Mathematics in Early Childhood and Primary Education (3–8 ...

[Pages:166]Research Report No. 17

Mathematics in Early Childhood and Primary Education (3?8 years)

Definitions, Theories, Development and Progression

Elizabeth Dunphy, Th?r?se Dooley and Gerry Shiel With Deirdre Butler, Dolores Corcoran, Miriam Ryan and Joe Travers International Advisor: Professor Bob Perry

? NCCA 2014 ISSN 1649-3362

National Council for Curriculum and Assessment 35 Fitzwilliam Square, Dublin 2.

T: +353 1 661 7177 F: +353 1 661 7180 E: info@ncca.ie

ncca.ie

Mathematics in Early Childhood and Primary Education (3?8 years)

Definitions, Theories, Development and Progression

Elizabeth Dunphy, St. Patrick's College, Dublin Th?r?se Dooley, St. Patrick's College, Dublin Gerry Shiel, Educational Research Centre, Dublin Deirdre Butler, St. Patrick's College, Dublin Dolores Corcoran, St. Patrick's College, Dublin Miriam Ryan, St. Patrick's College, Dublin Joe Travers, St. Patrick's College, Dublin Professor Bob Perry, Charles Sturt University, Australia

Research conducted on behalf of the National Council for Curriculum and Assessment

Educational Research Centre

Foras Taighde Ar Oideachas

2014

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Research Report No. 17 Mathematics in Early Childhood and Primary Education (3?8 years)

Acronyms

AAMT Australian Association of Mathematics Teachers

Aistear The Early Childhood Curriculum Framework (2009)

CCSSM Common Core States Standards for Mathematics (United States)

CHAT Cultural historical activity theory

DEIS Delivering Equality of Opportunities in Schools

DES Department of Education and Skills (formerly Department of Education and Science )

DfEE Department for Education and Employment (United Kingdom)

EAL English as an Additional Language

ECA Early Childhood Australia

ENRP Early Numeracy Research Project (Victoria, Australia)

ERC Educational Research Centre

HLT Hypothetical Learning Trajectory

ICT Information and Communication Technology

KDU Key Developmental Understanding

LFIN Learning Framework in Number (Wright, Martland & Stafford, 2006)

LT

Learning Trajectory

NAEYC National Association for the Education of Young Children (United States)

NCCA National Council for Curriculum and Assessment

NCTM National Council of Teachers of Mathematics (United States)

NGA National Governors Association (United States)

NRC National Research Council (United States)

OECD Organisation for Economic Cooperation and Development

PISA Programme for International Student Assessment

PM Project Maths

PSC Primary School Curriculum (1999)

PSMC Primary School Mathematics Curriculum (1999)

RME Realistic Mathematics Education

RTI Response to Intervention (United States Initiative)

STEM Science, Technology, Engineering and Mathematics

TAL Tussendoelen Annex Leerlijinen (in Dutch); Intermediate Attainment Targets (in English)

TIMSS Trends in International Mathematics and Science Study

Table of Contents

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Table of Contents

Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 A View of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Definitions of Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Theoretical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Language and Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Defining Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The Development of Children's Mathematical Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Assessing and Planning for Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Addressing Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Key Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

A View of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Developing Mathematics Education in Ireland for Children Aged 3?8 Years . . . . . . . . . . . . . . 20 Curriculum Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Performance Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Policy Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Linguistic and Social Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Overview of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 1: Defining Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The Foundations of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A Definition of Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Defining Mathematics Education for Children Aged 3?8 Years . . . . . . . . . . . . . . . . . . . . . . . . 36 A Key Aim of Mathematics Education: Mathematical Proficiency . . . . . . . . . . . . . . . . . . . . . . 37 Mathematization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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Research Report No. 17 Mathematics in Early Childhood and Primary Education (3?8 years)

Chapter 2: Theoretical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Sociocultural Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A Cultural-Historical Activity Theory Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A Situative Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Cognitive Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Constructivist Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Constructionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A Redeveloped Primary School Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Implications for Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 3: Language, Communication and Mathematics . . . . . . . . . . . . . . . . . . . . 57 The Role of Language in Developing Mathematical Knowledge . . . . . . . . . . . . . . . . . . . . . . . 59 Adult Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 The Nature and Scope of Mathematical Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Establishing a Math-Talk Culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Learning Mathematical Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Variation in Language Skills and Impact on Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 4: Defining Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A Coherent Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Specifying Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Overarching Goals for Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Higher-Order Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Engaging with Powerful Mathematical Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Exploring the Big Ideas in Mathematics Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 The Structure of Curriculum Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Breaking Down the Goals: Critical Transitions within Mathematical Domains . . . . . . . . . . . . . . 75 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter 5: The Development of Children's Mathematical Thinking . . . . . . . . . . . . . 79 A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 From Stages of Development to Levels of Sophistication in Thinking . . . . . . . . . . . . . . . . . . . 81

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Table of Contents

Developing Children's Mathematical Thinking: Three Approaches . . . . . . . . . . . . . . . . . . . . . 81 The First Approach: Working with Children's Thinking and Understanding (RME) . . . . . . . . . . . 82 Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 The Teacher's Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 The Second Approach: Teacher-Generated Hypothetical Learning Trajectories (Simon) . . . . . . . . 85 The Third Approach: Pre-Specified Developmental Progressions as a Basis for Learning Trajectories (Sarama and Clements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Comparing the Three Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Definitions and Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Recognising Diverse Routes in Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Recognising Developmental Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Curriculum Development and the Role of Learning Trajectories . . . . . . . . . . . . . . . . . . . . . . . 94 Supporting Teachers in Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Supporting Learning for Pre-Service Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 6: Assessing and Planning for Progression . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Assessing Mathematics Learning in Early Childhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Formative Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Conceptual Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Conversations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Pedagogical Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Supporting Children's Progression with Formative Assessment . . . . . . . . . . . . . . . . . . . . . . . . 104 Diagnostic and Summative Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Screening/Diagnostic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Standardised Norm-Referenced Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Planning for Progression Using Assessment Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Immersion Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Children with Special Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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Research Report No. 17 Mathematics in Early Childhood and Primary Education (3?8 years)

Chapter 7: Addressing Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Identification of Learning Difficulties in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Exceptional Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Intellectual and Developmental Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Specific Difficulties in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Developmental Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Mathematically Talented Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Cultural Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Ethnicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Socioeconomic Disadvantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Chapter 8: Key Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

The executive summaries of reports No. 17 and No. 18 are available online at ncca.ie/primarymaths. The online versions include some hyperlinks which appear

as text on dotted lines in this print copy.

Acknowledgements

The authors thank the National Council for Curriculum and Assessment for commissioning and supporting this report. We are very thankful to Arlene Forster and Aoife Kelly of the NCCA for providing detailed feedback on earlier drafts of the report. The authors are also indebted to Professor Bob Perry, Charles Sturt University, Australia who read early drafts of the report and who provided expert advice on various issues addressed in the report.

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