Research Report No. 17 Mathematics in Early Childhood and Primary Education (3–8 years) Definitions, Theories, Development and Progression Elizabeth Dunphy, Thérèse 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: [email protected] www.ncca.ie Mathematics in Early Childhood and Primary Education (3–8 years) Definitions, Theories, Development and Progression Elizabeth Dunphy, St. Patrick’s College, Dublin Thérèse 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 Educational Research Centre Foras Taighde Ar Oideachas Research conducted on behalf of the National Council for Curriculum and Assessment 2014 2 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 3 Table of Contents 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 4 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 5 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 ................................................