Part 1: Setting the Scene
1. Understanding School Mathematics
What is mathematics?
Goals of school mathematics
Affordances and constraints
2. Learning Mathematics
What does it mean to learn mathematics?
Learning and understanding mathematics
Developing your own theory of mathematics learning
3. Teaching Mathematics
What does it mean to teach mathematics?
Connections among beliefs
How can we know we are teaching?
Knowledge for teaching mathematics
Effective mathematics teaching
Part 2: Understanding the Challenges and Opportunities
4. Thinking Mathematically
Learning and doing mathematics
Making a start with mathematical thinking
General processes for problem solving and reasoning
Helping learners to think mathematically
5. Communicating Mathematically
The language of mathematics
Language and culture
Communicating in the mathematics classroom
6. Representing Mathematically
What are mathematical representations?
The importance of mathematical language and recording
Using representations to build abstract thinking
Choosing and using materials and models
Choosing materials and models for the classroom
Multi-representational learning environments
7. Assessing and Reporting
Assessment is about testing, right?
Assessment of learning
Assessment for learning
8. Understanding Diversity
Who are diverse learners?
Language of diversity
Diversifying the curriculum
Supporting diverse learners
Part 3: Exploring the Big Ideas in Mathematics
9. Numeracy in the Curriculum
What is numeracy?
Numeracy across the curriculum
10. Developing a Sense of Number and Algebra
Understanding number sense
Number sense in practice
Developing a sense of number
11. Developing a Sense of Measurement and Geometry
Linking measurement and geometry
What is measurement?
Developing measurement sense
How geometry is learned
12. Developing a Sense of Statistics and Probability
What is statistics?
What is probability?
Part 4: Laying the Basis for F–4 Mathematics
13. Algebraic Thinking: F–4
What is pattern and structure?
Why is pattern and structure important?
Early algebraic thinking
14. Number Ideas and Strategies: F–2
The origins of number
Research on early number learning
Playing with number
The numbers 0 to 10
A sense of numbers beyond 10
Scaffolding solution strategies
15. Place Value: F–4
Prerequisite ideas and strategies
Understanding tens and ones
Introducing three-digit numeration
Developing four-digit numeration
Extending to tens of thousands and beyond
16. Additive Thinking: F–4
Why additive thinking?
The development of additive thinking
Contexts for addition and subtraction
Additive solution strategies
17. Multiplicative Thinking: F–4
What is multiplicative thinking?
Why is multiplicative thinking important?
Initial ideas, representations and strategies
Building number fact knowledge and confidence
18. Fractions and Decimal Fractions: F–4
Making sense of fractions
Developing fraction knowledge and confidence
Introducing decimal fractions
19. Measurement Concepts and Strategies: F–4
Why is teaching measurement important?
Measurement concepts in the curriculum
Measurement learning sequence
Approaches to developing an understanding of length
Approaches to developing an understanding of time
20. Geometric Thinking: F–4
Classifying spatial objects
Relationships between spatial objects
Developing dynamic imagery
21. Statistics and Probability: F–4
Grappling with uncertainty
The development of students’ thinking about probability
Part 5: Extending Mathematics to the Middle Years: 5–9
22. Number: Fractions, Decimals and Reals: 5–9
Building the number line
Extending our place-value system
Density of the number line
23. Additive Thinking: 5–9
Ways of working with addition and subtraction
24. Multiplicative Thinking and Proportional Reasoning: 5–9
Meanings for multiplication and division
Working with an extended range of numbers
What is proportional reasoning?
Addressing the multiplicative gap
25. Algebraic Thinking: 5–9
What is algebraic thinking?
Why is algebra important?
Arithmetic, algebraic thinking and problem structure
Meaningful use of symbols
Model approach—using the length model
Equivalence and equations
Introducing the distributive law
Simplifying expressions and equations
26. Measurement Concepts and Strategies: 5–9
Extending measurement concepts
Developing area formulae
Volume and capacity
27. Geometric Thinking: 5–9
Working with spatial objects
Learning geometry in the middle years
28. Statistics and Probability: 5–9
Describing chance events
Part 6: Entering the Profession
29. Becoming a Professional Teacher of Mathematics
Standards for mathematics teaching
Final words of advice
Dianne Siemon: Professor of Mathematics Education, School of Education, RMIT
Kim Beswick: Professor of Mathematics Education, School of Education, University of Tasmania
Kathy Brady: Head of the Student Learning Centre, Flinders University
Julie Clark: Associate Professor of Mathematics Education, School of Education, Flinders University
Rhonda Faragher: Senior Lecturer, Faculty of Education and Arts, Australian Catholic University
Elizabeth Warren: Professor in Mathematics Education, Faculty of Education and Arts, Australian Catholic University
Margarita Breed: Secondary school mathematics educator and former lecturer, School of Education, RMIT