Helping Children Learn Mathematics, 3rd Australian Edition

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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