E. H. Moore's Early Twentieth-Century Program for Reform in Mathematics Education David Lindsay Roberts 1. INTRODUCTION. The purposeof this articleis to examinebriefly the natureand consequencesof an early twentieth-centuryprogram to reorientthe Americanmath- ematicalcurriculum, led by mathematicianEliakim Hastings Moore (1862-1932) of the Universityof Chicago.Moore's efforts, I conclude,were not ultimatelyvery suc- cessful, and the reasonsfor this failureare worthpondering. Like WilliamMueller's recentarticle [16], which drawsattention to the spiriteddebates regarding American mathematicseducation at the end of the nineteenthcentury, the presentarticle may re- mind some readersof morerecent educational controversies. I don't discouragesuch thinking,but cautionthat the lessons of historyare not likely to be simpleones. E. H. Mooreis especiallyintriguing as a pedagogicalpromoter because of his high placein the historyof Americanmathematics. He is a centralsource for muchof twen- tiethcentury American mathematical research activity. As a grossmeasure of Moore's influencewe may note the remarkablefact thatmore than twice as manyPh.D. mathe- maticianswere produced by his Universityof Chicagomathematics department during his tenurethan were producedby any othersingle institutionin the UnitedStates. In- deed, the Universityof Chicagoproduced nearly 20% of all Americanmathematics Ph.D.s duringthe period[22, p. 366]. Moreover,among Moore's doctoral students at Chicagowere suchluminaries as GeorgeDavid Birkhoff (1884-1944), OswaldVeblen (1880-1960), LeonardEugene Dickson (1874-1954), and RobertLee Moore (1882- 1974). This quartet,each one of whom became a presidentof the AmericanMathe- maticalSociety (AMS) as well as a memberof the NationalAcademy of Sciences, was especiallyfruitful in producingadditional distinguished researchers, at Harvard, Princeton,the Universityof Chicago, and the Universityof Texas, respectively.For details of the researchaccomplishments of Moore and his progenythe readermay consultthe excellentbook by KarenParshall and David Rowe on the formationof the Americanmathematical research community [18, pp. 323-426]. It is also clearthat Moore was a formidableacademic politician. Especially striking testimonyon this point was providedby OskarBolza (1857-1942), a Germanmath- ematicianwho came to the United States seeking academicopportunities, and who joined Moore'sdepartment when the Universityof Chicagoopened its doorsin 1892. Moorewas only thirty,five yearsBolza's junior, and Bolza was dubiousat firstabout this untriedperson as departmenthead. But Bolza foundhimself witnessingnot only the emergenceof a powerfulmathematics department at Chicagobut of a newly vig- orous Americanmathematical community as a whole. In his autobiography,written manyyears later after his returnto Germany,Bolza had no doubtwho was responsible for these developments.It was "E. H. Moore mit seinem aggressivenEnthusiasmus und seinem Organisationstalent"[3, p. 32], a characterizationthat readily leaps the languagebarrier. Remarkably,amidst his vigorous activities on behalf of the research commu- nity, Moore found time to engage with educationalreform in both the colleges and the secondaryschools. His most notable pedagogical statementcame in his retir- ing addressas presidentof the AMS in December 1902, publishedin March 1903 October2001] E. H. MOORE'S REFORM PROGRAM 689 [14, pp. 401-416]. This addresscreated much comment initially, and has contin- ued to stir interestin the years since. The NationalCouncil of Teachersof Math- ematics (NCTM) reprintedMoore's talk in its very first yearbook, published in 1926 [24, pp. 32-57]. A more recent referenceto Moore's addresscan be found in this MONTHLYfor December1997 [9, p. 955]. But whatexactly did Moorestand for andwhat became of his ideas? 2. CONTEXT OF MOORE'S EDUCATIONAL PROPOSALS. E. H. Moore shouldfirst of all be situatedwithin the contextof the declineof the so-calledclassical curriculumof Greek,Latin, and mathematics at the end of the nineteenthcentury. This curriculum,which had dominatedthe colleges and many of the secondaryschools, was frequentlyjustified in termsof "mentaldiscipline" and "faculty psychology". The mind was held to be composedof variousfaculties, such as the powers of memory, reasoning,observation, and will. These faculties could be strengthenedby mental exercise,much as muscles could be strengthenedby physicalexercise. Mathematics, it was claimed,was crucialfor promotingcertain key mentalfaculties [27, pp. 22-23], [2, pp. 78-85]. Thus we find an educatorin 1893 opining on the virtue of studying arithmetic: It is refreshingto know that there is one subject which [the student]must masterfor himself slowly, sometimes painfully, and always with much labor. If arithmeticstands in the way of drawingor music, of Germanor Latin, of book-keeping or civics, let us be thankfulthat it is an obstacle which will develop the sound mental muscle that is needed for the work further on [8, p. 135]. Moreadvanced mathematics was also valuedas a mentaldiscipline. This is epitomized in a statementmade by A. LawrenceLowell (1856-1943), who claimedto remember with specialfondness his undergraduatestudies at Harvardwith mathematicianBen- jamin Peirce (1809-1880) in the 1870s, beforeLowell moved on to become a lawyer andeventually president of Harvard: To those of us who have not pursuedthe study of mathematicssince college days the substance of what he taughtus has faded away, but the methods of thought,the attitudeof mind and the mode of approachhave remainedprecious possessions [29, pp. 40-41]. Pleasingas suchsentiments may have been to manymathematicians in the late nine- teenthcentury, there were some who were growingskeptical. They recognized that the mentaldiscipline thesis could be a barrierto introducingmore advanced topics into the school curriculum.Why shouldstudents study algebra when they could exercisetheir mentalmuscles simply by grindingaway at arithmeticyear afteryear? Furthermore, a numberof mathematiciansbegan to join othereducators in assertingthat true mental disciplinecould come only in a subjectin which the studentwas genuinelyinterested. Thusthe MathematicsConference of the celebratedCommittee of Tenof 1893,chaired by Simon Newcomb (1835-1909) and includingHenry Burchard Fine (1858-1928) andFlorian Cajori (1859-1930), madethe following statement: The opinion is widely prevalentthat even if the subjectsare totally forgotten,a valuablemental discipline is acquiredby the efforts made to masterthem. While the Conference admits that, consideredin itself this discipline has a certainvalue, it feels that such a discipline is greatly inferior to that which may be gained by a different class of exercises, and bears the same relation to a really improving discipline that lifting exercises in an ill-ventilated room bear to games in the open air. The movements of a race horse afford a better model of improving exercise than those of the ox in a tread-mill[2, pp. 133-34]. 690 ?nTHE MATHEMATICALASSOCIATION OF AMERICA [Monthly 108 But by emphasizingsuch distinctionsof disciplinaryvalue, mathematicianswere openingan avenuefor fundamentalattacks on theirown subject.If algebrawas better disciplinethan arithmetic,perhaps physics or biology, for example,would be better still. Indeed,a steadybuildup of argumentsfor the superiordisciplinary value of the naturalsciences had been going on since mid-centuryin both Britainand America, with mathematicsoften subjectto some degree of attackin the process. Some sci- entists saw mathematicsas characterizedby numbingdrill, in starkcontrast to the invigorationof scientific study,wherein the studentstood face to face with nature, in the field or in the laboratory.David StarrJordan (1851-1931), biologist and first presidentof StanfordUniversity, described students coming to science "by escape from Latinand calculuswith the eagernessof colts broughtfrom the barnto a spring pasture."[7, p. 358] 3. MOORE'S 1902 ADDRESS. It was into this environmentthat E. H. Mooreven- turedin his 1902 addressas retiringpresident of the AMS, andit is thusnot surprising to find him proposingto capitalizeon the rising prestigeof science and engineering by tying mathematicalinstruction closely to those endeavors,both in the secondary schools and in the early college courses.Moore had come to the conclusionthat the trainingof futurescientists and engineers had become a crucialjustification for teach- ing mathematics,and that it was destinedto becomemore so in the future.He champi- oned the ideas of Englishengineering educator John Perry (1850-1920), who strongly urgedthat the rigorousaxiomatic development of mathematicsbe de-emphasizedin favor of teachingit as an inductivescience. For example,beginning geometry stu- dents shouldtest propositionsfrom Euclidby carefulmeasurements using graphpa- per.Perry's influence can be foundin Moore'sinsistence that mathematical rigor was a relativerather than an absoluteconcept: "Sufficientunto the day is the precision thereof",declared Moore. Both Mooreand Perry sought to bringstudents very rapidly to the employmentof powerfulmathematical concepts, without insisting that all sup- portingtechnical details be mastered[14, p. 41 1]. Moore decriedvarious "chasms" he observedin Americanmathematical activity: *betweenpure and appliedmathematics; between research mathematicians and school teachers;between the public's foggy perceptionof mathematicsand "the very high positionin generalesteem and appreciativeinterest which it assuredlydeserves." [14,