Mathematics Education and Technology-Rethinking the Terrain (eBook)

Hoyles_FM.pdf 1
Hoyles_Ch01.pdf 13
Chapter 1 13
Introduction 13
1.1 Introduction 13
1.2 Background and Challenges to ICMI study 17 14
1.3 The Study Conference 15
1.4 Summary of the Book 17
Section 1: Design of Learning Environments and Curricula 17
Section 2: Learning and Assessing Mathematics with and Through Digital Technologies 18
Section 3: Teachers and Technology 20
Section 4: Implementation of Curricula: Issues of Access and Equity 21
Section 5: Future Directions 22
1.5 Conclusion 23
References 23
Hoyles_Ch02.pdf 24
Chapter 2 25
Introduction to Section 1 25
Hoyles_Ch03.pdf 28
Chapter 3 28
Designing Software for Mathematical Engagement through Modeling 28
3.1 Introduction 28
3.2 Case Study 1: Graphs ‘n Glyphs: Animation Software for Mathematics Learning 30
3.2.1 Aims and Description of the Software 30
3.2.2 Main Issues in Software Design 31
3.2.3 Notable Characteristics of the Software 33
3.2.4 Major Achievements 37
3.2.5 Major Challenges 39
3.2.5.1 Reflections on Design 40
3.3 Case Study 2: Lunar Lander, a Prototype Web-Based Space Travel Games Construction Kit 41
3.3.1 Aims and Description of the Software 41
3.3.2 The Activity Sequence 43
3.3.3 Main Issues in Software Design 45
3.3.4 Notable Characteristics of the Software 46
3.3.5 Snapshots of Learning 47
3.3.6 Challenges and Reflections on Design 49
3.4 Conclusions 50
References 52
Hoyles_Ch04.pdf 55
Chapter 4 55
Designing Digital Technologies and Learning Activities for Different Geometries 55
4.1 Geometry, Technology, and Teaching and Learning 55
4.2 Working with Different Geometries on the Flat Screen 56
4.3 Designing Digital Technologies for Different Geometries 58
4.3.1 2D Dynamic Geometry Environments 58
4.3.2 Software for 3D Geometry 60
4.3.3 Software for Various Non-Euclidean Geometries 62
4.4 Designing Learning Activities to Engage Students with Different Geometries 63
4.5 Shaping, and Being Shaped by, Digital Technologies 65
4.5.1 Coda 66
4.5.2 Notes 66
References 67
Hoyles_Ch05.pdf 69
Chapter 5 69
Implementing Digital Technologies at a National Scale 69
5.1 Introduction 69
5.2 Overview of the Projects 70
5.2.1 Enciclomedia 71
5.2.2 M@t.abel 72
5.2.3 Isfahan Mathematics House: E-Content 72
5.2.4 Mathematics 9 and 10 with The Geometer’s Sketchpad 73
5.2.5 Sketchpad for Young Learners 74
5.3 Comparing and Contrasting the Projects 75
5.3.1 Curriculum Content 75
5.3.2 Teaching Practices 77
5.3.3 Activity Design 78
5.4 Emerging Themes Across the Projects 81
5.4.1 Shifts in Audience: Moving Toward More Teacher Participation 81
5.4.2 Shifts in Value: From Pragmatic to Epistemic 81
5.5 Concluding Remarks 84
5.6 Looking Forward 84
References 85
Hoyles_Ch06.pdf 87
Chapter 6 88
Introduction to Section 2 88
6.1 The Points of Departure 88
6.2 A Guided Tour Through the Chapters 89
6.3 Looking Back at the Original Issues 91
6.4 Concluding Remarks 93
References 94
Hoyles_Ch07.pdf 95
Chapter 7 95
Integrating Technology into Mathematics Education: Theoretical Perspectives 95
7.1 Introduction 95
7.2 Looking Back 96
7.2.1 The Evolution of Technology and Its Use in the Mathematics Education Community 97
7.2.2 The Emergence of Theory from the Integration of Technology Within Mathematics Education 98
7.2.2.1 Tutor, Tool, Tutee 99
7.2.2.2 White Box – Black Box 99
7.2.2.3 Microworlds and Constructionism 100
7.2.2.4 Amplifier – Reorganizer 101
7.2.3 Theoretical Ideas Emanating from the Literature on Mathematical Learning 101
7.2.3.1 Process-Object 102
7.2.3.2 Visual Thinking vs. Analytical Thinking 102
7.2.3.3 Representational Issues 103
7.2.4 From Past to Present 104
7.3 Current Developments 104
7.3.1 Learning Theories from Mathematical Didactics 105
7.3.1.1 From Scaffolding and Abstraction to Webbing and Situated Abstraction 106
7.3.1.2 Theory of Didactical Situations: the Concept of Milieu 108
7.3.1.3 Perceptuo-Motor Activity in Mathematical Learning 110
7.3.1.4 Discussion 112
7.3.2 Instrumentation 112
7.3.2.1 Artifact and Instrument 114
7.3.2.2 Instrumental Genesis 114
7.3.2.3 Schemes and Techniques 115
7.3.2.4 Examples 116
7.3.2.5 Orchestration 118
7.3.2.6 Affordances, Constraints, Perspectives 119
7.3.3 Mediation and Semiotic Mediation 119
7.3.3.1 Representation and the Semiotic Approach 120
7.3.3.2 Mediation 121
7.3.3.3 Mediation According to a Semiotic Approach 122
7.3.3.4 Examples of Semiotic Mediation 124
7.4 Summary and Future Developments 126
7.4.1 Summary 126
7.4.2 Technological Developments 127
7.4.3 Theoretical Developments 127
7.4.4 The Remath Integrative Theoretical Framework 128
References 131
Hoyles_Ch08.pdf 139
8 139
Mathematical Knowledge and Practices Resulting from Access to Digital Technologies 139
8.1 Overview of the Chapter 139
8.1.1 Preface 140
8.2 Mathematical Knowledge in a Technological World 142
8.2.1 What Is Mathematical Knowledge? 142
8.2.2 The Influence of Technology on the Nature of Mathematical Knowledge 143
8.2.3 Mathematical Knowledge: Operational and Notational Aspects 145
8.2.4 Contexts for Learning Mathematics 146
8.2.5 A New Learning Ecology 147
8.2.6 Example Cases of Effective Technologies 148
8.2.6.1 The Fractions Project: Using Technology with High Levels of “Cognitive Fidelity” 149
8.2.6.2 The SimCalc Project: Introducing the Mathematics of Change in Middle School – Technology with High Levels of “Mathem 151
8.2.6.3 Dynamic Geometry Environments 153
8.2.7 Summary of Students’ Mathematical Knowledge in a Technological World 156
8.3 Mathematical Knowledge“Within” Technologies 156
8.3.1 Numbers and Arithmetic 156
8.3.2 CAS and Problem Spotting 157
8.3.3 Geometry with Linear Algebra 157
8.3.4 Who Has to Know What About the Underlying Mathematical Assumptions and Processes of Spreadsheets, DGEs, Statistical Pa 158
8.4 New Mathematical Practices 159
8.4.1 Link Between Knowledge and Practice 159
8.4.2 Interactions Among Students, Teachers, Tasks, and Technologies: Shifts in Empowerment 161
8.4.3 Role of Feedback in Practice 164
8.4.4 Example Technologies that Promote New Mathematical Practices 165
8.4.4.1 New Mathematical Practices in Dynamic Geometry Environments 165
8.4.4.2 Technologies that Encourage New Practices in Statistics 168
8.4.4.3 Children’s Mathematical Practices Using Robotics and Digital Games 170
8.4.5 Summary of New Mathematical Practices Made Possible with Technology 172
8.5 Final Words: An Adaptation of Our Didactical Tetrahedron 174
References 175
Hoyles_Ch09.pdf 184
Chapter 9 184
The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts 184
9.1 Introduction 184
9.2 Theoretical Overview 186
9.2.1 On Learning Trajectories 186
9.2.2 The Possible Influence and Mediating Role of Digital Technologies on Learning and Learning Trajectories 187
9.2.3 Digital Technological Environments as Domains of Abstraction 188
9.2.4 Hypothetical Learning Trajectories in DT Environments: Building on the Microworld Idea and Design 189
9.3 Affordances of Digital Technologies that Might Influence Learning Trajectories, and Considerations for the Design of HLT 189
9.3.1 Technical Aspects 191
9.3.1.1 The Choice of the Technological Tool(s) and Their Design 191
9.3.1.2 The Role of Representations 191
9.3.1.3 The Computational and Dynamic Capabilities of DT 192
9.3.1.4 The Networking Capabilities of DT 193
9.3.2 Pedagogical and Contextual Aspects (Task Design, the Role of the Teacher and the Didactical Context) 193
9.3.2.1 The Pedagogical Setting 193
9.3.2.2 Context of Inquiry of the Activity 194
9.3.2.3 Level of Openness of a DT-Based Activity 194
9.3.2.4 Sequencing of Tasks Within an Activity 195
9.3.2.5 Mathematical Content 196
9.3.2.6 The Possibility of Earlier Engagement (Shifts in Trajectories) 196
9.3.2.7 The Teacher’s Role and the Importance of Appropriate Intervention 197
9.3.3 The Learner Perspective 197
9.3.3.1 Affect and DT: The Role of Engagement and Motivation for Learning 197
9.3.3.2 The Role of the Feedback from DT 198
9.3.4 Introduction to the Following Sections 198
9.4 An Example of the Design of a Hypothetical Learning Trajectory Through Exploratory Tasks 199
9.4.1 Context of Inquiry 199
9.4.2 Mathematical Content 200
9.4.3 The Choice of the Technological Tool 200
9.4.4 Level of Openness 201
9.4.5 Representations 201
9.4.6 Sequencing of Tasks within the Activity 202
9.4.7 Comments 202
9.5 Learning Trajectories Within and Across Various Platforms: An Example with Dynamic Geometry and CAS 202
9.5.1 The Construction of a Dynamic Representationof the Problem 203
9.5.2 From Geometry to Algebra 206
9.5.3 Discussion 207
9.6 Emergence of Learning Trajectories from the Engagement with DT: An Example with Spreadsheets 208
9.7 Shifts in Trajectories: Possibilities of Earlier Engagement with Powerful Ideas Afforded by DT, and the Development of I 211
9.7.1 Using DT for Developing Intuitive Thinking 212
9.7.1.1 Early Accessto Powerful Mathematical Ideas: An Example of Early Algebra 214
The Case of Rodrigo 215
The Case of Ana Karen 216
9.7.1.2 The Role of Spreadsheets in the Transition Towards the Algebraic Method for Solving Word Problems 216
9.7.2 Early Access to Powerful Mathematical Ideas: Exploring Infinity-Related Notions 219
9.7.2.1 A Logo Microworld for the Exploration of Infinite Processes 219
9.7.2.2 Exploration of Infinite Sequences, Series and the Cardinality of Infinite Setswith ToonTalk 221
9.7.3 Early Access to Powerful Mathematical Ideas: Long-Term Impact 223
9.8 Concluding Remarks 224
References 225
Hoyles_Ch10.pdf 232
Chapter 10 232
Micro-level Automatic Assessment Supported by Digital Technologies 232
10.1 Introduction 232
10.2 Principle of CAA 234
10.2.1 How to Implement an Assessment? 234
10.2.2 Structure in the Tasks or Students Generated Mathematical Objects 237
10.2.2.1 Handling Algebraic Expressions 238
10.2.2.2 Handling Geometric Figures 240
10.2.3 Generation of Feedback for Students and for Teachers 243
10.2.3.1 Qualitative Feedback for Formative Assessment 243
10.2.3.2 Cohort Achievement Data 244
10.3 Results of Actual CAA Use 245
10.3.1 New Ways of Undertaking Mathematical Tasks 247
10.3.2 Limits and Difficulties of Using CAA 248
10.3.3 Interpreting Students’ Solutions in Simple Geometry Tasks 250
10.4 Conclusions and the Future 253
References 254
Hoyles_Ch11.pdf 256
Chapter 11 256
Technology, Communication, and Collaboration: Re-thinking Communities of Inquiry, Learning and Practice 256
11.1 Introduction 257
11.1.1 Social Perspectives on Learning 257
11.1.2 Socio-Constructivism 258
11.1.3 Socio-Culturalism 258
11.1.4 Communities of Practice 259
11.1.5 Voice and Discourse 261
11.1.6 Distributed Cognition 262
11.2 The Growth in Social Perspectives on Teaching and Learning with Technology 263
11.2.1 Early Accounts 264
11.2.2 A New Millennium 264
11.2.3 Current Climate 265
11.2.4 The Role of Technology in Collaborative Mathematical Practice 265
11.3 Different Technological Typologies for Fostering Communication, Collaboration, and Communities of Inquiry 267
11.3.1 Technologies Designed for Both Mathematics and Collaboration 268
11.3.1.1 Internet-Based Networks 268
11.3.1.2 Classroom-Based Networks 269
11.3.1.3 Non-networked Software 270
11.3.2 Technologies Designed for Mathematics but Not Specifically for Collaboration 271
11.3.3 Technologies Designed for Collaboration but Not Necessarily Mathematics 273
11.3.3.1 Enhancing the Learning Environment 274
11.3.3.2 Distance Learning 275
11.3.4 Technologies Designed for Neither Mathematics nor Collaboration 276
11.4 Future Developments 277
11.4.1 New Forms of Communities of Learners 277
11.4.1.1 Amplifying, Enhancing, Broadening Classroom-Based Communities 277
11.4.1.2 Online Communities: Virtual Communities of Learners 278
11.4.2 Extending the Role of the Teacher 279
11.4.3 New Forms of Voice and Discourse 280
11.4.4 Other Issues 281
11.4.4.1 The Case of Marginalised Members of a Community 281
11.4.4.2 Emergent Uses of Technology 281
11.4.4.3 Unit of Analysis 282
11.5 Conclusion and Final Remarks 282
References 283
Hoyles_Ch12.pdf 290
Chapter 12 291
Introduction to Section 3 291
12.1 Introduction 291
References 295
Hoyles_Ch13.pdf 297
Chapter 13 297
Working with Teachers: Context and Culture 297
13.1 Introduction 298
13.2 Context, Culture and Teachers’ Practices 298
13.3 Case 1: Using Inquiry Cycles in Activity Design 301
13.4 Case 2: Half-Baked Microworlds as Catalysts for Instrumentalisation 304
13.5 Case 3: Communal Design of a Tool for Statistical Explorations 308
13.5.1 Designing as Learning: From Means and Spread to Distribution as a Space of Possible Values 309
13.5.2 Distributing the Instrumental Genesis Process? 311
13.6 Reflections on the Case Studies 312
References 313
Hoyles_Ch14.pdf 315
Chapter 14 315
Teachers and Teaching: Theoretical Perspectives and Issues Concerning Classroom Implementation 315
14.1 Theoretical Perspectives 316
14.1.1 Instrumental Genesis 317
14.1.2 Zones and Affordances 320
14.1.3 Instrumentation Theory Applied to a Zone/Affordances Excerpt 322
14.1.4 Zone Theory and Affordances Applied to an Instrumental Genesis Excerpt 323
14.1.5 Complexity Theory 323
14.1.6 Theoretical Perspectives: What Is the Teacher’s Role in Technology Integration? 325
14.1.7 Factors Influencing Technology Integration in Schools 325
Vignette #1: Novice (Pre-service) Teacher 326
Vignette #2: Experienced Teacher 327
14.1.8 Factors Influencing Technology Integration in University Mathematics Departments 327
14.2 Classroom Implementation 328
14.2.1 Defining Criteria for Effective Use 328
14.2.2 Identifying Change 329
14.3 Future Visions 330
References 331
Hoyles_Ch15.pdf 333
Chapter 15 333
Teacher Education Courses in Mathematics and Technology: Analyzing Views and Options 333
15.1 Introduction 333
15.2 Examples of Teacher Development Courses 335
15.2.1 In Service Teacher Development Course “Mathematics Investigations” (MathInquiry) (Based on Alagic 2006) 335
15.2.2 A Bachelor of Education Course at the Institute of Education for Women in India (BecIEW) (Based on Das 2006) 336
15.2.3 Mieux Apprendre la Géométrie avec l’Informatique1 (MAGI) (Based on Assude et al. 2006) 336
15.2.4 A Teacher Development Course for Prospective Primary Mathematics Teachers (TdcPt) (Based on Hunscheidt and Peter-Koop 336
15.2.5 The Bachelor of Education “ITeach Laptop Learning Program” (BEdITeach) (Based on Jarvis 2006) 337
15.3 Characterizing the Varied Views that Underpin Teacher Development Programs in Technology 337
15.3.1 Views Concerning the Implementation of Technology in the Classroom and in Teacher Education 337
15.3.2 Views About Changes in Teachers’ Role, Activity and Practices Underpinning a Course 339
15.3.3 Views About How to Prepare Teachers 340
15.4 Identifying the Various Practical Decisions Related to Course Organisation 342
15.4.1 Contents Proposed in the Courses 342
15.4.1.1 Content 1: The Impact of Technology on Mathematics and the Resulting Evolution of the Curriculum 343
15.4.1.2 Content 2: The Potential of Computer Applications for New Alternatives in Mathematics Learning 344
15.4.1.3 Content 3: The Ideas of Instrumental Genesis and Intertwined Mathematical and Instrumental Knowledge 344
15.4.1.4 Content 4: Creating New Tasks and Making Them Work Together with Older Tasks 344
15.4.1.5 Content 5: New Teaching Abilities 345
15.4.1.6 Content 6: Introducing Technology into a Professional Context 345
15.4.2 Teacher Educator Strategies 345
15.4.2.1 Strategy 1: Demonstration (Showing How to Achieve a Specific Goal) 346
15.4.2.2 Strategy 2: Role Playing (Teacher as a Student) 346
15.4.2.3 Strategy 3: In Practice (Teacher as Reflective Practitioners) 347
15.4.2.4 Strategy 4: Learning in Communities 347
15.5 Conclusion 347
References 348
Hoyles_Ch16.pdf 350
Chapter 16 351
Introduction to Section 4 351
16.1 Introduction 351
16.2 Mathematics Curricula 352
16.2.1 Intended, Implemented and Attained Curricula 352
16.2.2 The Mathematics Curricula of Different Countries 353
16.2.3 The Mathematics Curricula of Different Sectors 355
16.2.4 Factors Influencing Implemented Curricula 356
16.3 Access and Equity 358
16.4 Conclusion 359
References 360
Hoyles_Ch17.pdf 363
Chapter 17 363
Some Regional Developments in Access and Implementation of Digital Technologies and ICT 363
17.1 Introduction 364
17.2 Macro Perspective on Education for the Twenty-First Century by Linda S. Posadas 364
17.3 Regional Reports 366
17.3.1 Case 1: Russia by Alexei Semenov 366
17.3.2 Case 2: Hong Kong (A Special Administrative Region of China) by Allen Leung 366
17.3.3 Case 3: Vietnam by Nguyen Chi Thanh 368
17.3.4 Case 4: South Africa (and Some Developments in Sub-Saharan Africa) by Cyril Julie 370
17.3.5 Case 5: Latin-America by Ana Isabel Sacristán 373
17.3.5.1 Latin-American Countries with Mainly Small-Scale Efforts of Integration of Digital Technologies Due to Individual I 374
17.3.5.2 Latin-American Countries with Large-Scale, Either Government-, or Privately-Sponsored, Projects 375
Brazil 375
Costa Rica 376
Mexico 377
Colombia 379
Chile 380
Venezuela 381
17.4 Conclusions 382
References 383
Hoyles_Ch18.pdf 386
Chapter 18 386
Technology for Mathematics Education: Equity, Access and Agency 386
18.1 Introduction 386
18.2 Definitions of Equity Including Access and Agency 387
18.2.1 Access 388
18.2.2 Resources for Equity 388
18.2.3 Equitable Pedagogies 389
18.2.4 Equitable Outcomes 390
18.2.5 Agency 391
18.3 Research Studies 392
18.3.1 Equity and Mathematics Learning with Technology 392
18.3.2 Equity and Attitudes, Beliefs, and Values Associated with Technology Use for Mathematics Learning 395
18.3.3 Resources for Mathematics Learning with Technology 397
18.3.4 Access to Mathematical Learning with Technology 398
18.3.5 Agency as an Outcome of Mathematical Learning with Technology 399
18.4 Conclusion 400
References 401
Hoyles_Ch19.pdf 405
Chapter 19 405
Factors Influencing Implementation of Technology-Rich Mathematics Curriculum and Practices 405
19.1 Introduction 405
19.2 Typology of Factors 406
19.2.1 The Social, Political, Economical and Cultural Level 406
19.2.2 The Mathematical and Epistemological Level 408
19.2.3 A School or an Institutional Level 410
19.2.4 The Classroom and Didactical Level 411
19.2.5 Multi-level Factors 412
19.3 Explaining the Problem 413
19.4 Conclusion 416
References 417
Hoyles_Ch20.pdf 420
Chapter 20 421
Introduction to Section 5 421
Hoyles_Ch21.pdf 423
Chapter 21 423
Design for Transformative Practices 423
21.1 Introduction 423
21.2 Potentialities of Dynamic Software for Teaching Challenging But Difficult Topics 424
21.2.1 Making Traditionally Difficult Topics Appear More Straightforward 424
21.2.1.1 Calculus: Illustrating Integrals, Areas and Volumes 424
21.2.1.2 Data Treatment: Understanding the Central Limit Theorem 425
21.2.2 Topics That Could be Re-introduced to Mainstream Post-16 Teaching 425
21.2.2.1 Differential Equations: Seeing What’s Going On 425
21.2.2.2 Bringing Back the Study of 3D Lines and Planes 426
21.2.3 Making Teaching More Effective and More Fun 426
21.3 Software for Mathematical Explorations: Attempting to Make a Curricular Agenda Visible 426
21.3.1 Clearing the Confusion Regarding the Role of Technology 427
21.3.2 From Bodily Actions to Symbolizing and Meaning Production 428
21.4 Attention to Detail: Broadening Our Design Language 429
21.4.1 Design Detail Counts 429
21.4.2 Well-Developed Design Discourses from Which to Draw 430
21.4.3 Paradigms of Embodied Interaction 431
21.5 Designing a 3D Dynamic and Interactive Environment 432
21.5.1 A Vision: Technology to Operate an Epistemological Shift 434
References 434
Hoyles_Ch22.pdf 436
Chapter 22 436
Connectivity and Virtual Networks for Learning 436
22.2 Developing Microworlds for On-LineCollaborative Learning 438
22.2.1 Background Issues 438
22.2.2 A Further Example 439
22.2.3 Some Reflections and Observations 440
22.2.4 Some Concluding Remarks 441
22.3 Connectivity: New Challenges for the Ideas of Webbing and Orchestrations 441
22.3.1 Introduction 441
22.3.2 Some Elements on the Experiment 443
22.3.3 Some Results 444
22.3.4 Questions to Be Considered in More Depth 446
22.3.5 Some More General Considerations 447
22.4 Concurrent Connectivity: Using Netlogo’s Hubnet Module to Enact Classroom Participatory Simulations 449
22.5 Designing for Exploiting Connectivity Across Classrooms 452
22.5.1 The Playground Project 452
22.5.2 The Weblabs Project 455
22.5.3 Concluding Remarks 457
References 457
Hoyles_Ch23.pdf 460
Chapter 23 460
The Future of Teaching and Learning Mathematics with Digital Technologies 460
23.1 Introduction 460
23.2 A Personal Journey with Digital Technologies 461
23.2.1 From Programming to Visualization and Experimentation: A First University Experience 461
23.2.2 Working with Low Achievers in Geometry with Logo Technology 462
23.2.3 The CAS Experience 463
23.2.4 From Microworlds and Open Software to Tutorial and on Line Resources 465
23.2.5 European Cooperation and Theoretical Connections 466
23.3 Towards a Vision: Some Crucial Directions 467
23.3.1 The Theoretical Perspective 467
23.3.2 The Teacher Perspective 468
23.3.3 The Institutional and Curricular Perspectives 469
23.3.4 Collaboration and Connectivity 470
23.3.5 Equity and Accessibility 470
23.4 Some Concluding Comments 471
References 471
Hoyles_Index.pdf 473
Hoyles_Author Index.pdf 473
Hoyles_Index.pdf 480