A Mathematician Comes of Age

Contents......Page 10
Preface......Page 14
Acknowledgements......Page 18
1.0 Chapter Overview......Page 20
1.1 Back Story......Page 21
1.2 First Principles......Page 23
1.3 Next Steps......Page 25
1.5 How DoesMathematics Differ from Other Fields?......Page 27
1.6 A Little History......Page 29
1.7 Examples of Mathematical Maturity......Page 30
1.8 Mathematical Maturity and Fear of Failure......Page 33
1.9 Levels of Mathematical Maturity......Page 35
1.10 The Changing Nature of Mathematical Maturity......Page 37
1.11 On Proof and Progress in Mathematics......Page 38
1.12 More on Brahe, Kepler, and Napier......Page 42
Math Concepts......Page 46
2.1 Problems that Can Exhibit Mathematical Maturity......Page 47
2.2 Approximate Solutions......Page 50
2.3 Computers and Calculators......Page 52
2.4 Proving Little Theorems, Proving Big Theorems......Page 54
2.5 Mistakes......Page 56
2.7 Modeling......Page 58
3.0 Chapter Overview......Page 60
3.1 Teaching Reform......Page 61
3.2 Math Maturity and Math Teachers......Page 64
3.3 Uri Treisman’s Teaching Techniques......Page 65
3.4 How Can We Improve the Education of Math Students?......Page 67
3.5 The Development of the Mathematics Curriculum......Page 69
3.6 Learning on the Internet......Page 71
3.7 Capstone Experiences......Page 72
3.8 Challenging Your Students......Page 73
3.9 The Sports Analogy......Page 74
3.10 Pacing......Page 75
3.11 Problems for Students......Page 76
3.12 Student Research Problems......Page 79
3.13 The Ph.D. Advisor......Page 80
4.1 Chapter Overview......Page 82
4.2 Math Anxiety......Page 83
4.3 Mathematics and Diversions from Mathematics......Page 85
4.4 National Standards......Page 86
4.5 The Myers-Briggs Index......Page 88
4.6 Intelligence Tests......Page 90
4.7 Asperger’s Syndrome......Page 92
4.8 The ABD......Page 93
4.9 Men and Women......Page 94
5.0 Chapter Overview......Page 96
5.2 Maturity vs. Immaturity......Page 97
5.3 Rote Learning vs. Learning for Understanding......Page 98
5.4 How Do Students Learn? How Do Students Think?......Page 99
5.5 How Do Students Become Motivated?......Page 101
5.6 Learning to Recover from Mistakes......Page 103
5.7 Will These Ideas Travel Well?......Page 104
5.8 Mathematical Maturity vs. Mathematical Inquisitiveness......Page 105
5.9 Mathematical Maturity is Not the Same as Knowledge......Page 108
5.10 Critical Thinking Skills......Page 109
5.11 Ideas of Piaget......Page 112
5.12 Reading and Thinking......Page 114
5.13 The Role of Writing......Page 115
5.14 Tenacity and Delayed Gratification......Page 116
5.15 Types of Intelligence......Page 119
5.16 Psychological Conditions......Page 120
5.17 What are Our Students’ Values? What are Our Students’ Goals?......Page 121
5.18 Intuition vs. Rigor......Page 123
6.0 Chapter Overview......Page 126
6.2 Key Attributes of Mathematical Maturity......Page 127
6.3 A Mathematician’s Miscellany......Page 129
7.0 Chapter Overview......Page 134
7.1 Who Needs Mathematical Maturity?......Page 135
7.2 The Role of Mathematical Maturity in Our World......Page 136
The Tree of Mathematical Maturity......Page 138
Etymology of the Word “Maturity”......Page 140
Bibliography......Page 142
Index......Page 148
About the Author......Page 156