Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example 51 Third example 52 A method for recognizing hidden geometrical thinking 54 Cultural and pedagogical value 55 References 56 5 Chapter 5: On culture, geometrical thinking and 57 mathematics education Some social and cultural aspects of mathematics education in 58 Third World countries Towards a cultural-mathematical reaffirmation 61 Examples of ‘cultural conscientialization’ of future 63 mathematics teachers Study of alternate axiomatic constructions of Euclidean 63 geometry in teacher training An alternate construction of regular polygons 67 From woven buttons to the ‘Theorem of Pythagoras’ 72 From traditional fish traps to alternate circular functions, 75 football and the generation of (semi)regular polyhedra a. Tiling patterns and the formulation of conjectures 76 b. An alternate circular function 78 c. Footballs and polyhedra 79 Concluding remarks 81 References 82 Chapter 6: A widespread decorative motif and the 85 Pythagorean theorem A widespread decorative motif 85 Discovering the Pythagorean Theorem 88 A first proof 91 An infinity of proofs 93 Pappos’ theorem 95 Example 96 References 97 Chapter 7: ‘Pythagoras’, similar triangles and the 99 “elephants’-defence”-pattern of the (Ba)Kuba (Central Africa) Introduction 99 A particular case of the Pythagorean proposition 100 The Mongo and Mwoong patterns 100 6 To arrive at the Pythagorean proposition 102 An ornamental variant and a generalization of the Pythagorean 104 proposition A combination of the Pythagorean proposition with its 105 generalization A geometrical alternative 106 Other geometrical alternatives 110 Once more ‘Pythagoras’ 112 Final considerations 112 References 112 Chapter 8: On possible uses of traditional Angolan sand 113 drawings in the mathematics classroom Introduction 113 The drawing tradition of the Tchokwe 114 Arithmetical relationships 117 First example 117 Arithmetical progressions 117 A Pythagorean Triplet 123 Geometrical ideas 124 Axial symmetry 124 Double bilateral and point symmetry 124 Rotational symmetry 126 Similarity 126 Geometrical determination of the greatest common 129 divisor of two natural numbers Towards the Euclidean algorithm 133 Euler graphs 133 Concluding remarks 138 Postscript 140 Acknowledgements 140 References 141 7 Chapter 9: Exploration of the mathematical potential of 145 ‘sona’: an example of stimulating cultural awareness in mathematics teacher education Introduction: necessity of culture-oriented education 145 Ethnomathematical research and teacher education 146 The (lu)sona tradition 147 Didactical experimentation 149 Further exploration of the mathematical potential of sona in 151 teacher education Composition rules 151 Systematic construction of monolinear drawings 152 How many lines 154 First example 154 Second example 157 Underlying black-and-white designs 163 Concluding remarks 168 Chapter 10: Technology, art, games and mathematics 175 education: An example Chapter 11: On mathematics in the history of Africa south 181 of the Sahara Introduction 182 Why study the history of mathematics in Africa south of the 183 Sahara? The beginnings 185 Numeration systems and number symbolism 186 Games, riddles and puzzles 189 Geometry 192 Art and symmetries 192 Networks, graphs or ‘sand drawings 195 Geometry and architecture 197 Uncovering ‘hidden’ geometrical ideas 198 Studies in related disciplines 200 Africa south of the Sahara, North Africa and the outside world 202 8 Slave trade and colonisation 205 Post-Independence 206 Acknowledgements 207 References 207 Appendices 231 Chapter 12: Conditions and strategies for emancipatory 233 mathematics education in underdeveloped countries Point of departure: mathematics education cannot be neutral 233 Mathematics education for emancipation. How? 234 Problemising reality in classroom situations 235 Creating confidence 239 A. Cultural strategies 239 B. Social strategies 245 C. Individual-collective strategies 245 Concluding remarks 248 References 249 Chapter 13: African slave and calculating prodigy: 251 Bicentenary of the death of Thomas Fuller Appendix 1: “Account of a wonderful talent for arithmetical 263 calculation, in an African slave, living in Virginia” (1788) Appendix 2: “Died – NEGRO TOM, the famous African 265 Calculator, aged 80 years” (1790) References 266 About the author 271 Books in English authored by Paulus Gerdes 273 Books in Portuguese authored by Paulus Gerdes 279 9 Cover of the original edition by the University of Stockholm 10 Preface PREFACE The book Ethnomathematics and Education in Africa was originally concluded in 1992 and published in January 1995 by the Institute of International Education (IIE) of the University of Stockholm (Sweden) within the framework of the institutional cooperation that had started in 1991 between the IEE and the Higher Pedagogical Institute / Pedagogical University (Instituto Superior Pedagógico / Universidade Pedagógica) in Mozambique. The book contained a collection of papers presented at conferences in Brazil, Kenya, Lesotho, Mozambique, Norway, and Spain and/or published in international journals such as For the Learning of Mathematics, Educational Studies in Mathematics and Historia Mathematica. The papers date from the period 1986-1992. The book has been out of print for many years. Many colleagues contacted me to know if the book was still available. The realisation of the 5th International Conference on Ethnomathematics in Mozambique in July 2014 constitutes another motif for re-editing the book, that dates from the years of the emergence – in various continents – of ethnomathematics as a research field. The new edition of the book Ethnomathematics and Education in Africa reproduces the eleven chapters of the original edition and contains two chapters not included in the first edition. The new chapters are entitled “Conditions and strategies for emancipatory mathematics education in underdeveloped countries” and “African slave and calculating prodigy: Bicentenary of the death of Thomas Fuller.” 11 Paulus Gerdes: ETHNOMATHEMATICS AND EDUCATION IN AFRICA The first of these two papers is the revised version of an invited paper presented at the Caribbean Conference on Mathematics for the Benefit of the Communities and its Reflection in the Curriculum, organised by the Inter-American Committee on Mathematical Education, Paramaribo, Suriname, October 18-21, 1982. It was published in For the Learning of Mathematics, Montreal, Vol. 5, No. 1, 15-20 (February 1985) and reproduced in the book Mathematics Education: Major Themes in Education, Routledge, London, 2010, Volume 1, 179-189, edited by Alan J. Bishop from Monash University, Melbourne, Australia. The second paper, on Thomas Fuller (1710-1790), was written 1 together with the late John Fauvel (1947-2001), and published in Historia Mathematica, New York, Vol. 17, 141-151 (1990). The new edition of Ethnomathematics and Education in Africa will be available both in printed form and as an eBook. Paulus Gerdes ([email protected]) ISTEG, Boane, Mozambique January 2014 1 Faculty of Mathematics, The Open University, Milton Keynes, England; President of the British Society for the History of Mathematics (1991-1994); Chair of the International Study Group on the Relations between History and Pedagogy of Mathematics (HPM) (1992-1996). 12 Chapter 1 Chapter 1 INTRODUCTION The Secretary of the International Study Group on Ethnomathematics, Patrick Scott (University of New Mexico, USA) explained during the hearing ‘What can we expect from Ethnomathematics?’ (6th International Congress on Mathematics Education, Budapest, 1988) that there are “two major models in ethnomathematics: the D’Ambrosio–Gerdes model (ethnomathematics in Third World Countries to bridge the gap between the home culture and ‘modern’ science) and the Zaslavsky–Ascher model (ethnomathematics in First World countries to bring the world into the classroom and to have an appreciation of the mathematics of other 1 cultures).” Both models may be considered as complementary: both industrialised