The SimCalc Vision and Contributions: Democratizing Access to Important Mathematics

 (Advances in Mathematics Education, 4)

Stephen J. Hegedus e Jeremy Roschelle

 Springer | 2013 | 471 páginas | rar - pdf | 9 Mb

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This volume provides essential guidance for transforming mathematics learning in schools through the use of innovative technology, pedagogy, and curriculum. It presents clear, rigorous evidence of the impact technology can have in improving students learning of important yet complex mathematical concepts - and goes beyond a focus on technology alone to clearly explain how teacher professional development, pedagogy, curriculum, and student participation and identity each play an essential role in transforming mathematics classrooms with technology. Further, evidence of effectiveness is complemented by insightful case studies of how key factors lead to enhancing learning, including the contributions of  design research, classroom discourse, and meaningful assessment. 

The volume organizes over 15 years of sustained research by multiple investigators in different states and countries who together developed an approach called "SimCalc" that radically transforms how Algebra and Calculus are taught.
The SimCalc program engages students around simulated motions, such as races on a soccer field, and builds understanding using visual representations such as graphs, and familiar representations such as stories to help students to develop meaning for more abstract mathematical symbols. Further, the SimCalc program leverages classroom wireless networks to increase participation by all students in doing, talking about, and reflecting on mathematics. Unlike many technology programs, SimCalc research shows the benefits of balanced attention to curriculum, pedagogy, teacher professional development, assessment and technology - and has proven effectiveness results at the scale of hundreds of schools and classrooms.
Combining the findings of multiple investigators in one accessible volume reveals the depth and breadth of the research program, and engages readers interested in:
*        Designing innovative curriculum, software, and professional development
·         Effective uses of technology to improve mathematics education
*        Creating integrated systems of teaching that transform mathematics classrooms
*        Scaling up new pedagogies to hundreds of schools and classrooms
*       Conducting research that really matters for the future of mathematics learning

*        Engaging students in deeply learning the important concepts in mathematics

Contents

Part I Philosophy & Background

Ubiratan D’Ambrosio

Introduction: Major Themes, Technologies, and Timeline  . 5

Jeremy Roschelle and Stephen Hegedus

The Mathematics of Change and Variation from a Millennial Perspective: New Content, New Context .. 13

James J. Kaput and Jeremy Roschelle

From Static to Dynamic Mathematics: Historical and Representational Perspectives . .. 27

Luis Moreno-Armella and Stephen Hegedus

Intersecting Representation and Communication Infrastructures .. . 47

Stephen Hegedus and Luis Moreno-Armella

Part II Aspects of Design

Reflections on Significant Developments in Designing SimCalc Software . 65

James Burke, Stephen Hegedus, and Ryan Robidoux

Designing for Generative Activities: Expanding Spaces for Learning and Teaching . . . . 85

Nancy Ares

SimCalc and the Networked Classroom . . .. . 99

Corey Brady, Tobin White, Sarah Davis, and Stephen Hegedus

Part III Impacts from Large-Scale Research

Jinfa Cai

SimCalc at Scale: Three Studies Examine the Integration of Technology, Curriculum, and Professional Development for Advancing Middle School Mathematics .  . 125

Jeremy Roschelle and Nicole Shechtman

Learning and Participation in High School Classrooms . . 145

Sara Dalton and Stephen Hegedus

Development of Student and Teacher Assessments in the Scaling Up SimCalc Project  . 167

Nicole Shechtman, Geneva Haertel, Jeremy Roschelle, Jennifer Knudsen, and Corinne Singleton

Sustainable Use of Dynamic Representational Environments: Toward a District-Wide Adoption of SimCalc-Based Materials . . 183

Phil Vahey, George J. Roy, and Vivian Fueyo

Impact of Classroom Connectivity on Learning and Participation .  203

Stephen Hegedus, Luis Moreno-Armella, Sara Dalton, Arden Brookstein, and John Tapper

Part IV Impacts from Small-Scale Research

Mathematical Discourse as a Process that Mediates Learning in SimCalc Classrooms . . 233

Jessica Pierson Bishop

Scaling Up Innovative Mathematics in the Middle Grades: Case Studies of “Good Enough” Enactments .  . . 251

Susan B. Empson, Steven Greenstein, Luz Maldonado, and Jeremy Roschelle

Changing from the Inside Out: SimCalc Teacher Changes in Beliefs and Practices . . 271

John Tapper

Connection Making: Capitalizing on the Affordances of Dynamic Representations Through Mathematically Relevant Questioning . . 285

Chandra Hawley Orrill

“They Need to Be Solid in Standard Skills First”: How Standards Can Become the Upper Bound  299

Margaret Dickey-Kurdziolek and Deborah Tatar

Part V International Contributions

Developing and Enhancing Elementary School Students’ Higher Order Mathematical Thinking with SimCalc .  319

Demetra Pitta-Pantazi, Paraskevi Sophocleous, and Constantinos Christou Adapting SimCalc to Different School Mathematics Cultures: A Case Study from Brazil . . 341

Rosana Nogueira de Lima, Lulu Healy, and Tânia M.M. Campos

Mathematical Modeling with SimCalc: Enhancing Students’ Complex Problem Solving Skills Using a Modeling Approach . . . 363

Nicholas G. Mousoulides

Approaching Calculus with SimCalc: Linking Derivative and Antiderivative . . . 383

Patricia Salinas

Part VI Extensions, Commentaries, & Future Visions

You Can Lead a Horse to Water: Issues in Deepening Learning Through Deepening Teaching .. 403

John Mason

Modeling as a Means for Making Powerful Ideas Accessible to Children at an Early Age . 419

Richard Lesh, Lyn English, Serife Sevis, and Chanda Riggs

The Kaputian Program and Its Relation to DNR-Based Instruction: A Common Commitment to the Development of Mathematics with Meaning .  437

Guershon Harel

The Evolution of Technology and the Mathematics of Change and Variation: Using Human Perceptions and Emotions to Make Sense of Powerful Ideas . .449

David Tall

Conversation About SimCalc, Its Evolution and Lessons Along the Way . 463

Eric Hamilton and Nora Sabelli

Index . . . 475

Wizards, aliens, and starships : physics and math in fantasy and science fiction

Charles L. Adler

Princeton University Press | 2014 | 392 páginas | rar - epub | 2,05 Mb

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Wizards, Aliens, and Starships will speak to anyone wanting to know about the correct--and incorrect--science of science fiction and fantasy.

From teleportation and space elevators to alien contact and interstellar travel, science fiction and fantasy writers have come up with some brilliant and innovative ideas. Yet how plausible are these ideas--for instance, could Mr. Weasley's flying car in the Harry Potter books really exist? Which concepts might actually happen, and which ones wouldn't work at all? Wizards, Aliens, and Starships delves into the most extraordinary details in science fiction and fantasy--such as time warps, shape changing, rocket launches, and illumination by floating candle--and shows readers the physics and math behind the phenomena.

With simple mathematical models, and in most cases using no more than high school algebra, Charles Adler ranges across a plethora of remarkable imaginings, from the works of Ursula K. Le Guin to Star Trek and Avatar, to explore what might become reality. Adler explains why fantasy in the Harry Potter and Dresden Files novels cannot adhere strictly to scientific laws, and when magic might make scientific sense in the muggle world. He examines space travel and wonders why it isn't cheaper and more common today. Adler also discusses exoplanets and how the search for alien life has shifted from radio communications to space-based telescopes. He concludes by investigating the future survival of humanity and other intelligent races. Throughout, he cites an abundance of science fiction and fantasy authors, and includes concise descriptions of stories as well as an appendix on Newton's laws of motion.

CONTENTS

1 PLAYING THE GAME 1

1.1 The Purpose of the Book 1

1.2 The Assumptions I Make 3

1.3 Organization 4

1.4 The Mathematics and Physics You Need 5

1.5 Energy and Power 6

I POTTER PHYSICS 11

2 HARRY POTTER AND THE GREAT CONSERVATION LAWS 13

2.1 The Taxonomy of Fantasy 13

2.2 Transfiguration and the Conservation of Mass 14

2.3 Disapparition and the Conservation of Momentum 16

2.4 Reparo and the Second Law of Thermodynamics 21

3 WHY HOGWARTS IS SO DARK 27

3.1 Magic versus Technology 27

3.2 Illumination 28

4 FANTASTIC BEASTS AND HOW TO DISPROVE THEM 38

4.1 Hic sunt Dracones 38

4.2 How to Build a Giant 39

4.3 Kleiber’s Law, Part 1: Mermaids 45

4.4 Kleiber’s Law, Part 2: Owls, Dragons, Hippogriffs, and Other Flying Beasts 49

II SPACE TRAVEL 57

5 WHY COMPUTERS GET BETTER AND CARS CAN’T (MUCH) 59

5.1 The Future of Transportation 59

5.2 The Reality of Space Travel 61

5.3 The Energetics of Computation 63

5.4 The Energetics of the Regular and the Flying Car 64

5.5 Suborbital Flights 68

6 VACATIONS IN SPACE 71

6.1 The Future in Science Fiction: Cheap, Easy Space Travel? 71

6.2 Orbital Mechanics 74

6.3 Halfway to Anywhere: The Energetics of Spaceflight 74

6.4 Financing Space Travel 82

7 SPACE COLONIES 86

7.1 Habitats in Space 86

7.2 O’Neill Colonies 87

7.3 Matters of Gravity 89

7.4 Artificial “Gravity” on a Space Station 93

7.5 The Lagrange Points 103

7.6 Off-Earth Ecology and Energy Issues 106

7.7 The Sticker Price 112

8 THE SPACE ELEVATOR 115

8.1 Ascending into Orbit 115

8.2 The Physics of Geosynchronous Orbits 116

8.3 What Is a Space Elevator, and Why Would We Want One? 118

8.4 Why Buildings Stand Up—or Fall Down 119

8.5 Stresses and Strains: Carbon Nanotubes 122

8.6 Energy, “Climbers,” Lasers, and Propulsion 123

8.7 How Likely Is It? 125

8.8 The Unapproximated Elevator 127

9 MANNED INTERPLANETARY TRAVEL 130

9.1 It’s Not an Ocean Voyage or a Plane Ride 130

9.2 Kepler’s Three Laws 131

9.3 The Hohmann Transfer Orbit 134

9.4 Delta v and All That 136

9.5 Getting Back 137

9.6 Gravitational Slingshots and Chaotic Orbits 138

9.7 Costs 142

10 ADVANCED PROPULSION SYSTEMS 145

10.1 Getting There Quickly 145

10.2 Why Chemical Propulsion Won’t Work 146

10.3 The Most Famous Formula in Physics 147

10.4 Advanced Propulsion Ideas 148

10.5 Old “Bang-Bang”: The Orion Drive 153

10.6 Prospects for Interplanetary Travel 155

11 SPECULATIVE PROPULSION SYSTEMS 157

11.1 More Speculative Propulsion Systems 157

11.2 Mass Ratios for Matter-Antimatter Propulsion Systems 168

11.3 Radiation Problems 173

12 INTERSTELLAR TRAVEL AND RELATIVITY 176

12.1 Time Enough for Anything 176

12.2 Was Einstein Right? 178

12.3 Some Subtleties 182

12.4 Constant Acceleration in Relativity 184

13 FASTER-THAN-LIGHT TRAVEL AND TIME TRAVEL 188

13.1 The Realistic Answer 188

13.2 The Unrealistic Answer 188

13.3 Why FTL Means Time Travel 190

13.4 The General Theory 193

13.5 Gravitational Time Dilation and Black Holes 195

13.6 Wormholes and Exotic Matter 198

13.7 The Grandfather Paradox and Other Oddities 205

III WORLDS AND ALIENS 215

14 DESIGNING A HABITABLE PLANET 217

14.1 Adler’s Mantra 218

14.2 Type of Star 221

14.3 Planetary Distance from Its Star 226

14.4 The Greenhouse Effect 229

14.5 Orbital Eccentricity 232

14.6 Planetary Size and Atmospheric Retention 233

14.7 The Anna Karenina Principle and Habitable Planets 237

14.8 Imponderables 239

15 THE SCIENTIFIC SEARCH FOR SPOCK 242

15.1 Exoplanets and Exoplants 242

15.2 Doppler Technique 246

15.3 Transits and the Kepler Mission 249

15.4 The Spectral Signatures of Life 250

15.5 Alien Photosynthesis 251

16 THE MATHEMATICS OF TALKING WITH ALIENS 255

16.1 Three Views of Alien Intelligences 255

16.2 Motivation for Alien Contact 259

16.3 Drake-Equation Models and the Mathematics of Alien Contact 267

IV YEAR GOOGOL 273

17 THE SHORT-TERM SURVIVAL OF HUMANITY 275

17.1 This Is the Way the World Will End 275

17.2 The Short-Term: Man-Made Catastrophes 275

18 WORLD-BUILDING 292

18.1 Terraforming 292

18.2 Characteristics of Mars 294

18.3 Temperature and the Martian Atmosphere 295

18.4 Atmospheric Oxygen 299

18.5 Economics 301

19 DYSON SPHERES AND RINGWORLDS 303

19.1 Dyson’s Sphere 303

19.2 The Dyson Net 305

19.3 Niven’s Ringworld 311

19.4 The Ringworld, GPS, and Ehrenfest’s Paradox 318

19.5 The Ringworld Is Unstable! 320

19.6 Getting There from Here—and Do We Need To? 324

20 ADVANCED CIVILIZATIONS AND THE KARDASHEV SCALE 326

20.1 The Kardashev Scale 326

20.2 Our Type 0.7 Civilization 327

20.3 Type I Civilizations 329

20.4 Moving Upward 331

20.5 Type II Civilizations 332

20.6 Type III Civilizations 334

21 A GOOGOL YEARS 336

21.1 The Future of the Future 336

21.2 The “Short Term”: Up to 500 Million Years or so 336

21.3 The “Medium Term”: Up to about 1013 Years 338

21.4 The “Long Term”: Up to a Googol Years 341

21.5 Black Hole–Powered Civilizations 344

21.6 Protons Decay—or Do They? 346

21.7 A Googol Years—All the Black Holes Evaporate 346

21.8 Our Last Bow 349

Acknowledgments 351

Appendix: Newton’s Three Laws of Motion 353

Bibliography 359

Index 371

Goedel's Way: Exploits into an undecidable world

Gregory Chaitin, Francisco A Doria e Newton C.A. da Costa

CRC Press | 2011 | 162 páginas | pdf | 1,1 Mb

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Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein’s general relativity, as he proved that Einstein’s theory admits time machines. 
The Gödel incompleteness theorem - one cannot prove nor disprove all true mathematical sentences in the usual formal mathematical systems- is frequently presented in textbooks as something that happens in the rarefied realm of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life.
This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book’s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer science.
See also: http://www.youtube.com/watch?v=REy9noY5Sg8

Contents
1. Gödel, Turing 
2. Complexity, randomness 
3. A list of problems 
4. The halting function and its avatars 
5. Entropy, P vs. NP
6. Forays into uncharted landscapes.

The Child’s Conception of Geometry

 

Jean Piaget, Barbel Inhelder e Alina Szeminska

International Library of Psychology

Routledge |1999 - 2ª edição | 420 páginas | rar - pdf |7,9 Mb

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The Child’s Conception of Geometry examines the development of geometric concepts in young children.

This volume from Piaget’s laboratory in Geneva deals primarily with the development of notions of measurement and geometrical concepts like coordinates, angles, and areas. It is a companion piece to The Child’s Conception of Space.

CONTENTS
Preface page vii
PART ONE - INTRODUCTION
I Change of Position 3
II Spontaneous Measurement 27
PART TWO - CONSERVATION AND MEASUREMENT OF LENGTH
III Reconstructing Relations of Distance 69
IV Change of Position and the Conservation of Length 90
V The Conservation and Measurement of Length 104
VI Subdividing a Straight Line 128
PART THREE - RECTANGULAR COORDINATES, ANGLES AND CURVES
VII Locating a Point in Two or Three Dimensional Space 153
VIII Angular Measurement 173
IX Two Problems of Geometrical Loci: the Straight Line and the Circle 209
X Representation of Circles, Mechanical and Composite Curves 226
PART FOUR - AREAS AND SOLIDS
XI The Conservation and Measurement of an Area and Subtracting Smaller Congruent Areas from Larger Congruent Areas 
XII Subdivision of Areas and the Concept of Fractions 302
XIII Doubling an Area or a Volume 336
XIV The Conservation and Measurement of Volume 354
PART FIVE CONCLUSIONS
XV The Construction of Euclidean Space: Three Levels 389
Index 409

Mathematics of the Transcendental: Onto-logy and being-there

 

Alain Badiou, A.J. Bartlett e Alex Ling

Bloomsbury Academic | 2014 | 291 páginas | rar - pdf | 2,74 Mb

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In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of Category Theory, demonstrating their internal logic and veracity, their derivation and distinction from Set Theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. This important book combines both his elaboration of the disjunctive synthesis between ontology and onto-logy (the discourses of being as such and being-appearing) from the perspective of Category Theory and the categorial basis of his philosophical conception of 'being there'.
Hitherto unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of Category Theory. The book is an essential aid to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.

TABLE OF CONTENTS
Translators’ Introduction: The Categorial Imperative 1
PART ONE TOPOS, OR LOGICS OF ONTO-LOGY: AN INTRODUCTION FOR PHILOSOPHERS 11
1 General Aim 13
2 Preliminary Definitions 17
3 The Size of a Category 21
4 Limit and Universality 27
5 Some Fundamental Concepts 29
6 Duality 37
7 Isomorphism 41
8 Exponentiation 45
9 Universe, 1: Closed Cartesian Categories 51
10 Structures of Immanence, 1: Philosophical Considerations 55
11 Structures of Immanence, 2: Sub-Object 59
12 Structures of Immanence, 3: Elements of an Object 63
13 ‘Elementary’ Clarification of Exponentiation 67
14 Central Object (or Sub-Object Classifier) 71
15 The True, the False, Negation and More 77
16 The Central Object as Linguistic Power 85
17 Universe, 2: The Concept of Topos 89
18 Ontology of the Void and Difference 95
19 Mono., Epi., Equ., and Other Arrows 99
20 Topoi as Logical Places 113
21 Internal Algebra of 1 123
22 Ontology of the Void and Excluded Middle 141
23 A Minimal Classical Model 147
24 A Minimal Non-Classical Model 151
PART TWO BEING THERE: MATHEMATICS OF THE TRANSCENDENTAL 163
Introduction 165
A. Transcendental Structures 171
B. Transcendental Connections 183
B.1. Connections between the transcendental and set-theoretic ontology: Boolean algebras 183
B.2. Connections between the transcendental and logic in its ordinary sense (propositional logic and first order predicate logic) 195
B.3. Connection between the transcendental and the general theory of localizations: Topology 202
C. Theory of Appearing and Objectivity 217
D. Transcendental Projections: Theory of Localization 235
E. Theory of Relations: Situation as Universe 249
Appendix: On Three Different Concepts of Identity Between Two
Multiples or Two Beings 265
Translator’s Endnotes 269
Index 277

150 Puzzles in Crypt-Arithmetic

Maxey Brooke

Dover | 1963 | 77 páginas | pdf | 1,1 Mb

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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