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A Century of Advancing Mathematics A Century of Advancing Mathematics c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2015936096 Print ISBN: 978-0-88385-588-1 Electronic ISBN: 978-1-61444-522-7 Printed in the United States of America Current Printing (last digit): 10987654321 A Century of Advancing Mathematics Stephen F. Kennedy, Editor Associate Editors Donald J. Albers Gerald L. Alexanderson Della Dumbaugh Frank A. Farris Deanna B. Haunsperger Paul Zorn ® Published and distributed by The Mathematical Association of America Contents Preface ........................................................................ ix Part I Mathematical Developments 1 The Hyperbolic Revolution: From Topology to Geometry, and Back .................. 3 Francis Bonahon A CenturyofComplexDynamics ................................................ 15 Daniel Alexander and Robert L. Devaney Map-ColoringProblems ........................................................ 35 Robin Wilson SixMilestonesinGeometry ..................................................... 51 Frank Morgan Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-GenerationExplosionof Combinatorics ............................ 65 Eric S. Egge WhatIstheBestApproachtoCountingPrimes? .................................. 83 Andrew Granville A CenturyofEllipticCurves ................................................... 117 Joseph H. Silverman Part II Historical Developments 133 The Mathematical Association of America: Its First 100 Years .................... 135 David E. Zitarelli The Stratification of the American Mathematical Community: The Mathematical Association of America and the American Mathematical Society, 1915–1925 . 159 Karen Hunger Parshall Time and Place: Sustainingthe American Mathematical Community ............... 177 Della Dumbaugh Abstract (Modern) Algebra in America 1870–1950:A Brief Account .............. 191 Israel Kleiner Part III Pedagogical Developments 217 The History of the Undergraduate Program in Mathematics in theUnitedStates ..... 219 Alan Tucker v vi A Century of Advancing Mathematics Inquiry-BasedLearningThroughthe LifeoftheMAA ............................ 239 Michael Starbird A PassporttoPleasure ......................................................... 253 Bob Kaplan and Ellen Kaplan Strength in Numbers: Broadening the View of the Mathematics Major ............. 257 Rhonda Hughes A Historyof Undergraduate Research in Mathematics ............................ 263 Joseph A. Gallian The CalculusReformMovement: A Personal Account ............................ 275 Paul Zorn Introducing ex ................................................................ 283 Gilbert Strang Part IV Computational Developments 295 Computational Experiences in the Pre-Electronic Days .. ... ...................... 297 Philip J. Davis A CenturyofVisualization:One Geometer’s View ............................... 301 Thomas F. Banchoff TheFutureofMathematics:1965to2065 ....................................... 313 Jonathan M. Borwein Part V Culture and Communities 331 Philosophy of Mathematics: What Has Happened Since G¨odel’s Results? ........... 333 Bonnie Gold Twelve Classics People who Love Mathematics Should Know; or, “Whatdoyoumean, youhaven’t read E.T.Bell? ............................ 351 Gerald L. Alexanderson The Dramatic Life of Mathematics: A Centennial History of the Intersection of Mathematics and Theater in a Prologue, Three Acts, and an Epilogue .......... 365 Stephen D. Abbott 2007:TheYearofEuler ....................................................... 379 William Dunham The PutnamCompetition:Origin,Lore, Structure ................................ 387 Leonard F. Klosinski GettingInvolvedwiththeMAA:APath Less Traveled ........................... 393 Ezra “Bud” Brown HenryL.Alder ............................................................... 397 Donald J. Albers and Gerald L. Alexanderson LidaK.Barrett ............................................................... 401 Kenneth A. Ross RalphP.Boas ................................................................ 405 Daniel Zelinsky Contents vii LeonardGillman—Reminiscences ............................................ 407 Martha J. Siegel PaulHalmos:NoApologies .................................................... 411 John Ewing IvanNiven ................................................................... 415 Kenneth A. Ross GeorgeP´olyaandtheMAA .................................................... 421 Gerald L. Alexanderson Preface Searching, in the early years of the twentieth century, for a source of support for the fledg- ing and financially unstable American Mathematical Monthly Herbert Ellsworth Slaught conceived the idea of a new society. Our MAA was founded in 1915 to serve as a home for the Monthly. The full details of that story are contained in David Zitarelli’s contribution to this volume beginning on page 135. Within a decade we had published our first book, Calculus of Variations by G. A. Bliss. We now publishthree world-class print journals, one magazine, a newsmagazine, eight book series, and a host of online material. Our identity, from our inceptionrising from the Monthly’sneed for support up to the present-day centen- nial celebration, is firmly rooted in extraordinary exposition of mathematics. This makes it fitting to produce a volume of essays by some of the great mathematicians and expositors writing today as part of that celebration. But, Slaught’s vision was larger and the MAA’s mission has always been much more than just producing great expository mathematics. That mission—to advance the mathe- matical sciences, especially at the collegiate level—has been reflected in the Association’s deep engagement with pedagogical practices and public policy issues, with the history and philosophy of mathematics, and with the connections between mathematics and the arts and other sciences. It seems only natural for this volume to sample from the entire broad cultural sweep of mathematics throughout the century of the MAA’s existence. To organize this broad collection of material, the present volume is divided thematically into five sections. First, of course, comes the mathematics. The MAA century began, mathematically speaking, at the tail end of the move towards axiomatization and formalism that started in the nineteenth century. That movement, finished off by Kurt G¨odel’s stunner in the early thirties, gave way to a movement towards deep abstraction and ever increasing distance be- tween, so-called, pure and applied mathematics. The introduction of computing machines has, over the past decades, heightened interest in heuristic and algorithmicthinking and en- ergized areas like chaotic dynamics and combinatorics where computing power can reveal new phenomena and prime new intuitions. Our section on mathematical developments includes some legendary, and some new, voices in mathematical exposition. Bob Devaney and Dan Alexander chronicle the early history of complex dynamics, touring the highlights up to the present day. Eric Egge takes on the challengingtask of givingus a lookat the explosionof interest in combinatoricsover the last few decades, focusing on the astounding developments surrounding the Stanley- Wilf Conjecture. Along the lines of famous conjectures that have fallen—Joe Silverman will show us some of the details that unraveled Fermat’s Last Theorem and Robin Wil- son takes us on a historico-mathematical magic ride through the story of the Four-Color ix x A Century of Advancing Mathematics Theorem. Frank Morgan chooses his favorite half-dozen geometric milestones of the last few decades and Andrew Granville gives us a new way to think about the Prime Number Theorem after a deep, and deeply historical, tour through our efforts to count the primes. Finally, Francis Bonahon pays tribute to the contributions of Bill Thurston in unearthing the deep connections between geometry, particularly hyperbolic geometry, and topology. From the very beginning the MAA has been about more than just world-class expo- sition; within three months of our founding the organization had constituted the National Committee on Mathematics Requirements charged with investigating the teaching of sec- ondary and post-secondary mathematics, making curricular recommendations, and working towards improving the teaching of mathematics. It is fitting that we devote a section of this volume to the pedagogy of mathematics. A commitment to effective teaching and learning of mathematics is in the organization’s DNA; the perceived lack of such a commitment from our sister organization contributed to the movement that led to the MAA’s found- ing. Alan Tucker leads off with a fascinating history of the evolution of the undergraduate mathematics major. Joe Gallian, the founding father of the mathematics REU, recounts the early history of that pedagogical innovation. Michael Starbird writes historically and per- suasively about inquiry-based pedagogy, while Bob and Ellen Kaplan—the founders of the original Math Circle—write, passionately and lyrically, on the same topic. Rhonda Hughes offers advice on attracting underrepresented groups to our subject. Gil Strang tells us how we ought to teach the exponential function; and Paul Zorn, a key player in the Calculus Reform movement, tells us what that movement hoped to achieve. Zorn’s contributionhighlightshow calculus teaching was changed, even given new life, by computing technology. Egge and Alexander & Devaney made clear that massive compu- tational power wrought seismic
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