Mathematical Reflections: In a Room with Many Mirrors

Couverture
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

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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.

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