Thinking mathematically (2a. ed.)

"Thinking Mathematically "is perfect for anyone who wants to develop their powers to think mathematically, whether at school, at university or just out of interest. This book is invaluable for anyone who wishes to promote mathematical thinking in others or for anyone who has always wondered what lies at the core of mathematics. "Thinking Mathematically" reveals the processes at the heart of mathematics and demonstrates how to encourage and develop them. Extremely practical, it involves the reader in questions so that subsequent discussions speak to immediate experience. Cover......Page 1 Thinking Mathematically......Page 2 Contents......Page 6 Introduction to First Edition......Page 9 Introduction to Second Edition......Page 12 Specializing......Page 18 Generalizing......Page 25 Writing yourself notes......Page 26 Review and preview......Page 38 Reference......Page 40 Phases of work......Page 41 Three phases......Page 42 The Entry phase......Page 43 Entry 1: What do I KNOW?......Page 44 Entry 2: What do I WANT?......Page 47 Entry 3: What can I INTRODUCE?......Page 49 The Attack phase......Page 52 The Review phase......Page 53 Review 1: CHECK the resolution......Page 54 Review 3: EXTEND to a wider context......Page 55 Practising Review......Page 57 Review summarized......Page 59 The three phases summarized......Page 60 Reference......Page 61 Being STUCK......Page 62 Summary......Page 73 What is conjecturing?......Page 75 Conjecture: backbone of a resolution......Page 79 How do conjectures arise?......Page 87 Discovering pattern......Page 90 Summary......Page 93 Structure......Page 95 Seeking structural links......Page 99 When has a conjecture been justified?......Page 103 Developing an internal enemy......Page 107 Summary......Page 111 Reference......Page 112 Still STUCK?......Page 113 Distilling and mulling......Page 114 Specializing and generalizing......Page 116 Hidden assumptions......Page 118 Summary......Page 120 References......Page 121 Developing an internal monitor......Page 122 Roles of a monitor......Page 123 Emotional snapshots......Page 125 Getting started......Page 126 Getting involved......Page 128 Mulling......Page 129 Keeping going......Page 131 Insight......Page 132 Being sceptical......Page 134 Summary......Page 135 On becoming your own questioner......Page 137 A spectrum of questions......Page 138 Some ‘questionable’ circumstances......Page 139 Noticing......Page 144 Obstacles to a questioning attitude......Page 146 Summary......Page 148 Reference......Page 149 Developing mathematical thinking......Page 150 Improving mathematical thinking......Page 151 Provoking mathematical thinking......Page 154 Supporting mathematical thinking......Page 156 Sustaining mathematical thinking......Page 157 Summary......Page 161 Reference......Page 162 Something to think about......Page 163 References......Page 197 Thinking mathematically in curriculum topics......Page 198 Place value and arithmetic algorithms......Page 199 Factors and primes......Page 201 Fractions and percentages......Page 205 Ratios and rates......Page 208 Equations......Page 213 Patterns and algebra......Page 215 Graphs and functions......Page 219 Functions and calculus......Page 223 Sequences and iteration......Page 227 Mathematical induction......Page 230 Abstract algebra......Page 232 Perimeter, area and volume......Page 237 Geometrical reasoning......Page 240 Reasoning......Page 243 References......Page 247 Natural powers and processes......Page 248 Mathematical themes......Page 253 Mathematical worlds......Page 255 Attention......Page 256 Summary......Page 257 Bibliography......Page 258 Subject index......Page 260 Index of questions......Page 264