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International Handbook of Mathematical Learning Difficulties - From the Laboratory to the Classroom
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International Handbook of Mathematical Learning Difficulties - From the Laboratory to the Classroom
von: Annemarie Fritz, Vitor Geraldi Haase, Pekka Räsänen
Springer-Verlag, 2019
ISBN: 9783319971483
834 Seiten, Download: 15215 KB
Format:  PDF
geeignet für: Apple iPad, Android Tablet PC's Online-Lesen PC, MAC, Laptop

Typ: A (einfacher Zugriff)


eBook anfordern

  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  

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