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Ethnomathematics

New Approaches to its Theory and
Application
Dissertation
zur Erlangung des Grad eines
Doktors der P¨ adagogik (Dr. Paed.)
Fakult¨ at fur¨ Mathematik
Universit¨ at Bielefeld
vorgelegt von
Ursula Andrea Barbara Verdugo Rohrer
geboren in Campinas, Brasilien
im Mai 2010Gedruckt auf alterungsbest¨ andigem Papier ISO 9706.
2Contents
Table of Contents 3
List of Figures 7
List of Tables 9
Acknowledgments 11
Introduction 13
I Theory and Historiography of Ethnomathematics 17
1 The Emergence of ethnomathematics and its Contexts 19
1.1 Mathematics as Cultural Element: First Approaches . . . . . . . 19
1.1.1 Empirical Results: Luquet, Fettweis, Raum . . . . . . . . . 23
1.1.2 Theoretical Results: Keyser, Wilder, White . . . . . . . . 38
1.1.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.2 ContributionsofCognitivePsychologyandConstructivistApproaches
in Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.2.1 Jean Piaget . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2.2 Jerome S. Bruner . . . . . . . . . . . . . . . . . . . . . . . 49
1.2.3 Ernst von Glasersfeld . . . . . . . . . . . . . . . . . . . . . 52
1.2.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.3 The Convergence to Ethnomathematics . . . . . . . . . . . . . . . 55
1.4 Definitions and Conceptions of Ethnomathematics . . . . . . . . . 59
1.4.1 Ethnomathematical Movement. . . . . . . . . . . . . . . . 64
1.5 Critiques of Ethnomathematics . . . . . . . . . . . . . . . . . . . 69
1.5.1 Reactions and Ethnomathematical Research in Europe . . 72
32 Building a Theory of Ethnomathematics 77
2.1 Concepts of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.1.1 Kuhn’s Theory of Scientific Development . . . . . . . . . . 79
2.2 Ethnomathematics: a Kuhnian Theory? . . . . . . . . . . . . . . 81
2.3 Ethnomathematics and Interdisciplinarity . . . . . . . . . . . . . 83
2.4 Challenges of Ethnomathematics to Mathematics . . . . . . . . . 85
3 Research Methods in 91
3.1 Some Important Theories and Methods in the Social Sciences . . 91
3.1.1 Ethnography . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1.2 Ethnology . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.3 Ethnoscience . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.2 Research Methods in Ethnomathematics . . . . . . . . . . . . . . 102
3.3 Ethnographic Research in Africa . . . . . . . . . . . . . . . . . . . 104
3.3.1 Ethnomathematics in Southern African Countries . . . . . 106
II Ethnomathematical Research 109
4 Theoretical Framework: The golden Ratio, Le Corbusier and
Albrecht Dürer 111
4.1 The golden section . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Le Corbusier and the modulor . . . . . . . . . . . . . . . . . . . . 117
4.3 Albrecht Dürer and die menschliche Proportion . . . . . . . . . . 121
4.4 Comparison of Dürer’s proportions and Le Corbusier’s modulor . 124
4.5 Facial features and the golden ratio . . . . . . . . . . . . . . . . . 125
4.6 t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 The Makonde Sculptors of Mozambique 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 About the Makonde and their history . . . . . . . . . . . . . . . . 132
5.3 The Makonde and the Ebony Wood . . . . . . . . . . . . . . . . . 135
5.3.1 Ujamaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.3.2 Shetani . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.3 Sculpture in relief . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.4 Masks - Mapiko . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Artist or Artisan . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
45.6 Data Collection - Sculptures . . . . . . . . . . . . . . . . . . . . . 149
5.6.1 Mulher by Martins Bernardo. . . . . . . . . . . . . . . . . 151
5.6.2 Senhora by Arcanjo Bachela . . . . . . . . . . . . . . . . . 154
5.6.3 Macacos by Martins Bernardo . . . . . . . . . . . . . . . . 155
5.6.4 Likhomba - Mask by Pakholo Laza . . . . . . . . . . . . . 156
5.6.5 Ujamaa-Família by Júlio Carlos . . . . . . . . . . . . . . . 159
5.6.6 Interview with Eduardo Patrício Namembe . . . . . . . . . 160
5.7 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.7.1 Fitting Le Corbusier’s modulor to Makonde Sculptures . . 163
5.7.2 Comparison of Dürer’s and Makonde proportions . . . . . 166
5.7.3 Facial Features and the Golden Section . . . . . . . . . . . 166
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.8.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 The teuto-Brazilians of Friburgo, Brazil 173
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 The Foundation of Friburgo - Historical Facts . . . . . . . . . . . 177
6.2.1 The School of Friburgo . . . . . . . . . . . . . . . . . . . . 180
6.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.4 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4.1 Textbooks - Centro de Memória, Unicamp . . . . . . . . . 185
6.4.2 Interviews in Friburgo . . . . . . . . . . . . . . . . . . . . 191
6.5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.5.1 Textbooks in Brazil and Germany . . . . . . . . . . . . . . 201
6.5.2 Ethnomathematical Analysis of Interviews . . . . . . . . . 222
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.6.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7 Final Conclusions 233
Primary Sources 237
Bibliography 239
A Personalakte - Ewald Fettweis 257
B Field Notes: Museum of Ethnology - Nampula, Mozambique 259
B.1 Written collected data . . . . . . . . . . . . . . . . . . . . . . . . 259
5B.2 Interview Transcript . . . . . . . . . . . . . . . . . . . . . . . . . 270
B.3 Enlarged Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
C FieldNotes: CentrodeMemória,Unicamp,andFriburgo,Camp-
inas - Brazil 289
C.1 Transcripton of the article Friedburg by Richard Guebel . . . . . . 289
C.2 Request and Authorization for the Development of the Field Re-
search in Friburgo and at the Centro de Memória of Unicamp . . 292
C.3 Interview Transcripts . . . . . . . . . . . . . . . . . . . . . . . . . 296
6List of Figures
1.1 Portrait of Ewald Fettweis . . . . . . . . . . . . . . . . . . . . . . 28
1.2 Examples of sipatsi (Mozambique) . . . . . . . . . . . . . . . . . 62
1.3 Mathematical exercises for Waimiri-Atroari (Brazil) . . . . . . . . 64
2.1 Ethnomathematics diagram (1) . . . . . . . . . . . . . . . . . . . 84
2.2 (2) . . . . . . . . . . . . . . . . . . . 85
4.1 Straight line of DEMR . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Genealogy of a drone . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 The Vitruvian Man . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4 How to obtain the modulor . . . . . . . . . . . . . . . . . . . . . . 119
4.5 The modulor of Le Corbusier . . . . . . . . . . . . . . . . . . . . 119
4.6 The human body drawn by Dürer . . . . . . . . . . . . . . . . . . 123
4.7 Sketch of facial features . . . . . . . . . . . . . . . . . . . . . . . 126
5.1 Map of Mozambique . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2 Map of Mueda’s plateau . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Tatooed Makonde women . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Compact and non-compact ujamaa sculptures . . . . . . . . . . . 139
5.5 Shetani sculptures. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.6 Women. . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.7 Maasai and Ethiopian sculptures . . . . . . . . . . . . . . . . . . 144
5.8 Masks sculptures . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.9 Sculpture copied from picture . . . . . . . . . . . . . . . . . . . . 150
5.10 Tools of a Makonde sculptor . . . . . . . . . . . . . . . . . . . . . 151
5.11 Sculpting a woman (1) . . . . . . . . . . . . . . . . . . . . . . . . 152
5.12 a woman (2) . . . . . . . . . . . . . . . . . . . . . . . . 153
5.13 Sculpting a woman (3) . . . . . . . . . . . . . . . . . . . . . . . . 154
5.14 monkeys (1) . . . . . . . . . . . . . . . . . . . . . . . . 155
5.15 Sculpting monkeys (2) . . . . . . . . . . . . . . . . . . . . . . . . 157
75.16 Sculpture of mask (1) . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.17 of mask (2) . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.18 Sculpture of ujamaa . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.19 modulor’s scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.20 Facial features (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.21 Facial (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1 Political map of Brazil . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2 Picture of Fazenda Sete Quedas . . . . . . . . . . . . . . . . . . . 175
6.3 Maps of the state of São Paulo and Campinas . . . . . . . . . . . 176
6.4 Map of Comunidade Helvetia and Friburgo . . . . . . . . . . . . . 177
6.5 Church of Friburgo . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.6 Cemetery of Friburgo . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.7 Events in Friburgo . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.8 School of Friburgo . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.9 The school of Friburgo and the local authorities by 1923 . . . . . 182
6.10 Small business Sítio Mirim of Friburgo . . . . . . . . . . . . . . . 192
6.11 Grapevines of Hedinho . . . . . . . . . . . . . . . . . . . . . . . . 194
6.12 Rootstock procedure . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.13 Grape bunches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.14 Wooden boxes for grape bunches . . . . . . . . . . . . . . . . . . 198
6.15 Baskets and cardboard boxes for grape bunches . . . . . . . . . . 198
6.16 Cover of Guia do Medidor de Terrenos . . . . . . . . . . . . . . . 224
6.17 Alqueire table of Guia do Medidor de Terrenos . . . . . . . . . . . 225
6.18 Manger diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.1 Modulor of Maasai man . . . . . . . . . . . . . . . . . . . . . . . 282
B.2 Modulor of Maasai woman . . . . . . . . . . . . . . . . . . . . . . 283
B.3 Modulor of woman (1) . . . . . . . . . . . . . . . . . . . . . . . . 284
B.4 Modulor of woman (2) . . . . . . . . . . . . . . . . . . . . . . . . 285
B.5 Facial features of boy and woman . . . . . . . . . . . . . . . . . . 286
B.6 Facial of two women . . . . . . . . . . . . . . . . . . . . . 287
C.1 Authorization from Sociedade Escolar do Bairro Friburgo . . . . . 295
8List of Tables
4.1 Cardinality of female and male bees . . . . . . . . . . . . . . . . . 115
4.2 The red and blue series of the modulor . . . . . . . . . . . . . . . 120
4.3 Synthesis of some of Dürer’s tables. . . . . . . . . . . . . . . . . . 124
4.4 Body measures by Le Corbusier and Dürer . . . . . . . . . . . . . 125
4.5 Proportions of body measures . . . . . . . . . . . . . . . . . . . . 125
4.6 t-test values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1 Distances given by facial features . . . . . . . . . . . . . . . . . . 169
5.2 Statistical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1 List of digitized books . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2 Kleikamp’s table of units (1898) . . . . . . . . . . . . . . . . . . . 204
6.3 Büchler’s table of Brazilian units (1932) . . . . . . . . . . . . . . 205
6.4 Büchler’s table of metric units 1932 . . . . . . . . . . . . . . . . . 206
6.5 Table of unit systems used by 1925 . . . . . . . . . . . . . . . . . 207
6.6 Some of Büttner’s school books . . . . . . . . . . . . . . . . . . . 210
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