Teaching Mathematics

Part 1: Setting the Scene1. Understanding School MathematicsIntroductionWhat is mathematics?Goals of school mathematicsAffordances and constraintsConclusion2. Learning MathematicsIntroductionWhat does it mean to learn mathematics?Learning and understanding mathematicsDeveloping your own theory of mathematics learning3. Teaching MathematicsIntroductionWhat does it mean to teach mathematics?Connections among beliefsHow can we know we are teaching?Knowledge for teaching mathematicsEffective mathematics teachingPart 2: Understanding the Challenges and Opportunities4. Thinking MathematicallyLearning and doing mathematicsMaking a start with mathematical thinkingGeneral processes for problem solving and reasoningHelping learners to think mathematicallyConclusion5. Communicating MathematicallyIntroductionThe language of mathematicsLanguage and cultureCommunicating in the mathematics classroomConclusion6. Representing MathematicallyWhat are mathematical representations?Traditional representationsThe importance of mathematical language and recordingUsing representations to build abstract thinkingChoosing and using materials and modelsChoosing materials and models for the classroomMulti-representational learning environmentsConclusion7. Assessing and ReportingAssessment is about testing, right?Assessment of learningAssessment for learningReportingConclusion8. Understanding DiversityWho are diverse learners?Language of diversityDiversifying the curriculumSupporting diverse learnersConclusionPart 3: Exploring the Big Ideas in Mathematics9. Numeracy in the CurriculumWhat is numeracy?Numeracy across the curriculumCritical numeracyConclusion10. Developing a Sense of Number and AlgebraUnderstanding number senseNumber sense in practiceDeveloping a sense of numberConclusion11. Developing a Sense of Measurement and GeometryLinking measurement and geometryWhat is measurement?Developing measurement senseGeometrySpatial senseHow geometry is learnedConclusion12. Developing a Sense of Statistics and ProbabilityIntroductionStatistical literacyWhat is statistics?What is probability?ConclusionPart 4: Laying the Basis for F-4 Mathematics13. Algebraic Thinking: F-4What is pattern and structure?Why is pattern and structure important?Early algebraic thinkingFunctional thinkingConclusion14. Number Ideas and Strategies: F-2The origins of numberResearch on early number learningPlaying with numberThe numbers 0 to 10A sense of numbers beyond 10Scaffolding solution strategiesConclusion15. Place Value: F-4Prerequisite ideas and strategiesUnderstanding tens and onesIntroducing three-digit numerationDeveloping four-digit numerationExtending to tens of thousands and beyondConclusion16. Additive Thinking: F-4Why additive thinking?The development of additive thinkingContexts for addition and subtractionAdditive solution strategiesProblem solvingConclusion17. Multiplicative Thinking: F-4IntroductionWhat is multiplicative thinking?Why is multiplicative thinking important?Initial ideas, representations and strategiesBuilding number fact knowledge and confidenceComputation strategiesProblem solvingConclusion18. Fractions and Decimal Fractions: F-4IntroductionMaking sense of fractionsDeveloping fraction knowledge and confidenceIntroducing decimal fractionsConsolidating understandingConclusion19. Measurement Concepts and Strategies: F-4Why is teaching measurement important?Measurement concepts in the curriculumMeasurement learning sequenceApproaches to developing an understanding of lengthApproaches to developing an understanding of timeConclusion20. Geometric Thinking: F-4Classifying spatial objectsRelationships between spatial objectsDeveloping dynamic imageryLocationGeometric reasoningConclusion21. Statistics and Probability: F-4IntroductionGrappling with uncertaintyThe development of students' thinking about probabilityRepresenting dataUnderstanding distributionsPart 5: Extending Mathematics to the Middle Years: 5-9 22. Number: Fractions, Decimals and Reals: 5-9Building the number lineWhole numbersExtending our place-value systemIntegersScientific notationThe rationalsThe realsDensity of the number lineConclusion23. Additive Thinking: 5-9Ways of working with addition and subtractionAlgorithmsFractionsDecimalsIntegers24. Multiplicative Thinking and Proportional Reasoning: 5-9IntroductionMeanings for multiplication and divisionWorking with an extended range of numbersWhat is proportional reasoning?Addressing the multiplicative gapConclusion25. Algebraic Thinking: 5-9What is algebraic thinking?Why is algebra important?Arithmetic, algebraic thinking and problem structureMeaningful use of symbolsModel approach-using the length modelEquivalence and equationsAlgebraic lawsIntroducing the distributive lawSimplifying expressions and equationsFunctional thinkingConclusion26. Measurement Concepts and Strategies: 5-9Extending measurement conceptsAreaDeveloping area formulaeVolume and capacityMassMoneyConclusion27. Geometric Thinking: 5-9Working with spatial objectsGeometric proofTransformational geometryNon-Euclidean geometryLocationLearning geometry in the middle yearsConclusion28. Statistics and Probability: 5-9Data investigationData representationsData measuresVariationDescribing chance eventsConclusionPart 6: Entering the Profession 29. Becoming a Professional Teacher of MathematicsLooking forwardStandards for mathematics teachingFinal words of advice