the child’s environment. It features many subjects and guarantees equality of opportunity of access to higher levels of education and to vocational training” (Graca Machel, 1981, 4; italics p.g.) * “...the quality of teaching must guarantee at each stage... that the great majority of children pass to the next class each year” (Graca Machel, 1981, 5) * Education has to be polytechnic in character. There has to exist a close relationship between theory and practice “which is brought about by joining study with productive work and by the organic link between the school and the community”; “We put into practice the principle of linking theory with practice at all levels and in all subjects, organising correctly the planning of the curriculum, eliminating the bookish and encyclopaedic concept of education, and systematically and constantly applying knowledge acquired” (Graca Machel, 1981, 8) * “The success of compulsory education lies very much in the countryside, where the largest number of children are to be found” (Graca Machel, 1981, 6).

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What are the implications of these general guidelines for the development of mathematics curricula? Which objectives, contents, teaching methods and evaluation procedures are the most convenient? Before discussing these fundamental questions, let us briefly summarize some of the main problems of mathematics education in other developing countries that gained their national independence before Mozambique. What lessons can be drawn from their experiences?

2. The fight against mathematical underdevelopment: some lessons from other Third World countries

In 1980, still 33% of the world population was illiterate. Although there had occurred a dramatic explosion in the student population in many African countries over the last twenty five years, the mean illiteracy rate for Africa was still 66% in 1980. Overcrowded classrooms and the shortage of qualified mathematics teachers – who are, by the way, relatively low paid and who enjoy a low social status compared with profession graduates – and lack of teaching materials, contributed towards the lowering of levels of attainment (cf. Eshiwani, 1979; Nebres, 1979, 1983; El Tom, 1984). In the case of mathematics education, this tendency has been reinforced by hasty curriculum transplantation.

Curriculum transplantation

The African Mathematics Program (AMP), financed by the Ford-Foundation and USAID, started in 1962. More than 60 ‘New Math’ books have been produced under this programme. Although 102 out of 186 textbook writers have been Africans, the contents and methods of these books are dominated by advocates of the North American School Mathematics Study Group (SMSG). By consequence, the AMP-materials are very theoretical, applications are avoided and the language used is strongly formalised. These books had been tested by well-qualified teachers in a few experimental schools. However, when they were introduced on a large scale, the results were rather doubtful (cf. Swetz, 1975). Teachers, 95 Paulus Gerdes

parents and politicians protested. E.g., in 1977, the Nigerian government decided to ‘abolish’ the teaching of ‘modern mathematics’ and to return to ‘traditional mathematics’ (see Shirley, 1980; Ohuche, 1978); Kenya’s President, Arap Moi, announced in 1982 the abolition of ‘new mathematics’, because it was “part of an imperialist plot to keep down Kenyans” (Anon, 1982, 28).

Selective perspective

Curriculum transplantation is not specific for Africa, but constitutes a general phenomenon of mathematical underdevelopment (cf. e.g. Nebres, 1979, 1983, 1984). With the transplantation of curricula their perspective was also copied: “(primary) mathematics is seen only as a stepping stone towards secondary mathematics, which in turn is seen as a preparation for university education” (Broomes & Kuperes, 1983, 709; cf. e.g. Freudenthal, 1979, 321). Mathematics education is therefore structured in the interests of a social elite. To the majority of children, mathematics looks rather useless. The president of the Interamerican Committee on Mathematics Education, Ubiratan D’Ambrosio, characterizes this ‘traditional pattern’ of mathematical education as follows: “... the traditional pattern... carries an over—emphasis on the value of mathematics by itself, with a clear tendency towards overstating a somewhat romantic importance of mathematics as the builder of clear thinking, as the rigorous science par excellence, and an unconvincingly possible practical value” (D’Ambrosio, 1979 a, 37). Math anxiety is wide spread; especially for sons and daughters of peasants and labourers mathematics enjoys little popularity. Mathematics education serves the selection of elites: “Mathematics is universally recognised as the most effective education filter” (E1 Tom, 1984, 3), as Mohammed El Tom underlines. D’Ambrosio agrees “... mathematics has been used as a barrier to social access, reinforcing the power structure which prevails in the societies (of

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the Third World). No other subject in school serves so well this purpose of reinforcement of power structure as does mathematics. And the main tool for this negative aspect of mathematics education is evaluation” (D’Ambrosio, 1983, 363).

Necessity of adaptation to local culture and needs

The Secretary-General of the African Commission for Mathematical Instruction, George Eshiwani, stressed the necessity to reformulate the goals of mathematics education “in relation to local culture and needs” (Eshiwani, 1979, 346). The International Conference on ‘Developing Mathematics in Third World Countries’, held in Khartoum, 1978, advocated how the objectives of teaching mathematics at the primary level may be attuned to local conditions. Among other recommendations, the Conference concluded that: “Pupils should learn mathematics as far as possible through active practical experience and with learning aids drawn from the environment. Generalizations and structures should come out of pupil’s experiences, rather than from formal assertions from the teacher” (El Tom, 1979 c, 182). Still open stays the question, that Eshiwani (implicitly) raised “which local needs?” should be taken into account when developing a mathematics curriculum. Broomes and Kuperes answered in 1980 this fundamental question in the following way. For the majority of children from the Third World (maybe more than 80%, p.g.) the primary school is the endpoint of their formal schooling; and 85% of these children live in the rural areas. Therefore, the mathematics curriculum has to be defined in accordance with the interests and the desires of these rural communities: “...the mathematics school curriculum...would require methods of teaching and learning modelled upon the attitudes, skills, and work habits that are desired by the community” (Broomes, 1982, 25). But desired by whom in the community, is a principal question still to be answered.

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Profound criticisms of actual mathematics education

In their study of the mathematics learning difficulties of the Kpelle (Liberia), Gay and Cole concluded, that there do not exist any inherent difficulties: what happened in the classroom, was that the contents did not make any sense from the point of view of Kpelle- culture; moreover the methods used were primarily rote memory and harsh discipline (Gay & Cole, 1967, 6). Experiments showed that Kpelle illiterate adults performed better than North American adults, when solving problems, like the estimation of numbers of cups of rice in a container, that belong to their ‘indigenous mathematics’ (Gay & Cole, 1967, 66). For Gay and Cole, the principal question to be answered is: “How can we teach effectively, respecting the old while bringing in the new, in as humane and efficient a way possible” (Gay & Cole, 1967, 7). All right, but, we may ask, who does determine the new that has to be taught? Later psychological research confirms not only the results of Gay and Cole, but reveals also some other important aspects of actual mathematics learning. Young Baoulé and Dioula children from Ivory Coast performed initially, because of language difficulty and unfamiliarity with schooling, at a somewhat lower level than North American children on arithmetic tasks. However, after a few years of schooling, these African and North American children performed at roughly the same levels on virtually all tasks (Ginsburg et al., 1981 a, 30). Moreover, both schooled and unschooled Dioula (merchant), as well as schooled subjects from the Baoulé population (agricultural) “demonstrated an understanding of number conservation and displayed accurate counting procedures” (Posner & Baroody, 1979, 493). And unschooled Dioula achieved a high level of accuracy in mental addition: their daily commercial activities stimulated the development of their cognitive skills (Ginsburg et al., 1981 b, 174, 175; cf. Petito, 1982; Petito & Ginsburg, 1982). Schooled subjects, on the other hand, tend to transfer to mental calculation problems the standard algorithms acquired in school, that are often not so practical 98 Mathematics Education in Mozambique

(cf. Ginsburg et al., 1981 b, 176). The development of this ‘informal arithmetic’ is dependent on cultural factors, e.g. unschooled Baoulé children – they live in rural areas and are not engaged in commercial or comparable activities – performed at a lower level than unschooled Dioula children on mental additions (Posner, 1982). Still more profound doubts about the effectiveness of school mathematics teaching are raised by Latin-American researchers. Eduardo Luna (Dominican Republic) posed the question if it is possible, that the practical mathematical knowledge that children acquired outside the school is ‘repressed’ and ‘confused’ in the school (Luna, 1983, 4). Not only possible, but this happens frequently, showed the Brazilians Carraher and Schliemann: children, who knew, before they went to school, how to solve creatively arithmetical problems they encountered in daily life, e.g. at the marketplace, could, later in the school, not solve the same problems, i.e. not solve with the methods taught in arithmetic class (Carraher et al., 1982). D’Ambrosio concludes (compare with his remarks on mathematics education and the reproduction of oppressing power structures) that “‘learned’ matheracy eliminates the so-called ‘spontaneous’ matheracy” (D’Ambrosio, 1982, 3; 1984, 6), i.e. “An individual who manages perfectly well numbers, operations, ‘geometric forms and notions, when facing a completely new and formal approach to the same facts and needs creates a psychological blockade which grows as a barrier between the different modes of numerical and geometrical thought” (D’Ambrosio, 1982, 4; 1984, 6,7; italics p. g.). What happens in the school, is that “The former, let us say, spontaneous, abilities (are) downgraded, repressed and forgotten, while the learned ones (are not being) assimilated, either as a consequence of a learning blockage, or of an early dropout,..” (D’Ambrosio, 1984, 8; 1982, 4; italics, p. g.). For this reason,

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“the early stages of mathematics education (offer) a very efficient way of instilling the sense of failure, of dependency in the children” (D’Ambrosio, 1982, 4; 1984, 7). How can this psychological blockade be avoided? How can this ‘pushing aside’ and ‘wiping out’ of spontaneous (D’Ambrosio), natural (Carraher et al.), informal (Posner et al.), indigenous (Gay & Cole), folk (Mellin-Olsen), or non-standard (paraphrasing Labov; Labov 1972, cf. Carraher et al., 1982, 81) mathematics be avoided? Gay and Cole became convinced that it is necessary to investigate first the ‘indigenous mathematics’, in order to be able to build effective bridges from this ‘indigenous mathematics’ to the new mathematics to be introduced in the school (Gay & Cole, 1967, 1): “...the teacher should begin with materials of the indigenous culture, leading the child to use them in a creative way” (Gay & Cole, 1967, 94), and from there advance to the new school mathematics. The question remains: which school mathematics and for what? The Tanzanian curriculum specialist Mmari stresses, that: “...there are traditional mathematics methods still being used in Tanzania...A good teacher can utilize this situation to underline the universal truths of the mathematical concepts” (Mmari, 1978, 313). And how could the good teacher achieve this? Jacobsen answers: “The (African) people that are building the houses are not using mathematics; they’re doing it traditionally...if we can bring out the scientific structure of why it’s done, then you can teach science that way” (quoted by Nebres, 1984, 4). For D’Ambrosio, it becomes necessary “... to generate ways of understanding, and methods for the incorporation and compatibilisation of known and current popular practices into the curriculum. In other words, in the case of mathematics, recognition and incorporation of ethnomathematics into the curriculum” (D’Ambrosio, 1984, 10), “...this...requires the development of quite difficult anthropological research methods relating to mathematics...

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anthropological mathematics ... constitutes an essential research theme in Third World countries...as the underlying ground upon which we can develop curriculum in a relevant way” (D’Ambrosio, 1985, 47).

3. Towards a socialist mathematics-education-policy

The general education policy in Mozambique is directed to the construction of a socialist society (see section 1). On the basis of the experiences of other developing countries (as described in section 2), the following necessary elements of a socialist mathematics-education- policy in Mozambique can be formulated: a. The mathematics curriculum of the seven year primary school has to display the correct perspective. About 70% of the pupils will not continue, immediately after completing primary school, their formal schooling and most of these students live in the rural areas. Therefore, not only mathematical skills that permit continuation in secondary schools, but foremost skills that enable the school leaver to solve daily practical problems, have to be developed. Every student – also with only primary school – should have the basic knowledge to be able to contribute to the transformation of society, especially to be capable of raising production and productivity in agriculture. The achieved curriculum must be so effective, that less schooled peasants and other workers understand the practica1-productive, transformative value of school education for their children and for the improvement of their own living conditions. b. Any curriculum reform has only success, when – at each of its stages – it is wholeheartedly supported by the teachers and by the ‘public in general’. Therefore, the debate on curriculum- reform has to be as democratic as possible: well-organized involvement of the mass of teachers, parents and users. The ‘hidden’ (ethno)mathematics into the curriculum has to result from incorporation of ‘informal’ and of other forms of concrete experiences, from research done by the teachers (see c).

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New curriculum has to be tested in distinct experimental schools, with different conditions, like schools in the rural areas and in the cities. Broad discussion on the possible, negative vestiges of traditional-feudal and colonial-capitalist education in actual mathematics teaching is necessary (Gerdes, 1981 b, 472; see Appendix 1). c. It may not be the consequence of formal, administrative measures that “the great majority of children pass to the next class each year”. On the contrary, it has to be the result of good teaching: * Teachers should use dynamic methods, guaranteeing that all children in the class are busy with meaningful mathematical activities, that all children are learning mathematics by doing it. * Teachers should give all possible stimulus/motivation to those children, who need it more, both directly and in groupwork. * Teachers should dominate techniques – triad: practice- theory-practice – how to abstract from practical situations, which are familiar to the children; how to proceed from concrete operations to mathematical operations and constructions, and how to use the learned mathematical concepts and strategies in new situations. * Teachers should learn to recognize and incorporate all sorts of ‘informal’ or ‘hidden’ mathematics ‘known’ by the pupils and their parents. The development of these qualities calls for corresponding (pre- and in-service) training activities. d. In order to combat the lack of popularity of mathematics, it is important to improve the climate of mathematical learning by extra-curricular activities. Student-clubs, public lectures, publication of booklets and journals, mathematics contests etc. are ways to popularise mathematical knowledge, that are suggested by experiences elsewhere, particularly in other socialist countries (of. Gerdes, 1984 b).

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Taking into account these necessary elements (a, b, c) of a socialist mathematics-education-policy in Mozambique, the National Institute for the Development of Education (INDE) elaborates the mathematics-curriculum for primary and general secondary school. It is not an easy task. On the contrary, the time pressure is very high: in order to guarantee the gradual – year by year – introduction of the National System of Education, it is necessary to produce every year the pupil books and teacher guides for the next class to be introduced. Note that there did not exist any textbooks in the years before. In the new books, one starts by posing one or more (mathematically) related practical problems; then one proceeds to a theoretical elaboration and develops the corresponding mathematical skills (this second ‘formal’ phase is still underestimated by many teachers, who think it is only necessary to operate with concrete objects) and at the end of each theme there are presented many different practical problems where the learned mathematical techniques can be applied. These practical problems are not artificial nor borrowed / copied from ‘European’ curricula; on the contrary, the materials of other ministries on the development of the rural and industrial areas of the country, on healthcare (e.g. the calendar of vaccination), agriculture (e.g. calculation of sowing-seed and seed—time), building, production cooperatives, food distribution etc. serve as the base for the elaboration of these practical exercises. On the other hand, the small mathematics section of INDE organizes and supports the in-service training of all teachers and of many educational administrators in view of the new curriculum. This new curriculum does not fall from heaven. The first National Seminar on the Teaching of Mathematics held in Maputo in 1980 – with the participation of teachers of all levels, literacy workers and adult educators, representatives of all ministries – made a thorough analysis of the current mathematics teaching practices (see Gerdes, 1981 b, 470 ff.). This analysis is continued and concretized by INDE’s detailed study of former curriculum and elaboration of the new curriculum. In pre-service-training of teachers (see Gerdes, 1991), it is also tried to implement elements of this socialist mathematics-education- policy (especially c). The Faculty of Education of the Eduardo Mondlane University organizes since 1981 mathematics competitions at a national scale and publishes the mathematics-education journal 103 Paulus Gerdes

TLANU and booklets, e.g. on uses of mathematics for raising agricultural production (e.g. Gerdes, 1983). In the next section, we shall offer some ideas on further purposes and possibilities of the before mentioned necessary recognition and incorporation of ‘hidden’ mathematics.

4. Incorporation of mathematical traditions into the curriculum

D’Ambrosio stressed the need for incorporation of ethnomathematics into the curriculum in order to avoid a psychological blockade. In former colonized countries, there exists also a related cultural blockade to be eliminated. “Colonization – in the words of President Samora Machel –is the greatest destroyer of culture that humanity has ever known. African society and its culture were crushed, and when they survived they were co-opted so that they could be more easily emptied of their content. This was done in two distinct ways. One was the utilization of institutions in order to support colonial exploitation...The other was the ‘folklorizing’ of culture, its reduction to more or less picturesque habits and customs, to impose in their place the values of colonialism” (Samora Machel, 1978, 400). “Colonial education appears in this context as a process of denying the national character, alienating the Mozambican from his country and his origin and, in exacerbating his dependence on abroad, forcing him to be ashamed of his people and his culture” (Samora Machel, 1978, 401). In the specific case of mathematics, this science was presented as an exclusively white men’s creation and ability; the mathematical capacities of the colonized peoples were negated or reduced to rote memorization; the African and American-Indian mathematical traditions became ignored or despised. A cultural rebirth is indispensable, as President Samora Machel underlines: “...1ong suppressed manifestations of culture (have to) regain their place” (Samora Machel, 1978, 402).

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In this cultural rebirth, in this combat against racial and colonial prejudice, a cultural-mathematical-reaffirmation plays a part: it is necessary to encourage an understanding that the Mozambican people has been capable of developing mathematics in the past, and therefore – regaining cultural confidence (cf. Gerdes, 1982, 1985 a) – will be able, now and in the future, to develop and use creatively mathematics. We may conclude that the incorporation of mathematical traditions into the curriculum will contribute not only to the elimination of individual and social psychological blockade, but also of the related cultural blockade. Now, here raises an important question: which mathematical traditions? In order to be able to incorporate popular (mathematical) practices, it is first of all necessary to recognize their mathematical character. In this sense, D’Ambrosio speaks about the need to broaden our understanding of what mathematics is (D’Ambrosio, 1985, 45). Traditional counting methods, e.g. by means of knots in strings, and counting systems are easily recognized as mathematics. But what about geometrical thinking? Traditional houses have conical roofs and circular or rectangular bases. Rectangular mats are rolled up into cylinders. Baskets possess circular borders. Fish traps display hexagonal holes. Etc. Could these examples only figure in the mathematics-lesson as illustrations of geometrical notions? Only as illustrations? This is a rather fundamental question that has recently also been posed by Howson, Nebres and Wilson in their study on ‘School mathematics in the 1990s’: “There has been increasing talk, particularly with respect to developing countries, of ‘ethnomathematics’, i.e. mathematical activities identified within the everyday life of societies. Thus, for example, a variety of types of symmetries are used for decoration in all cultures, numerous constructions are erected which illustrate mathematical laws. To what extent are these activities really ‘mathematical’? What is it which makes the activities ‘mathematical’ rather than, say, ‘capable of mathematical elaboration or legitimisation’?” (Howson et al., 1985, 15).

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Please, answer for yourself.

First example

May I give you an example? Maybe the example could change your view. Take two strips of paper into your hands. How do you have to fold them around another in order to be able to weave them further (see Figures 1 and 2)? What has to be the initial angle between the two strips? Vary the angle? What do you discover?

Figure 1

Figure 2

Only one special angle makes further plaiting possible (see Figures 3, 4 and 5). Two types of strips can be woven in this way (see Figures 6 and 7).

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Figure 3 Figure 4

Figure 5

Figure 6

Figure 7

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The strip pattern in Figure 7 admits changes in direction, like the “circling around” in Figure 8. It is exactly this possibility that makes this strip weaving process very useful. E.g. Mozambican artisans use this method for making their straw hats (See Photograph 1).

Figure 8

A Mozambican straw hat Photograph 1

Now, I repeat the question. The result can only be used in the mathematics lesson as an illustration of geometrical notions? What is your answer? When discovering the strip weaving method, did you do mathematics? Did you analyse the effects of the variation of the angle between the two initial strips of paper? 108 Mathematics Education in Mozambique

Lets go further. What can be said about that particular necessary angle between the two strips? Observe the resulting strip. That particular angle goes three times into a straight angle (Figure 9).

Figure 9

The little triangles possess three of those angles, etc. Which other geometrical knowledge can be obtained? (See e.g. Figure 10).

Figure 10

You did mathematics and the product of your activity did invite you to further reflection, to further mathematical activity and thinking ... I started my lecture by confronting you with some questions. Now I repeat one of them: do you know how to construct an angle that measures 60 degrees, only using the strips of paper I distributed to you? Now, your answer will be: yes. So, you learned mathematics from our Mozambican artisans. You learned mathematics by doing it: “Learning mathematics by doing it, means not just handling objects, but also thinking about the handling, and reflecting on the processes and products.” (Bishop & Goffree, 1984, 11).

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Before giving another related example of learning mathematics by doing it, let me explain to you in a few words our basic method for recognizing hidden mathematics. In our analysis of the geometrical forms of traditional objects, like baskets, mats. pots, houses, fish-traps, we posed the question: why do these material products possess the form they have? We learned the usual production techniques and tried to vary the forms. It came out that the form of these objects is almost never arbitrary, but generally represents many practical advantages and is, quite a lot of times, the only possible solution of a production problem, like in the case of the example I just gave you. The traditional form reflects accumulated experience and wisdom, expresses knowledge. Not only biological or physical knowledge about the materials that are used, but also mathematical knowledge (the first results of this research are summarized in our book (Gerdes. 1985 b).

Second example

Consider the following practical problem. In many situations it is disadvantageous to have a densely woven basket, e.g. when transporting little birds in a basket, these animals must have the possibility to breath. So it is useful to have a basket with holes. A basket with holes will also be less heavy. Etc. Can you weave a basket with holes? Like in Figure 11?

Figure 11

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The holes are fixed? More or less flexible? This may be admitted? Why not? How to solve the problem? Maybe by weaving in more than two directions? What happens when you introduce supporting strands? E.g. in a diagonal way? See Figure 12.

Figure 12

How do you have to introduce them so that the holes become fixed? Is it possible to adapt the three directions in such a way that they become more “equal”? The resultant regular hexagonal pattern is exactly the one Mozambican peasants use for their light transportation baskets and fishermen for their fish traps (See Photograph 2). Did we do mathematics? You still doubt? Please postpone your judgement for a while. Let us solve another practical production problem.

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Photograph 2

Third example

How can you fasten a border to the walls of a basket, when both border and walls are made out of the same material? Try to solve this problem fir yourself. Take two equal strips of paper into your hands, and consider one of them as part of the border, the other as belonging to the wall. How should one join them together? Should we join the border- and wall strips like in Figure 13?

Figure 13

No ... It is necessary to wrap once more the wall strip around the border strip. In the following way (Figure 14)? 112 Mathematics Education in Mozambique

Figure 14

Not so? How then? As in Figure 15?

Figure 15

But what happens when you are pressing the wall strip (see Figure 15)? How to avoid this problem? What has to be the initial angle between the border- and wall-strip? (Figure 16).

Figure 16

Let us complete the border and wall. What happens? See Figure 17.

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Figure 17

Other possibilities? Introducing more horizontal strips ... what now? Once again a hexagonal pattern appears (Figure 18).

Figure 18

What other geometrical knowledge can be obtained? Possibility of a hexagonal tiling pattern (Figure 19). Etc.

Figure 19

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Did you do mathematics? Let us try to draw some conclusions from these few examples (many other examples can be given, see Gerdes 1985 b). a. There exists ‘hidden’ or ‘frozen’ mathematics. The artisan, who merely imitates a known production technique, is not doing a lot of mathematics. But the artisan(s) who discovered the technique, did mathematics, was / were thinking mathematically. b. By unfreezing this frozen mathematics, by rediscovering hidden mathematics in our material culture, we show indeed that our (like every other) people did mathematics. In other words by making conscious hidden mathematics, we contribute to cultural self-confidence, to the aforementioned necessary cultural- mathematical reaffirmation. c. Defrosting frozen mathematics can serve as a starting point for doing and elaborating mathematics in the classroom. Traditional (possibly monopolized) production techniques will be understood in their crucial aspects by all children. In this way knowledge is becoming more democratic. At the same time. by using (in this sense) traditional production techniques from all over the country, knowledge is becoming less ‘tribal’, less regional: knowledge or culture are becoming national (important in a process of nation building as in the case of Mozambique). d. ‘Defrosting frozen mathematics’ forces us to reflect on the relationship between mathematical thinking and material production, between doing mathematics and technology: where do (first) mathematical ideas come from? However, the problem still remains of how this rediscovered mathematics can best be integrated into the curriculum.

REFERENCES

D’Ambrosio, U. (1979), Adequate mathematics for Third World countries: Consideranda and strategies, in El Tom (1979), 33-46. D’Ambrosio, U. (1982), Mathematics for the rich and poor countries: similarities and differences, CARIMATHS, Paramaribo.

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D’Ambrosio, U. (1983), Successes and failures of mathematics curricula in the past two decades: a developing society viewpoint in a holistic framework, in M. Zweng et al. (1983), 362-364. D’Ambrosio, U. (1984), The intercultural transmission of mathematical knowledge: effects on mathematical education, UNICAMP, Campinas. D’Ambrosio, U. (1985), Ethnomathematics and its place in the history and pedagogy of mathematics, For the Learning of Mathematics, 5, 1, 44-48. Anon. (1982), Culture in dependent Kenya, Journal of African Marxists, 2, 22-37. Bishop, A. & Goffree, F. (1984), Classroom organisation and dynamics. In: B. Christiansen, A. G. Howson, M. Otte (Eds). Perspectives on mathematics education. In press, D. Reidel. Dordrecht. [Published in 1986, pp. 309-365]. Broomes, D. (1982), The mathematics curriculum at secondary level in the Carribean context, CARIMATHS, Paramaribo. Broomes, D. & Kuperes, P. (1983), Problems of defining the mathematics curriculum in rural communities, in M. Zweng et al. (1983), 708-711. Carrahar, T., Carraher, D. and Schliemann, A. (1982), Na vida, dez, na escola, zero; os contextos culturais de aprendizagem da matemática, Cadernos de Pesquisa, São Paulo, 42, 79-86 El Tom, M. (Ed.) (1979), Developing mathematics in Third World Countries, proceedings of the International Conference held in Khartoum, March 6-9, 1978, North-Holland Publishing Company, Amsterdam, 1979. El Tom, M. (1984), The role of Third World University Mathematics Institutions in promoting mathematics, ICME-V, Adelaide, 1984. Eshiwani, G. (1979), The goals of mathematics teaching in Africa: a need for re-examination, Prospects, IX, 3, 346-352. Freudenthal, H. (1979), New math or new education?, Prospects, IX, 3. 321-331.

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Ganhão, F. (1978). The Struggle continues: Mozambique’s revolutionary experience in education, Development Dialogue, 2, 25-36. Gay, J. & Cole, M. (1967), The New Mathematics and an Old Culture. A study of learning among the Kpelle of Liberia, Holt, Rinehart and Winston, New York. Gerdes, P. (1981), Changing mathematics education in Mozambique, Educational Studies in Mathematics, 12, 455-477. Gerdes, P. (1982a), Exemplos de aplicações da matemática na agricultura e na veterinária, Tlanu-brochura 3, Universidade Eduardo Mondlane, Maputo. Gerdes, P. (1982b), Mathematics for the benefit of the people, CARIMATHS, Paramaribo. Gerdes, P. (1984a), On cultural aspects of evaluation in mathematical education, ICME V, Adelaide. Gerdes, P. (1984b), The first mathematics Olympiads in Mozambique, Educational Studies in Mathematics, 15, 149-172. Gerdes, P. (1985a), Conditions and strategies for emancipatory mathematics education in underdeveloped countries, For the Learning of Mathematics, 5, 1, 15-21. Gerdes, P. (1985b), Gesellschaftliche Tätigkeit und die mögliche Herkunft einiger früher geometrischer Begriffe und Relationen, Dresden (to be published). [Published in German, Portuguese, and English. Latest edition in English: Ethnogeometry: Awakening of Geometrical Thought in Early Culture, ISTEG, Boane & Lulu, Morrisville NC, 2014, 210 pp. (Foreword: Dirk Struik, Massachusetts Institute of Technology, Cambridge MA)]. Giglioli, P. (Ed.) (1972), Language and social context, Penguin Books, Harmondsworth. Ginsburg, H., Posner, J. & Russell, R. (1981a), The development of knowledge concerning written arithmetic: a cross-cultural study, International Journal of Psychology, 16, 13-34.

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Ginsburg, H., Posner, J. & Russell, R. (1981b), The development of mental addition as a function of schooling and culture, Journal of Cross-Cultural Psychology, 12, 2, 163-178. Howson, A., Nebres, B. & Wilson, B. (1985), School mathematics in the 1990s, ICMI, Southampton Labov, W. (1972), The logic of non-standard English, in Giglioli, 179- 215. Luna, E. (1983), Analisis curricular y contexto sociocultural, Santiago. Machel, Graça (1981), The National System of Education, Mozambique-Information-Agency, Information-bulletin, 66. Machel, Samora (1978), Knowledge and science should be for the total liberation of man, Race & Class, XIX, 4, 399-404. Mellin-Olsen, S. (1984), The politization of mathematics education, Bergen College of Education. [Published in 1987: The politics of mathematics education, Reidel, Dordrecht]. Mmari, G. (1978), The United Republic of Tanzania: mathematics for social transformation, in Swetz, 301-350. Nebres, B. (1979), Research and higher education in mathematics: the Philippine experience, in El Tom, 1979, 67-80. Nebres, B. (1983), Problems of mathematical education in and for changing societies: problems in Southeast Asian countries, Regional Conference on Mathematical Education, Tokyo Nebres, B. (1984), The problem of universal mathematics education in developing countries, ICME-V, Adelaide.

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Appendix 1

During the first National Seminar on the Teaching of Mathematics the influence of the traditional and colonial society on the actual teaching of mathematics were analysed: That there are still so many authoritarian teachers who hardly permit a doubt or a suggestion from their students; that there are older teachers who do not accept the experience and the help of younger ones, even when these are their superiors or when they have a better training; that certain fundamental mathematical concepts are presented as absolute truths without being explained or demonstrated: and that many students receive passively the information transmitted by the teacher (and that this passivity is most marked in girls) was explained as a negative heritage of traditional-feudal society, just as the persistent presence of superstitious ideas which prevent teachers from understanding the origin of the development of mathematics. That mathematics teaching continues to be too detached from its applications, and to be based on memorisation, was explained as one of the deep scars left by the colonial society. Under the Portuguese system, teaching was by rote. Students memorised for exams, but never learned to think mathematically. This was in the interest or the colonists who wanted to create only some low-level clerks who could carry out orders accurately but not be able to use any creative thinking. These vestiges of traditional- feudal and colonial-capitalist societies constitute serious blocks to the creation of the new, of the love of learning and of scientific curiosity. i.e. blocks to some of the most important objectives of socialist education.

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Cover of the 1986 edition of the book “Mathematics and Culture”

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Books in English authored by Paulus Gerdes

* Geometry from Africa: Mathematical and Educational Explorations, The Mathematical Association of America, Washington DC, 1999, 210 pp. (Outstanding Academic Book 2000, Choice Magazine) * Mathematics Education in Mozambique, 12 pp.  * Ethnomathematics and Education in Africa, 270 pp. # * Ethnogeometry: Awakening of Geometrical Thought in Early Culture, 210 pp. # * Sona Geometry from Angola: Mathematics of an African Tradition, 232 pp. (colour edition) # * Sona Geometry from Angola. Volume 2: Educational and mathematical explorations of African designs on the sand, 220 pp. # * Lusona – Geometrical Recreations from Africa: Problems and Solutions, 216 pp. (Colour edition) # * Drawings from Angola: Living Mathematics, 72 pp. (Children’s book) (Colour edition) # * Women, Culture and Geometry in Southern Africa, 276 pp. # (Special Commendation, Noma Award for Publishing in Africa 1996) * Otthava: Making Baskets and Doing Geometry in the Makhuwa Culture in the Northeast of Mozambique, 290 pp. (full colour edition) # * Tinhlèlò, Interweaving Art and Mathematics: Colourful Circular Basket Trays from the South of Mozambique, 132 pp. (Colour edition) # * Hats from Mozambique, 52 pp. (Colour edition) #

 Distributed by Lulu, Morrisville NC, USA: http://www.lulu.com/spotlight/pgerdes 121 Paulus Gerdes

* Sipatsi: Basketry and Geometry in the Tonga Culture of Inhambane (Mozambique, Africa), 422 pp. & Sipatsi Images in Colour: A Supplement, 56 pp. # * African Basketry: A Gallery of Twill-Plaited Designs and Patterns, 220 pp. # * African Pythagoras: A Study in Culture and Mathematics Education, 124 pp. (Colour edition) # * Basketry, Geometry, and Symmetry in Africa and the Americas, E- book, [http://www.mi.sanu.ac.yu/vismath/, ‘Special E-book issue’ (2004)]. * Lunda Geometry: Mirror Curves, Designs, Knots, Polyominoes, Patterns, Symmetries, 198 pp. # * Adventures in the World of Matrices, Nova Science Publishers, New York, 2008, 196 pp. * Mathematics in African History and Cultures. An annotated Bibliography (co-author Ahmed Djebbar), African Mathematical Union, 430 pp. # * History of Mathematics in Africa: AMUCHMA 25 Years (co-author Ahmed Djebbar) (Volume 1: 1986-1999; Volume 2: 2000-2011), African Mathematical Union, 924 pp. # * African Doctorates in Mathematics: A Catalogue, African Mathematical Union, 383 pp. # * 1000 Doctoral Theses by Mozambicans or about Mozambique, Academy of Sciences of Mozambique, 2013, 236 pp. # * Geometry and Basketry of the Bora in the Peruvian Amazon, 176 pp. (full colour edition) # * Examples of applied mathematics in agriculture and veterinary science, 68 pp. (booklet) #

Puzzle books: * Enjoy puzzling with biLLies, 248 pp. # * More puzzle fun with biLLies, 76 pp. # * Puzzle fun with biLLies, Lulu, 76 pp. # * The Bisos Game: Puzzles and Diversions, 72 pp. #

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Books edited: * Explorations in Ethnomathematics and Ethnoscience in Mozambique, Universidade Pedagógica, 104 pp. # * Mathematics, Education and Society (co-editors: Cristine Keitel, Alan Bishop, Peter Damerow), UNESCO, Paris, 1989, 193 pp.

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