# Mathematical Reflections: In a Room with Many Mirrors

Springer Science & Business Media, 31 juil. 1998 - 352 pages
Focusing Your Attention The purpose of this book is Cat least) twofold. First, we want to show you what mathematics is, what it is about, and how it is done-by those who do it successfully. We are, in fact, trying to give effect to what we call, in Section 9.3, our basic principle of mathematical instruction, asserting that "mathematics must be taught so that students comprehend how and why mathematics is qone by those who do it successfully./I However, our second purpose is quite as important. We want to attract you-and, through you, future readers-to mathematics. There is general agreement in the (so-called) civilized world that mathematics is important, but only a very small minority of those who make contact with mathematics in their early education would describe it as delightful. We want to correct the false impression of mathematics as a combination of skill and drudgery, and to re inforce for our readers a picture of mathematics as an exciting, stimulating and engrossing activity; as a world of accessible ideas rather than a world of incomprehensible techniques; as an area of continued interest and investigation and not a set of procedures set in stone.

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### Table des matières

 Going Down the Drain 3 12 Cobwebs 11 13 Consolidation 17 14 Fibonacci Strikes 22 15 Dénouement 24 Final Break 25 Answers for Final Break 26 A Far Nicer Arithmetic 27
 54 Up the Wall 168 Final Break 180 References 184 Answers for Final Break 186 Pascal Euler Triangles Windmills 187 62 The Binomial Theorem 189 63 The Pascal Triangle and Windmill 198 64 The Pascal Flower and the Generalized Star of David 213

 Getting Ready for the Fun 35 Some Beautiful Mathematics 39 The Same But Different 46 25 Primes Codes and Security 51 Tricks of the Trade 56 Final Break 60 Answers for Final Break 61 Fibonacci and Lucas Numbers 63 32 The Explanation Begins 65 33 Divisibility Properties 74 34 The Number Trick Finally Explained 78 35 More About Divisibility 80 36 A Little Geometry 83 Final Break 87 Answers for Final Break 88 PaperFolding and Number Theory 89 Folding 2Periods Regular Polygons 93 43 Folding Numbers 105 44 Some Mathematical Tidbits 119 45 General Folding Procedures 125 46 The QuasiOrder Theorem 130 A Little Solid Geometry 140 Final Break 143 Quilts and Other RealWorld Decorative Geometry 145 52 Variations 155 53 Round and Round 164
 65 Eulerian Numbers and Weighted Sums 219 66 Even Deeper Mysteries 240 References 249 Hair and Beyond 251 72 The Biggest Number 255 73 The Big Infinity 257 74 Other Sets of Cardinality n₀ 263 75 Schröder and Bernstein 270 76 Cardinal Arthmetic 271 77 Even More infinities? 272 Final Break 275 References 276 Answers for Final Break 277 An Introduction to the Mathematics of Fractal Geometry 279 82 Intuitive Notion of SelfSimilarity 282 83 The Tent Map and the Logistic Map 292 84 Some More Sophisticated Material 300 Final Break 316 References 319 Answers for Final Break 321 Some of Our Own Reflections 325 91 General Principles 326 92 Specific Principles 331 Principles of Mathematical Pedagogy References 338 Index 341

### Fréquemment cités

Page xi - Certain fixed quantities which appear very often have standard names; thus, the ratio of the circumference of a circle to its diameter is always denoted by if.

### Références à ce livre

 A First Course in Geometric Topology and Differential GeometryEthan D. BlochAperçu limité - 1997
 SymmetryHans WalserAperçu limité - 2000
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