International Handbook of Mathematical Learning Difficulties - From the Laboratory to the Classroom

Dedication

Foreword

Acknowledgements

Contents

Contributors

About the Editors

Chapter 1: Introduction

References

Part I: Development of Number Understanding: Different Theoretical Perspectives

Chapter 2: Neurocognitive Perspective on Numerical Development

Introduction

The Triple-Code Model of Numerical Processing and the Mental Number Line

The Approximate Number System

Number Words and Verbal Counting

Visual-Arabic Code

Place Value and Number Syntax

Experimental Effects of Numerical Processing

Subitizing vs. Counting in Dot Enumeration

Ratio Effect in Non-symbolic Number Comparison

Distance Effect in Symbolic Number Comparison

Size-Congruity Effect in Symbolic Comparison

Compatibility Effect in Comparison of Two-Digit Numbers

SNARC Effect

Numbers in the Brain

Implications for Instruction and Intervention

References

Chapter 3: Everyday Context and Mathematical Learning: On the Role of Spontaneous Mathematical Focusing Tendencies in the Development of Numeracy

Introduction

Early Development of Numeracy

Early Approximate and Exact Number Recognition

Subitizing and Counting

Basic Arithmetic Skills

Children’s Mathematical Activities in School and Home

Role of Children’s Own Practice in Numeracy Development

How to Measure SFON?

Findings of SFON Studies

Beyond Mere Numerosity: The Development of Relational Reasoning as the Foundation for Rational Number Knowledge

Spontaneous Focusing on Quantitative Relations

Self-Initiated Practice and Number Sense

Discussion

References

Chapter 4: Competence Models as a Basis for Defining, Understanding, and Diagnosing Students’ Mathematical Competences

Competence Models as Normative Definitions of Educational Goals

Competence Models to Understand and Evaluate Students’ Learning

Level I (Lowest Level): Basic Technical Knowledge (Routine Procedures Based on Elementary Conceptual Knowledge)

Level II: Basic Use of Elementary Knowledge (Routine Procedures Within a Clearly Defined Context)

Level III: Recognition and Utilization of Relationships Within a Familiar Context (Both Mathematical and Factual)

Level IV: Secure and Flexible Utilization of Conceptual Knowledge and Procedures Within the Curricular Scope

Level V: Modeling Complex Problems and Independent Development of Adequate Strategies

Competence Models to Better Understand the Difficulty of Mathematical Problems: Examples

Competence Models as Tools to Support Teachers’ Diagnostic Processes

Advancing Mathematical Competence Models: The Role of Student Errors

Desiderata

References

Chapter 5: Mathematical Performance among the Poor: Comparative Performance across Developing Countries

Introduction

Background

Methodology and Data

Comparing Social Gradients Across Contexts

Conclusion

Appendix

References

Chapter 6: Didactics as a Source and Remedy of Mathematical Learning Difficulties

A Lack of Certain Arithmetical Abilities or a Certain Way of Doing Arithmetic?

Computing by Counting: What Else Could a Child Do to Solve a Basic Task?

Direct Fact Retrieval

Deriving Unknown Facts from Known Facts

Numbers as Compositions of Other Numbers

Evidence on the Impact of Instructional Efforts Focused on Noncounting Strategies

International Comparisons

Longitudinal and Cross-Sectional Data and Related Theories

Intervention and Field Studies

Overcoming Computing by Counting as a Didactic Challenge

Learning Difficulties, Teaching Difficulties, and the Role of Education Policies

References

Chapter 7: Development of Number Understanding: Different Theoretical Perspectives

Introduction

What Kind of Perspectives on Learning Mathematics Have Developed Most During the Last Decade?

Have Some Views About MLD Dominated the Discussion?

Have Some Perspectives Got Too Little Attention in General Discussion?

Can We Compare the Results from Studies on Dyscalculia from Different Countries to Each Other?

How Far Are We in Understanding the Mathematical Brain?

What Are the Key Questions to Focus on Next to Improve the Understanding of the Mathematical Brain?

Are There Some Breakthroughs in Science that You Think Would Change Our Picture in the Near Future?

What Is the Role of Spontaneous Focusing on Numerosity (SFON) in MLD?

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?

What Are the Roles of Informal and Formal Learning in Mathematics?

What is the Role of Socioeconomic Status in the Development of Math Skills

What Is the Interplay Between Different Perspectives of Numerical Development? Do They Talk to Each Other?

How Could We Improve the Discussion Between Different Views?

Will Science Change Math Education in the Near Future?

References

Part II: Mathematical Learning and Its Difficulties Around the World

Chapter 8: Mathematical Learning and Its Difficulties: The Case of Nordic Countries

Sweden

Norway

Iceland

Finland

Denmark

Summing Up

References

Chapter 9: Mathematical Learning and Its Difficulties in the Middle European Countries

The Big Picture

Educational Policies on MLD

Theories and Educational Practice

What Is the Role of Research Guiding the Practice?

References

Chapter 10: Mathematical Learning and Its Difficulties in Eastern European Countries

Eastern European Mathematics Education as Defined by Geographical, Historical, and Political Factors

Constraints and Promises of Recent Decades in Eastern European Mathematics Education

Lessons from International System-Level Surveys

Strengths and Weaknesses as Measured by International Surveys

Socioeconomic Background and Mathematics Achievement

Some Current Features and Tendencies in Eastern European Mathematics Education

Looking into Classrooms: Methodological Challenges

Fostering Students’ Mathematics Learning Talent Development, Remedial Education, School Readiness, and Attitudes

Talent Development and Participation in the International Mathematics Olympiad

School Readiness in Mathematics

Conclusion

References

Chapter 11: Mathematical Learning and Its Difficulties in Southern European Countries

Introduction

Educational Policies in Southern Europe

Definition of Mathematics Learning Difficulties, and Assessment and Diagnostic Criteria

Assessment of Mathematics Learning Difficulties in Italy

Assessment of Mathematics Learning Difficulties in Greece

Assessment of Mathematics Learning Difficulties in Spain

Assessment of Mathematics Learning Difficulties in France

Intervention: Theories, Research, and Educational Practice

Conclusions

References

Chapter 12: Mathematical Learning and Its Difficulties in the United States: Current Issues in Screening and Intervention

Mathematical Learning and Its Difficulties in the United States: Best Practices for Screening and Intervention

Early Number Competencies

Early Number Competencies Predict Future Mathematics Success, and Deficiencies in Number Concepts Underlie Many Mathematical Learning Difficulties

Core Number Competencies for Early Screening Involve Knowledge of Number, Number Relations, and Number Operations

Deficits in Number Sense Can Be Reliably Identified Through Early Screening, and Interventions Based on Screening Lead to Improved Mathematics Achievement in School

Fractions

Fraction Knowledge in the Intermediate Grades Predicts Algebra Success in Secondary School, and Weaknesses with Fractions Characterize Middle School Students with Mathematical Learning Difficulties

Fractions Are Especially Hard for Children with MLD

Because they Lack Magnitude Understanding, Students with MLD Struggle to Place Fractions on a Number Line

Fraction Difficulties Can Be Reliably Identified by Fourth Grade

Fraction Difficulties Can Be Improved Through Meaningful Interventions that Center on the Number Line

Conclusion

References

Chapter 13: Mathematical Learning and Its Difficulties in Latin-American Countries

Introduction

About the Region

Theories and Educational Practice

Mathematical Learning Disabilities in Latin American Countries

Mathematical Learning Disabilities in Brazil

Research on Mathematical Learning Disabilities

Future of Mathematical Learning Disabilities in Latin American Countries

Conclusions

References

Chapter 14: Mathematics Learning and Its Difficulties: The Cases of Chile and Uruguay

Introduction

Mathematics Learning Achievement

International Assessment

National Assessment

Educational Policies Addressing MLD and Educational Practice

Chile

Uruguay

Research into MLD

Chile

Uruguay

Conclusions

References

Chapter 15: Mathematical Learning and Its Difficulties in Southern Africa

Introduction

Theoretical Framing

Identified Problem and Research Questions

Methods

Results and Discussion of Findings

Lesotho

Malawi

South Africa

Zimbabwe

Case Study of Mathematical Inclusion in a Full-Service School in South Africa

What Was Done to Support Teachers?

Staff Professional Development

Responding to Annual National Assessments (ANAs)

Sharing Lessons

Were There Any Changes in Mathematics Learner Outcomes?

Conclusion

References

Chapter 16: Mathematical Learning and Its Difficulties in Australia

Australia: The Big Picture

Australia: Educational Policies and MLD

Australia: Theories and Educational Practice

Definitions in MLD in Australian States and Territories

Neuroscience and MLD/Dyscalculia in Australia

References

Chapter 17: Mathematical Learning and Its Difficulties in Taiwan: Insights from Educational Practice

Introduction

The Cultural Background

National Differences in Mathematical Learning

Educational Policies for Learning Difficulties in Taiwan

Diagnosis and Assessment Tool for Mathematical Learning Difficulties

Summary and Conclusion

Reference

Chapter 18: Mathematical Learning and Its Difficulties in Israel

Introduction

General Description: Population and Diversity

General Education and Mathematics Education in Israel

International Educational Tests in Math in Israel

Diagnosis of Mathematical Learning Disabilities in the Israeli School System

Current Changes in the Diagnosis and Treatment of MLD in Israel

Teaching Accommodations for Children Suffering from MLD in Israel

Diagnosis of MLD in Universities in Israel

Conclusion

References

Chapter 19: Learning Difficulties and Disabilities in Mathematics: Indian Scenario

Introduction

Education in India—New Initiatives

Initiatives for the Education of Children with Special Needs

Definition of Specific Learning Disability

Prevalence of Children with Special Needs in India

Teacher Preparation Courses in the Area of Learning Disabilities

Management of Specific Learning Disability in Schools in India

National Institute of Open Schooling

Learning Indicators/Outcomes and National Achievement Survey

Research on Learning Disabilities in India

Identification of the Prevalence of Learning Disabilities in Mathematics in India

Research on Learning Difficulties and Disabilities in Mathematics in India

Conclusion

References

Chapter 20: Adding all up: Mathematical Learning Difficulties Around the World

Math Achievement Around the World

Gender Issues

Heritage of the Soviet Regime

Intranational Diversity

Achievement-Motivation Gap

Definition of Special Needs in Math

Support at School for Children with Severe Math Difficulties

Teacher Training

Toward Evidence-Based Education

Key Issues and Trends

References

Part III: Mathematical Learning Difficulties and Its Cognitive, Motivational and Emotional Underpinnings

Chapter 21: Genetics of Dyscalculia 1: In Search of Genes

Introduction

Clinical Epidemiology of Developmental Dyscalculia

Genetic Susceptibility to Dyscalculia

Familial Aggregation in Dyscalculia

Heritability of Dyscalculia

Gene-Finding Strategies

Genome-Wide Association Studies

Candidate Genes from Comorbidities

Perspectives

References

Chapter 22: Genetics of Dyscalculia 2: In Search of Endophenotypes

Introduction

Cognitive Endophenotypes of Dyscalculia

Basic Number Processing

Phonological Processing

Visuospatial and Visuoconstructional Abilities

Working Memory

Chromosomal Abnormalities

Dyscalculia in Turner Syndrome

Dyscalculia in Klinefelter Syndrome

Genomic Disorders

Dyscalculia in 22q11.2 Deletion Syndromes

Dyscalculia in Williams Syndrome

Monogenic Conditions

Dyscalculia in Fragile X Syndrome and FMR1 Premutations

From the Lab to the Classroom

References

Chapter 23: Neurobiological Origins of Mathematical Learning Disabilities or Dyscalculia: A Review of Brain Imaging Data

Introduction

Brain Activity During Numerical Magnitude Processing and Arithmetic

Numerical Magnitude Processing

Arithmetic

Structural Brain Imaging

Connectivity

Effects of Remedial Interventions on Brain Activity

Discussion

Conclusion

References

Chapter 24: Comorbidity and Differential Diagnosis of Dyscalculia and ADHD

Introduction

What Is Comorbidity?

Why Are Comorbidity Rates for Neurodevelopmental Disorders So High?

What Can Be Causes for Difficulties in Mathematics?

Why Is It Important to Distinguish Between Primary and Secondary MLD?

What Are Difficulties for a Respective Differential Diagnosis?

Which Error Types Are Not Specific to Primary MLD?

Objectives of the Current Study

Materials and Methods

Participants

Assessment

Error Categories

Analyses

Results

Descriptive Statistics

Convergent and Discriminant Validity of the Postulated More Specific Clinical Cut-Off

Differences in Calculation Error Types Between Secondary and Possible Primary MLD

Differences in Counting Error Types Between Secondary and Possible Primary MLD

Discussion

Validation of the Postulated Clinical Cut-Off for the Basis-Math Overall Score

Specific and Unspecific Error Types

Limitations of This Study

Conclusions

References

Chapter 25: Working Memory and Mathematical Learning

Introduction

Working Memory (WM): A Domain-General Precursor of Mathematical Learning

Contribution of WM Components to Mathematical Learning

Working Memory, Word Problems, and Calculation

Executive Functions of Central Executive Component of WM and Their Role in Mathematics

Working Memory Training

Conclusion

References

Chapter 26: The Relation Between Spatial Reasoning and Mathematical Achievement in Children with Mathematical Learning Difficulties

Introduction

Numerical Magnitude and Spatial Reasoning in Typically Developing Children

Spatial Reasoning in Children with MD

Spatial Training to Support Children with MD

Conclusions

References

Chapter 27: The Language Dimension of Mathematical Difficulties

Language Factors on Different Levels and Their Connection to Mathematics Achievement

Differences Between Everyday and Academic Language on Word, Sentence, and Text/Discourse Level

Disentangling Language Obstacles on Word, Sentence, Text, and Discourse Levels and Their Connection to Mathematics Achievements

Obstacles on the Word Level

Obstacles on the Sentence and Text Level

Language Factors in the Achievement of Specific Groups

Second-Language Learners

Students with Learning Disabilities in Mathematics and Reading

Students with Specific Language Impairment and Mathematics Learning

Language Dimensions in Learning Processes

Language as a Learning Medium, Learning Prerequisite, and Learning Goal

Discourse Practices as a Construct to Capture Language Demands on the Discourse Level

Discourse Practices and Discourse Competence in Mathematics Classrooms

General and Topic-Specific Lexical Means for Different Mathematical Discourse Practices

Approaches for Fostering Students’ Language Proficiency in Mathematics

Enhancing Discourse Practices: Qualitative Output Hypotheses

Enhancing Conceptual Knowledge: Relating Registers and Representations

Specifying Mathematical and Language Goals: The SIOP Model

Combining Conceptual and Lexical Learning Trajectories: Macro-Scaffolding

Including Home Languages: Activating Students’ Multilingual Repertoires

Conclusion

References

Chapter 28: Motivational and Math Anxiety Perspective for Mathematical Learning and Learning Difficulties

Introduction

Opportunity–Propensity Model

Motivation

Definition of the Construct

Math Anxiety

Conclusions and Implications

References

Chapter 29: Mathematics and Emotions: The Case of Math Anxiety

Introduction

Math Anxiety as a Construct

Math Anxiety and Motivation

Antecedents of Math Anxiety

Genetics

Age

Gender

Culture

Teachers

Parents

Peers

Math Achievement

Cognitive Mechanisms

Working Memory

Numerical Abilities

Visuospatial Abilities

Neurobiological Underpinnings of Math Anxiety

Assessment of Math Anxiety

Interventions for Math Anxiety: From the Lab to the Classroom

Conclusion

References

Obs. References marked with # refer to self-report questionnaires presented in Tables 29.1, 29.2, and 29.3.

Chapter 30: Cognitive and Motivational Underpinnings of Mathematical Learning Difficulties: A Discussion

Chapter 21: Carvalho and Haase

Chapter 22: Haase and Carvalho

Chapter 23: DeSmedt, Peters, and Ghesquière

Chapter 24: Krinzinger

Chapter 25: Passolunghi and Costa

Chapter 26: Resnick, Newcombe, and Jordan

Chapter 27: Prediger, Erath, and Opitz

Chapter 28: Baten, Pixner, and Desoete

Chapter 29: Haase, Guimarães, and Wood

Common Themes

Concluding Remarks

References

Part IV: Understanding the Basics: Building Conceptual Knowledge and Characterizing Obstacles to the Development of Arithmetic Skills

Chapter 31: Counting and Basic Numerical Skills

Number Sense

Small Number Representations

Approximate Number Representations

Summary

Number Language

Knower Levels

Discrete Quantification

Numerosity

Summary

Counting Principles

Cardinality Principle

Successor Function

Summary

Facilitating the Acquisition of Exact Number Concepts

Facilitating the Acquisition of Individual Number Words

Facilitating the Acquisition of the Cardinality Principle

Broad-Scale Intervention

Numerically Based Toys

Number Language

Summary

References

Chapter 32: Multi-digit Addition, Subtraction, Multiplication, and Division Strategies

Multi-digit Arithmetic Solution Strategies

Multi-digit Addition and Subtraction Strategies

Strategies Framework

Children’s Strategy Use: Empirical Findings

Obstacles in Development

Multi-digit Multiplication and Division Strategies

Strategies Framework

Children’s Strategy Use: Empirical Findings

Obstacles in Development

Discussion

References

Chapter 33: Development of a Sustainable Place Value Understanding

Introduction

Properties of Place Value Systems

Place Value Understanding

Procedural Place Value Understanding

Conceptual Place Value Understanding

Difficulties in Place Value Understanding

Development of Place Value Understanding

Nonstructured Numbers

Identifying Decimal Units

Ordinal Aspect of Place Value Understanding

Cardinal Aspect of Place Value Understanding

Integration of Cardinal and Ordinal Aspects

Nonsustainable Concepts

Our Own Model

Predecadic Level

Level I: Place Values

Level II: Tens-Units Relation with Visual Support

Level III: Tens–Units Relation Without Visual Support

Level IV: General Decimal-Bundling-Unit Relations

Empirical Research

Conclusion

Barriers in the Development of a Sustainable Place Value Understanding

Educational Implications

Future Perspectives

References

Chapter 34: Understanding Rational Numbers – Obstacles for Learners With and Without Mathematical Learning Difficulties

Introduction

Learning of Rational Numbers: Learning a New Concept

Dual Processes in Rational Number Problems: The Natural Number Bias

Obstacles for Learners with Mathematical Learning Difficulties

How to Support Learners: Evidence from Intervention Studies

Conclusions and Perspectives

References

Chapter 35: Using Schema-Based Instruction to Improve Students’ Mathematical Word Problem Solving Performance

Mathematical Word Problem Solving

Theoretical Framework for Understanding How Schema-Based Instruction Is Beneficial to Word Problem Solving Performance

What Are the Unique Features of SBI and How Does It Contribute to Word Problem Solving Performance?

Teaching Word Problem Solving Using SBI: Empirical Evidence from Intervention Studies

Studies 1 and 2: Supporting Evidence for SBI Compared to Traditional Instruction

Studies 3 and 4: Supporting Evidence for SBI Compared to Standards-Based Instruction

Remaining Challenges

References

Chapter 36: Geometrical Conceptualization

Characterizing School Geometry

Three Approaches to School Geometry

G1. The Geometry of Concrete Objects

G2. The Geometry of Graphically Justified Ideal Plane Figures and Solids

G3. Quasi-axiomatic Geometry

The van Hiele Theory about the Stages of Development in Geometrical Thinking

Level 1 (Visualizing)

Level 2 (Analyzing Properties)

Level 3 (Ordering Properties)

Level 4 (Formal Deduction)

Level 5 (Understanding Axiomatic Systems)

About the Characteristics of Geometric Concept Formation

Basic Skills in Geometry

Classifying and Designating Figures

The Skills of Definition and the Clarification of Concepts

The Skills of Proving

Towards a Dialogue of the Traditional and the Dynamic Geometry

Geometry and Learning Difficulties

Summary

Bibliography

Part V: Mathematical Learning Difficulties: Approaches to Recognition and Intervention

Chapter 37: Assessing Mathematical Competence and Performance: Quality Characteristics, Approaches, and Research Trends

Introduction

Quality Characteristics

Categories of Classification

Norm-Referenced Versus Not-Norm-Referenced Tests

Individual Versus Group Testing

Paper-and-Pencil Tests Versus Interviews Versus Computer-Based Tests

Chronological Versus Educational Age–Oriented Tests

Speed Versus Power Tests

Principles of Task Selection

Outline of Different Approaches

Curriculum-Based Measures

Approaches Based on Neuropsychological Theories

Approaches Based on Developmental Psychology Theories

Research Trends

References

Chapter 38: Diagnostics of Dyscalculia

Differential Diagnosis of Dyscalculia

Criterion 1: To Determine the Presence and Severity of the Math Problem

Criterion 2: To Determine the Math Problem Related to the Personal Abilities

Criterion 3: To Determine Obstinacy of the Mathematical Problem

Process Research

Learnability

Math Problems in Early Education

From Problems at a Young Age to Dyscalculia

Conclusion

Appendix

The Five Steps of Math Help

References

Chapter 39: Three Frameworks for Assessing Responsiveness to Instruction as a Means of Identifying Mathematical Learning Disabilities

Systemic RTI Reform

Embedded RTI

Dynamic Assessment

Comparisons across the Three Frameworks

References

Chapter 40: Technology-Based Diagnostic Assessments for Identifying Early Mathematical Learning Difficulties

Introduction

Advantages and Possibilities of Technology-Based Assessment: The Move from Summative to Diagnostic Assessment to Realise Efficient Testing for Personalised Learning

Theoretical Foundations of Framework Development: A Three-Dimensional Model of Mathematical Knowledge

A Three-Dimensional Model of Students’ Knowledge for Diagnostic Assessment in Early Education

Creating an Assessment System: Online Platform Building and Innovative Item Writing

Mathematical Reasoning Items

Mathematical Literacy Items

Items that Assess Disciplinary Mathematics Knowledge

Field Trial and Empirical Validation of the Theoretical Model

Applicability of the Diagnostic System in Everyday School Practice

Scaling and Item Difficulty

Dimensionality and Structural Validity

Conclusions and Further Research and Development

References

Chapter 41: Small Group Interventions for Children Aged 5–9 Years Old with Mathematical Learning Difficulties

Introduction

Learning Difficulties in Mathematics

Intervention

The Features of Effective Instruction for Children with Mathematical Learning Difficulties

Responsiveness to Intervention Practice in Supporting Children with Learning Difficulties

Finnish Web Services for Educators

Studies with ThinkMath Intervention Programs

Conclusion

References

Chapter 42: Perspectives to Technology-Enhanced Learning and Teaching in Mathematical Learning Difficulties

Global Inequalities in Access to Learning Technologies

Online Learning, Virtual Worlds, and Social Learning Environments

Availability: The Surge of Learning Games

Usage: Does Using TEL Tools Help to Produce Better Learning?

Affective and Motivational Factors

Contents: What Is Inside the Intervention Games for MLD?

Training Number Sense

From the Classrooms to the Lab

Final Word

References

Chapter 43: Executive Function and Early Mathematical Learning Difficulties

Executive Function and Early Math Learning Difficulties

The Role of Cognitive Executive Function

The Role of Emotional Executive Function

The Executive Function of Children with Special Needs

The Role of Subject-Matter Knowledge

Teaching Executive Function

Relationships Between EF and Math

Relationships Between EF and Math Learning

Exploring Causality in the Relationship Between EF and Math Learning

Causation: Experimental Studies of EF and Math Interventions

Checking Whether Teaching EF Causes Math Achievement

Alternative Approaches, Especially for Children with Learning Difficulties

Teaching Math Can Cause Both Math Learning and EF Development

Math Activities that May Develop EF

Conclusions

References

Chapter 44: Children’s Mathematical Learning Difficulties: Some Contributory Factors and Interventions

National and Cultural Factors: What Do We Learn from International Comparisons?

Might International Differences in Teaching Methods Affect Performance?

Socio-economic Differences

The Role of Attitudes and Emotions

Interventions for Mathematical Difficulties

Whole-Class Approaches

Light-Touch Individualized and Small-Group Interventions

Highly Intensive Interventions

Numbers Count

What Makes Interventions Effective?

References

Chapter 45: Beyond the “Third Method” for the Assessment of Developmental Dyscalculia: Implications for Research and Practice

Challenges for Educational Policy and Practice

References

Chapter 46: Challenges and Future Perspectives

We Need Research from Genes to Behavior to Build Bridges Between Them

Educational Neuroscience: Where Are We?

What Is Learning Arithmetic from a Neuroscientific Perspective?

Focus on Early Development

Lack of Tools for Screening and Monitoring Learning

Monitoring-Based Framework for Interventions in Schools

The Challenges of the Response-to-Intervention Approach

Professional Development for Teachers

The Scaffold of Teaching Math Content at School

Construction of Curricula in a Tension Between the Two Poles of Individual Prerequisites and Normative Guidelines

Reforming Math Education in the Twenty-First Century

References

Index