Challenging Mathematics In and Beyond the Classroom (eBook)

Preface 6
References 7
School years 7
Contents 9
The Authors 11
Introduction 14
0.1 Challenging: a human activity 14
0.2 Challenges and education 16
0.3 Debilitating and enabling challenges 18
0.4 What is a challenge? 18
References 22
Challenging Problems: Mathematical Contents and Sources 23
1.1 Introduction 23
1.2 Challenges within the regular classroom regime 24
1.2.1 Challenge from observation 26
1.2.2 Challenge from a textbook problem 27
1.2.3 Increasing fluency with fractions 28
1.2.4 Engaging with algebra 29
1.2.5 Pedagogies to help development 31
1.2.6 Combinatorics 31
1.2.7 Geometry 32
1.2.8 Other settings for school challenges 34
1.3 Challenges in popular culture 38
1.3.1 Another schoolyard problem 40
1.3.2 A Russian problem 40
1.3.3 The Microsoft problem 41
1.3.4 A problem from children’s literature 42
1.3.5 A probabilistic element 43
1.3.6 Concluding comments 43
1.4 Challenges from inclusive and other teacher-supported contests 43
1.4.1 Diophantine equations 44
1.4.2 Pigeonhole principle 45
1.4.3 Discrete optimization and graph theory 46
1.4.4 Cases 47
1.4.5 Proof by contradiction 47
1.4.6 Enumeration 48
1.4.7 Invariance 49
1.4.8 Inverse thinking 49
1.4.9 Coloring problems 50
1.4.10 Concluding comments 51
1.5 Challenges from Olympiad contests: Students independent of classroom teacher 51
1.6 Content and context 57
1.6.1 Three groups of requirements for assignments 57
1.6.2 Challenges in classrooms: identifying patterns in their appearance 59
1.6.3 The psychology of the art of writing problems as a research problem 60
1.6.4 Using different areas of mathematics in different contexts 60
1.6.5 The structure of problems and the form of their presentation as a means of responding to context and transforming it 61
1.6.6 The issue of mathematics teacher education 61
1.6.7 Conclusion 62
References 62
Challenges Beyond the Classroom-Sources and Organizational Issues 64
2.1 Introduction 64
2.1.2 Working as individuals and in teams 66
2.1.2 Involvement of teachers 66
2.2 Environments for challenging mathematics 66
2.2.1 Mathematics competitions 68
2.2.1.1 Inclusive competitions 70
2.2.1.2 Different types of competition 71
2.2.1.3 Some general comments 74
2.2.2 Mathematics journals, books and other published materials (including Internet) 75
2.2.3 Research-like activities, conferences, mathematics festivals 77
2.2.3.1 Jugend Forscht (youth quests), Germany and Switzerland 78
2.2.3.2 Research Science Institute (RSI), USA 78
2.2.3.3 High School Students’ Institute for Mathematics and Informatics, Bulgaria 79
2.2.3.4 Mathematics festivals, Iran 80
2.2.4 Mathematical exhibitions 80
2.2.4.1 Historical background 81
2.2.4.2 Examples of exhibitions 83
2.2.5 Mathematics houses 86
2.2.6 Mathematics lectures 87
2.2.7 Mentoring mathematical minds 88
2.2.8 Mathematics camps, summer schools 88
2.2.8.1 International Mathematics Tournament of Towns summer camp 89
2.2.8.2 International mathematics kangaroo summer camps 89
2.2.8.3 Summer School Festival UM+ 89
2.2.8.4 The Canadian seminar 89
2.2.8.5 Isfahan summer camps 90
2.2.8.6 The Institute for Advanced Study in USA 90
2.2.9 Correspondence programs 90
2.2.10 Web sites 93
2.2.11 Public lectures, columns in newspapers, magazines, movies, books, general purpose journals 93
2.2.12 Math days, open houses, promotional events for school students at universities 94
2.2.13 Mathematics fairs 94
2.2.13.1 Canadian Andy Liu model 95
2.2.13.2 A mathematical house for younger children (Years 1 to 5) 95
2.2.13.3 Mathematics day at universities 95
2.2.13.4 Long night of mathematics at the high school, Karlsruhe 96
2.2.13.5 India 96
2.2.14 Mathematical quizzes 96
2.2.14.1 The mathematical organization Archimedes 97
2.3 Concluding remarks: challenging infrastructure-a powerful motivational factor 97
2.4 Appendix 98
2.4.1 Iran: what is a Mathematics House? 99
2.4.1.1 History 99
2.4.1.2 Audiences 100
2.4.1.3 Activities 100
2.4.1.4 Activities for high school students 101
2.4.1.5 Activities for university students 101
2.4.1.6 Activities for teachers 101
2.4.1.7 Other activities 102
2.4.1.8 Library 102
2.4.1.9 Laboratories 102
2.4.1.10 Achievements 102
2.4.2 Serbia: the mathematics organization Archimedes 103
2.4.2.1 Activities 103
2.4.2.2 Lessons learned 104
References 106
Technological Environments beyond the Classroom 108
3.1 Introduction 108
3.2 Technology and challenging mathematics beyond the classroom: tool of learning and fun 110
3.3 What kind of challenging mathematics beyond the classroom can be supported by technology? 113
3.3.1 Types of digital tools and their support for challenging mathematics 113
3.3.1.1 A context for thinking skills and digital tools 113
3.3.1.2 Challenging mathematics in this context 114
3.3.2 Two approaches to challenging mathematics by hypermedia learning 115
3.3.2.1 Learning through design 115
3.3.2.2 Learner-tailored instruction 116
3.4 Making mathematics beyond the classroom more challenging 117
3.4.1 Software to support mathematical investigation 118
3.4.2 Numerical working spaces 120
3.4.2.1 NWS1. Goal adaptable databases 121
3.4.2.2 NWS2. Communication and exchange 122
3.4.2.3 NWS3. Freedom for learning mathematics 123
3.4.2.4 NWS4. Local, regional and worldwide expansion 124
3.4.3 Issues related to cost and maintenance 124
3.4.4 Psychological difficulties 124
3.5 Challenging mathematics beyond the classroom using Internet-based learning environments 125
3.5.1 Problem of the week challenge 125
3.5.2 Collaborative problem solving: challenge and historical context 128
3.5.2.1 A Babylonian problem 129
3.5.2.2 L’Agora de Pythagore: virtual communityof young mathematical philosophers 130
3.6 Effect on practices of teachers 130
3.6.1 Mathenpoche 131
3.6.2 WIMS 132
3.6.3 PUBLIREM 133
3.6.4 La main à la pâte 133
3.7 General discussion 134
3.7.1 Conception 135
3.7.2 Forms 135
3.7.3 Content 136
3.7.4 Implementation 137
3.7.5 Research 137
3.8 Conclusion 139
References 140
Challenging Tasks and Mathematics Learning 143
4.1 Introduction 143
4.1.5 A goal for challenging mathematical problems 143
4.1.5 Importance of schemas in mathematical problem solving 144
4.1.5 Mathematical tasks, exercises, and challenging problems 145
4.1.5 Use of challenging problems to promote schema construction 146
4.2 Categories of challenging mathematics problems 147
4.3 Challenging mathematics problems and schema development 148
4.3.1 Strands of challenging mathematical tasks 148
4.3.2 Examples from a strand of challenging mathematical tasks 149
4.3.3 Mathematical analysis 150
4.3.4 Cognitive analysis 151
4.3.5 Students’ work on problems from a strand of challenging tasks 151
4.3.5.1 How many pizzas are there with four different toppings? 151
4.3.5.2 Linking the pizza problem, the towers problem, and Pascal’s triangle 152
4.3.5.3 Solving the taxicab problem 153
4.3.5.4 Discussion-strand and schema 155
4.4 Other examples and contexts for challenging mathematics problems 156
4.4.1 Example: Number producer 156
4.4.2 Example: Pattern sequence 160
4.4.3 Examples: Probability 163
4.4.4 Examples: weekly problems 166
4.5 Example: the challenge of a contradiction and schema adjustment 170
4.5.1 Inconsistency, contradiction and cognitive development 170
4.5.2 What do you do if you have to prove that 1 = 2? and other paradoxes 171
4.5.3 Brief comments on the paradoxes in Problems 2 to 4 173
4.5.4 Analysis of Problem 1 174
4.5.5 Concluding remarks 175
4.6 Conclusion 176
References 178
Mathematics in Context: Focusing on Students 181
5.1 Introductory comments 181
5.2 Discussion of contexts for challenges 183
5.2.1 Highlight on long-term studies 185
5.2.1.1 Ingénierie didactique 186
5.2.1.2 Activity theory 186
5.2.2 Conclusion 187
5.3 Case studies 187
5.3.1 The Laboratory of Mathematical Machines 187
5.3.1.1 Learning environment 188
5.3.1.2 Duration 189
5.3.1.3 Instruments 189
5.3.1.4 Pedagogical methods 189
5.3.2 Seeding mathematical challenges at morning assembly at a school in India 191
5.3.3 Heaps of sand: what we can do with sand in and beyond the classroom with a mathematical aim 193
5.3.4 SalsaJ: astronomical software 196
5.3.5 Mathematical challenges around Orsay 197
5.3.6 Challenging gifted high school students 199
5.3.7 Maths à Modeler: research situations for teaching mathematics 202
5.3.7.1 Research situations for the classroom (RSC): a definition 202
5.3.7.2 Hunting the beast! 204
5.3.8 Mathematics and art 208
5.3.9 Lawn constructions 210
References 212
Teacher Development and Mathematical Challenge 214
6.1 Introduction 214
6.1.1 What is mathematics and what are mathematical challenges? 214
6.1.1 Why are challenging mathematics problems important in school? 216
6.1.1 What do challenging mathematical problems for school classrooms look like? 217
6.1.3.1 The Six Circles problem 218
6.1.3.2 Decimal grid task 219
6.1.3.3 Triangle of odd numbers 221
6.1.3.4 The Dirichlet Principle 224
6.1.3 What barriers might prevent teachers from using challenging problems? 225
6.2 Research 226
6.2.1 What do we know about the effect of teachers’ knowledge and beliefs on the teaching and learning of challenging mathematics? 226
6.2.2 What other factors are important to teaching and learning challenging mathematics? 228
6.2.2.1 Motivation 228
6.2.2.2 Brain development 229
6.2.2.3 Zone of proximal development 229
6.2.3 What other research is needed? 230
6.3 Effective pedagogy 230
6.3.1 What is the role of the teacher in a class where challenging problems are used? 230
6.3.2 What is effective pedagogy for classrooms using challenging mathematical problems? 232
6.4 Teacher preparation 235
6.4.1 What is the role of professional development in encouraging classes with challenging mathematical problems? 235
6.4.1.1 Modeling 235
6.4.1.2 Didactical content 237
6.4.1.3 Practica 239
6.4.1.4 Mathematical coaching 239
6.4.1.5 Outstanding student work 239
6.4.1.6 Teacher-innovator model 239
6.4.2 Some in-service and pre-service programs 240
6.4.2.1 A Chinese experience 240
6.4.2.2 A German experience 242
6.4.2.3 A New Zealand experience 243
6.4.2.4 An American experience 244
6.5 Summary 245
6.5.1 Overview 245
6.5.1.1 Fundamental principles 245
6.5.1.2 Aims 245
6.5.1.3 Modeling 246
References 246
Challenging Mathematics: Classroom Practices 252
7.1 Challenging mathematics-the essence of mathematics classrooms 252
7.1.4 Why do we need challenges in regular classrooms? 254
7.1.4 How often should challenges be used and for whom? 256
7.2 Designing challenging mathematics for classrooms 257
7.2.1 Setting the scene 257
7.2.1.1 Nature of mathematical understandings expected to be deepened 257
7.2.1.2 The gap between what is being proposed and present practice 257
7.2.1.3 Clarifying changes in expectations 261
7.2.2 Task design 262
7.2.2.1 Rephrasing as a means of tweaking a task for different grade levels 263
7.2.2.2 Does the nature of the task change with increasing grade level? 266
7.2.2.3 Do we need new topics for challenges or can we find them within the existing curriculum? 267
7.2.2.4 How can technology be incorporated into task design to facilitate the use of mathematical challenges? 268
7.3 Designing classrooms for mathematics challenges 269
7.3.1 How do we teach students strategically to address a challenge? 269
7.3.2 How can we make sense of the pedagogical challenge of having students appreciate challenge in mathematics? 271
7.3.3 The role of textbooks 272
7.3.4 Managing the challenge 273
7.3.5 How can teachers introduce mathematical challenges into the regular classroom? 274
7.4 Designing research for challenging mathematics classroom practices 275
7.4.1 Fruitful research designs for examining challenges 275
7.4.2 Design-based research 275
7.4.2.1 An Australian Example 276
7.4.2.2 A Canadian example: can students think like Archimedes? 279
7.4.3 Japanese lesson study 281
7.4.4 Teaching experiments or teaching-research 281
7.4.4.1 A North American example 282
7.4.4.2 A Russian example 284
7.5 Conclusion 285
References 286
Curriculum and Assessment that Provide Challenge in Mathematics 293
8.1 Introduction 293
8.2 The case studies 294
8.2.1 The case of Singapore: primary school level 294
8.2.1.1 Background and curriculum 294
8.2.1.2 National examination for primary schools 295
8.2.1.3 Test items 295
8.2.1.4 Discussion 297
8.2.2 The case of Norway: lower secondary education 299
8.2.2.1 Background and curriculum 299
8.2.2.2 Traditional examination for lower secondary schools 301
8.2.2.3 The written exams 302
8.2.2.4 The oral exams 303
8.2.2.5 Course grade and final grade 304
8.2.2.6 Discussion 304
8.2.3 The case of Brazil: upper primary and lower secondary levels 304
8.2.3.1 Curriculum and background 304
8.2.3.2 The Brazilian Mathematics Olympiad for public schools 306
8.2.3.3 Discussion 309
8.2.4 The case of Iran: upper secondary education and beyond 309
8.2.4.1 Curriculum and background 309
8.2.4.2 The range of assessment in upper secondary level 310
8.2.4.3 The national university entrance examination 310
8.2.4.4 High school students mathematical Olympiad 314
8.2.4.5 Assessment of the gifted high school students to enter special schools 314
8.2.4.6 University students mathematics competition 315
8.2.4.7 University Students International Scientific Olympiad in Mathematics 315
8.3 Assessment and learning, assessment and challenge, assessment and curriculum 315
8.3.1 The role of assessment: assessment and learning 315
8.3.2 The role of assessment: assessment and challenge 316
8.3.3 The role of assessment: assessment and curriculum 317
8.3.4 Competitions and curriculum 319
8.4 Knowledge gaps and future research questions in this domain 319
8.4.1 Examining opposing views of assessment and their relationship to challenging mathematics 319
8.4.2 What is the role of challenging mathematics in the relationship between assessment and learning? 320
8.4.3 What is the nature of good classroom assessment? 321
8.4.4 What are the differences in focus for challenging mathematics in the light of assessment to determine ability and assessment to determine achievement? 321
8.4.5 What are the pedagogical differences and effects attained by enrichment and challenge? 322
References 322
Concluding Remarks 324
Acknowledgements 326
Chapter contributions 326
Organization support 327
Plenary speakers 328
IPC conference 328
Study Conference 328
Proofreading and other technical support 329
Author Index 331
Subject Index 336