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