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Helping Children Learn Mathematics
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Table of Contents

Chapter 1 School mathematics in a changing world 1 Introduction 2 1.1 What is mathematics? 3 1.2 What determines the mathematics being taught? 4 1.3 Where can you turn? 12 Chapter 2 Helping children learn mathematics with understanding 20 Introduction 21 2.1 How can we support the diverse learners in our classrooms? 21 2.2 Meaningful connections between procedural and conceptual knowledge 27 2.3 How do children learn mathematics? 29 2.4 How can we help children make sense of mathematics? 34 Chapter 3 Planning and teaching 50 Introduction 51 3.1 Effective planning and preparation for teaching: Using strategic questions to inform teaching practice 51 3.2 Planning for effective teaching 66 3.3 Levels of planning 67 3.4 Planning different types of lessons 68 3.5 Meeting the needs of all students 75 3.6 Assessment and analysis in planning 80 Chapter 4 Enhancing learning and teaching through assessment and feedback 87 Introduction 88 4.1 Enhancing learning and teaching 89 4.2 Gathering information on student learning 92 4.3 Ways to assess students' learning and dispositions 94 4.4 Keeping records and communicating about assessments 109 Chapter 5 Processes of doing mathematics 121 Introduction 122 5.1 Understanding 123 5.2 Fluency 124 5.3 Problem solving 125 5.4 Reasoning and proof 129 5.5 Communication 134 5.6 Connections 135 5.7 Representations 138 Chapter 6 Helping children with problem solving 148 Introduction 149 6.1 What is a problem and what is problem solving? 150 6.2 Teaching mathematics through problem solving 151 6.3 Strategies for problem solving 162 6.4 The importance of looking back 170 6.5 Helping all students with problem solving 172 Chapter 7 Counting and number sense in early childhood and primary years 181 Introduction 182 7.1 Developing number sense 183 7.2 Counting principles 197 7.3 Counting strategies 200 7.4 Cardinal, ordinal and nominal numbers 210 7.5 Writing numerals 211 Chapter 8 Extending number sense: Place value 220 Introduction 221 8.1 Our numeration system 223 8.2 Nature of place value 224 8.3 Beginning place value 231 8.4 Consolidating place value 234 8.5 Extending place value 241 8.6 Reading and writing numbers 247 8.7 Rounding 251 Chapter 9 Operations: Meanings and basic facts 260 Introduction 261 9.1 Helping children develop number sense and computational fluency 263 9.2 Developing meanings for the operations 266 9.3 Mathematical properties 274 9.4 Overview of learning the basic facts 275 9.5 Thinking strategies for basic facts 283 Chapter 10 Mental computation, calculators and estimation 307 Introduction 308 10.1 Calculators 312 10.2 Mental computation 316 10.3 Estimation 323 Chapter 11 Solving problems with written strategies 339 Introduction 340 11.1 Emergent understanding and experiences 342 11.2 Addition 343 11.3 Subtraction 350 11.4 Multiplication 357 11.5 Division 364 11.6 Finding the balance between practice and proficiency 371 Chapter 12 Fractions and decimals: Meanings and operations 380 Introduction 381 12.1 Conceptual development of fractions 383 12.2 Operations with fractions 402 12.3 Conceptual development of decimals 410 12.4 Operations with decimals 414 Chapter 13 Ratio, proportion and percentages: Meanings and applications 423 Introduction 424 13.1 Ratios 426 13.2 Proportions 429 13.3 Percentages 436 Chapter 14 Extending students with number theory 451 Introduction 452 14.1 Number theory in primary school mathematics 452 14.2 Number theory topics for primary school students 458 14.3 Other number theory topics 471 Chapter 15 Pattern and algebraic thinking 482 Introduction 483 15.1 Problems, patterns and relations 485 15.2 Language and symbols of algebra 494 15.3 Modelling, generalising and justifying 496 Chapter 16 Geometry 517 Introduction 518 16.1 The geometry of two-dimensional shapes and three-dimensional objects 521 16.2 Location, position and spatial relationships 545 16.3 Transformations 549 16.4 Visualisation and spatial reasoning 552 Chapter 17 Measurement 560 Introduction 561 17.1 The measurement process 564 17.2 Identifying attributes and comparing 565 17.3 Measurement concepts for all units 578 17.4 Measuring with informal units 581 17.5 Measuring with formal units 584 17.6 Applications including formulae 593 17.7 Comparing and converting measurements 598 17.8 Estimating measurements 600 17.9 Connecting attributes 603 Chapter 18 Data analysis, statistics and probability 613 Introduction 615 18.1 Formulating questions for data collection 618 18.2 Organising and representing data 621 18.3 Analysing data: Descriptive statistics 632 18.4 Interpreting results 639 18.5 Probability 642 Appendix A 658 Appendix B 670 Appendix C 679 Index 707

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