Dover | 1963 | 77 páginas | pdf | 1,1 Mb
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domingo, 30 de março de 2014
Harmony of the World - 75 Years of Mathematics Magazine
Gerald L. Alexanderson e Peter Ross
Mathematical Association of America | 2007 | 302 páginas | rar - pdf |3,8 Mb
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Who would expect to find in the pages of Mathematics Magazine the first full treatment of one of the more important and oft-cited twentieth century theorems in analysis, the Stone-Weierstrass Theorem in an article by Marshall Stone himself? Where else would one look for proofs of trigonometric identities using commutative ring theory? Or one of the earliest and best expository articles on the then new Jones knot polynomials, an article that won the prestigious Chauvenet Prize? Or an amusing article purporting to show that the value of has been time dependent over the years? These and much more are in this collection of the best from Mathematics Magazine. Readers are inundated with new material in the many mathematical journals. Gems from past issues of Mathematics Magazine or the Monthly or the College Mathematics Journal are read with pleasure when they appear but get pushed into the background when the next issues arrive. So from time to time it is rewarding to go back and see just what marvelous material has been published over many years, articles now to some extent forgotten. There is history of mathematics (algebraic numbers, inequalities, probability and the Lebesgue integral, quaternions, Pólya s enumeration theorem, and group theory) and stories of mathematicians (Hypatia, Gauss, E. T. Bell, Hamilton, and Euler). The list of authors is star-studded: E. T. Bell, Otto Neugebauer, D. H. Lehmer, Morris Kline, Einar Hille, Richard Bellman, Judith Grabiner, Paul Erdos, B. L. van der Waerden, Paul R. Halmos, Doris Schattschneider, J. J. Burckhardt, Branko Grunbaum, and many more. Eight of the articles included have received the Carl B. Allendoerfer or Lester R. Ford Awards.
Contents
The name of each article is followed by a notation indicating the field of mathematics from which it comes: (Al) algebra, (AM) applied mathematics, (An) analysis, (CG) combinatorics and graph theory, (G) geometry, (H) history, (L) logic, (M) miscellaneous, (NT) number theory, (PS) probability/statistics, and (T) topology.
Introduction . .vii
A Brief History of Mathematics Magazine . . xi
Part I: The First Fifteen Years .. .1
Perfect Numbers, Zena Garrett (NT) .. .3
Rejected Papers of Three Famous Mathematicians, Arnold Emch (H) . 5
Review of Men of Mathematics, G. Waldo Dunnington (H) .. 9
Oslo under the Integral Sign, G. Waldo Dunnington (H) . 11
Vigeland’s Monument to Abel in Oslo, G. Waldo Dunnington (H) . .19
The History of Mathematics, Otto Neugebauer (H) . .. .23
Numerical Notations and Their Influence on Mathematics, D. H. Lehmer (NT) . .29
Part II: The 1940s .33
The Generalized Weierstrass Approximation Theorem, Marshall H. Stone (An) . . .35
Hypatia of Alexandria, A. W. Richeson (H) .. . 45
Gauss and the Early Development of Algebraic Numbers, E. T. Bell (Al) . . 51
Part III: The 1950s . . 69
The Harmony of the World, Morris Kline (M) .. 71
What Mathematics Has Meant to Me, E. T. Bell (M) .. .79
Mathematics and Mathematicians from Abel to Zermelo, Einar Hille (H) . 81
Inequalities, Richard Bellman (An) . . 95
A Number System with an Irrational Base, George Bergman (NT) . 99
Part IV: The 1960s .107
Generalizations of Theorems about Triangles, Carl B. Allendoerfer (G) . .109
A Radical Suggestion, Roy J. Dowling (NT) . .115
Topology and Analysis, R. C. Buck (An, T). 117
The Sequence fsin ng, C. Stanley Ogilvy (An) . .. . 121
Probability Theory and the Lebesgue Integral, Truman Botts (PS) . . . 123
On Round Pegs in Square Holes and Square Pegs in Round Holes, David Singmaster (G) . . 129
t : 1832–1879, Underwood Dudley (M) . . . . 133
Part V: The 1970s . . 135
Trigonometric Identities, Andy R. Magid (Al) . . .137
A Property of 70, Paul Erdos (NT) . . . .139
Hamilton’s Discovery of Quaternions, B. L. van der Waerden (Al, H) . . 143
Geometric Extremum Problems, G. D. Chakerian and L. H. Lange (G . .151
Polya’s Enumeration Theorem by Example, Alan Tucker (CG) . 161
Logic from A to G, Paul R. Halmos (L) . . . . 169
Tiling the Plane with Congruent Pentagons, Doris Schattschneider (G) .. . 175
Unstable Polyhedral Structures, Michael Goldberg (G) . . 191
Part VI: The 1980s . .197
Leonhard Euler, 1707–1783, J. J. Burckhardt (H) . . .199
Love Affairs and Differential Equations, Steven H. Strogatz (An) . 211
The Evolution of Group Theory, Israel Kleiner (Al) . . . 213
Design of an Oscillating Sprinkler, Bart Braden (AM) . . . 229
The Centrality of Mathematics in the History of Western Thought, Judith V. Grabiner (M) . ..237
Geometry Strikes Again, Branko Gr¨unbaum (G) .. 247
Why Your Classes Are Larger than “Average”, David Hemenway (PS) . 255
The New Polynomial Invariants of Knots and Links, W. B. R. Lickorish and Kenneth C. Millett (CG, T) . 257
Briefly Noted .. 273
The Problem Section . . . 279
Index . . . .281
About the Editors . . . 287
sábado, 29 de março de 2014
Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics
Ulianov Montano
Springer | 2014 | 224 páginas | rar - pdf | 1,1 Mb
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This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations are prevalent, however, rigorous investigation of them, of mathematical beauty, is much less common. The volume integrates the basic elements of aesthetics, as it has been developed over the last 200 years, with recent findings in neuropsychology as well as a good knowledge of mathematics.
The volume begins with a discussion of the reasons to interpret mathematical beauty in a literal or non-literal fashion, which also serves to survey historical and contemporary approaches to mathematical beauty. The author concludes that literal approaches are much more coherent and fruitful, however, much is yet to be done. In this respect two chapters are devoted to the revision and improvement of McAllister’s theory of the role of beauty in science. These antecedents are used as a foundation to formulate a naturalistic aesthetic theory. The central idea of the theory is that aesthetic phenomena should be seen as constituting a complex dynamical system which the author calls the aesthetic as processtheory.
The theory comprises explications of three central topics: aesthetic experience (in mathematics), aesthetic value and aesthetic judgment. The theory is applied in the final part of the volume and is used to account for the three most salient and often used aesthetic terms often used in mathematics: beautiful, elegant and ugly. This application of the theory serves to illustrate the theory in action, but also to further discuss and develop some details and to showcase the theory’s explanatory capabilities.
Contents
Introduction.
Part 1. Antecedents.
Chapter 1. On Non-literal Approaches.
Chapter 2. Beautiful, Literally.
Chapter 3. Ugly, Literally.
Chapter 4. Problems of the Aesthetic Induction.
Chapter 5. Naturalizing the Aesthetic Induction.
Part 2. An Aesthetics of Mathematics.
Chapter 6. Introduction to a Naturalistic Aesthetic Theory.
Chapter 7. Aesthetic Experience.
Chapter 8. Aesthetic Value.
Chapter 9. Aesthetic Judgement I: Concept.
Chapter 10. Aesthetic Judgement II: Functions.
Chapter 11. Mathematical Aesthetic Judgements.
Part 3. Applications.
Chapter 12. Case Analysis I: Beauty.
Chapter 13. Case Analysis II: Elegance.
Chapter 14. Case Analysis III: Ugliness, Revisited.
Chapter 15. Issues of Mathematical Beauty, Revisited.
A History of Astronomy
London Methuen 1907
online: archive.org
Routledge | 2013 | 434 páginas | rar - pdf | 5,52 Mb
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A History of Astronomy, first published in 1907, offers a comprehensive introduction to the steady development of the science since its inception in the ancient world up to the momentous progress of the nineteenth century. It includes biographical material relating to the most famous names in the study of astronomy – Copernicus, Galileo, Newton, Herschel – and their contributions, clear and accessible discussions of key discoveries, as well as detailing the incremental steps in technology with which many of the turning points in astronomy were intimately bound up.
CONTENTS
CHAP. PAGE
I. EARLY NOTIONS
II. THE EASTERN NATIONS OF ANTIQUITY 8
III. THE GREEKS 14
IV. THE ARABS 25
V. THE REVIVAL-COPERNICUS-TYCHO BRAHE 28
VI. KEPLER-GALl LEO 39
'VII. NEWTON 47
VIII. NEWTON'S SUCCESSORS: LAPLACE 53
IX. FLAMSTEED-HALLEy-BRADLEy-HERSCHEL 63
X. THE EARLY NINETEENTH CENTURy-NEPTUNE 73
XI. HERSCHEL-BESSEL-STRUVE • 83
XII. COMETS • 96
XIII. THE SUN-EcLIPSES-PARALLAX 103
XIV. GENERAL ASTRONOMY AND CELESTIAL MECHANICS 118
XV. OBSERVATORIES AND INSTRUMENTS • 132
XVI. ADJUSTMENT OF OBSERVATIONS. PERSONAL ERRORS 141
XVII. THE SUN 146
XVIII. SOLAR SPECTROSCOPY 159
XIX. SOLAR ECLIPSES-SPECTROSCOPY 169
XX. THE MOON 183
XXI. THE EARTH 192
XXII. THE INTERIOR PLANETS 201
XXIII. MARS 209
XXIV. MINOR PLANETS 219
XXV. THE MAJOR PLANETS 226
XXVI. THE SOLAR SYSTEM • 24I
XXVII. COMETS, METEORS, ZODIACAL LIGHT 247
XXVIII. THE STARS-CATALOGUES-PROPER MOTION-PARALLAX-MAGNITUDE 27I
XXIX. DOUBLE STARS 292
XXX. VARIABLE STARS 303
XXXI. CLUSTERS-NEBULIE-MILKY WAY. 318
XXXII. STELLAR SPECTROSCOPY 327
XXXIII. CONCLUSION • 340
Learning Mathematics and Logo
The MIT Press | 1992 | 492 páginas | pdf | 22,4 Mb
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These original essays summarize a decade of fruitful research and curriculum development using the LISP-derived language Logo. They discuss a range of issues in the areas of curriculum, learning, and mathematics, illustrating the ways in which Logo continues to provide a rich learning environment, one that allows pupil autonomy within challenging mathematical settings.
Essays in the first section discuss the link between Logo and the school mathematics curriculum, focusing on the ways in which pupils' Logo activities relate to and are influenced by the ideas they encounter in the context of school algebra and geometry.
In the second section the contributions take up pedagogical styles and strategies. They tackle such cognitive and metacognitive questions as, What range of learning styles can the Logo setting accommodate? How can teachers make sense of pupils' preferred strategies? And how can teachers help students to reflect on the strategies they are using?
Returning to the mathematical structures, essays in the third section consider a variety of mathematical ideas, drawing connections between mathematics and computing and showing the ways in which constructing Logo programs helps or does not help to illuminate the underlying mathematics.
Contents
Contributors vii
Series Foreword ix
Foreword by Seymour Papert xi
Preface xvii
Acknowledgments xxiii
LOGO IN THE CURRICULUM
Introduction to Part I 3
The Notion of Variable in the Context of Turtle Graphics 11
Joel Hillel
2 What Is Algebraic about Programming in Logo? 37
Rosamund Sutherland
3 Conceptually Defined Turtles 55
Herbert Loethe
4 The Turtle Metaphor as a ToOl for Children's Geometry 97
Chronis Kynigos
5 A Logo Microworld for Transformation Geometry 127
Laurie D. Edwards
II STYLES AND STRATEGIES
Introduction to Part II 159
6 LEGO-Logo: A Vehicle for Learning 165
Sylvia Weir
7 On Intra- and Interlndividual Differences in Children's Learning Styles 191
Tamara Lemerise
8 Mathematics in a Logo Environment: A Recursive Look at a Complex Phenomenon 223
Thomas E. Kieren
9 Between Logo and Mathematics: A Road of Tunnels and Bridges 247
Jean-Luc Gurtner
III EXPRESSING MATHEMATICAL STRUCTURES
Introduction to Part ", 27 1
10 Processes: A Dynamical Integration of Computer Science into Mathematical Education 279
Bruno Vitale
Of Geometry, Turtles, and Groups
Uri Leron and Rina Zazkis
Patterns, Permutations, and Groups
Trevor Fletcher
Avoiding Recursion 393
Brian Harvey
IV AFTERWORD
Introduction to Part IV 429
Looking Back and Looking Forward
Richard Noss and Celia Hoyles
Index 47 1
Eureka! Math Fun From Many Angles
David Lewis
Perigee Trade | 1955 | 199 páginas | pdf | 4,8 Mb
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Presents a variety of puzzles, problems, and paradoxes that test the reader's skills in logic and knowledge of mathematics
Presents a variety of puzzles, problems, and paradoxes that test the reader's skills in logic and knowledge of mathematics
Contents
Fun and Games
1. Twisted Topology
2. A Bag of Tricks and Treats
3. A Score of Games
4. The Magic's There
5. Rubiquity
Nifty Numerics
6. Palindromesemordnilap
7. A Pole Vaulter
8. A Timely Switch
Fallacies and Logic
9. If This Is Not a Chapter, My Name Is Raymond Smullyan
10. Thrice Befuddled
11. Better Mixed-Up Than Lost
And Even Dissection of Solids
12. Archimedes Anderson and the Case of the Sinister Plot 93
13. How to Dissect a Square and Other Marvels of Modern Biology 96
14. Geometer's Heaven 105
15. Hole in the Sphere 108
16. Convexstasy 110
17. Great Unsolved Problems 113
18. Out of This World 115
Photons Are Light Matter, Too
19. Archimedes Anderson and the Gambling Candidate 119
20. Once Upon a Time. . . 122
21. A Problem Fly 123
22. The Leading Series of Pisa 124
23. The Early Something Catches the Whatever 127
Shortcuts
24. A Speedier System of Solving 129
25. A Letter Home 135
26. In Which We Are Initiated Into the Secret
Society of Square Root Solvers
27. Heads and Legs
28. Noble Bases
29. A Division in Ancient Rye
30. Getting at the Root of the Problem
31. Or is it 32? Remumbt:r Nembers
Neat Numbers
32. Prime Time 151
33. A Sense of Balance 159
34. Perfect Numbers and Some Not-So-Perfect Numbers 161
Cranium Crackers and Cheese: Problems to Munch On
35. Classy Problems 165
36. LEITERS + DIGITS = FRUSTRATION 174
FUNdamental Ratios
37. Expand Your Mind
38. E? Ah!
39. A Section of Gold
40. A Bundled-Up Buyer
41. A Piece of Pi
Bibliography
Writing Strategies for Mathematics
(Reading and Writing Strategies)
Trisha Brummer e Sarah Kartchner Clark
Shell Education | 2013 - 2ª edição | 259 páginas | rar - PDF | 6,4 Mb
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1ª edição - 2008
Help students write mathematics content! This 2nd edition resource was created to support Common Core State Standards, provides an in-depth research base about literacy instruction, and includes key strategies to help students write and comprehend mathematics content. Designed in an easy-to-use format, this resource offers details approaches and activities with classroom examples by grade ranges and includes graphic organizer templates and digital resources to help teachers implement quickly and easily. Specific suggestions for differentiating instruction are also provided to help English language learners, gifted students, and students reading below grade level.
Table of Contents
Introduction
What Is Writing? . . . . . . . . . . . . . . . . . . . . . . . . . 5
Motivating Students to Write . . . . . . . . . . . . . . . 10
The Writing Process . . . . . . . . . . . . . . . . . . . . . 16
Writing Across the Curriculum . . . . . . . . . . . . . 20
Writing Instruction . . . . . . . . . . . . . . . . . . . . . . 25
How to Use This Book . . . . . . . . . . . . . . . . . . . 28
Correlation to Standards . . . . . . . . . . . . . . . . . . 29
Part 1: Writing to Learn
Developing Vocabulary
Developing Vocabulary Overview . . . . . . . . . . . 31
Word Wall . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Frayer Model . . . . . . . . . . . . . . . . . . . . . . . . 39
Concept of De nition Map . . . . . . . . . . . . . 43
List-Group-Label . . . . . . . . . . . . . . . . . . . . . 48
Vocabulary Self-Collection . . . . . . . . . . . . . 53
Possible Sentences . . . . . . . . . . . . . . . . . . . . 55
Word Trails . . . . . . . . . . . . . . . . . . . . . . . . . 59
Previewing and Reviewing
Previewing and Reviewing Overview . . . . . . . . 63
KWL Chart . . . . . . . . . . . . . . . . . . . . . . . . . 66
Think Sheet . . . . . . . . . . . . . . . . . . . . . . . . . 72
Free-Association Brainstorming . . . . . . . . . 76
Probable Passages . . . . . . . . . . . . . . . . . . . . 80
Guided Free Write . . . . . . . . . . . . . . . . . . . . 85
End-of-Class Re ection . . . . . . . . . . . . . . . . 89
Reader-Response Writing Chart . . . . . . . . . 92
Journal Writing
Journal Writing Overview . . . . . . . . . . . . . . . . . 96
Vocabulary Journal . . . . . . . . . . . . . . . . . . . 99
Dialogue Journal . . . . . . . . . . . . . . . . . . . . 103
Highlighted Journal . . . . . . . . . . . . . . . . . . 106
Key Phrase Journal . . . . . . . . . . . . . . . . . . 109
Double-Entry Journal . . . . . . . . . . . . . . . . 112
Critical Incident Journal . . . . . . . . . . . . . . 116
Three-Part Journal . . . . . . . . . . . . . . . . . . 119
Note-Taking
Note-Taking Overview . . . . . . . . . . . . . . . . . . 123
Cornell Note-Taking System . . . . . . . . . . . 125
Note-Taking System for Learning . . . . . . . 129
T-List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Using Diagrams and Maps
Using Diagrams and Maps Overview . . . . . . . 137
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Venn Diagram . . . . . . . . . . . . . . . . . . . . . . 143
Triangular Venn Diagram . . . . . . . . . . . . . 147
Cause-and-Effect Map . . . . . . . . . . . . . . . . 151
Semantic Word Map . . . . . . . . . . . . . . . . . 155
Concept Map . . . . . . . . . . . . . . . . . . . . . . . 159
Problem-Solution Map. . . . . . . . . . . . . . . . 162
Time Order Map . . . . . . . . . . . . . . . . . . . . 168
Part 2: Writing to Apply
Authoring
Authoring Overview . . . . . . . . . . . . . . . . . . . . 173
Guided Writing Procedure . . . . . . . . . . . . 175
Reading-Writing Workbench . . . . . . . . . . . 178
Author’s Chair . . . . . . . . . . . . . . . . . . . . . . 182
Read, Encode, Annotate, Ponder . . . . . . . . 186
Summarizing
Summarizing Overview . . . . . . . . . . . . . . . . . 191
GIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Key Words . . . . . . . . . . . . . . . . . . . . . . . . . 197
Guided Reading and Summarizing Procedure .. . . . . 199
Applying Knowledge
Applying Knowledge Overview . . . . . . . . . . . 204
Summary-Writing Microtheme . . . . . . . . . 208
Thesis-Support Microtheme . . . . . . . . . . . 210
Data-Provided Microtheme . . . . . . . . . . . . 212
Quandary-Posing Microtheme . . . . . . . . . 214
RAFT Assignment . . . . . . . . . . . . . . . . . . 216
Business Letter . . . . . . . . . . . . . . . . . . . . . 218
Friendly Letter . . . . . . . . . . . . . . . . . . . . . . 220
Data Report . . . . . . . . . . . . . . . . . . . . . . . . 222
Newspaper Article . . . . . . . . . . . . . . . . . . . 224
Mathematics Fiction Story . . . . . . . . . . . . 226
Research Report . . . . . . . . . . . . . . . . . . . . 228
Part 3: Assessing Writing
Assessing Writing
Assessing Writing Overview . . . . . . . . . . . . . . 230
Holistic Assessment . . . . . . . . . . . . . . . . . . 233
Analytic Assessment . . . . . . . . . . . . . . . . . 235
Primary Trait Assessment . . . . . . . . . . . . . 237
Self-Assessment . . . . . . . . . . . . . . . . . . . . . 239
Peer Assessment . . . . . . . . . . . . . . . . . . . . 242
Teacher Conference . . . . . . . . . . . . . . . . . . 244
Appendix A: Additional Resources . . . . . . . 246
Appendix B: References Cited . . . . . . . . . . . 247
Appendix C: Suggestions for Further Reading . . . 253
Appendix D: Contents of the Digital Resource CD . . 254
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