Challenging Mathematics In and Beyond the Classroom

The 16th ICMI Study
1. Auflage 2009
ISBN: 978-0-387-09603-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 337 Seiten

In the mid 1980s, the International Commission on Mathematical Instruction (ICMI) inaugurated a series of studies in mathematics education by comm- sioning one on the influence of technology and informatics on mathematics and its teaching. These studies are designed to thoroughly explore topics of c- temporary interest, by gathering together a group of experts who prepare a Study Volume that provides a considered assessment of the current state and a guide to further developments. Studies have embraced a range of issues, some central, such as the teaching of algebra, some closely related, such as the impact of history and psychology, and some looking at mathematics education from a particular perspective, such as cultural differences between East and West. These studies have been commissioned at the rate of about one per year. Once the ICMI Executive decides on the topic, one or two chairs are selected and then, in consultation with them, an International Program Committee (IPC) of about 12 experts is formed. The IPC then meets and prepares a Discussion Document that sets forth the issues and invites interested parties to submit papers. These papers are the basis for invitations to a Study Conference, at which the various dimensions of the topic are explored and a book, the Study Volume, is sketched out. The book is then put together in collaboration, mainly using electronic communication. The entire process typically takes about six years.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;6
1.1;References;7
1.2;School years;7
2;Contents;9
3;The Authors;11
4;Introduction;14
4.1;0.1 Challenging: a human activity;14
4.2;0.2 Challenges and education;16
4.3;0.3 Debilitating and enabling challenges;18
4.4;0.4 What is a challenge?;18
4.5;References;22
5;Challenging Problems: Mathematical Contents and Sources;23
5.1;1.1 Introduction;23
5.2;1.2 Challenges within the regular classroom regime;24
5.2.1;1.2.1 Challenge from observation;26
5.2.2;1.2.2 Challenge from a textbook problem;27
5.2.3;1.2.3 Increasing fluency with fractions;28
5.2.4;1.2.4 Engaging with algebra;29
5.2.5;1.2.5 Pedagogies to help development;31
5.2.6;1.2.6 Combinatorics;31
5.2.7;1.2.7 Geometry;32
5.2.8;1.2.8 Other settings for school challenges;34
5.3;1.3 Challenges in popular culture;38
5.3.1;1.3.1 Another schoolyard problem;40
5.3.2;1.3.2 A Russian problem;40
5.3.3;1.3.3 The Microsoft problem;41
5.3.4;1.3.4 A problem from children’s literature;42
5.3.5;1.3.5 A probabilistic element;43
5.3.6;1.3.6 Concluding comments;43
5.4;1.4 Challenges from inclusive and other teacher-supported contests;43
5.4.1;1.4.1 Diophantine equations;44
5.4.2;1.4.2 Pigeonhole principle;45
5.4.3;1.4.3 Discrete optimization and graph theory ;46
5.4.4;1.4.4 Cases ;47
5.4.5;1.4.5 Proof by contradiction;47
5.4.6;1.4.6 Enumeration;48
5.4.7;1.4.7 Invariance ;49
5.4.8;1.4.8 Inverse thinking ;49
5.4.9;1.4.9 Coloring problems;50
5.4.10;1.4.10 Concluding comments;51
5.5;1.5 Challenges from Olympiad contests: Students independent of classroom teacher;51
5.6;1.6 Content and context;57
5.6.1;1.6.1 Three groups of requirements for assignments;57
5.6.2;1.6.2 Challenges in classrooms: identifying patterns in their appearance;59
5.6.3;1.6.3 The psychology of the art of writing problems as a research problem;60
5.6.4;1.6.4 Using different areas of mathematics in different contexts;60
5.6.5;1.6.5 The structure of problems and the form of their presentation as a means of responding to context and transforming it;61
5.6.6;1.6.6 The issue of mathematics teacher education;61
5.6.7;1.6.7 Conclusion;62
5.7;References;62
6;Challenges Beyond the Classroom-Sources and Organizational Issues;64
6.1;2.1 Introduction;64
6.1.1;2.1.2 Working as individuals and in teams;66
6.1.2;2.1.2 Involvement of teachers;66
6.2;2.2 Environments for challenging mathematics;66
6.2.1;2.2.1 Mathematics competitions;68
6.2.1.1;2.2.1.1 Inclusive competitions;70
6.2.1.2;2.2.1.2 Different types of competition;71
6.2.1.3;2.2.1.3 Some general comments;74
6.2.2;2.2.2 Mathematics journals, books and other published materials (including Internet);75
6.2.3;2.2.3 Research-like activities, conferences, mathematics festivals;77
6.2.3.1;2.2.3.1 Jugend Forscht (youth quests), Germany and Switzerland;78
6.2.3.2;2.2.3.2 Research Science Institute (RSI), USA ;78
6.2.3.3;2.2.3.3 High School Students’ Institute for Mathematics and Informatics, Bulgaria;79
6.2.3.4;2.2.3.4 Mathematics festivals, Iran;80
6.2.4;2.2.4 Mathematical exhibitions;80
6.2.4.1;2.2.4.1 Historical background;81
6.2.4.2;2.2.4.2 Examples of exhibitions;83
6.2.5;2.2.5 Mathematics houses;86
6.2.6;2.2.6 Mathematics lectures;87
6.2.7;2.2.7 Mentoring mathematical minds;88
6.2.8;2.2.8 Mathematics camps, summer schools;88
6.2.8.1;2.2.8.1 International Mathematics Tournament of Towns summer c89
6.2.8.2;2.2.8.2 International mathematics kangaroo summer camps;89
6.2.8.3;2.2.8.3 Summer School Festival UM+;89
6.2.8.4;2.2.8.4 The Canadian seminar;89
6.2.8.5;2.2.8.5 Isfahan summer camps;90
6.2.8.6;2.2.8.6 The Institute for Advanced Study in USA;90
6.2.9;2.2.9 Correspondence programs;90
6.2.10;2.2.10 Web sites;93
6.2.11;2.2.11 Public lectures, columns in newspapers, magazines, movies, books, general purpose journals;93
6.2.12;2.2.12 Math days, open houses, promotional events for school students at universities;94
6.2.13;2.2.13 Mathematics fairs ;94
6.2.13.1;2.2.13.1 Canadian Andy Liu model;95
6.2.13.2;2.2.13.2 A mathematical house for younger children (Years 1 to 5);95
6.2.13.3;2.2.13.3 Mathematics day at universities;95
6.2.13.4;2.2.13.4 Long night of mathematics at the high school, Karlsruhe;96
6.2.13.5;2.2.13.5 India;96
6.2.14;2.2.14 Mathematical quizzes;96
6.2.14.1;2.2.14.1 The mathematical organization Archimedes ;97
6.3;2.3 Concluding remarks: challenging infrastructure-a powerful motivational factor;97
6.4;2.4 Appendix;98
6.4.1;2.4.1 Iran: what is a Mathematics House?;99
6.4.1.1;2.4.1.1 History;99
6.4.1.2;2.4.1.2 Audiences;100
6.4.1.3;2.4.1.3 Activities;100
6.4.1.4;2.4.1.4 Activities for high school students;101
6.4.1.5;2.4.1.5 Activities for university students;101
6.4.1.6;2.4.1.6 Activities for teachers;101
6.4.1.7;2.4.1.7 Other activities;102
6.4.1.8;2.4.1.8 Library;102
6.4.1.9;2.4.1.9 Laboratories;102
6.4.1.10;2.4.1.10 Achievements;102
6.4.2;2.4.2 Serbia: the mathematics organization Archimedes ;103
6.4.2.1;2.4.2.1 Activities;103
6.4.2.2;2.4.2.2 Lessons learned;104
6.5;References;106
7;Technological Environments beyond the Classroom;108
7.1;3.1 Introduction;108
7.2;3.2 Technology and challenging mathematics beyond the classroom: tool of learning and fun;110
7.3;3.3 What kind of challenging mathematics beyond the classroom can be supported by technology?;113
7.3.1;3.3.1 Types of digital tools and their support for challenging mathematics;113
7.3.1.1;3.3.1.1 A context for thinking skills and digital tools;113
7.3.1.2;3.3.1.2 Challenging mathematics in this context;114
7.3.2;3.3.2 Two approaches to challenging mathematics by hypermedia learning;115
7.3.2.1;3.3.2.1 Learning through design;115
7.3.2.2;3.3.2.2 Learner-tailored instruction;116
7.4;3.4 Making mathematics beyond the classroom more challenging;117
7.4.1;3.4.1 Software to support mathematical investigation;118
7.4.2;3.4.2 Numerical working spaces;120
7.4.2.1;3.4.2.1 NWS1. Goal adaptable databases;121
7.4.2.2;3.4.2.2 NWS2. Communication and exchange;122
7.4.2.3;3.4.2.3 NWS3. Freedom for learning mathematics;123
7.4.2.4;3.4.2.4 NWS4. Local, regional and worldwide expansion;124
7.4.3;3.4.3 Issues related to cost and maintenance;124
7.4.4;3.4.4 Psychological difficulties;124
7.5;3.5 Challenging mathematics beyond the classroom using Internet-based learning environments;125
7.5.1;3.5.1 Problem of the week challenge;125
7.5.2;3.5.2 Collaborative problem solving: challenge and historical context;128
7.5.2.1;3.5.2.1 A Babylonian problem;129
7.5.2.2;3.5.2.2 L’Agora de Pythagore: virtual communityof young mathematical philosophers;130
7.6;3.6 Effect on practices of teachers;130
7.6.1;3.6.1 Mathenpoche;131
7.6.2;3.6.2 WIMS;132
7.6.3;3.6.3 PUBLIREM;133
7.6.4;3.6.4 La main à la pâte;133
7.7;3.7 General discussion;134
7.7.1;3.7.1 Conception;135
7.7.2;3.7.2 Forms;135
7.7.3;3.7.3 Content;136
7.7.4;3.7.4 Implementation;137
7.7.5;3.7.5 Research;137
7.8;3.8 Conclusion;139
7.9;References;140
8;Challenging Tasks and Mathematics Learning;143
8.1;4.1 Introduction;143
8.1.1;4.1.5 A goal for challenging mathematical problems;143
8.1.2;4.1.5 Importance of schemas in mathematical problem solving;144
8.1.3;4.1.5 Mathematical tasks, exercises, and challenging problems;145
8.1.4;4.1.5 Use of challenging problems to promote schema construction;146
8.2;4.2 Categories of challenging mathematics problems;147
8.3;4.3 Challenging mathematics problems and schema development;148
8.3.1;4.3.1 Strands of challenging mathematical tasks;148
8.3.2;4.3.2 Examples from a strand of challenging mathematical tasks;149
8.3.3;4.3.3 Mathematical analysis;150
8.3.4;4.3.4 Cognitive analysis;151
8.3.5;4.3.5 Students’ work on problems from a strand of challenging tasks;151
8.3.5.1;4.3.5.1 How many pizzas are there with four different toppings?;151
8.3.5.2;4.3.5.2 Linking the pizza problem, the towers problem, and Pascal’s triangle;152
8.3.5.3;4.3.5.3 Solving the taxicab problem;153
8.3.5.4;4.3.5.4 Discussion-strand and schema;155
8.4;4.4 Other examples and contexts for challenging mathematics problems;156
8.4.1;4.4.1 Example: Number producer;156
8.4.2;4.4.2 Example: Pattern sequence;160
8.4.3;4.4.3 Examples: Probability;163
8.4.4;4.4.4 Examples: weekly problems;166
8.5;4.5 Example: the challenge of a contradiction and schema adjustment;170
8.5.1;4.5.1 Inconsistency, contradiction and cognitive development;170
8.5.2;4.5.2 What do you do if you have to prove that 1 = 2? and other paradoxes;171
8.5.3;4.5.3 Brief comments on the paradoxes in Problems 2 to 4;173
8.5.4;4.5.4 Analysis of Problem 1;174
8.5.5;4.5.5 Concluding remarks;175
8.6;4.6 Conclusion;176
8.7;References;178
9;Mathematics in Context: Focusing on Students;181
9.1;5.1 Introductory comments;181
9.2;5.2 Discussion of contexts for challenges;183
9.2.1;5.2.1 Highlight on long-term studies;185
9.2.1.1;5.2.1.1 Ingénierie didactique;186
9.2.1.2;5.2.1.2 Activity theory ;186
9.2.2;5.2.2 Conclusion;187
9.3;5.3 Case studies;187
9.3.1;5.3.1 The Laboratory of Mathematical Machines;187
9.3.1.1;5.3.1.1 Learning environment;188
9.3.1.2;5.3.1.2 Duration;189
9.3.1.3;5.3.1.3 Instruments;189
9.3.1.4;5.3.1.4 Pedagogical methods;189
9.3.2;5.3.2 Seeding mathematical challenges at morning assembly at a school in India;191
9.3.3;5.3.3 Heaps of sand: what we can do with sand in and beyond the classroom with a mathematical aim;193
9.3.4;5.3.4 SalsaJ: astronomical software;196
9.3.5;5.3.5 Mathematical challenges around Orsay;197
9.3.6;5.3.6 Challenging gifted high school students;199
9.3.7;5.3.7 Maths à Modeler: research situations for teaching mathematics;202
9.3.7.1;5.3.7.1 Research situations for the classroom (RSC): a definition;202
9.3.7.2;5.3.7.2 Hunting the beast!;204
9.3.8;5.3.8 Mathematics and art;208
9.3.9;5.3.9 Lawn constructions;210
9.4; References;212
10;Teacher Development and Mathematical Challenge;214
10.1;6.1 Introduction;214
10.1.1;6.1.1 What is mathematics and what are mathematical challenges?;214
10.1.2;6.1.1 Why are challenging mathematics problems important in school?;216
10.1.3;6.1.1 What do challenging mathematical problems for school classrooms look like?;217
10.1.3.1;6.1.3.1 The Six Circles problem ;218
10.1.3.2;6.1.3.2 Decimal grid task;219
10.1.3.3;6.1.3.3 Triangle of odd numbers ;221
10.1.3.4;6.1.3.4 The Dirichlet Principle;224
10.1.4;6.1.3 What barriers might prevent teachers from using challenging problems?;225
10.2;6.2 Research;226
10.2.1;6.2.1 What do we know about the effect of teachers’ knowledge and beliefs on the teaching and learning of challenging mathematics?;226
10.2.2;6.2.2 What other factors are important to teaching and learning challenging mathematics?;228
10.2.2.1;6.2.2.1 Motivation;228
10.2.2.2;6.2.2.2 Brain development;229
10.2.2.3;6.2.2.3 Zone of proximal development;229
10.2.3;6.2.3 What other research is needed?;230
10.3;6.3 Effective pedagogy;230
10.3.1;6.3.1 What is the role of the teacher in a class where challenging problems are used?;230
10.3.2;6.3.2 What is effective pedagogy for classrooms using challenging mathematical problems?;232
10.4;6.4 Teacher preparation;235
10.4.1;6.4.1 What is the role of professional development in encouraging classes with challenging mathematical problems?;235
10.4.1.1;6.4.1.1 Modeling;235
10.4.1.2;6.4.1.2 Didactical content;237
10.4.1.3;6.4.1.3 Practica;239
10.4.1.4;6.4.1.4 Mathematical coaching ;239
10.4.1.5;6.4.1.5 Outstanding student work;239
10.4.1.6;6.4.1.6 Teacher-innovator model;239
10.4.2;6.4.2 Some in-service and pre-service programs;240
10.4.2.1;6.4.2.1 A Chinese experience;240
10.4.2.2;6.4.2.2 A German experience;242
10.4.2.3;6.4.2.3 A New Zealand experience;243
10.4.2.4;6.4.2.4 An American experience;244
10.5;6.5 Summary;245
10.5.1;6.5.1 Overview;245
10.5.1.1;6.5.1.1 Fundamental principles;245
10.5.1.2;6.5.1.2 Aims;245
10.5.1.3;6.5.1.3 Modeling;246
10.6;References;246
11;Challenging Mathematics: Classroom Practices;252
11.1;7.1 Challenging mathematics-the essence of mathematics classrooms;252
11.1.1;7.1.4 Why do we need challenges in regular classrooms?;254
11.1.2;7.1.4 How often should challenges be used and for whom?;256
11.2;7.2 Designing challenging mathematics for classrooms;257
11.2.1;7.2.1 Setting the scene;257
11.2.1.1;7.2.1.1 Nature of mathematical understandings expected to be deepened;257
11.2.1.2;7.2.1.2 The gap between what is being proposed and present practice;257
11.2.1.3;7.2.1.3 Clarifying changes in expectations;261
11.2.2;7.2.2 Task design;262
11.2.2.1;7.2.2.1 Rephrasing as a means of tweaking a task for different grade levels;263
11.2.2.2;7.2.2.2 Does the nature of the task change with increasing grade level?;266
11.2.2.3;7.2.2.3 Do we need new topics for challenges or can we find them within the existing curriculum?;267
11.2.2.4;7.2.2.4 How can technology be incorporated into task design to facilitate the use of mathematical challenges?;268
11.3;7.3 Designing classrooms for mathematics challenges;269
11.3.1;7.3.1 How do we teach students strategically to address a challenge?;269
11.3.2;7.3.2 How can we make sense of the pedagogical challenge of having students appreciate challenge in mathematics?;271
11.3.3;7.3.3 The role of textbooks;272
11.3.4;7.3.4 Managing the challenge;273
11.3.5;7.3.5 How can teachers introduce mathematical challenges into the regular classroom?;274
11.4;7.4 Designing research for challenging mathematics classroom practices;275
11.4.1;7.4.1 Fruitful research designs for examining challenges;275
11.4.2;7.4.2 Design-based research;275
11.4.2.1;7.4.2.1 An Australian Example;276
11.4.2.2;7.4.2.2 A Canadian example: can students think like Archimedes?;279
11.4.3;7.4.3 Japanese lesson study;281
11.4.4;7.4.4 Teaching experiments or teaching-research;281
11.4.4.1;7.4.4.1 A North American example;282
11.4.4.2;7.4.4.2 A Russian example;284
11.5;7.5 Conclusion;285
11.6;References;286
12;Curriculum and Assessment that Provide Challenge in Mathematics;293
12.1;8.1 Introduction;293
12.2;8.2 The case studies;294
12.2.1;8.2.1 The case of Singapore: primary school level;294
12.2.1.1;8.2.1.1 Background and curriculum;294
12.2.1.2;8.2.1.2 National examination for primary schools;295
12.2.1.3;8.2.1.3 Test items;295
12.2.1.4;8.2.1.4 Discussion;297
12.2.2;8.2.2 The case of Norway: lower secondary education;299
12.2.2.1;8.2.2.1 Background and curriculum;299
12.2.2.2;8.2.2.2 Traditional examination for lower secondary schools;301
12.2.2.3;8.2.2.3 The written exams;302
12.2.2.4;8.2.2.4 The oral exams;303
12.2.2.5;8.2.2.5 Course grade and final grade;304
12.2.2.6;8.2.2.6 Discussion;304
12.2.3;8.2.3 The case of Brazil: upper primary and lower secondary levels;304
12.2.3.1;8.2.3.1 Curriculum and background;304
12.2.3.2;8.2.3.2 The Brazilian Mathematics Olympiad for public schools;306
12.2.3.3;8.2.3.3 Discussion;309
12.2.4;8.2.4 The case of Iran: upper secondary education and beyond;309
12.2.4.1;8.2.4.1 Curriculum and background;309
12.2.4.2;8.2.4.2 The range of assessment in upper secondary level;310
12.2.4.3;8.2.4.3 The national university entrance examination;310
12.2.4.4;8.2.4.4 High school students mathematical Olympiad;314
12.2.4.5;8.2.4.5 Assessment of the gifted high school students to enter special schools;314
12.2.4.6;8.2.4.6 University students mathematics competition;315
12.2.4.7;8.2.4.7 University Students International Scientific Olympiad in Mathematics;315
12.3;8.3 Assessment and learning, assessment and challenge, assessment and curriculum;315
12.3.1;8.3.1 The role of assessment: assessment and learning;315
12.3.2;8.3.2 The role of assessment: assessment and challenge;316
12.3.3;8.3.3 The role of assessment: assessment and curriculum;317
12.3.4;8.3.4 Competitions and curriculum;319
12.4;8.4 Knowledge gaps and future research questions in this domain;319
12.4.1;8.4.1 Examining opposing views of assessment and their relationship to challenging mathematics;319
12.4.2;8.4.2 What is the role of challenging mathematics in the relationship between assessment and learning?;320
12.4.3;8.4.3 What is the nature of good classroom assessment?;321
12.4.4;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
12.4.5;8.4.5 What are the pedagogical differences and effects attained by enrichment and challenge?;322
12.5;References;322
13;Concluding Remarks;324
14;Acknowledgements;326
14.1;Chapter contributions;326
14.2;Organization support;327
14.3;Plenary speakers;328
14.4;IPC conference;328
14.5;Study Conference;328
14.6;Proofreading and other technical support;329
15;Author Index;331
16;Subject Index;336