International Handbook of Mathematical Learning Difficulties (eBook)

Annemarie Fritz-Stratmann is full professor of Psychology at the University of Duisburg-Essen, Germany, where she runs a research and counseling center for children with learning difficulties. She received a PhD in Psychology from the University of Cologne and has habilitation in psychology of special education and rehabilitation from the University of Dortmund. Since 2015 she is Distinguished Visiting professor at the University of Johannesburg. In the past 25 years her research turned to children with mathematical learning difficulties. Here, the focus of her scientific work was the empirical validation of a development model of key numerical concepts and arithmetic skills from age 4 to 8. Based on this model some diagnostic assessments (MARKO-Series) and training programs for pre-school and elementary school-children were developed. Recently her Dr. Fritz  extended her interests to mathematical problems in secondary education and on math anxiety. She acts as a consultant to several scientific journals.  Vitor Geraldi Haase is full professor of Psychology at the Federal University of Minas Gerais (UFMG), Brazil. He graduated in medicine, did his medical residency in pediatric neurology, has an M.A. in applied linguistics and a Ph.D. in medical psychology (Ludwig-Maximilians-Universität zu München). Working at UFMG since 1995, he heads the Laboratory for Developmental Neuropsychology and Número, a clinic for math learning difficuties. He has been doing clinical work and research on numerical cognition applied to math learning difficulties for the last 10 years. The main focus of this research is the characterization of the molecular-genetic variability underlying the cognitive mechanisms associated with math learning difficulties and math anxiety. Dr. Haase is the author of more than 85 scientific articles and 42 book chapters; editor of three multi-authored books and consultant to several scientific journals. He has a research productivity grant from the Brazilian Council of Scientific and Technological Research (CNPq).Pekka Räsänen is a clinical neuropsychologist and a principal investigator in Niilo Mäki Institute, Jyväskylä, Finland. After his graduation he has worked as a junior researcher in the Finnish Academy of Science, and as an interim associate professor of psychology in the University of Lapland. Since 1997 he has worked as a researcher and as an executive vice director in the Niilo Mäki Institute, which is the largest and most influential research unit on learning disabilities in Finland offering also clinical services, further education and a publication unit. The main focus of his research has been assessment and interventions of mathematical learning disabilities. He has written and co-authored over 50 academic publications and developed most of the standardised tests of mathematical disabilities used in Finland. He has also conducted studies on early grade mathematics learning in sub-Saharan Africa and Latin America. Together with his colleagues, he has developed intervention programs on language and mathematics for early education, as well as adaptive computer-assisted tests and rehabilitation programs for mathematical learning disabilities. 

Dedication 5
Foreword 6
Acknowledgements 9
Contents 10
Contributors 14
About the Editors 20
Chapter 1: Introduction 22
References 27
Part I: Development of Number Understanding: Different Theoretical Perspectives 28
Chapter 2: Neurocognitive Perspective on Numerical Development 29
Introduction 29
The Triple-Code Model of Numerical Processing and the Mental Number Line 29
The Approximate Number System 30
Number Words and Verbal Counting 30
Visual-Arabic Code 31
Place Value and Number Syntax 32
Experimental Effects of Numerical Processing 34
Subitizing vs. Counting in Dot Enumeration 34
Ratio Effect in Non-symbolic Number Comparison 35
Distance Effect in Symbolic Number Comparison 35
Size-Congruity Effect in Symbolic Comparison 36
Compatibility Effect in Comparison of Two-Digit Numbers 36
SNARC Effect 37
Numbers in the Brain 38
Implications for Instruction and Intervention 39
References 40
Chapter 3: Everyday Context and Mathematical Learning: On the Role of Spontaneous Mathematical Focusing Tendencies in the Development of Numeracy 45
Introduction 45
Early Development of Numeracy 45
Early Approximate and Exact Number Recognition 46
Subitizing and Counting 46
Basic Arithmetic Skills 47
Children’s Mathematical Activities in School and Home 48
Role of Children’s Own Practice in Numeracy Development 49
How to Measure SFON? 50
Findings of SFON Studies 51
Beyond Mere Numerosity: The Development of Relational Reasoning as the Foundation for Rational Number Knowledge 52
Spontaneous Focusing on Quantitative Relations 54
Self-Initiated Practice and Number Sense 55
Discussion 56
References 58
Chapter 4: Competence Models as a Basis for Defining, Understanding, and Diagnosing Students’ Mathematical Competences 63
Competence Models as Normative Definitions of Educational Goals 63
Competence Models to Understand and Evaluate Students’ Learning 65
Level I (Lowest Level): Basic Technical Knowledge (Routine Procedures Based on Elementary Conceptual Knowledge) 66
Level II: Basic Use of Elementary Knowledge (Routine Procedures Within a Clearly Defined Context) 66
Level III: Recognition and Utilization of Relationships Within a Familiar Context (Both Mathematical and Factual) 66
Level IV: Secure and Flexible Utilization of Conceptual Knowledge and Procedures Within the Curricular Scope 67
Level V: Modeling Complex Problems and Independent Development of Adequate Strategies 67
Competence Models to Better Understand the Difficulty of Mathematical Problems: Examples 68
Competence Models as Tools to Support Teachers’ Diagnostic Processes 70
Advancing Mathematical Competence Models: The Role of Student Errors 72
Desiderata 73
References 74
Chapter 5: Mathematical Performance among the Poor: Comparative Performance across Developing Countries 77
Introduction 77
Background 78
Methodology and Data 80
Comparing Social Gradients Across Contexts 83
Conclusion 88
Appendix 89
References 89
Chapter 6: Didactics as a Source and Remedy of Mathematical Learning Difficulties 92
A Lack of Certain Arithmetical Abilities or a Certain Way of Doing Arithmetic? 92
Computing by Counting: What Else Could a Child Do to Solve a Basic Task? 93
Direct Fact Retrieval 94
Deriving Unknown Facts from Known Facts 94
Numbers as Compositions of Other Numbers 95
Evidence on the Impact of Instructional Efforts Focused on Noncounting Strategies 97
International Comparisons 97
Longitudinal and Cross-Sectional Data and Related Theories 98
Intervention and Field Studies 101
Overcoming Computing by Counting as a Didactic Challenge 102
Learning Difficulties, Teaching Difficulties, and the Role of Education Policies 104
References 105
Chapter 7: Development of Number Understanding: Different Theoretical Perspectives 109
Introduction 109
What Kind of Perspectives on Learning Mathematics Have Developed Most During the Last Decade? 109
Have Some Views About MLD Dominated the Discussion? 110
Have Some Perspectives Got Too Little Attention in General Discussion? 111
Can We Compare the Results from Studies on Dyscalculia from Different Countries to Each Other? 113
How Far Are We in Understanding the Mathematical Brain? 114
What Are the Key Questions to Focus on Next to Improve the Understanding of the Mathematical Brain? 114
Are There Some Breakthroughs in Science that You Think Would Change Our Picture in the Near Future? 115
What Is the Role of Spontaneous Focusing on Numerosity (SFON) in MLD? 116
Can a Child Be at Different Levels in Different Math Contents in the Way Described by Reiss or Is the Development More Based on Some General Factors? 117
What Are the Roles of Informal and Formal Learning in Mathematics? 117
What is the Role of Socioeconomic Status in the Development of Math Skills 118
What Is the Interplay Between Different Perspectives of Numerical Development? Do They Talk to Each Other? 119
How Could We Improve the Discussion Between Different Views? 119
Will Science Change Math Education in the Near Future? 120
References 120
Part II: Mathematical Learning and Its Difficulties Around the World 123
Chapter 8: Mathematical Learning and Its Difficulties: The Case of Nordic Countries 124
Sweden 129
Norway 130
Iceland 132
Finland 133
Denmark 135
Summing Up 137
References 139
Chapter 9: Mathematical Learning and Its Difficulties in the Middle European Countries 143
The Big Picture 143
Educational Policies on MLD 145
Theories and Educational Practice 147
What Is the Role of Research Guiding the Practice? 153
References 155
Chapter 10: Mathematical Learning and Its Difficulties in Eastern European Countries 160
Eastern European Mathematics Education as Defined by Geographical, Historical, and Political Factors 160
Constraints and Promises of Recent Decades in Eastern European Mathematics Education 161
Lessons from International System-Level Surveys 163
Strengths and Weaknesses as Measured by International Surveys 164
Socioeconomic Background and Mathematics Achievement 167
Some Current Features and Tendencies in Eastern European Mathematics Education 170
Looking into Classrooms: Methodological Challenges 170
Fostering Students’ Mathematics Learning Talent Development, Remedial Education, School Readiness, and Attitudes 172
Talent Development and Participation in the International Mathematics Olympiad 173
School Readiness in Mathematics 174
Conclusion 175
References 176
Chapter 11: Mathematical Learning and Its Difficulties in Southern European Countries 179
Introduction 179
Educational Policies in Southern Europe 180
Definition of Mathematics Learning Difficulties, and Assessment and Diagnostic Criteria 183
Assessment of Mathematics Learning Difficulties in Italy 187
Assessment of Mathematics Learning Difficulties in Greece 188
Assessment of Mathematics Learning Difficulties in Spain 188
Assessment of Mathematics Learning Difficulties in France 189
Intervention: Theories, Research, and Educational Practice 190
Conclusions 192
References 193
Chapter 12: Mathematical Learning and Its Difficulties in the United States: Current Issues in Screening and Intervention 197
Mathematical Learning and Its Difficulties in the United States: Best Practices for Screening and Intervention 197
Early Number Competencies 200
Early Number Competencies Predict Future Mathematics Success, and Deficiencies in Number Concepts Underlie Many Mathematical Learning Difficulties 200
Core Number Competencies for Early Screening Involve Knowledge of Number, Number Relations, and Number Operations 201
Deficits in Number Sense Can Be Reliably Identified Through Early Screening, and Interventions Based on Screening Lead to Improved Mathematics Achievement in School 203
Fractions 204
Fraction Knowledge in the Intermediate Grades Predicts Algebra Success in Secondary School, and Weaknesses with Fractions Characterize Middle School Students with Mathematical Learning Difficulties 204
Fractions Are Especially Hard for Children with MLD 205
Because they Lack Magnitude Understanding, Students with MLD Struggle to Place Fractions on a Number Line 206
Fraction Difficulties Can Be Reliably Identified by Fourth Grade 206
Fraction Difficulties Can Be Improved Through Meaningful Interventions that Center on the Number Line 206
Conclusion 208
References 209
Chapter 13: Mathematical Learning and Its Difficulties in Latin-American Countries 214
Introduction 214
About the Region 216
Theories and Educational Practice 218
Mathematical Learning Disabilities in Latin American Countries 218
Mathematical Learning Disabilities in Brazil 219
Research on Mathematical Learning Disabilities 220
Future of Mathematical Learning Disabilities in Latin American Countries 222
Conclusions 222
References 223
Chapter 14: Mathematics Learning and Its Difficulties: The Cases of Chile and Uruguay 226
Introduction 226
Mathematics Learning Achievement 227
International Assessment 227
National Assessment 230
Educational Policies Addressing MLD and Educational Practice 231
Chile 231
Uruguay 234
Research into MLD 236
Chile 236
Uruguay 237
Conclusions 238
References 239
Chapter 15: Mathematical Learning and Its Difficulties in Southern Africa 244
Introduction 244
Theoretical Framing 245
Identified Problem and Research Questions 246
Methods 248
Results and Discussion of Findings 248
Lesotho 248
Malawi 250
South Africa 251
Zimbabwe 253
Case Study of Mathematical Inclusion in a Full-Service School in South Africa 256
What Was Done to Support Teachers? 257
Staff Professional Development 258
Responding to Annual National Assessments (ANAs) 259
Sharing Lessons 260
Were There Any Changes in Mathematics Learner Outcomes? 260
Conclusion 261
References 262
Chapter 16: Mathematical Learning and Its Difficulties in Australia 265
Australia: The Big Picture 265
Australia: Educational Policies and MLD 266
Australia: Theories and Educational Practice 267
Definitions in MLD in Australian States and Territories 269
Neuroscience and MLD/Dyscalculia in Australia 273
References 275
Chapter 17: Mathematical Learning and Its Difficulties in Taiwan: Insights from Educational Practice 277
Introduction 277
The Cultural Background 278
National Differences in Mathematical Learning 279
Educational Policies for Learning Difficulties in Taiwan 283
Diagnosis and Assessment Tool for Mathematical Learning Difficulties 285
Summary and Conclusion 287
Reference 288
Chapter 18: Mathematical Learning and Its Difficulties in Israel 291
Introduction 291
General Description: Population and Diversity 292
General Education and Mathematics Education in Israel 294
International Educational Tests in Math in Israel 296
Diagnosis of Mathematical Learning Disabilities in the Israeli School System 296
Current Changes in the Diagnosis and Treatment of MLD in Israel 299
Teaching Accommodations for Children Suffering from MLD in Israel 300
Diagnosis of MLD in Universities in Israel 301
Conclusion 302
References 304
Chapter 19: Learning Difficulties and Disabilities in Mathematics: Indian Scenario 307
Introduction 307
Education in India—New Initiatives 308
Initiatives for the Education of Children with Special Needs 308
Definition of Specific Learning Disability 309
Prevalence of Children with Special Needs in India 309
Teacher Preparation Courses in the Area of Learning Disabilities 310
Management of Specific Learning Disability in Schools in India 311
National Institute of Open Schooling 311
Learning Indicators/Outcomes and National Achievement Survey 313
Research on Learning Disabilities in India 316
Identification of the Prevalence of Learning Disabilities in Mathematics in India 316
Research on Learning Difficulties and Disabilities in Mathematics in India 317
Conclusion 320
References 320
Chapter 20: Adding all up: Mathematical Learning Difficulties Around the World 323
Math Achievement Around the World 324
Gender Issues 326
Heritage of the Soviet Regime 328
Intranational Diversity 328
Achievement-Motivation Gap 329
Definition of Special Needs in Math 329
Support at School for Children with Severe Math Difficulties 330
Teacher Training 331
Toward Evidence-Based Education 332
Key Issues and Trends 333
References 334
Part III: Mathematical Learning Difficulties and Its Cognitive, Motivational and Emotional Underpinnings 338
Chapter 21: Genetics of Dyscalculia 1: In Search of Genes 339
Introduction 339
Clinical Epidemiology of Developmental Dyscalculia 341
Genetic Susceptibility to Dyscalculia 343
Familial Aggregation in Dyscalculia 344
Heritability of Dyscalculia 344
Gene-Finding Strategies 345
Genome-Wide Association Studies 345
Candidate Genes from Comorbidities 348
Perspectives 349
References 350
Chapter 22: Genetics of Dyscalculia 2: In Search of Endophenotypes 354
Introduction 354
Cognitive Endophenotypes of Dyscalculia 354
Basic Number Processing 355
Phonological Processing 357
Visuospatial and Visuoconstructional Abilities 357
Working Memory 357
Chromosomal Abnormalities 358
Dyscalculia in Turner Syndrome 358
Dyscalculia in Klinefelter Syndrome 360
Genomic Disorders 360
Dyscalculia in 22q11.2 Deletion Syndromes 361
Dyscalculia in Williams Syndrome 362
Monogenic Conditions 364
Dyscalculia in Fragile X Syndrome and FMR1 Premutations 364
From the Lab to the Classroom 365
References 366
Chapter 23: Neurobiological Origins of Mathematical Learning Disabilities or Dyscalculia: A Review of Brain Imaging Data 375
Introduction 375
Brain Activity During Numerical Magnitude Processing and Arithmetic 377
Numerical Magnitude Processing 377
Arithmetic 379
Structural Brain Imaging 383
Connectivity 383
Effects of Remedial Interventions on Brain Activity 385
Discussion 385
Conclusion 387
References 387
Chapter 24: Comorbidity and Differential Diagnosis of Dyscalculia and ADHD 393
Introduction 393
What Is Comorbidity? 393
Why Are Comorbidity Rates for Neurodevelopmental Disorders So High? 394
What Can Be Causes for Difficulties in Mathematics? 395
Why Is It Important to Distinguish Between Primary and Secondary MLD? 396
What Are Difficulties for a Respective Differential Diagnosis? 397
Which Error Types Are Not Specific to Primary MLD? 398
Objectives of the Current Study 400
Materials and Methods 400
Participants 400
Assessment 401
Error Categories 402
Analyses 402
Results 403
Descriptive Statistics 403
Convergent and Discriminant Validity of the Postulated More Specific Clinical Cut-Off 403
Differences in Calculation Error Types Between Secondary and Possible Primary MLD 405
Differences in Counting Error Types Between Secondary and Possible Primary MLD 406
Discussion 407
Validation of the Postulated Clinical Cut-Off for the Basis-Math Overall Score 407
Specific and Unspecific Error Types 408
Limitations of This Study 409
Conclusions 409
References 410
Chapter 25: Working Memory and Mathematical Learning 414
Introduction 414
Working Memory (WM): A Domain-General Precursor of Mathematical Learning 415
Contribution of WM Components to Mathematical Learning 417
Working Memory, Word Problems, and Calculation 418
Executive Functions of Central Executive Component of WM and Their Role in Mathematics 420
Working Memory Training 422
Conclusion 424
References 425
Chapter 26: The Relation Between Spatial Reasoning and Mathematical Achievement in Children with Mathematical Learning Difficulties 429
Introduction 429
Numerical Magnitude and Spatial Reasoning in Typically Developing Children 432
Spatial Reasoning in Children with MD 433
Spatial Training to Support Children with MD 434
Conclusions 436
References 437
Chapter 27: The Language Dimension of Mathematical Difficulties 442
Language Factors on Different Levels and Their Connection to Mathematics Achievement 442
Differences Between Everyday and Academic Language on Word, Sentence, and Text/Discourse Level 443
Disentangling Language Obstacles on Word, Sentence, Text, and Discourse Levels and Their Connection to Mathematics Achievements 444
Obstacles on the Word Level 444
Obstacles on the Sentence and Text Level 445
Language Factors in the Achievement of Specific Groups 446
Second-Language Learners 446
Students with Learning Disabilities in Mathematics and Reading 446
Students with Specific Language Impairment and Mathematics Learning 447
Language Dimensions in Learning Processes 448
Language as a Learning Medium, Learning Prerequisite, and Learning Goal 448
Discourse Practices as a Construct to Capture Language Demands on the Discourse Level 449
Discourse Practices and Discourse Competence in Mathematics Classrooms 449
General and Topic-Specific Lexical Means for Different Mathematical Discourse Practices 451
Approaches for Fostering Students’ Language Proficiency in Mathematics 452
Enhancing Discourse Practices: Qualitative Output Hypotheses 452
Enhancing Conceptual Knowledge: Relating Registers and Representations 452
Specifying Mathematical and Language Goals: The SIOP Model 453
Combining Conceptual and Lexical Learning Trajectories: Macro-Scaffolding 454
Including Home Languages: Activating Students’ Multilingual Repertoires 454
Conclusion 455
References 456
Chapter 28: Motivational and Math Anxiety Perspective for Mathematical Learning and Learning Difficulties 461
Introduction 461
Opportunity–Propensity Model 462
Motivation 463
Definition of the Construct 463
Math Anxiety 466
Conclusions and Implications 468
References 468
Chapter 29: Mathematics and Emotions: The Case of Math Anxiety 472
Introduction 472
Math Anxiety as a Construct 473
Math Anxiety and Motivation 474
Antecedents of Math Anxiety 475
Genetics 475
Age 476
Gender 476
Culture 477
Teachers 478
Parents 478
Peers 479
Math Achievement 479
Cognitive Mechanisms 480
Working Memory 480
Numerical Abilities 482
Visuospatial Abilities 482
Neurobiological Underpinnings of Math Anxiety 482
Assessment of Math Anxiety 483
Interventions for Math Anxiety: From the Lab to the Classroom 493
Conclusion 495
References 496
Obs. References marked with # refer to self-report questionnaires presented in Tables 29.1, 29.2, and 29.3. 496
Chapter 30: Cognitive and Motivational Underpinnings of Mathematical Learning Difficulties: A Discussion 507
Chapter 21: Carvalho and Haase 508
Chapter 22: Haase and Carvalho 508
Chapter 23: DeSmedt, Peters, and Ghesquière 509
Chapter 24: Krinzinger 511
Chapter 25: Passolunghi and Costa 512
Chapter 26: Resnick, Newcombe, and Jordan 514
Chapter 27: Prediger, Erath, and Opitz 515
Chapter 28: Baten, Pixner, and Desoete 516
Chapter 29: Haase, Guimarães, and Wood 517
Common Themes 518
Concluding Remarks 519
References 520
Part IV: Understanding the Basics: Building Conceptual Knowledge and Characterizing Obstacles to the Development of Arithmetic Skills 521
Chapter 31: Counting and Basic Numerical Skills 522
Number Sense 523
Small Number Representations 523
Approximate Number Representations 524
Summary 525
Number Language 525
Knower Levels 526
Discrete Quantification 528
Numerosity 530
Summary 531
Counting Principles 532
Cardinality Principle 532
Successor Function 534
Summary 535
Facilitating the Acquisition of Exact Number Concepts 535
Facilitating the Acquisition of Individual Number Words 535
Facilitating the Acquisition of the Cardinality Principle 537
Broad-Scale Intervention 537
Numerically Based Toys 538
Number Language 539
Summary 540
References 540
Chapter 32: Multi-digit Addition, Subtraction, Multiplication, and Division Strategies 544
Multi-digit Arithmetic Solution Strategies 545
Multi-digit Addition and Subtraction Strategies 547
Strategies Framework 547
Children’s Strategy Use: Empirical Findings 548
Obstacles in Development 550
Multi-digit Multiplication and Division Strategies 552
Strategies Framework 552
Children’s Strategy Use: Empirical Findings 554
Obstacles in Development 555
Discussion 556
References 559
Chapter 33: Development of a Sustainable Place Value Understanding 562
Introduction 562
Properties of Place Value Systems 563
Place Value Understanding 564
Procedural Place Value Understanding 565
Conceptual Place Value Understanding 565
Difficulties in Place Value Understanding 566
Development of Place Value Understanding 567
Nonstructured Numbers 568
Identifying Decimal Units 569
Ordinal Aspect of Place Value Understanding 569
Cardinal Aspect of Place Value Understanding 570
Integration of Cardinal and Ordinal Aspects 570
Nonsustainable Concepts 570
Our Own Model 571
Predecadic Level 571
Level I: Place Values 572
Level II: Tens-Units Relation with Visual Support 572
Level III: Tens–Units Relation Without Visual Support 573
Level IV: General Decimal-Bundling-Unit Relations 574
Empirical Research 575
Conclusion 575
Barriers in the Development of a Sustainable Place Value Understanding 576
Educational Implications 577
Future Perspectives 578
References 578
Chapter 34: Understanding Rational Numbers – Obstacles for Learners With and Without Mathematical Learning Difficulties 581
Introduction 581
Learning of Rational Numbers: Learning a New Concept 582
Dual Processes in Rational Number Problems: The Natural Number Bias 584
Obstacles for Learners with Mathematical Learning Difficulties 586
How to Support Learners: Evidence from Intervention Studies 588
Conclusions and Perspectives 590
References 591
Chapter 35: Using Schema-Based Instruction to Improve Students’ Mathematical Word Problem Solving Performance 595
Mathematical Word Problem Solving 595
Theoretical Framework for Understanding How Schema-Based Instruction Is Beneficial to Word Problem Solving Performance 597
What Are the Unique Features of SBI and How Does It Contribute to Word Problem Solving Performance? 598
Teaching Word Problem Solving Using SBI: Empirical Evidence from Intervention Studies 603
Studies 1 and 2: Supporting Evidence for SBI Compared to Traditional Instruction 603
Studies 3 and 4: Supporting Evidence for SBI Compared to Standards-Based Instruction 604
Remaining Challenges 605
References 606
Chapter 36: Geometrical Conceptualization 610
Characterizing School Geometry 610
Three Approaches to School Geometry 611
G1. The Geometry of Concrete Objects 612
G2. The Geometry of Graphically Justified Ideal Plane Figures and Solids 612
G3. Quasi-axiomatic Geometry 612
The van Hiele Theory about the Stages of Development in Geometrical Thinking 613
Level 1 (Visualizing) 613
Level 2 (Analyzing Properties) 613
Level 3 (Ordering Properties) 614
Level 4 (Formal Deduction) 614
Level 5 (Understanding Axiomatic Systems) 614
About the Characteristics of Geometric Concept Formation 616
Basic Skills in Geometry 617
Classifying and Designating Figures 617
The Skills of Definition and the Clarification of Concepts 618
The Skills of Proving 621
Towards a Dialogue of the Traditional and the Dynamic Geometry 624
Geometry and Learning Difficulties 625
Summary 626
Bibliography 627
Part V: Mathematical Learning Difficulties: Approaches to Recognition and Intervention 630
Chapter 37: Assessing Mathematical Competence and Performance: Quality Characteristics, Approaches, and Research Trends 631
Introduction 631
Quality Characteristics 632
Categories of Classification 632
Norm-Referenced Versus Not-Norm-Referenced Tests 633
Individual Versus Group Testing 633
Paper-and-Pencil Tests Versus Interviews Versus Computer-Based Tests 633
Chronological Versus Educational Age–Oriented Tests 634
Speed Versus Power Tests 634
Principles of Task Selection 634
Outline of Different Approaches 635
Curriculum-Based Measures 635
Approaches Based on Neuropsychological Theories 636
Approaches Based on Developmental Psychology Theories 643
Research Trends 645
References 647
Chapter 38: Diagnostics of Dyscalculia 650
Differential Diagnosis of Dyscalculia 652
Criterion 1: To Determine the Presence and Severity of the Math Problem 652
Criterion 2: To Determine the Math Problem Related to the Personal Abilities 654
Criterion 3: To Determine Obstinacy of the Mathematical Problem 655
Process Research 657
Learnability 658
Math Problems in Early Education 658
From Problems at a Young Age to Dyscalculia 660
Conclusion 661
Appendix 662
The Five Steps of Math Help 662
References 664
Chapter 39: Three Frameworks for Assessing Responsiveness to Instruction as a Means of Identifying Mathematical Learning Disabilities 666
Systemic RTI Reform 668
Embedded RTI 670
Dynamic Assessment 673
Comparisons across the Three Frameworks 675
References 677
Chapter 40: Technology-Based Diagnostic Assessments for Identifying Early Mathematical Learning Difficulties 679
Introduction 679
Advantages and Possibilities of Technology-Based Assessment: The Move from Summative to Diagnostic Assessment to Realise Efficient Testing for Personalised Learning 681
Theoretical Foundations of Framework Development: A Three-Dimensional Model of Mathematical Knowledge 683
A Three-Dimensional Model of Students’ Knowledge for Diagnostic Assessment in Early Education 683
Creating an Assessment System: Online Platform Building and Innovative Item Writing 687
Mathematical Reasoning Items 688
Mathematical Literacy Items 690
Items that Assess Disciplinary Mathematics Knowledge 692
Field Trial and Empirical Validation of the Theoretical Model 693
Applicability of the Diagnostic System in Everyday School Practice 695
Scaling and Item Difficulty 695
Dimensionality and Structural Validity 697
Conclusions and Further Research and Development 699
References 700
Chapter 41: Small Group Interventions for Children Aged 5–9 Years Old with Mathematical Learning Difficulties 704
Introduction 704
Learning Difficulties in Mathematics 704
Intervention 705
The Features of Effective Instruction for Children with Mathematical Learning Difficulties 707
Responsiveness to Intervention Practice in Supporting Children with Learning Difficulties 716
Finnish Web Services for Educators 717
Studies with ThinkMath Intervention Programs 718
Conclusion 721
References 721
Chapter 42: Perspectives to Technology-Enhanced Learning and Teaching in Mathematical Learning Difficulties 727
Global Inequalities in Access to Learning Technologies 729
Online Learning, Virtual Worlds, and Social Learning Environments 730
Availability: The Surge of Learning Games 732
Usage: Does Using TEL Tools Help to Produce Better Learning? 733
Affective and Motivational Factors 735
Contents: What Is Inside the Intervention Games for MLD? 736
Training Number Sense 737
From the Classrooms to the Lab 742
Final Word 743
References 744
Chapter 43: Executive Function and Early Mathematical Learning Difficulties 749
Executive Function and Early Math Learning Difficulties 749
The Role of Cognitive Executive Function 749
The Role of Emotional Executive Function 750
The Executive Function of Children with Special Needs 751
The Role of Subject-Matter Knowledge 751
Teaching Executive Function 752
Relationships Between EF and Math 753
Relationships Between EF and Math Learning 753
Exploring Causality in the Relationship Between EF and Math Learning 755
Causation: Experimental Studies of EF and Math Interventions 756
Checking Whether Teaching EF Causes Math Achievement 756
Alternative Approaches, Especially for Children with Learning Difficulties 757
Teaching Math Can Cause Both Math Learning and EF Development 757
Math Activities that May Develop EF 758
Conclusions 759
References 759
Chapter 44: Children’s Mathematical Learning Difficulties: Some Contributory Factors and Interventions 766
National and Cultural Factors: What Do We Learn from International Comparisons? 766
Might International Differences in Teaching Methods Affect Performance? 767
Socio-economic Differences 768
The Role of Attitudes and Emotions 769
Interventions for Mathematical Difficulties 771
Whole-Class Approaches 771
Light-Touch Individualized and Small-Group Interventions 772
Highly Intensive Interventions 773
Numbers Count 774
What Makes Interventions Effective? 776
References 777
Chapter 45: Beyond the “Third Method” for the Assessment of Developmental Dyscalculia: Implications for Research and Practice 781
Challenges for Educational Policy and Practice 787
References 788
Chapter 46: Challenges and Future Perspectives 791
We Need Research from Genes to Behavior to Build Bridges Between Them 792
Educational Neuroscience: Where Are We? 793
What Is Learning Arithmetic from a Neuroscientific Perspective? 795
Focus on Early Development 798
Lack of Tools for Screening and Monitoring Learning 801
Monitoring-Based Framework for Interventions in Schools 803
The Challenges of the Response-to-Intervention Approach 805
Professional Development for Teachers 806
The Scaffold of Teaching Math Content at School 807
Construction of Curricula in a Tension Between the Two Poles of Individual Prerequisites and Normative Guidelines 810
Reforming Math Education in the Twenty-First Century 812
References 814
Index 820