Teaching Mathematics with Insight : The Identification, Diagnosis and Remediation of Young Children's Mathematical Errors

the Initial Teacher Training National Curriculum Says That Student Teachers Should Be Trained To Analyse Pupil's Errors In Maths And Act Accordingly. This Is The Only Book That Supports Teachers' Analysis Of Mathematical Errors And Helps Them Predict Potential Problems And Propose Solutions For Themselves. Written In An Accessible Style, teaching Mathematics With Insight Guides The Primary And Early Years Teacher, And The Student Teacher Through A Series Of Processes That Will Enable Them To Become More Effective And Enlightened Teachers Of Early Mathematics. The Processes Include: Unravelling The Complexities Of A Concept, For Example Subtraction, By Considering Its Component Parts And The Knowledge Required To Acquire Understanding; Watching Children Work To Observe Common Mistakes And Analyse The Underlying Misconceptions; Discussing The Concepts With Other Adults. Education, Politics & IR, Sociology & Social Policy Book Cover 1 Half-ile 2 Dedication 3 Title 4 Copyright 5 Contents 6 List of figures 7 Introduction and acknowledgments 8 Introduction 8 Acknowledgments 10 References 10 Chapter 1 Making mathematical errors 11 Child: Experience 11 Child: Expertise 12 Child: Mathematical knowledge and understanding 13 Child: Imagination and creativity 13 Child: Mood 14 Child: Attitude and confidence 14 Task: Mathematical complexity 15 Task: Presentational complexity 15 Task: Translational complexity 15 Teacher: Attitude and confidence 16 Teacher: Mood 17 Teacher: Imagination and creativity 17 Teacher: Knowledge 18 Teacher: Expertise 18 Teacher: Experience 18 References 19 Chapter 2 Starting with place value 20 The basic concept 20 Prerequisite skills 21 Counting 21 Reading and writing numbers 22 Manipulating numbers 23 Zero 23 Concluding remarks 23 References 23 Chapter 3 Why is place value so challenging to teach? 24 Counting in English 25 Misapprehension that an ability to recite numbers implies understanding 25 Lack of experience 25 Abstract teaching 26 Position of digit 28 Premature introduction 28 Zero 28 Reading large numbers 29 Vertical layout 29 Some thoughts on brainstorming 29 Implications for action 30 Language difficulties 31 Counting with meaning rather than recitation 31 Working with larger quantities 31 Making meaning 32 References 33 Chapter 4 Exploring the analytical process in greater depth 35 Step 1: Unravelling the mathematical complexities of a topic 35 Step 2: Considering psychological complexities 36 Step 3: Teacher observation 37 Step 4: Child observation and discussion 38 Step 5: Discussions with adults 39 Step 6: Researching the literature 40 Concluding remarks 40 References 40 Chapter 5 What is subtraction? 42 Some basic concepts 42 Postscript 45 References 45 Chapter 6 Unravelling the complexities of subtraction 48 Intrinsic mathematical problems 49 Intrinsic psychological problems 50 Extrinsic home-related problems 51 Extrinsic school-related problems 51 Concluding remarks 52 References 53 Chapter 7 What is time? 54 Passing the time 54 Passing even more time 55 Diachronic thinking 56 Time measurements 56 Concluding remarks 58 References 58 Chapter 8 The problems of teaching time 59 Problems with the passage of time 60 Telling the time 61 Concluding remarks 64 Reference 64 Chapter 9 Thinking about shape 65 Brainstorming 66 Intrinsic mathematical problems 66 Intrinsic psychological problems 69 Teaching shape 71 Representation 72 Active learning 73 Encouraging language 73 Orientation 74 Properties 74 Concluding remarks 75 References 75 Chapter 10 Learning more about children in classrooms 77 Verbalising thoughts 77 Insight into young children’s ‘mathematical’ thinking 79 Task 81 Behind the scenes in mathematics sessions 82 Implications 83 References 84 Chapter 11 Concluding remarks 85 References 86 Index 87 This text aims to guide the primary and early years teacher, and the student teacher through a series of processes that will enable them to become more effective and enlightened teacher of early mathematics.