Cover
Title page
Copyright page
Brief contents
Contents
About the authors
Preface
Features of this text
About this text
Chapter 1 School mathematics in a changing world
Chapter 1 concept map
Introduction
1.1 What is mathematics?
1.2 What determines the mathematics being taught?
Needs of the subject
Needs of the child
Needs of society
Your role
1.3 Where can you turn?
National guidelines for school mathematics
State and local guidelines
Research
Cultural and international resources
History
Textbooks and other materials
Electronic materials
Professional organisations
Professional development
Other teachers
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
References
Acknowledgements
Chapter 2 Helping children learn mathematics with understanding
Chapter 2 concept map
Introduction
2.1 How can we support the diverse learners in our classrooms?
Creating a positive learning environment
Avoiding negative experiences that increase anxiety
Establishing clear expectations
Treating all students as equally likely to have aptitude for mathematics
Helping students improve their ability to retain mathematical knowledge and skills
2.2 Meaningful connections between procedural and conceptual knowledge
2.3 How do children learn mathematics?
Exploring issues with behaviourism
Constructing understanding
2.4 How can we help children make sense of mathematics?
Recommendation 1: Teach to the developmental characteristics of students
Recommendation 2: Actively involve students
Recommendation 3: Move learning from concrete to abstract
Recommendation 4: Use communication to encourage understanding
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 3 Planning and teaching
Chapter 3 concept map
Introduction
3.1 Effective planning and preparation for teaching: Using strategic questions to inform teaching practice
What mathematics content knowledge and pedagogical content knowledge do I know and need?
How will I differentiate my teaching to ensure that all students are learning?
What do my students already know?
What kinds of tasks will I give my students?
How will I encourage my students to talk, what kinds of questions will I ask, and how will I group my students?
What materials will my students and I use?
3.2 Planning for effective teaching
The importance of planning
3.3 Levels of planning
Planning for the year
Planning for units
Planning for daily lessons
3.4 Planning different types of lessons
Investigative lessons
Direct instruction lessons
Explorations
3.5 Meeting the needs of all students
Teaching aboriginal and torres strait islander students
Teaching students from other cultures
Teaching English-language learners
Teaching students with identified special needs
3.6 Assessment and analysis in planning
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 4 Enhancing learning and teaching through assessment and feedback
Chapter 4 concept map
Introduction
4.1 Enhancing learning and teaching
Assessment for learning
Assessment as learning
4.2 Gathering information on student learning
Assessment of learning
Making teaching and learning decisions
Monitoring student progress
Evaluating student achievement
4.3 Ways to assess students’ learning and dispositions
Observation
Questioning
Interviewing
Performance tasks
Self-assessment and peer assessment
Work samples
Portfolios
Writing
Teacher-designed paper-and-pencil tests
Standardised achievement tests
4.4 Keeping records and communicating about assessments
Recording the information
Communicating the information
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 5 Processes of doing mathematics
Chapter 5 concept map
Introduction
5.1 Understanding
5.2 Fluency
5.3 Problem solving
5.4 Reasoning and proof
Reasoning is about making generalisations
Reasoning leads to a web of generalisations
Reasoning leads to mathematical memory built on relationships
Learning through reasoning requires making mistakes and learning from them
5.5 Communication
5.6 Connections
5.7 Representations
Creating and using representations
Selecting, applying and translating among representations
Using representations to model and interpret phenomena
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 6 Helping children with problem solving
Chapter 6 concept map
Introduction
6.1 What is a problem and what is problem solving?
6.2 Teaching mathematics through problem solving
Factors for success in problem solving
Choosing appropriate problems
Finding problems
Having students pose problems
Using calculators, computers and tablets
6.3 Strategies for problem solving
Act it out
Make a drawing or diagram
Look for a pattern
Construct a table
Guess, check and improve
Work backward
Solve a similar but simpler problem
6.4 The importance of looking back
Looking back at the problem
Looking back at the answer
Looking back at the solution process
Looking back at one’s own thinking
6.5 Helping all students with problem solving
Managing time
Managing classroom routines
Managing student needs
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 7 Counting and number sense in early childhood and primary years
Chapter 7 concept map
Introduction
7.1 Developing number sense
Prenumber concepts
Early number development
7.2 Counting principles
Counting stages
7.3 Counting strategies
Counting practice
Developing number benchmarks
Understanding numbers one to ten
7.4 Cardinal, ordinal and nominal numbers
7.5 Writing numerals
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 8 Extending number sense: place value
Chapter 8 concept map
Introduction
8.1 Our numeration system
8.2 Nature of place value
Modelling ungrouped and pre-grouped materials
Modelling proportional and nonproportional materials
Grouping and trading
8.3 Beginning place value
A place to start
8.4 Consolidating place value
Regrouping and renaming
8.5 Extending place value
Counting and patterns
8.6 Reading and writing numbers
8.7 Rounding
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 9 Operations: meanings and basic facts
Chapter 9 concept map
Introduction
9.1 Helping children develop number sense and computational fluency
Facility with counting
Experience with a variety of concrete situations
Familiarity with many problem contexts
Experience in talking and writing about mathematical ideas
9.2 Developing meanings for the operations
Addition and subtraction
Multiplication and division
9.3 Mathematical properties
9.4 Overview of learning the basic facts
Start where the children are
Build understanding of the basic facts
Focus on how to remember facts
9.5 Thinking strategies for basic facts
Thinking strategies for addition facts
Thinking strategies for subtraction facts
Thinking strategies for multiplication facts
Thinking strategies for division facts
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 10 Mental computation, calculators and estimation
Chapter 10 concept map
Introduction
10.1 Calculators
Using calculators requires thinking
Using calculators can raise student achievement
Calculators are not always the fastest way of doing computations
Calculators are useful for more than doing computations
10.2 Mental computation
Strategies and techniques for mental computation
Encouraging mental computation
10.3 Estimation
Background for estimating
Front-end estimation
Adjusting
Compatible numbers
Flexible rounding
Clustering
Choosing estimation strategies
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 11 Solving problems with written strategies
Chapter 11 concept map
Introduction
11.1 Emergent understanding and experiences
Using materials
Using place value
11.2 Addition
Standard addition algorithm
11.3 Subtraction
Standard subtraction algorithm
Partial-difference subtraction algorithm
11.4 Multiplication
Multiplication with one-digit multipliers
Partial-products multiplication algorithm
Lattice multiplication algorithm
Multiplication by 10 and multiples of 10
Multiplication with zeros
Multiplication with two-digit multipliers
Multiplication with large numbers
11.5 Division
Division with one-digit divisors
Distributive algorithm
Subtractive algorithm
Division with two-digit divisors
Making sense of division with remainders
11.6 Finding the balance between practice and proficiency
Using inverse operations to check answers
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 12 Fractions and decimals: meanings and operations
Chapter 12 concept map
Introduction
12.1 Conceptual development of fractions
Three meanings of fractions
Models of the part–whole meaning
Making sense of fractions
Ordering fractions and equivalent fractions
Mixed numbers and improper fractions
12.2 Operations with fractions
Addition and subtraction
Multiplication
12.3 Conceptual development of decimals
Relationship to common fractions
Relationship to place value
Ordering and rounding decimals
12.4 Operations with decimals
Addition and subtraction
Multiplication and division
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 13 Ratio, proportion and percentages: meanings and applications
Chapter 13 concept map
Introduction
13.1 Ratios
13.2 Proportions
13.3 Percentages
Understanding percentages
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature Resource
References
Acknowledgements
Chapter 14 Extending students with number theory
Chapter 14 concept map
Introduction
14.1 Number theory in primary school mathematics
14.2 Number theory topics for primary school students
Odds and evens
Factors and multiples
Prime factorisation
Divisibility
14.3 Other number theory topics
Polygonal numbers
Relatively prime pairs of numbers
Modular arithmetic
Pascal’s triangle
Pythagorean triples
Fibonacci sequence
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 15 Pattern and algebraic thinking
Chapter 15 concept map
Introduction
15.1 Problems, patterns and relations
Problems
Patterns
Relations
15.2 Language and symbols of algebra
Equality and inequality
Variables
Expressions and equations
15.3 Modelling, generalising and justifying
Routine problems
Nonroutine problems and patterns
Relations: functions
Relations: properties of numbers
Another look at modelling, generalising and justifying
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 16 Geometry
Chapter 16 concept map
Introduction
16.1 The geometry of two-dimensional shapes and three-dimensional objects
Three-dimensional objects
16.2 Location, position and spatial relationships
16.3 Transformations
Congruence
16.4 Visualisation and spatial reasoning
Using geometric physical and pictorial materials
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 17 Measurement
Chapter 17 concept map
Introduction
17.1 The measurement process
17.2 Identifying attributes and comparing
Length
Capacity
Mass (weight)
Area
Volume
Angle
Time
Temperature
Other attributes
17.3 Measurement concepts for all units
17.4 Measuring with informal units
Length
Area
Volume and capacity
Mass
Time
Temperature
Angle
17.5 Measuring with formal units
Length
Area
Volume and capacity
Mass
Time
Temperature
Angle
Scaled instruments
17.6 Applications including formulae
Rectangles
Parallelogram
Triangle
Trapezium
The combination of multiple shapes
Circumference and area of a circle
Volume and capacity
Time
Problem solving with measurement
17.7 Comparing and converting measurements
Conversions
17.8 Estimating measurements
17.9 Connecting attributes
Area and shape
Volume and solids
Perimeter and area
Volume and surface area
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Chapter 18 Data analysis, statistics and probability
Chapter 18 concept map
Introduction
Children encounter ideas of statistics and probability outside school every day
Data analysis, statistics and probability provide connections to other mathematics topics or school subjects
Data analysis, statistics and probability provide opportunities for computational activity in a meaningful context
Data analysis, statistics and probability provide opportunities for developing critical-thinking skills
18.1 Formulating questions for data collection
Surveys
Experiments
Simulations
18.2 Organising and representing data
Quick and easy graphing methods
Plots
More detailed graphing methods
Bar graphs, column graphs and histograms
Pie graphs
Line graphs
Graphical roundup
18.3 Analysing data: descriptive statistics
Measures of central tendency or averages
Measures of variation
18.4 Interpreting results
Data sense
Misleading graphs
Communicating results
18.5 Probability
Probability of an event
Randomness
Independence of events
Misconceptions about probability
Making connections
A glance at where we’ve been
Things to do: From what you’ve read
Things to do: Going beyond this text
Children’s literature connections
References
Acknowledgements
Appendix A
Standards and Expectations, National Council of Teachers of Mathematics, 2000
Number and Operations
Algebra
Geometry
Measurement
Data analysis and Probability
Problem Solving
Standard
Reasoning and Proof
Standard
Communication
Standard
Connections
Standard
Representation
Standard
Appendix B
Curriculum Focal Points, National Council of Teachers of Mathematics, 2006
Curriculum Focal Points for Mathematics in Prekindergarten through Grade 8
Curriculum Focal Points and Connections for Prekindergarten
Curriculum Focal Points and Connections for Kindergarten
Curriculum Focal Points and Connections for Grade 1
Curriculum Focal Points and Connections for Grade 2
Curriculum Focal Points and Connections for Grade 3
Curriculum Focal Points and Connections for Grade 4
Curriculum Focal Points and Connections for Grade 5
Curriculum Focal Points and Connections for Grade 6
Appendix C
Masters
1. Attribute pieces
2. Cuisenaire rods
3. Base-ten blocks
4. Pattern blocks
5. Five- and ten-frames
6. Hundred charts
7. Variations of hundred charts
8. Basic addition and multiplication facts
9. 0–9 Cards
10. Blank place-value chart
11. Trading mat for different lands
12. Powers of 10 (place-value chart)
13. Decimal or percentage paper
14. Fraction bars
15A. Fraction models and spinners
15B. Fraction models and spinners
16. Rulers
17. Geoboard template
18. Geoboard recording paper
19. Centimetre dot paper
20. Isometric paper
21. Centimetre grid paper
22. Geometric design paper
23. Equilateral triangle paper
24. Tangram
25. Circle point paper
Index
EULA