Encyclopedia of mathematics education

Foreword......Page 5
Preface to Second Edition......Page 7
Preface to First Edition......Page 9
List of Topics......Page 11
About the Section Editors......Page 17
Contributors......Page 21
  Ability Grouping Practices in Different Countries......Page 33
  Ability Grouping, Achievement, and Equity......Page 34
  References......Page 36
  Historical and Epistemological Landmarks......Page 37
   Structures......Page 38
   Structuralist Thinking and Praxeologies......Page 39
  References......Page 40
  The Emergence of the Theoretical Framework......Page 41
  Model and Methodology......Page 42
  Research Directions......Page 43
  References......Page 44
  Elaboration......Page 45
  Perspectives......Page 46
  Cross-References......Page 47
  The Mental Structures and Mechanisms of APOS Theory......Page 48
  Studies That Use APOS Theory......Page 49
  Results......Page 50
  References......Page 51
  Minimum Unit......Page 52
  Structure of Activity......Page 53
  Subjectification and Personality......Page 54
  References......Page 55
  Definitions: Contested Meanings in an Emerging Field......Page 56
  Characteristics of an Emerging Field......Page 57
  Adults as Workers......Page 59
  Teaching, Learning, and Professional Development......Page 60
  References......Page 61
  From Anxiety and Problem-Solving to Affective Systems......Page 64
  The Role of Emotional States in Self-Regulation......Page 65
  Creating an Emotionally Supportive Learning Environment......Page 66
  References......Page 67
  Definition......Page 68
  Evolving Perspectives on School Algebra and Its Research Over the Years......Page 69
  What Does Research Tell Us About the Learning of School Algebra?......Page 70
  What Research Says About the Teaching of Algebra......Page 73
  For Further Study and Reflection......Page 74
  References......Page 75
  Definition......Page 76
  Students´ Understanding of Algorithmic Structures and Languages......Page 77
  Algorithmics and Programming Competencies......Page 78
  References......Page 79
  Definition......Page 80
  References......Page 81
  Foundations: Real Numbers......Page 82
  The Calculus-Analysis Transition at University......Page 83
  References......Page 84
  Introduction......Page 85
  Relations to Objects......Page 86
  The Theory of Praxeologies......Page 87
  Conditions and Constraints......Page 88
  The (Possibly) Didactic......Page 90
  Study Paradigms......Page 91
  References......Page 92
  Characteristics of Argumentation......Page 93
  Mathematics Classrooms and Argumentation......Page 95
  Approaches to Argumentation in Mathematics Education......Page 96
  References......Page 97
  Certification of New Teachers......Page 98
  Research on Teacher Knowledge......Page 99
  References......Page 100
  Characteristics......Page 101
  References......Page 103
  Characteristics......Page 104
  References......Page 105
  Bilingualism......Page 107
  Research Findings......Page 108
  History......Page 109
  Cross-References......Page 110
  Characteristics......Page 111
  Cross-References......Page 112
  Historical Development......Page 113
  Influence on Mathematics Education......Page 114
  Contemporary Classroom Applications......Page 116
  References......Page 117
  Calculus Curriculum......Page 119
  Early Research in Learning Calculus: The Cognitive Difficulties......Page 120
  The Role of Technology......Page 121
  The Role of Historical Perspective and Other Approaches......Page 122
  The Transition Between Secondary and Tertiary Education......Page 123
  New Directions of Research......Page 124
  References......Page 125
 Collaborative Learning in Mathematics Education......Page 126
  Origins......Page 127
  Foundations......Page 128
  Method......Page 129
  Commognitive Theory of the Development of Mathematical Discourses......Page 130
  Commognitive Theory of Factors that Shape the Learning of Mathematics......Page 131
  Contributions of Commognitive Research: Past and Future......Page 132
  References......Page 133
  Characteristics......Page 134
  References......Page 135
  How CoP Connects to Developments in Theories of Learning Mathematics......Page 136
  Perspectives on Issues in Different Cultures/Places......Page 137
  Cross-References......Page 138
  Background......Page 139
   Learning as Participation in Practices......Page 140
  Issues for Future Research......Page 141
  Characteristics......Page 142
  References......Page 144
  Complexity as an Epistemological Discourse......Page 145
  Complexity as a Historical Discourse......Page 146
  Complexity as a Disciplinary Discourse......Page 147
  Cross-References......Page 148
  Introduction......Page 149
  Defining CT and AT and Relating Them......Page 150
  State of Research on CT/AT......Page 151
  Curricular Aspects and Emerging Implications for Mathematics Education......Page 152
 Acknowledgment......Page 153
  References......Page 154
  Characteristics......Page 155
  Background......Page 159
  Radical and Social Constructivism in Mathematics Education......Page 161
  A Short List: Impact of Constructivism in Mathematics Education......Page 163
  References......Page 164
  Introduction......Page 166
   Meanings of ``Experiment´´......Page 167
  Meanings of Teaching in a Teaching Experiment......Page 168
   The Role of a Witness of the Teaching Episodes......Page 169
  Appendix: Example Studies Using Teaching Experiment Methodology......Page 170
  References......Page 171
  CE´s Background Assumptions......Page 173
  Principles......Page 174
  A Method of Documenting Practice and Research on Practice: The PTHAS......Page 175
  References......Page 176
  Creativity......Page 177
  Giftedness and Ability......Page 178
  Giftedness and Creativity......Page 179
  Empirical Research......Page 180
  Conceptual Relationships......Page 181
  Giftedness and Creativity in Psychology and Neuroscience......Page 182
  Teaching for Creativity......Page 183
  References......Page 184
   Critical Education......Page 186
   Some Issues in Critical Mathematics Education......Page 187
   Some Notions in Critical Mathematics Education......Page 188
   Critical Mathematics Education for the Future......Page 189
  References......Page 190
  Definition......Page 191
   Critical Thinking and Mathematical Reasoning......Page 192
   Critical Thinking and Applications of Mathematics......Page 193
   Further Unresolved Issues......Page 194
  References......Page 195
  Diversity and Uses of Cultural Mathematical Tools......Page 196
  Diversity and Cultural and Mathematical Identities......Page 197
  Diversity and Teachers´ Social Representations of Cultural Differences......Page 198
  References......Page 199
  Introduction......Page 200
  Perceptions of Mathematical Success......Page 201
  References......Page 203
  Definition of Curriculum Resources......Page 204
   Design and Quality of Curriculum Resources......Page 205
   The ``Use´´ of Curriculum Resources......Page 206
  Cross-References......Page 207
  References......Page 208
  Changing Views on Teaching Statistics Over the Years......Page 209
  What Does Research Tell Us About Teaching and Learning Statistics?......Page 210
  Teaching and Learning......Page 211
  References......Page 212
  Counting......Page 213
  Conclusion......Page 214
  Different Theoretical Perspectives on What a Deductive Inference Is......Page 215
  Purposes of Deductive Reasoning and Proof......Page 217
  Students´ Proving Abilities, Conceptions, and Perceptions......Page 218
   What Do Students and Teachers Think the Role of Proof Is?......Page 219
   How Can We Have Proof Serve the Functions of Providing Insight, Promoting Discovery, and Illustrating Methods?......Page 220
  References......Page 221
  Characteristics......Page 222
  References......Page 223
  The Origins and Need for Design Research......Page 224
   Interventionist and Iterative......Page 225
  Some Issues and Challenges......Page 226
   Sources of Inspiration......Page 227
  References......Page 228
  Definition......Page 229
   Paradoxes of the Didactical Contract......Page 230
   Observations of Reactions of Teachers to Difficulties......Page 231
   Predicting and Explaining Certain Long-Term Effects......Page 232
   Extensions......Page 233
  Definition......Page 234
  DE as a Research Methodology......Page 235
  Realizations......Page 236
  Challenges and Perspectives......Page 237
  Didactical Situation......Page 238
   Teaching Methods......Page 239
   The Project of a Mathematical Science: Didactique......Page 240
   Types of Mathematical Knowledge, Reference Knowledge (Savoirs)......Page 241
  Types of Mathematical Situations Characteristic of Activities, of Pieces of Knowledge, and of Pieces of Mathematical Learning......Page 242
  Some of the Results of Research on Didactical Situation......Page 243
  Research Perspectives......Page 244
  References......Page 245
  Scope......Page 246
  Enlargement of the Object of Study......Page 247
  The Need for Researchers´ Own Epistemological Models......Page 248
  References......Page 249
  Relation with Realistic Mathematics Education......Page 250
  Mathematics-Related Analyses Constituting the Didactics of Mathematics......Page 251
  Learning......Page 252
  Teaching......Page 253
  References......Page 254
  Introduction......Page 255
  Poststructural Approaches......Page 257
  Mathematical Discourse, Thinking, and Learning......Page 258
  Definition......Page 259
   To Integrate Discrete Mathematics into the School Curriculum: A Current Challenge......Page 260
    Accessible Problems and Concepts......Page 261
    Several Ways of Questioning, Proving, and Modeling......Page 262
  Interesting Perspectives for Research in Mathematics Education......Page 264
  References......Page 265
  Foundations......Page 266
  Strands......Page 267
  Methods......Page 268
  Introduction......Page 269
  Sources of the Approach......Page 270
  The Documentational Approach to Didactics: A Holistic Approach to Teachers´ Work......Page 271
  Deepening the Approach: Schemes and Systems......Page 273
  Reflexive Investigation: A Developing Methodological Construct......Page 275
  Perspectives for Further Evolutions......Page 277
  References......Page 278
  Characteristics......Page 279
  References......Page 280
   Operating on Unknowns......Page 281
   Reasoning About Physical Quantities and Measures......Page 282
  References......Page 283
  History......Page 284
  Ways of Teaching Mathematics to Young Children......Page 286
  Information and Communication Technology......Page 287
  Future Perspectives in Early Childhood Mathematics Education......Page 288
 Education of Facilitators (for Educators of Practicing Teachers)......Page 289
  Background......Page 290
  Mathematics Teacher Educators´ Knowledge, Skills, and Practice......Page 291
  Education and Development of Mathematics Teacher Educators......Page 292
  References......Page 293
 Education of Professional Development Providers (for Educators of Practicing Teachers)......Page 294
  Characteristics and Origin......Page 295
  Implementation and Adaptation......Page 297
  Some Definitional Differences in Mathematics Education......Page 298
  Dienes´ Contributions to Embodied Mathematics......Page 299
  Characteristics......Page 300
  References......Page 304
  Behavioral Engagement......Page 305
  Social Engagement and Opportunities to Engage......Page 306
  References......Page 307
 Epistemological Obstacles in Mathematics Education......Page 308
  References......Page 310
  Characteristics......Page 311
  Cross-References......Page 314
  Introduction......Page 315
  Conceptual Foundations of Ethnomathematics......Page 317
  Current Work in Ethnomathematics......Page 319
  Characteristics......Page 320
  Design-Based Assessment......Page 322
  References......Page 327
  Features......Page 329
  References......Page 330
  Lee Shulman......Page 331
  Mathematical Knowledge for Teaching......Page 332
  The Knowledge Quartet......Page 333
  References......Page 334
  Definition and Brief History......Page 335
  Overcoming Learning Difficulties......Page 336
  References......Page 337
  Historical Overview of Gender and Mathematics Education Research......Page 339
  Theoretical Considerations......Page 341
  Cross-References......Page 342
  Introduction to the Literature on Gesture......Page 343
  Definition of Gesture in Mathematics Education......Page 344
  References......Page 346
  Introduction......Page 347
  Definition of Mathematical Giftedness and Its Relativity......Page 349
   Mathematical Expertise and Mathematical Giftedness......Page 350
   Mathematical Creativity, Insight, and Mathematical Giftedness......Page 351
  Cognitive and Neurocognitive Characteristics Associated with Mathematical Giftedness......Page 352
  Affect and Personality......Page 353
  Development of Mathematical Ability......Page 354
  References......Page 355
  Definition......Page 358
  References......Page 360
  Characteristics......Page 362
  History and Epistemology of Mathematics: An Ever-Increasing Interest for New Conceptions and Practices in Mathematics Education......Page 364
  Epistemological Contributions......Page 366
  Cultural Contributions......Page 367
  History of Mathematics in Teachers´ Training......Page 368
  History as an Instrument for an Interdisciplinary Approach in Teaching......Page 369
  History of Mathematics in the Classroom......Page 370
  Current Concerns and Emergent Questions in the Field......Page 371
  References......Page 372
  General Education and Professional Training in Mathematics......Page 373
  Mathematics in Secondary Schooling......Page 374
  Curriculum......Page 375
  Textbooks for Mathematics......Page 376
  Mathematics in the Global World......Page 377
  Research into History of Mathematics Education as a Field......Page 378
  References......Page 379
  Main Text......Page 380
  Comparative Studies of School Mathematics......Page 381
  Becoming Scientific......Page 382
  Studying the Teaching of Mathematics......Page 383
  References......Page 384
  Definition......Page 385
  Explanation of the Construct......Page 386
  References......Page 387
  Introduction......Page 389
  Different Forms of Mathematics......Page 390
  Teacher Education in an Immigration Context......Page 391
  Immigrant Parents´ Perceptions of Mathematics Education......Page 392
  References......Page 393
 Immigrant Teachers in Mathematics Education......Page 395
  Classroom Practice......Page 397
  Mathematics......Page 398
  Introduction......Page 399
  Gaining Western Mathematics......Page 400
  Ensuring Access and Control of the Decision Making......Page 401
  References......Page 402
  What is Informal Learning?......Page 403
   Consciousness of Concepts Versus Non-consciousness Concepts......Page 405
   Status of Knowledge......Page 406
   Models of Formal Knowledge and Informal Learning in Mathematics......Page 407
  An Alternative Approach......Page 408
   Further Reading......Page 409
  Characteristics......Page 410
  References......Page 413
  Definition......Page 414
  Historical Background......Page 415
  IBME and Mathematics Education Research......Page 416
  Research on Teachers´ Practices Regarding IBME......Page 417
  References......Page 418
  Characteristics......Page 419
  References......Page 421
  Mediated Action as a Source of Tensions......Page 422
  Instrumentation/Instrumentalization Dialectic......Page 423
  Instrumentalization as a Key Feature of the Instrumental Approach to Didactics......Page 424
   Taking into Account the Evolution of Artifacts......Page 427
   Taking into Account the Teacher´ Role for Designing the Artifacts Environment......Page 428
   Rethinking Instrumentalization from the Teacher´s Perspective......Page 431
 Acknowledgment......Page 432
  References......Page 433
  Instrumentation and Instruction......Page 434
  The Critical Notion of Affordance......Page 435
  The Dialectical Relationships Between Artifact and Instrument......Page 436
  An Essential Dialectic Relationship between Instrumentation and Instrumentalization......Page 438
  From a Set of Artifacts to a System of Instruments: The Crucial Notion of ``Orchestration´´......Page 439
  Conclusion: From Student Instrumentation to Teacher Instrumentation......Page 440
  References......Page 441
  Definition......Page 442
  Perspectives for Future Research......Page 443
  References......Page 444
  Interdisciplinarity and Mathematics (Education)......Page 445
  References......Page 448
  What Are ICS?......Page 449
  Why Are ICS Important?......Page 450
  Some Issues with ICS......Page 451
  The Way Forward......Page 452
  References......Page 453
  Introduction......Page 454
  Interpretative Knowledge......Page 455
  Main Implications for Teacher Education......Page 457
  Definition......Page 458
  Characteristics, Approaches, and Role in Mathematics Education......Page 459
  Cross-References......Page 461
  References......Page 462
  Introduction......Page 464
  Learning Games, the Double Dialectics Reticence-Expression/Contract-Milieu, the Equilibration Process......Page 465
  Cooperative Engineering......Page 466
  References......Page 467
  Introduction......Page 469
  Theoretical Perspectives on Language Background in Mathematics Education......Page 470
  Language Background and Learning and Teaching Mathematics......Page 471
   Plurilingual Societies......Page 472
  Language Background and Researchers in Mathematics Education......Page 473
  References......Page 474
  Characteristics......Page 475
  Definition......Page 476
   Intellectual Autonomy......Page 477
   Classroom Environment......Page 478
   Mathematical Discourse......Page 479
  References......Page 481
  Introduction......Page 482
  Two Sorts of Mathematical Skills to Be Learned......Page 483
  The Specificity of Mathematics Learning Difficulties......Page 485
  References......Page 486
  Computer Scaffolded Learning......Page 487
  Computer-Supported Collaborative Learning......Page 490
  References......Page 491
  Origin......Page 492
  References......Page 494
  Characteristics......Page 495
  Characteristics......Page 497
  Exploring the Object of Learning......Page 498
  References......Page 499
  Historical Developments and Contexts......Page 500
  Lesson Study Adopted as a Model of Professional Development in Other Countries......Page 501
  Preliminary Work......Page 502
  Different Approaches to the Learning and Teaching of Linear Algebra Concepts......Page 503
  The Use of Realistic Problems and Models in the Teaching and Learning of Linear Algebra......Page 504
  References......Page 505
  Definition......Page 506
  Logic and Mathematical Language......Page 507
  The Role of Logic in Mathematical Proof, Argumentation, and Reasoning......Page 508
  Definition......Page 509
  Characteristics......Page 510
  Instruction in Mathematical Logic......Page 511
  Cross-References......Page 512
  References......Page 513
   Manipulatives and Mathematics Education......Page 514
   Critical Issues......Page 516
   Some Examples of Manipulatives and Tasks......Page 517
   Pascaline......Page 518
   Pantographs......Page 519
  References......Page 520
   The Evolvement of Mathematical Abilities......Page 521
   Characterizing Different Students on the Spectrum of Mathematical Abilities......Page 522
   Measuring and Evaluating Students´ Mathematical Ability......Page 523
  References......Page 524
  Developments......Page 525
  Characteristics......Page 526
  Stoffdidaktik......Page 527
  An Epistemological Program......Page 528
  Mathematical Knowledge for Teaching......Page 529
  Unresolved Issues......Page 530
  References......Page 531
  Definition/Introduction......Page 532
  Slope and Proportional Reasoning......Page 533
  Rate of Change......Page 534
  Covariance......Page 535
  Functions......Page 537
  Generalizing......Page 538
  Types of Mathematical Generalization......Page 539
  Strategies in Pattern Generalizing......Page 540
  Justification of Mathematical Generalization......Page 543
  References......Page 544
  Definition/Introduction......Page 547
  Theoretical Models Guiding Research on Geometry Learning......Page 548
  Proofs......Page 550
  Genres of Investigation......Page 551
  References......Page 554
   Piagetian Views......Page 557
   Non-Piagetian Views: Van Hiele Levels of Geometric Thinking......Page 559
   Piagetian Views......Page 560
   Non-Piagetian Views: Cognitively Guided Instruction......Page 562
  References......Page 564
  Characteristics......Page 565
  Introduction: What Is Mathematical Language?......Page 567
  Characteristics of Mathematical Language......Page 568
  Variations in Language and Thinking Mathematically......Page 569
  Introduction......Page 570
  The Mathematics Education Perspectives on MLD: The Issues of Prevention and Remediation......Page 571
  Communicating Across Fields......Page 573
  References......Page 574
  Definition......Page 575
  Characteristics and Delimitation......Page 576
  Motivations for Introducing Mathematical Literacy......Page 577
  Critique and Further Research......Page 578
  References......Page 579
  Characteristics......Page 580
  Theoretical Debate on Mathematical Modelling: Historical Development and Current State......Page 581
  The Modelling Process as Key Feature of Modelling Activities......Page 583
  Westerhever Lighthouse......Page 584
  Further Explorations and Extensions......Page 585
  Modelling Competencies and Their Promotion......Page 586
  Cross-References......Page 587
  Definition......Page 588
  Mathematical Proof......Page 589
  Argumentation and Proof......Page 590
  Practical Classroom Approaches......Page 591
  References......Page 592
  Definitions......Page 593
  Characteristics......Page 594
  Research......Page 596
  References......Page 598
  Background......Page 599
  Eliciting Evidence of Student Learning......Page 600
  Acting on Evidence of Student Learning......Page 601
  References......Page 602
  Curriculum......Page 603
  Curriculum Evaluation......Page 604
  Models of Mathematics Curriculum Evaluation......Page 605
  Mathematics Curriculum Evaluation and Large-Scale Reforms......Page 606
  References......Page 608
  Definition......Page 609
  References......Page 611
  Rationale......Page 612
  Teacher Learning Theory......Page 613
  Structure and Characteristics......Page 614
  Institutions......Page 615
  Who Teaches Future Teachers?......Page 616
  Professional Teacher´s Competences......Page 617
  References......Page 619
  Mathematics Teacher Educators: Definition......Page 620
  Mathematics Teacher Educators´ Learning Through Research......Page 621
  Mathematics Teacher Educators´ Learning Through Action Research and Intervention Research......Page 622
  References......Page 623
  Identity Research in Mathematics Education and MTI......Page 624
  Clusters of Research in MTI......Page 625
  References......Page 626
  Definition and Historical Background......Page 627
  Different Cultures Shaping Different Forms of Interaction Between Teachers and Curricula......Page 628
  Teachers and Curricula Within a Collaborative Perspective......Page 629
  Open Questions......Page 630
  References......Page 631
  There Is Mathematics Everywhere......Page 632
  Mathematics in Action......Page 633
  New Challenges......Page 634
  Definition......Page 635
  Characteristics......Page 636
  References......Page 637
  Why Do We Need a Dynamic Framework Relating to Teachers´ Practices?......Page 638
  Meta-Didactical Transposition......Page 640
   The Meta-Didactical Praxeologies......Page 641
   The Role of the Broker......Page 643
   The Double Dialectic......Page 644
   Applications, Integration, and Evolution of the Meta-Didactical Transposition Framework......Page 645
  Definition......Page 646
   Metaphors for Metaphor......Page 647
  Metaphors for Teaching and Learning......Page 648
  Examples of Metaphors for Multiplication......Page 649
  References......Page 650
  Definition......Page 652
  Characteristics......Page 653
 Models of In-Service Mathematics Teacher Education Professional Development......Page 654
  References......Page 657
  Characteristics......Page 659
  References......Page 660
  Attribution Theory......Page 662
  Affect......Page 663
  References......Page 664
  Definition of Mathematics Teacher Noticing......Page 665
  Contributions of Mathematics Teacher Noticing......Page 666
  The Empty Number Line......Page 667
  An Alternative Approach......Page 669
  Effect Studies......Page 670
  Introduction......Page 671
   Natural Numbers, Operations, and Estimation......Page 672
   Number Sense......Page 674
  Cross-References......Page 675
   Teaching of Rational Numbers......Page 676
  Irrational Numbers......Page 677
  References......Page 678
  Characteristics......Page 681
  Measuring PCK......Page 682
  References......Page 683
  History......Page 684
  Analytical Framework......Page 685
  Defusing the Debates......Page 687
  References......Page 688
  History......Page 689
  Strong Political Perspectives......Page 690
  Cross-References......Page 691
  Characteristics......Page 692
  Psychoanalytic Approaches......Page 694
  References......Page 695
  What Is Included in Preparation and Professional Development?......Page 696
  University Provision for Both New and Experienced Teachers of Mathematics......Page 697
  Teachers´ Learning through Engagement in Research and Development Projects......Page 698
  Mathematics Support......Page 699
  References......Page 700
  Probabilistic and Statistical Thinking......Page 701
  Probabilistic Thinking in Probability Education......Page 702
  Statistical Thinking in Statistics Education......Page 704
  The Data Deluge......Page 705
  References......Page 706
  Different Meanings of Probability......Page 708
  Probability in the School Curriculum......Page 709
  Intuitions and Misconceptions......Page 710
  Challenges in Teaching Probability......Page 711
  Introduction......Page 712
  Focusing of Problems in the Development of Mathematics......Page 713
  Mathematics Education and Problem-Solving Developments......Page 714
   Curriculum Proposals and Instruction......Page 715
   Digital Technologies and Mathematical Problem-Solving......Page 716
  Directions for Future Research......Page 717
  References......Page 718
  Characteristics......Page 719
  Data-Informed and Knowledge-Based Enquiry......Page 720
  Impact and Research......Page 721
  Characteristics......Page 722
  The Piagetian Approach: Research on Conceptions and Conceptual Change......Page 723
  Departing from Piaget: From Research on Concept Formation to Teaching Experiments......Page 724
  Open Issues......Page 725
  References......Page 726
  Characteristics......Page 728
  Definition......Page 730
  Mathematical Perspectives......Page 731
  Asking as Enquiring......Page 732
  Classroom Ethos......Page 733
  Internalizing Questions......Page 734
  References......Page 735
  The Onset of RME......Page 737
  The Core Teaching Principles of RME......Page 738
  Various Local Instruction Theories......Page 739
  Cross-References......Page 740
  Characteristics......Page 741
  Cross-References......Page 744
 Reflective Practitioner in Mathematics Education......Page 745
  References......Page 747
  Cognitive Requirements for Understanding Conversions......Page 748
  Cognitive Requirements for Understanding Treatments......Page 749
  Knowledge and Recognition in Mathematics Understanding and Learning......Page 750
  References......Page 751
   Probability and a First Approach to Risk......Page 752
   Definitions of Risk......Page 753
   Risk and Hazard......Page 754
   Knight´s Conception of Risk and Uncertainty......Page 755
   Types of Situations and Analytical Investigation of a Situation under Risk......Page 756
   Rational Approach......Page 757
   Mathematical Approach......Page 758
   Paradigmatic Contexts for Decisions under Uncertainty......Page 759
   Risk with Small Probabilities......Page 760
  Cross-References......Page 762
  References......Page 763
  Psychological Aspects of Risk......Page 764
   Probability and Utility......Page 765
   Inconsistencies: Psychologically and Otherwise......Page 766
   Heuristics and Biases: The Approach by Kahneman and Tversky......Page 767
   Twin Concepts of Probability and Risk......Page 768
   Probabilistic Literacy Is Risk Literacy......Page 769
   Tools to Foster Risk Literacy......Page 772
  Conclusions......Page 775
  References......Page 777
  Definition......Page 778
  Pedagogy of Risk......Page 779
 Rural and Remote Mathematics Education......Page 780
  Opportunities in Rural and Remote Settings......Page 781
  References......Page 782
   Tutoring Learning......Page 783
  Students´ Difficulties and Their Origins......Page 786
   The Proof Practices of Professional Mathematicians......Page 787
   Providing Help to the Students......Page 788
  References......Page 789
  Definitions and Background......Page 790
  Semiotic Lenses and Their Uses......Page 791
  Questions for Research on Semiosis in Learning and Teaching Mathematics......Page 792
  References......Page 793
  Main Problematic Issue of Service-Courses......Page 794
  Students´ Practices......Page 795
  Final Remark......Page 796
  References......Page 797
  Definition and Teaching Situation......Page 798
   Piaget......Page 799
   Freedom in Selecting Geometrical Context, Content, and Teaching/Learning Paradigms......Page 800
  High-School Geometry or Shapes in Space as Ingredients for Constructing a Theory......Page 801
  Cross-References......Page 802
  Introduction......Page 803
   Mathematics Classes in Single-Sex Schools......Page 804
   Mathematics Classes in Sex-Segregated Classes in Coeducational Schools......Page 805
   History......Page 806
   Knowing......Page 807
   Implication for Teaching......Page 808
  Definition......Page 809
  Sociopolitical Analyses......Page 810
  Explanatory and Analytical Frameworks......Page 811
  Research on Action or Intervention......Page 812
  References......Page 813
  Introduction......Page 814
  Socioepistemology: Theoretical Foundations, Main Concepts, and Specific Methodologies......Page 815
   Example of the Problematization of Knowledge in Use: Successive Variation......Page 817
  Main Research Questions and Results......Page 818
  References......Page 820
  Topic and History......Page 821
  Issues of Research......Page 822
  References......Page 825
   Sociomathematical Norms......Page 826
   Growth of the Concept......Page 827
   Common Issues......Page 828
  References......Page 829
  Characteristics......Page 830
  S