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THE CALCULUS THE CALCULUS A Genetic Approach OTTO TOEPLITZ New Foreword by David Bressoud Published in Association with the Mathematical Association of America The University of Chicago Press Chicago · London Thepresentbook isatranslation, edited aftertheauthor'sdeath byGottfried Kothe andtranslatedintoEnglishbyLuiseLange.TheGerman edition, DieEntwicklungder Infinitesimalrechnungw,as published by Springer-Verlag. TheUniversityof Chicago Press,Chicago 60637 TheUniversityof Chicago Press,ltd., London © 1963 byTheUniversityofChicago Foreword ©2007 byTheUniversityof Chicago All rightsreserved.Published2007 PrintedintheUnitedStatesofAmerica 16 15 14 13 12 11 10 09 08 07 2 3 4 5 ISBN-13: 978-0-226-80668-6 (paper) ISBN-10: 0-226-80668-5 (paper) Library ofCongressCataloging-in-Publication Data Toeplitz,Otto, 1881-1940. [Entwicklungder Infinitesimalrechnung. English] Thecalculus: ageneticapproach / OttoToeplitz;with anewforeword byDavid M. Bressoud. p. cm. Includesbibliographical referencesand index. ISBN-13: 978-0-226-80668-6 (pbk. :alk. paper) ISBN-10: 0-226-80668-5 (pbk. :alk. paper) 1. Calculus. 2. Processes, Infinite. I.Title. QA303.T64152007 515-dc22 2006034201 § Thepaper usedinthispublication meetstheminimumrequirementsofthe American National Standard for Information Sciences-Permanence of Paperfor PrintedLibraryMaterials, ANSI Z39.48-1992. FOREWORDTO THECALCULUS: A GENETICAPPROACHBYOTTOTOEPLITZ September 30, 2006 Otto Toeplitz isbestknown forhiscontributions tomathematics, but hewas also an avid student ofitshistory. He understood how useful this history could beinin forming and shaping the pedagogy ofmathematics. This book, the first part ofan uncompleted manuscript, presents his vision ofan historically informed pedagogy for the teaching ofcalculus. Though written inthe 1930s,it has much to tell usto day about how we might-even how weshould-teach calculus. Welive inan ageofagreat democratization ofcalculus. Acourse once reserved for an elite few is now moving into the standard college preparatory curriculum. This began inthe 1950s,but the movement has accelerated inthe past fewdecades as knowledge ofcalculus has come to be viewed as aprerequisite for admission to the best colleges and universities, almost irrespective ofthe field that will be stud ied.The pressures and opportunities created bythis popularization have resulted in two significant movements that have shaped our current calculus curriculum, the New Math ofthe 1950sand '60s, and the Calculus Reform movement ofthe 1980s and '90s. These movements took the curriculum invery different directions. The New Math was created inresponse tothe explosion indemand forscientists and engineers intheyears following World WarII.Toprepare these students forad vanced mathematics, the curriculum shifted tofocus onabstraction and rigor. This isthe period in which Riemann's definition ofthe integral entered the mainstream calculus curriculum, acurriculum that adopted many ofthe standards ofrigor that had been developed in the nineteenth century as mathematicians extricated them selves from the morass ofapparent contradictions revealed by the introduction of Fourier series. Oneofthe more reasoned responses tothe New Math was acollective statement byLipman Bers, Morris Kline, George P6lya, and Max Schiffer, cosigned by many others, that was published inTheAmerican Mathematical Monthly in 1962.1In this letter, they called for the use ofthe "genetic method:" "The best way to guide the mental development ofthe individual isto let him retrace the mental development ITheAmerican Mathematical Monthly, 1962,69:189-93. v vi FOREWORD ofthe race-retrace itsgreat lines,ofcourse, and not the thousand errors ofdetail." Icannot believe it was acoincidence that oneyear later the University ofChicago Press published the firstAmerican edition ofThe Calculus:A Genetic Approach. The Calculus Reform movement of the 1980swas born from the observation that too many students were confused and overwhelmed byan approach to calcu lusthat was still rooted in the rigor ofthe 1950sand '60s. In my experience, most calculus students genuinely want to understand the subject. But as students en counter concepts that donotmake sensetothem and asthey become confused, they fall back onmemorization. These students then emerge from the study ofcalculus with nothing morethan acapacity tohandle itsprocedures and algorithms, with lit tleawareness ofitsideas ortherange ofitsuses.In the 1980s,departments ofmath ematics werefacing criticism from other departments, especially departments inen gineering, that wewere failing toomany oftheir students, and those wecertified as knowing calculus infact had noidea how toapply itsconcepts inother classes.The Calculus Reform movement tried toachieve two goals:tocreate student awareness ofand ability towork directly with the concepts ofcalculus, and toincrease the ac cessibility ofcalculus, tomake iteasier formore students tolearn what they would need as they moved into subsequent coursework and careers. It created its own backlash. The argument commonly given against itsinnovations was that itweak ened the teaching ofcalculus, but much ofthe resistance came from the fact that it required more effort to teach calculus in ways that improve both accessibility and understanding. Today,the battles over how toteach calculus have receded. Most ofthe innova tive curricula created in the late 1980shave either disappeared or mutated into something that looks suspiciously like the competition. The movement did change what and how we teach: more opportunities for exploration, greater emphasis on the interpretation ofgraphical and tabular information, arecognition that the abil ityto read and communicate mathematical ideas issomething that must bedevel oped, more varied and interesting problems, arecognition ofwhen and how com puting technology can aid in the transmission of ideas and insights. At the same time, westilluseRiemann's definition ofthe integral, and there isalingering long ing for the rigor ofepsilons and deltas. The problems that initiated both the New Math and the Calculus Reform are stillwith us.Westillhave too fewstudents pre pared for the advanced mathematics that isneeded for many oftoday's technical fields.Toofewofthe students who attempt calculus will succeed in it. Too few of those who complete the calculus sequence understand how to transfer this knowl edgetoother disciplines. Toeplitz's The Calculus: A Genetic Approach is not a panacea now any more than itwas over fortyyears ago.But itbrings back tothe forean approach that has received toolittle attention: tolook at the origins ofthe subject for pedagogical in spiration. AsAlfred Putnam wrote inthe Preface tothe firstAmerican edition, this isnot atextbook. It isalsonot ahistory. Though Toeplitz knew the history, heisnot attempting toexplain the historical development ofcalculus. What hehascreated is adistillation ofkey concepts ofcalculus illustrated through many ofthe problems bywhich they arose. FOREWORD vii I agree with Putnam that there are three important audiences for this book, though Iwould nolongergroup them quite ashedidin 1963.The firstaudience con sists ofstudents, especially those who are not challenged by the common calculus curriculum, whocanturn tothis booktosupplement their learning ofcalculus. One ofthe penalties onepays for accessibility isaleveling ofthe curriculum. There are too few opportunities in our present calculus curricula for students with talent to wrestle with difficult ideas. Ifear that we losetoo many talented students to other disciplines because they are introduced to calculus too early in their academic ca reers and are never given the opportunity to explore its complexities. For the high schoolteacher who wonders what todowith astudent who has "finished" calculus asasophomore orjunior, forthe undergraduate director ofmathematics who won ders what kind ofacourse tooffer those students who enter collegewith credit for calculus but without thefoundation forhigher study that onewould wish, this book offers at least a partial solution. Its ideas and problems are difficult and enticing. Those who have worked through itwillemerge with adeep appreciation ofthe na ture and power ofcalculus. Putnam's second and third audiences, those who teach calculus and those preparing to teach secondary mathematics, have largely merged. Prospective sec ondary mathematics teachers must beprepared toteach calculus, which means they must have adepth ofunderstanding that goesbeyond the ability topass acourse of calculus. Toeplitz's book can provide that depth. The final audience consists ofthose who write the texts and struggle to create meaningful curricula forthe study ofcalculus. This book can help usbreak free of the current duality that limits our view ofthe calculus curriculum, the belief that calculus can beeither rigorous or accessible, but not both. It suggests an alternate route through the historical development ofdifficult ideas, gradually building the piecessothat they make sense.Toomany authors think they are using history when they insert potted accounts that attempt to personalize the topic under discussion. Real reliance onhistory should throw students into the midst ofthe confusion and exhilaration ofthe moment ofdiscovery.Iadmit that inmyown teaching Imay cel ebrate confusion more than Toeplitz would have tolerated, but he does identify many ofthekeyconceptual difficulties that onceconfronted mathematicians and to daystymieourstudents. Heconveys thehistorical roleofconflicting understandings aswellasthe exhilaration ofthe discovery ofsolutions. Mathematics consists oftheabstraction ofpattern, theoverlay ofabstracted pat terns ofdifferent origin that exhibit points ofsimilarity, and the extrapolation from the patterns that emerge from this overlay. The keytothis process isarich under standing of the conceptual patterns with which we work. The Calculus Reform mantra of"symbolic, graphical, numerical, and verbal" arose from the recognition that students need abroad viewofthe mathematics they learn ifthey are ever tobe able to do mathematics. The beauty ofToeplitz's little book isthat he forces pre ciselythis.broadening ofone's view ofcalculus. The firstchapter isanhistorical exploration ofthe concept oflimit. While itcul minates inthe epsilon definition, Toeplitz iscareful tolaythe groundwork, explain ing the Greek "method of exhaustion" and the role of the principle of continuity. viii FOREWORD Most significantly, hetakes great caretoexplain our understanding ofthe real num bersand howitcame tobe.Hedemonstrates how limits and the structure ofthe real numbers are intimately bound together. Epsilons and deltas are not handed down from above with acollection ofarcane rules. Rather, they emerge after considerable work on the concept oflimit, appearing as a convenient shorthand for some very deep ideas. This is one of the clearest examples of the fallacy of a dichotomy be tween rigor and accessibility. That dichotomy only exists when we decide to "do" limits inoneortwo classes. Toeplitz's approach suggests afundamental rethinking ofthe topic oflimits in calculus. Ifit isan important concept, then devote to itthe weeks itwilltake forstudents todevelop atrue understanding. Ifyou cannot afford that time, then maybe for this course it isnot as important as you thought it was. Though it would have been heresy to me earlier in my career, I have come to the conclusion that most students ofcalculus are best served byavoiding anydiscussion oflimits.?Itisthe students who have agoodunderstanding ofthe methods and uses ofcalculus who are ready to learn about limits, and they need atreatment such as Toeplitz provides. Chapter 2moves ontothe general problem ofarea and the definition ofthe def inite integral. I especially enjoy Toeplitz's brief section on the dangers ofinfinitesi mals. I find it refreshing that Toeplitz completely ignores the Riemann integral which was, after all, created for the investigation of functions that do not and should not arise inafirstyear ofcalculus. Instead Toeplitz relies onwhat might be called the Cauchy integral, taking limits ofwhat today are commonly called left and right-hand Riemann sums. Iagree with Toeplitz that this isthe correct integral definition tobeused in the firstyear ofcalculus. The fundamental theorem ofcalculus provides the theme for the third chapter, which isthe longest and richest. There isan extended section onNapier's tables of logarithms. Few students today are aware of such tables or the role they once played. But themathematics isbeautiful. Understanding the application ofthese ta bles and the complexity oftheir construction provides insight into exponential and logarithmic functions. Today's texts present these functions in the context ofexpo nential growth and decay. While that is their most important application, it pro vides only alimited view offunctions that are central to somuch ofmathematics. This chapter includes a discussion of the development of the relationship of dis tance, velocity,and acceleration. The difficulties Galileo encountered inconceiving, formulating, and then convincing others ofthese relationships often isunderappre ciated. Toeplitz pays Galileo his rightful due. My favorite part ofChapter 3isToeplitz's discussion in the last section, "Limi tations ofExplicit Integration." He clarifies a point which, when ignored, leads to confusion among our students. That isthe distinction between what hecalls "com- 2Foranillustration ofhowthiscanbedone,seetheclassictextCalculusMade Easy bySylvanus P.Thompson. Itisacommentary ontheholdlimitshaveonourcurrent curriculum that themost re centedition, St.Martin's Press,NewYork, 1998,includes additional chapters byMartin Gardner, one ofthem onlimits. FOREWORD ix putational functions," the standard repertoire built from roots, exponentials, sines, tangents, and logarithms, functions for which we can compute values to any pre assigned accuracy, and the "geometrical functions," those represented by agraph of a continuous, smooth curve. He rightly points out that the challenge issued by Fourier series was to leave the limitations ofcomputational functions and embrace the varied possibilities ofgeometrical functions. AsToeplitz says in the concluding lines ofthis chapter, Today's researchers have them both at their disposal. They use them sepa rately or in mutual interpenetration. For the student, however, it isdifficult to keep them apart; the textbooks he studies do not give him enough help, be cause they tend to blur rather than to sharpen the difference. The emphasis on graphical representation that received impetus from the Cal culus Reform movement has helped to promote student awareness ofthis distinc tion, but the two understandings offunction still draw too little direct attention. I have found ithelpful in my own classes toemphasize this distinction. For example, too many of my students enter my classes only knowing concavity as an abstract property determined by checking the sign of the second derivative. They are amazed to seethat itcan be used to describe geometric functions that are not given by any formula. Finally, we come to Chapter 4in which Toeplitz demonstrates how calculus en abled the solution ofthe great scientific problem ofthe seventeenth century, the ex planation ofhow itisthat wesitonaball revolving at 1,000miles per hour asithur tles through space at speeds, relative to our sun, ofover 65,000 miles per hour, yet we feel no sense of motion. Newton's Principia is a masterpiece. I teach an occa sional course on it, and wish that all calculus students, especially those who are preparing toteach, would learn to appreciate what Newton accomplished. Toeplitz gives usan excellent ifbrief overview ofNewton's work. He also explores the study ofthe pendulum, aremarkably rich source ofmathematical inspiration. There ismuch that all of us can learn about the teaching of calculus from this book, but I do not want to freight it with too much gravity. It is, above all, a de lightful and entertaining introduction tomathematical problems that have inspired the creation of calculus. Read it for the sheer enjoyment of well-crafted explana tions. Read itto learn something new. Read itto seeclassic problems in arich con text. But then take some time to ponder its lessons for how we teach calculus." DAVID M. BRESSOUD Macalester College St.Paul, Minnesota 3With thanks toPaul Zorn forhiscomments onadraft ofthis Foreword.
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