Mathematical Modelling in Education Research and Practice (eBook)

Series Preface 6
Contents 8
Chapter 1: Cultural, Social, Cognitive and Research Influences on Mathematical Modelling Education 13
1.1 Introduction 13
1.2 Innovative Practices in Modelling Education Research and Teaching 14
1.3 Research into, or Evaluation of, Teaching Practice 18
1.4 Pedagogical Issues for Teaching and Learning of Modelling 24
1.5 Influences of Technologies 31
1.6 Assessment in Schools and Universities 34
1.7 Applicability at Different Levels of Schooling, Vocational Education, and in Tertiary Education 34
1.8 Conclusion 38
References 39
Part I: Innovative Practices in Modelling Education Research and Teaching 45
Chapter 2: Mathematical Modelling as a Strategy for Building-Up Systems of Knowledge in Different Cultural Environments 46
2.1 Introduction 46
2.2 The Generation of Knowledge 47
2.3 How About Modelling? 53
References 55
Chapter 3: The Meaning of the Problem in a Mathematical Modelling Activity 56
3.1 Introduction 56
3.2 Problems, Mathematical Modelling and Meaning 57
3.3 Design of the Study 59
3.4 The Meaning of the Problem and the Generation of Interpretants in Mathematical Modelling Activities 59
3.5 Discussion and Implications for Teaching, Learning and Research 63
References 65
Chapter 4: Extending the Reach of the Models and Modelling Perspective: A Course-Sized Research Site 66
4.1 Introduction 67
4.2 Research Questions Addressed by the MMP 67
4.3 Claims About the Nature of Knowing and Learning 68
4.4 Research Tools and the Data Generated by Inquiry Within the MMP 70
4.5 Extending the Questions Expanding the Toolkit
4.6 Some Assumptions and Conjectures 71
4.6.1 Learning Progressions 71
4.6.1.1 Alternative Model #1: Learning as Finding One´s Way Around in a Terrain 72
4.6.1.2 Alternative Model #2: An Evolutionary Model for the Development of Ideas 73
4.6.2 Teaching Problem Solving and Heuristics 73
4.7 Implications for Design 75
4.8 Conclusion: Contributions of a Course-Sized Research Site 76
References 76
Chapter 5: Prescriptive Modelling - Challenges and Opportunities 78
5.1 Introduction 78
5.2 Examples 80
5.2.1 Example 1: BMI (Body Mass Index) 80
5.2.2 Example 2: A-Paper (DIN) Formats 82
5.2.3 Example 3: The Gini Coefficient of Income Inequality 85
5.2.4 Conclusions from the Examples 87
5.3 Teaching and Learning of Prescriptive Modelling 88
5.4 Challenges and Opportunities 89
References 90
Chapter 6: An Approach to Theory Based Modelling Tasks 91
6.1 Introduction 91
6.2 Theoretical Framework and Method 92
6.2.1 Task Criteria 92
6.2.2 Degree of Difficulty 93
6.3 Design of the Pilot Study 94
6.3.1 The Modelling Tasks 94
Toothpaste Item 95
Taj Mahal Item 95
Potato Item 96
6.4 Results 96
6.4.1 Toothpaste Item 97
6.4.2 Taj Mahal Item 98
6.4.3 Potato Item 98
6.5 Discussion 99
6.6 Outlook 100
References 101
Chapter 7: Facilitating Mathematisation in Modelling by Beginning Modellers in Secondary School 102
7.1 Introduction 102
7.2 Theoretical Frame 103
7.3 Empirical Evidence 105
7.4 Proposed Framework 106
7.5 Illustrative Example 107
7.6 Discussion and Conclusion 110
References 112
Chapter 8: Authenticity in Extra-curricular Mathematics Activities: Researching Authenticity as a Social Construct 114
8.1 Introduction 114
8.2 Theoretical Background 115
8.2.1 Authenticity as Simulation or Imitation 115
8.2.2 Authenticity as a Social Construct 116
8.3 Methods 117
8.4 Results 118
8.5 Conclusion and Discussion 121
References 122
Chapter 9: The Teaching Goal and Oriented Learning of Mathematical Modelling Courses 123
9.1 Introduction 123
9.2 Intuitions: Insights into the Nature of the Problem 124
Case 1: Lanchester Battle Model (Lucas 1983) 125
Case 2: More Means the Smoother Journey? (Braess´s Paradox 2008 Mind Your Decisions 2009)
9.3 Connections: Reconstruction of Personal Knowledge System 127
9.4 Innovation: Critical Thinking Ability 128
Case 3: Locks Packing Problem - Complaint Degree Tolerance 128
9.5 Choice: the Ability of Macro-coordination and Direction Control 129
Case 4: Cable Car Operation Plan - Realistic Alternative 130
9.6 Inductions: Improve Thinking Mode 131
Case 5: Forecasting Popularity - Induction of the Impact Factors (Roja et al. 2012) 131
9.7 Conclusions 132
References 132
Part II: Research into, or Evaluation of, Teaching and Learning 134
Chapter 10: Modelling Competencies: Past Development and Further Perspectives 135
10.1 Development of Modelling Competencies from the Past Until Today 135
10.1.1 Modelling Competencies at the Beginning of the Modelling Discussion 136
10.1.2 Role of Modelling Competencies in the Recent International Modelling Discussion 139
10.1.3 Four Strands of the Earlier and the Recent International Discussion on Modelling Competencies 141
10.2 Further Perspectives on Modelling Competencies: Evaluation of the Structure of Modelling Competencies and Their Promotion 145
10.2.1 Aims and Design of the Study 145
10.2.2 Results 149
10.2.3 Discussion of the Results and Looking Ahead 151
References 152
Chapter 11: How to Support Teachers to Give Feedback to Modelling Tasks Effectively? Results from a Teacher-Training-Study in ... 156
11.1 Introduction 156
11.2 The Idea of Formative Assessment 157
11.3 Mathematical Modelling as Part of Competency-Oriented Mathematics 158
11.4 Teacher Knowledge as Predictor for the Quality of Teaching and Students´ Achievement 158
11.5 A Teacher-Training-Study for In-service Teachers: Formative Assessment in Competency-Oriented Mathematics 159
11.5.1 Design of the Teacher-Training-Study 159
11.5.2 Content of the Teacher Training 160
11.5.3 Evaluating the Teacher Training: Tests on Teachers´ General Pedagogical Knowledge and Pedagogical Content Knowledge 161
11.5.4 First Results of the Teacher-Training-Study 162
11.6 Summary and Outlook 163
References 164
Chapter 12: A Reflection on Mathematical Modelling and Applications as a Field of Research: Theoretical Orientation and Divers... 166
12.1 Introduction 166
12.2 Method 167
12.2.1 Lens 1: The Orientation of the Field 169
12.2.2 Lens 2: The Diversity of Theory 170
12.3 Results 171
12.3.1 The Orientation of Publications 171
12.3.2 Diversity of Theory 171
12.4 Conclusions, Discussion and Implications 173
References 176
Chapter 13: Problem Solving Methods for Mathematical Modelling 177
13.1 Sub-competencies of Modelling Competencies 177
13.2 Problem Solving Methods 178
13.3 Study Design 182
Teeth Brushing Task 182
Flip-Flop Task 183
13.4 Results 183
13.5 Discussion and Conclusion 185
References 186
Chapter 14: Improving Mathematical Modelling by Fostering Measurement Sense: An Intervention Study with Pre-service Mathematic... 188
14.1 Introduction 188
14.2 Measurement Sense as Central Aspect of Handling Quantities 189
14.3 Measurement Sense in the Context of Mathematical Modelling 190
14.4 Improving Mathematical Modelling by Fostering Measurement Sense: An Intervention Study with Pre-service Mathematical Teac... 191
14.4.1 Design of the Intervention Study 191
14.4.2 Content of the Intervention 192
14.4.3 Evaluating the Intervention Study: Tests on Mathematical Pre-service Teachers´ Measurement Sense and Modelling Competen... 192
14.4.4 Selected Results of the Intervention Study 194
14.5 Summary and Conclusion 195
References 196
Chapter 15: How Do Students Share and Refine Models Through Dual Modelling Teaching: The Case of Students Who Do Not Solve Ind... 198
15.1 Dual Modelling Cycle Framework (DMCF) 199
15.2 Diversity of Student Models in Modelling Teaching Based on DMCF 200
15.2.1 The Teaching Material Based on DMCF 200
Oil Tank Task (Initial Task) 200
15.2.2 Outline of the Modelling Teaching Based on DMCF 200
15.2.3 Types of Student Solutions 203
15.3 Analysis of Responses of Students who Could not Solve Independently 203
15.3.1 Kawa´s Case 203
15.3.1.1 Lesson 1: First Trial of Oil Tank Task and Toilet Paper Tube Task 203
15.3.1.2 Lesson 2: Second Trial of Oil Tank Task Based on the Toilet Paper Tube Task 204
15.3.1.3 Lesson 3: Final Trial of Oil Tank Task Through a Class Presentation 204
15.3.2 Kato´s Case 205
15.3.2.1 Lesson 1: First Trial of Oil Tank Task and Toilet Paper Tube Task 205
15.3.2.2 Lesson 2: Second Trial of Oil Tank Task Based on the Toilet Paper Tube Task 205
15.3.2.3 Lesson 3: Final Trial of Oil Tank Task Through a Class Presentation 206
15.4 Discussion 206
15.5 Conclusion 208
References 208
Chapter 16: Exploring Interconnections Between Real-World and Application Tasks: Case Study from Singapore 210
16.1 Introduction 210
16.2 The Study 212
16.2.1 Sample and Sampling Process 213
16.2.2 Data Collection and Analysis 213
16.2.3 PW: Designing an Environmentally Friendly Building 214
16.3 Findings 214
16.3.1 Connections Between School-Based Disciplines 215
16.3.2 Connections Within Mathematics Topics 216
16.3.3 Connections Between School-Based Mathematics and the Real-World 217
16.3.4 Connections Between ISL and BEC Scores and Qualitative Findings 217
16.4 Discussion 218
16.5 Conclusion 219
References 219
Chapter 17: Mathematical Modelling Tasks and the Mathematical Thinking of Students 221
17.1 Introduction 221
17.2 Theoretical Assumptions 222
17.2.1 Transitions from Elementary Mathematical Thinking into Advanced Mathematical Thinking 222
17.2.2 Mathematical Modelling 224
17.3 Methodology 225
17.4 Results and Discussion 226
Diazepam in the Body Task 226
17.5 Final Remarks 229
References 230
Chapter 18: Measurement of Area and Volume in an Authentic Context: An Alternative Learning Experience Through Mathematical Mo... 231
18.1 Introduction 231
18.2 The Problem 232
18.3 Mathematical Modelling as a Learning Strategy in the Classroom 232
18.4 Research Methods 233
18.4.1 Context 233
18.4.2 Participants 233
18.4.3 Sources of Data Collection 234
18.5 Towards the Building of Mathematical Models 234
18.5.1 The Authentic Context: Flood Phenomenon and Social Impact 234
18.5.2 Delimitation of the Situations in the Context of the Floods 235
18.5.3 A Mathematical Model: Surface Area Versus Water Level 237
18.5.4 A Mathematical Model: Water Volume Versus Height of Water Level 238
18.6 Discussion 240
18.7 Conclusion 241
References 241
Chapter 19: Mathematical Modelling and Culture: An Empirical Study 243
19.1 Introduction 243
19.2 Roles of Mathematical Modelling in Culture 244
19.3 The Study 245
19.4 Some Findings 246
19.5 Discussion 249
19.6 Conclusions 251
References 252
Chapter 20: Mathematical Modelling of a Social Problem in Japan: The Income and Expenditure of an Electric Power Company 253
20.1 Introduction 253
20.2 Teaching Practices 254
Electric Power Company Task 254
20.2.1 The First Period 255
20.2.2 The Second Period 257
20.2.3 The Third Period 257
20.3 Evaluation of Students´ Beliefs 258
20.3.1 Utility of Mathematics in the Junior High School 258
20.3.2 Students´ Interest in This Lesson 260
20.3.3 Students´ Evaluation of the Pliability of Mathematics 261
20.4 Conclusion 261
References 263
Part III: Pedagogical Issues for Teaching and Learning 264
Chapter 21: The Place of Mathematical Modelling in the System of Mathematics Education: Perspective and Prospect 265
21.1 Introduction 265
21.2 The Effect of Modelling on the Relationship Between Secondary and Tertiary Education 266
21.2.1 Modelling in the High Schools: What Will Universities Do? 266
21.2.2 Modelling in the Universities: What Will High Schools Do? 266
21.3 Teacher Education: Preparing Teachers to Teach Mathematical Modelling 267
21.3.1 Teaching About the Modelling Process: Can It Involve New Mathematics? 267
21.3.2 Assessment: How Do You Judge the Success of a Model? 268
21.3.3 What Do Teachers Believe Modelling Is? 269
21.3.3.1 Mathematical Models 269
21.3.3.2 Mathematical Modelling 269
21.3.3.3 Mathematical Modelling in Education 269
21.4 The Relations Between Mathematics and Mathematics Education 270
21.4.1 A Modelling Problem 270
21.4.2 Modeling as Vehicle and Modelling as Content 271
21.4.3 Primary and Middle School 272
21.4.4 Secondary School 273
21.4.5 The Secondary: Tertiary Interface 273
21.5 Modelling as a Source of New Insights in Mathematics Itself 273
21.6 Coda 275
References 276
Chapter 22: Moving Within a Mathematical Modelling Map 277
22.1 Introducing the Mathematical Modelling Map 277
22.2 ``Moves´´ Within the Map 279
22.3 Conclusions 281
References 281
Chapter 23: Negotiating the Use of Mathematics in a Mathematical Modelling Project 283
23.1 Introduction 283
23.2 Context, Participants, Methodological Aspects and the Modelling Project 285
23.3 ``It Isn´t My Fault If I Already Have All the Data!´´ 287
23.4 The Negotiation Between Maria Estela and the Teacher: Data Through a Theoretical Lens 289
23.5 Final Remarks 290
References 291
Chapter 24: Moving Beyond a Single Modelling Activity 293
24.1 Introduction 293
24.2 Models, Modelling, and Model Development Sequences 294
24.3 Variation Theory 296
24.4 Model Development Sequences in the Light of Variation Theory 297
24.4.1 Principles for Designing Model Exploration Activities 298
24.4.1.1 An Example of Contrast Variation 298
24.4.1.2 An Example of Variation of Fusion 299
24.4.2 Principles for Designing Model Application Activities 299
24.4.2.1 An Example of Variation of Separation 300
24.4.2.2 An Example of Variation of Generalization 300
24.5 Discussion and Conclusions 301
References 302
Chapter 25: The Possibility of Interdisciplinary Integration Through Mathematical Modelling of Optical Phenomena 304
25.1 Theoretical Framework and Justification 305
25.2 Methodology 307
25.3 Workshop Details and Observations 307
25.3.1 Participants 307
25.3.2 Methods and Materials 308
25.3.2.1 Flat Mirror Movement 308
25.3.2.2 Combination of Flat Mirrors and Parallel Flat Mirrors 308
25.3.2.3 Mirror Rotation 310
25.4 Results Analysis 311
25.5 Final Considerations 313
References 314
Chapter 26: Activation of Student Prior Knowledge to Build Linear Models in the Context of Modelling Pre-paid Electricity Cons... 316
26.1 Introduction 316
26.2 Problem 317
26.3 Theoretical Considerations 318
26.4 Methodological Design 319
26.5 Findings 319
26.5.1 Context of Pre-paid Energy: A Context in the Student´s Life 319
26.5.2 Building of Linear Models Adjusted to the Context of Pre-paid Energy 320
26.5.3 Use of Models to Make the Household Economy More Stable 322
26.6 Discussion 323
26.7 Conclusions 324
References 324
Chapter 27: Mathematical Modellers´ Opinions on Mathematical Modelling in Upper Secondary Education 326
27.1 Introduction 326
27.2 Workplace Mathematics, Modelling and Modellers´ Opinions 328
27.3 Methodology 329
27.4 Results and Analysis 331
27.4.1 Source of Learning of Mathematical Modelling 331
27.4.2 Goals of Mathematics Education in Upper Secondary School 332
27.4.3 Goals of Modelling in Upper Secondary School 332
27.4.4 Examples from Practice Suitable for Use in Secondary School 333
27.4.5 Mathematical Modelling as a Part of a General Education 334
27.5 Discussion and Implications 334
References 335
Chapter 28: Modelling, Education, and the Epistemic Fallacy 337
28.1 Introduction 337
28.2 Epistemic Fallacy 339
28.3 Models of Modelling in Education 340
28.4 Modelling Critique 341
28.5 Authenticity 342
28.6 A Modelling Example 343
Population Prediction 343
28.7 Valsiner´s Zone Theory 344
28.8 A Curricular Imperative 346
References 347
Chapter 29: Reconsidering the Roles and Characteristics of Models in Mathematics Education 348
29.1 Research Background 348
29.2 Models in Mathematics Education and Their Role 349
29.2.1 Role 1: Model as a Hypothetical Working Space 350
29.2.2 Role 2: Model as a Physical/Mental Entity for Comparing and Contrasting 351
29.3 Two Roles of Models in Teaching of Modelling 352
29.3.1 Modelling Is Interpreted as Interactive Translations Among Plural Worlds, Not Simply Between Two Fixed Worlds 352
29.3.2 Models Having the Potential to Incorporate Scenarios Even Beyond the Initial Problem Situation 354
29.3.3 Where a Mathematical World Provides Entities for Comparing and Contrasting Different Design Standards 355
29.4 Modelling Competency Interpreted as How to Balance Between Two Roles of Models 356
References 357
Chapter 30: Developing Statistical Numeracy: The Model Must Make Sense 359
30.1 Introduction 359
30.2 Research Design 361
30.2.1 The Task 361
Student Exercise Task 362
30.2.2 Data Collection and Analysis 362
30.3 Results and Discussion 363
30.3.1 Teacher Models Give Opportunities for Discussion 363
30.3.2 Effects of Teaching Experience on Modelling 366
30.4 Conclusions 367
References 368
Chapter 31: Mathematical Modelling and Cognitive Load Theory: Approved or Disapproved? 370
31.1 Introduction 370
31.2 Cognitive Load Theory 372
31.3 Mathematics Didactics and Mathematical Modelling Education in the Dutch Handbook 373
31.4 Cognitive Load Theory and Mathematics Didactics 374
31.5 Cognitive Load Theory and Mathematical Modelling Education 375
31.6 Cognitive Load Theory and Problem Based Learning: A Relevant Debate 376
31.7 Conclusions and Discussion 378
References 379
Chapter 32: Social-critical Dimension of Mathematical Modelling 380
32.1 Introduction 380
32.2 Conceptualizing Social-critical Efficacy 381
32.3 Teaching for Social-critical Efficacy 382
32.4 Theoretical Basis for the Social-critical Dimension of Mathematical Modelling 382
32.4.1 Sociocultural Theory 383
32.4.2 Critical Theory of Knowledge 383
32.5 Determining an Epistemology of the Social-critical Dimension of Mathematical Modelling 384
River Pollutants Task 386
32.6 The Process of the Social-critical Dimension of Mathematical Modelling 388
32.7 Final Considerations 389
References 390
Chapter 33: Pedagogical Actions of Reflective Mathematical Modelling 391
33.1 Introduction 392
33.2 Mathematical Modelling and Didactical Transposition: Thinking Process 392
33.3 Context of the Study 394
33.4 Methodological Procedures 395
33.5 Presentation of Pedagogical Actions and Findings 395
33.5.1 First Pedagogical Action 395
33.5.2 Second Pedagogical Action 396
33.6 Discussion 397
33.7 Conclusion 399
References 399
Chapter 34: Context Categories in Mathematical Modelling in Fundamentals of Calculus Teaching 401
34.1 Introduction 401
34.2 Theoretical Framework 402
34.3 Methods for Evaluation 404
34.4 Results 405
34.5 Discussion and Considerations for Future Teaching 407
References 409
Chapter 35: Applied Mathematical Problem Solving: Principles for Designing Small Realistic Problems 411
35.1 Introduction 411
35.2 Real-World Problems Have Solutions That Are of Interest 412
35.3 Real-World Problems Require Exploration 413
35.4 Problems as Cases and Three Dimensions of Learning 414
35.5 Extending the Problem Solving Experience: Perspective and Variation 415
35.6 Examples of Mathematical Modelling Problems 417
35.7 Discussion of Related Work 418
35.8 Conclusions 419
References 420
Part IV: Influences of Technologies 422
Chapter 36: Visualisation Tactics for Solving Real World Tasks 423
36.1 Background 423
36.2 Affordances 425
36.3 The Study 425
Platypus Task 426
36.3.1 The Participants and TRTLE´s 426
36.3.2 Data Collection 427
36.4 Analysis 427
36.4.1 Data Display-Ability 427
36.4.2 Function View-Aability 428
36.4.3 Data Display-Ability and Function View-Ability 428
36.4.3.1 Multiple Data Display-Ability and Multiple Function View-Ability 429
36.4.4 Consideration of Multiple Models 430
36.5 Discussion and Conclusion 432
References 433
Chapter 37: Developing Modelling Competencies Through the Use of Technology 435
37.1 Introduction 435
37.2 Mathematical Modelling and the Development of Competencies 436
37.3 The Use of Technology and the Differential Equations 438
37.4 Experimental Situation: RC and RL Electric Circuits 438
37.5 Results 440
37.5.1 Assembly of the Electric Circuit 440
37.5.2 Generation of a Graph Using the Sensor or Simulator 441
37.5.3 Modelling Competencies Promoted 441
37.6 Conclusions and Discussion 442
References 443
Chapter 38: Model Analysis with Digital Technology: A ``Hybrid Approach´´ 445
38.1 Introduction 445
38.2 The Teaching Approach 446
38.3 The Study 447
38.4 Model Analysis and Mathematical Modelling: Some Reflections 448
38.5 The Role of Software Modellus 451
38.6 Final Considerations 454
References 454
Chapter 39: Collective Production with Mathematical Modelling in Digital Culture 456
39.1 Introduction 457
39.2 Production of Learning Objects: An Approach to Bring the Real World into the Classroom 459
39.3 Learning Object: Population Dynamics 460
39.4 Final Considerations 464
References 465
Part V: Assessment in Schools and Universities 466
Chapter 40: Learners´ Dealing with a Financial Applications-Like Problem in a High-Stakes School-Leaving Mathematics Examinati... 467
40.1 Introduction 467
40.2 Theoretical Machinery 468
40.3 The Context 470
QUESTION 7 471
40.4 Analysis of the Production of Answer Responses by Examinees 471
40.4.1 Translation to Familiar Symbolism 471
40.4.2 Anticipation of Adjustments to Be Made 472
40.4.3 U-Turning 472
40.5 Discussion and Conclusion 474
References 476
Chapter 41: Evidence of Reformulation of Situation Models: Modelling Tests Before and After a Modelling Class for Lower Second... 477
41.1 Situation Models Based on Individual Modelling 477
41.2 Case Study 478
41.2.1 The Modelling Pre-test 479
The Brightness Situation 479
41.2.2 Experimental Class 479
The Problem of Brightness Set Up by H.M. 480
41.2.3 The Modelling Post-test 481
41.3 Reformulation of Situation Models 481
41.3.1 Responses from A.S. Through Modelling Tests and Experimental Class 482
41.3.1.1 Responses from A.S. at the Pre-test 482
A Problem Set Up by A.S. in Task (1c) 482
41.3.1.2 Responses from A.S. at the Experimental Class 483
41.3.1.3 Responses from A.S. at the Post-test 483
A Problem Set Up by A.S. in Task (2c) 484
41.3.2 Responses from Y.R. Through Modelling Tests and Experimental Class 484
41.3.2.1 Responses from Y.R. at the Pre-test 484
The Problem Set Up by Y.R. in Task (1c) 485
41.3.2.2 Responses from Y.R. at Experimental Class 485
41.3.2.3 Responses from Y.R. at the Post-test 486
The Problem Set Up by Y.R. in Task (2c) 486
41.4 Discussion and Conclusion 487
References 488
Part VI: Applicability at Different Levels of Schooling, Vocational Education, and in Tertiary Education 489
Chapter 42: Mathematical Modelling in the Teaching of Statistics in Undergraduate Courses 490
42.1 Introduction 490
42.2 Critical Education, Statistics Education and Critical Statistics Education 491
42.3 Environment Description 494
42.4 Presentations and Results of Student Investigations 495
42.5 Analysis 497
42.5.1 About the Statistical Content 497
42.5.2 About the Statistical Competences 497
42.5.3 About Critical Education and Critical Statistics Education 498
42.5.4 About the Mathematical Modelling 499
42.6 Final Considerations 499
References 500
Chapter 43: Models and Modelling in an Integrated Physics and Mathematics Course 502
43.1 Introduction 502
43.2 Teaching Strategies 503
43.3 Methodology 504
43.3.1 Course Description 505
43.3.2 Final Project of the Course 505
Spring-Mass System Project 505
43.4 Results 506
43.5 Discussion and Conclusions 509
References 510
Chapter 44: Research-Based Modelling Teaching Activities: A Case of Mathematical Positioning with GNSS 512
44.1 Introduction 512
44.2 The Mechanism of Research-Based Mathematical Modelling Activities 514
44.2.1 The Practical Mathematical Modelling Program in NUDT 514
44.2.2 The Scheme for Research-based Mathematical Modelling 515
44.2.3 Cultivation of Modelling Competency 516
44.3 A Case: Multipath in Mathematical Positioning 517
44.3.1 Guiding Exploring Possible Solution to Problems 518
44.3.2 Directing Solution Selection 519
44.3.3 Developing Active and Critical Thinking 519
44.3.4 Improving the Ability of Data Analysis and Summarising 520
44.4 Conclusion 521
References 522
Chapter 45: Mathematical Texts in a Mathematical Modelling Learning Environment in Primary School 523
45.1 Introduction 523
45.2 Pedagogical Practice Based on the Theory of Basil Bernstein 525
45.3 Method, Participants and Research Context 526
45.4 Data Analysis 526
45.5 The Modelling Environment 526
45.6 Final Considerations 529
References 531
Chapter 46: A Differential Equations Course for Engineers Through Modelling and Technology 532
46.1 Introduction 532
46.2 Designing a New Proposal: The Case of a Differential Equations Course 533
46.2.1 Problems in the Teaching and Learning of DE 533
46.2.2 Re-designing a DE Course, a Collegiate Experience 534
46.2.3 A Mathematical Modelling Perspective in the DE Course 534
46.2.3.1 Development of Modelling Skills in a Differential Equations Course 535
46.3 Methodology in the Design and Implementation of Innovative Material in the DE Course 536
46.3.1 Using Technology in a DE Course 537
46.3.2 Designing Modelling Activities 537
46.3.2.1 Modelling the Temperature of Hot Water 537
46.3.2.2 Modelling the Mixing of Water and Salt in a Tank 538
46.3.2.3 Modelling the Change of Charge in an RC Circuit 539
46.4 Conclusion 541
References 542
Chapter 47: Contributions of Mathematical Modelling in Education of Youth and Adults 543
47.1 Introduction 543
47.2 Methodology and Participants 545
47.3 Activity: Study of Recycling of Aluminium Cans 546
47.4 Discussion and Final Considerations 550
References 551
Chapter 48: Pre-service Mathematics Teachers´ Experiences in Modelling Projects from a Socio-critical Modelling Perspective 553
48.1 Mathematical Modelling and Teacher Education 553
48.2 Methodological Approach 556
48.3 Results and Discussion 556
48.3.1 The Mathematical Modelling Projects 557
48.3.2 The Project About Trash and Recyclable Collection 558
48.4 Concluding Remarks 562
References 563
Chapter 49: A Mathematical Modelling Challenge Program for J.H.S. Students in Japan 565
49.1 Introduction 565
49.2 Aims and Design of Teaching Units 566
49.3 Framework of the Challenge Program in 2013 567
49.4 Description of the Modelling Examples 568
49.4.1 The Scientific Problem Course 568
PAScar Task A 568
PAScar Task B 568
49.4.2 The Social Problem Course 570
Home Electricity Generation Task 570
Electric Power Company Finances 570
49.5 Response of the Students 572
49.6 Conclusion 573
References 574
Part VII: Modelling and Applications in the Lived Environment 576
Chapter 50: Modelling the Wall: The Mathematics of the Curves on the Wall of Colégio Arquidiocesano in Ouro Preto 577
50.1 Introduction 577
50.2 Mathematical Modelling and Its Relation to the Community Context 579
50.2.1 Ethnomodelling: The Cultural Aspects of Mathematical Modelling 579
50.3 Modelling the Wall: Searching for Mathematical Models 580
50.3.1 Data Collection and Mathematical Models 581
50.4 Similarities and Differences Between Parabolas and Catenaries 582
50.4.1 The Specific Case of Suspension Bridges 584
50.5 Some Concluding Remarks About the Ethnomodelling Process 585
50.6 Final Considerations 586
References 586
Refereeing Process 588
Index 590