All positive action starts with criticism. Hans Freudenthal and the didactics of mathematics

Financing Statement......Page 5
Acknowledgements......Page 6
Contents......Page 7
List of Abbreviations......Page 11
Chapter-1......Page 14
Introduction......Page 14
A way to master this world......Page 14
1.1 Didactics of Mathematics and Hans Freudenthal: Definition of the Problem and Phrasing of the Question......Page 14
1.2 Don Quixote: The Freudenthal Myth?......Page 16
......Page 17
1.4 The Nature of the Study and the Historiography......Page 18
1.5 The Structure of This Book......Page 20
References......Page 21
Chapter-2......Page 23
Mathematics Education in Secondary Schools and Didactics of Mathematics in the Period Between the Two World Wars......Page 23
2.1 Secondary Education in the Period Between the Two World Wars......Page 24
2.1.1 The Origin of the School Types in Secondary Education......Page 24
2.1.2 Some School Types......Page 25
2.1.2.1 The HBS......Page 25
2.1.2.2 The Gymnasium......Page 26
2.1.2.3 The MMS......Page 26
2.1.2.4 The Lyceum......Page 27
2.1.3 The Competition between HBS and Gymnasium......Page 27
2.2 Discussions on the Mathematics Education at the VHMO......Page 28
2.2.1 The Initial Geometry Education and the Foundation of the Journal Euclides......Page 30
2.2.2 The Beth Committee and the Introduction of Differential and Integral Calculus......Page 34
2.2.3 The Controversy About Mechanics......Page 37
2.2.4 Educating the Mathematics Teacher......Page 39
2.2.5 New Insights and the Wiskunde Werkgroep (Mathematics Working Group)......Page 42
References......Page 44
Chapter-3......Page 48
......Page 48
......Page 48
3.2 Luckenwalde......Page 50
3.3 Berlin......Page 52
3.4 Amsterdam......Page 54
3.5 Utrecht......Page 62
References......Page 67
Chapter-4......Page 71
Didactics of Arithmetic......Page 71
......Page 72
4.2 Cause and Intention......Page 73
4.3 Teaching of Arithmetic in Primary Schools......Page 76
......Page 78
4.4.1 Preface......Page 79
4.4.2 Auxiliary Sciences......Page 79
4.4.3 Aim and Use of Teaching of Arithmetic......Page 84
......Page 86
References......Page 89
Chapter-5......Page 91
A New Start......Page 91
5.1 Educating......Page 97
5.1.1 Educating at Home......Page 97
......Page 98
......Page 100
......Page 105
......Page 108
5.1.6 Education: A Summary......Page 111
5.2 Higher Education......Page 112
5.2.1 Studium Generale......Page 113
5.2.2 The Teachers Training......Page 116
5.2.3 Student Wage......Page 121
5.2.4 Higher Education: A Ramshackle Parthenon or a House in Order?......Page 123
5.3 The Wiskunde Werkgroep (Mathematics Working Group)......Page 124
5.3.1 Activities of the Wiskunde Werkgroep......Page 126
......Page 128
......Page 134
......Page 135
References......Page 135
Chapter-6......Page 143
From Critical Outsider to True Authority......Page 143
6.1 Mathematics Education and the Education of the Intellectual Capacity......Page 146
6.2 A Body Under the Floorboards: The Mechanics Education......Page 152
6.3 Preparations for a New Curriculum......Page 158
6.4 Probability Theory and Statistics: A Text Book......Page 165
6.5 Paedagogums, Paeda Magicians and Scientists: The Teacher Training......Page 171
6.6 Freudenthal Internationally......Page 180
References......Page 182
Chapter-7......Page 188
......Page 188
A Learning Process......Page 188
7.1 Introduction: A Special PhD Project......Page 188
7.2 Freudenthal as Supervisor......Page 190
......Page 194
......Page 198
7.5 Analysis of a Learning Process: Reflection on Reflection......Page 207
7.6 To Conclude......Page 208
References......Page 209
Chapter-8......Page 214
Method Versus Content......Page 214
New Math and the Modernization  of Mathematics Education......Page 214
8.1 Introduction: Time for Modernization......Page 214
8.2 New Math......Page 216
8.2.1 The Gap Between Modern Mathematics and Mathematics Education......Page 216
8.2.2 Modernization of the Mathematics Education in the United States......Page 217
8.3 Royaumont: A Bridge Club With Unforeseen Consequences......Page 221
......Page 222
8.3.2 Royaumont Without Freudenthal: The Launch of New Math......Page 233
8.4 Freudenthal on Modern Mathematics and Its Meaning for Mathematics Education......Page 235
8.4.1 The Nature of Modern Mathematics......Page 235
8.4.2 Modern Mathematics for the Public at Large......Page 238
......Page 239
8.4.4 Fairy Tales and Dead Ends......Page 241
8.4.5 Modern Mathematics as the Solution?......Page 243
8.5 Modernization of Mathematics Education in the Netherlands......Page 245
8.5.1 Initiatives Inside and Outside of the Netherlands......Page 246
......Page 248
8.5.3 The Commissie Modernisering Leerplan Wiskunde......Page 251
8.5.4 A Professional Development Programme for Teachers......Page 253
8.5.5 A New Curriculum......Page 254
8.6 Geometry Education......Page 259
8.6.1 Freudenthal and Geometry Education......Page 261
8.6.2 Freudenthal on the Initial Geometry Education: Try It and See......Page 263
......Page 264
8.6.4 Modern Geometry in the Education According to Freudenthal......Page 266
8.7 Logic......Page 267
8.7.1 Exact Logic......Page 268
8.7.2 The Application of Modern Logic in Education......Page 271
8.8 Freudenthal and New Math: Conclusion......Page 275
8.8.1 A Lonely Opponent of New Math?......Page 275
8.8.2 Cooperate in Order to Adjust......Page 277
8.8.3 Knowledge as a Weapon in the Struggle for a Better Mathematics Education......Page 278
8.8.4 Freudenthal About the Aim of Mathematics Education......Page 280
References......Page 281
Chapter-9......Page 292
......Page 292
9.1 Introduction: Changes in the Scene of Action......Page 292
9.2 Educational Studies in Mathematics......Page 295
9.2.1 Not Exactly Bursting with Enthusiasm: The Launch......Page 296
9.2.2 Freudenthal as Guardian of the Level......Page 298
9.3 The Institute for the Development of Mathematics Education......Page 300
9.3.1 From CMLW to IOWO......Page 300
9.3.2 Freudenthal and the IOWO......Page 302
9.4 Exploring the World from the Paving Bricks to the Moon......Page 309
......Page 309
9.4.2 Observing as Grandfather Walking with the Grandchildren......Page 313
9.4.3 Granddad Hans: A Critical Comment......Page 315
9.4.4 Walking on the Railway Track: The Mathematics of a 3-Year Old......Page 319
9.4.5 Observing and the IOWO......Page 321
9.5 Observations as a Source......Page 323
9.5.1 Professor or Senile Grandfather?......Page 323
9.5.2 The Paradigm: The Ultimate Example......Page 325
9.5.3 Here is How Freudenthal Saw It: Concept of Number and Didactical Phenomenology......Page 328
9.5.3.1 The Number Concept in New Math......Page 329
......Page 330
......Page 333
9.5.4 The Right to Sound Mathematics for All......Page 336
9.6 Enfant Terrible......Page 342
9.6.1 Weeding......Page 342
9.6.2 Drumming on Empty Barrels......Page 344
9.6.3 Freudenthal on Piaget: Admiration and Merciless Criticism......Page 348
9.6.3.1 A Swiss Developmental Psychologist......Page 349
9.6.3.2 About Flimsy Experiments and Pseudo Mathematics......Page 351
9.7 The Task for the Future......Page 355
References......Page 357
Chapter-10......Page 369
Epilogue......Page 369
We Have Come Full Circle......Page 369
References......Page 373
Bibliography......Page 375
Index......Page 390