E. H. Moore's Early Twentieth-Century Program for Reform in Mathematics Education David Lindsay Roberts

Home , George David Birkhoff, Henry Burchard Fine, Leonard Eugene Dickson, Oskar Bolza, Robert Lee Moore, William Fogg Osgood

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. mathematicianswere 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 vigorous 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 community, 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 nineteenthcentury, 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 students shouldtest propositionsfrom Euclidby carefulmeasurements using graphpaper.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, pp. 405-408] The key to resolvingthese problems, according to Moore,was reformationof math- ematicalpedagogy at all levels, andthe centerpieceof his plan was whathe calledthe "laboratorymethod". In Moore'svision teacherswould work with studentsindividu- ally or in small cooperativegroups, depending on the needs and strengthsof the students.Beginning with "mattersof thoroughlyconcrete character", they wouldproceed by meansof graphs,models, coloredchalks and elementarydemonstrations of phys- ical phenomena,"to develop on the partof every studentthe true spiritof research, and an appreciation,practical as well as theoretic,of the fundamentalmethods of science."He furtherproposed "to organize the algebra,geometry, and physics of the secondaryschool into a thoroughlycoherent four years' course," thus doing away with the standardpartition of these subjectsinto "water-tightcompartments". Moore also ex- plainedhow problemsin practicalmathematics could be introducedto studentsso as to awakeninterest in abstractconcepts. Thus problemsinvolving measurement of phys- ical quantitiescould become exercises in the study of approximationand allowable errorbounds, eventually leading to the theoryof limits and irrationalnumbers; study of the areasunder specially selected curves could yield the notionof uniformconver- gence.Important results should be demonstratedin at leasttwo differentways. In those

October2001] E. H. MOORE'S REFORM PROGRAM 691 cases whereformal proof was appropriate"much of the proofshould be securedby the researchwork of the studentsthemselves." To accommodatethe rich level of activity hopedfor in his mathematicallaboratory, Moore proposed that two consecutiveclass periodsbe allocatedto it [14, pp. 409-413]. Finally,the studentsin Moore'slaboratory were to be muchengaged in manualactivity, a featurehe statedmost explicitlyin his privatecorrespondence: "Provision should be madefor the actualconstruction by the studentsof manyof the simplerdrawings, models andmechanisms." [13] Anotheraspect of Moore'sprogram is notable:he soughtto upliftthe professional standingof mathematicsteachers in the secondaryschools. In particularhe called on the AMS to expandits membershipby activelyrecruiting secondary school teachers to join its ranks[14, p. 414].

4. CONSEQUENCES. Mooreenjoyed several advantages as an educationalspokes- man:he was widely consideredto be one of the preeminentmathematicians of the country;further, he was associatedwith an institution,the Universityof Chicago,of considerableeducational influence; and finally,as Bolza noted,he was a skilled academic politician.Moore promoted his ideas strenuouslyfor aboutthree years. He ar- rangedto teachlaboratory method classes at the Universityof Chicago,and he taught the introductorycalculus course himself using this method.He encouragedsecondary schoolteachers to adoptsimilar methods. He andcolleagues from the Chicagomathe- maticsdepartment wrote articles directed at the secondaryschools [23, pp. 301-325]. In 1906, for example, Moore wrote a paperin which he elaboratedon the use of graphpaper and the functionconcept in secondaryschool instruction,and proposed thatnomography be emphasized[15]. At meetingsof school teachersof mathematics Mooreurged these teachersto form theirown associations.These associationswould become perhapsMoore's most substantialeducational legacy. The foundingof the NCTMin 1920 has clearroots in Moore'sefforts earlier in the century[23, pp. 364- 367]. But enthusiasmfor the substanceof Moore's programsoon began to wane, at Chicagoand elsewhere,and mathematicseducation did not develop as he had envisioned.Moore's laboratory method did not become the dominantmethod of instruction;a "thoroughlycoherent four years' course"of algebra,geometry, and physics did not become the normin the secondaryschools; mathematicalinstruction in the colleges and universitiesdid not undergoa majorreorientation to betterserve the needs of science and engineeringapplications. Nor did the AMS ever recruitlarge numbers of high school teachersas Moorehad urged[23, pp. 326-339]. Moore's special efforts at the Universityof Chicago also soon faded away. For two or three years the whole mathematicsdepartment pulled togetherto reformin- structionalmethods and to promotethe laboratorymethod in particular.But then the reactionset in. The laboratorymethod was found to be too labor intensive,and too complicatedto administer.In addition,sharp lines of stratificationbegan to assert themselvesamong the Chicagomathematical educators. Those who would carryon the agitatingand organizingand broadreform efforts were those who had elected to specializein education,such as HerbertEllsworth Slaught (1861-1937) and Jacob WilliamAlbert Young (1865-1948). Those like Mooreand Dickson who hadresearch ambitionsfaded into the background,educationally speaking [23, pp. 320-321]. Meanwhile,the secondaryschool environmentwas undergoingan upheaval.Be- tween 1890 and 1900 the numberof studentsgoing to publichigh schools increased morethan two andhalf times.Another doubling of enrollmentoccurred between 1900 and 1912, and yet anotherbetween 1912 and 1920. Withthese huge increasesin en- rollmentcame increasingpressure to tie educationof studentsmore closely to their

692 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly108 futureemployment. With the rise of this so-calledvocational education movement the key distinctionwas no longerbetween classical and modem educationalsubjects but betweenacademic and practical, and the academicsubjects were put on the defensive. Mathematicshad successfullycrossed from classical to modem,but had a muchmore difficulttime shakingthe disparaginglabel of "academic"[10, pp. 170-173; p. 236; p. 244]. Moreover,the notionof mentaldiscipline now came underfull-fledged attack from a range of psychologistsand educators.School subjectswere asked to justify them- selves as supporting"socially efficient" goals. Efficiency-mindededucators combed the curriculumfor useless topics.In one notablestudy, some 4,000 sixth, seventh,and eighth gradestudents were askedto follow their parentsaround for two weeks, col- lecting mathematical"problems" solved by the adultsin the course of business and householdtasks. Most adultswere found to use little exceptthe mostelementary arith- meticprocesses; proof, according to this line of thought,that much mathematics could be jettisonedfrom the curriculum.[28, p. 52] Mathematicaleducators eventually became greatly alarmed by these attackson their subject.One responsewas a furtherround of teacherassociation building around the time of the FirstWorld War, culminating in the foundingof the NCTMin 1920. Even the AMS was rousedto form a special committeein 1914, chargedwith determining "whetherany actionshould be takenby the Society in regardto the movementto dis- place mathematicsin the schools."[1] This committeepresented a one and one-half pagereport of verymodest suggestions, beginning with a Freudiantypographical error thatset the tone for the resultinginaction. In the very firstsentence it was proclaimed thatthe questionat issue was "whetherany action is desirableon the partof the Society in the matterof the movementagainst mathematicians in the schools."[21]Mathemati- cians did not wantto be in the schools;ergo, there-was no problemrequiring action. At this same time, 1914-1915, therewas a renewalof Moore'searlier call thatthe AMS engage in efforts to improvethe teachingof mathematics.The leading agita- tor was Moore'sChicago colleague and formerstudent Herbert Slaught; Moore was clearly supportive,but stayedbehind the scenes. Specifically,Slaught proposed that the AMS rescue the financiallydistressed Mathematical Monthly, and at the same time form a specialmembership class of "Associate"for those primarilyinterested in educationalissues. The AMS rejectedthese proposals,but at the same time suggested that those with interestssimilar to those of Slaughtought to form their own society. Slaughttook the hint, resultingin the foundationof the MathematicalAssociation of America(MAA) with the MONTHLYas its officialjournal. [12, pp. 19-20] One of the first actions taken by the MAA was to form a National Committee on MathematicalRequirements (NCMR) to study the secondaryschool curriculum. E. H. Moorewas one of the originalfive members,and it provedto be the last major educationalinitiative in whichhe engaged.The evidencesuggests that he let othersdo thebulk of the work,especially his brother-in-law,John Wesley Young (1879-1932) of DartmouthCollege. The report of this committeeappeared in 1923 in the formof a 600 page book entitled The Reorganization of Mathematics in Secondary Education [20]. This reporthas been hailed as a landmark[4], [6, pp. 208-209]. But if one looks at it in relationto the ambitionsof the segmentof the mathematicalcommunity that had soughtto follow the vision of E. H. Moore's 1902 address,one must come to a rather modestassessment. The NCMRdirectly followed Moore's lead in promotingthe ped- agogicalbenefits of graphicrepresentation and the functionconcept, and in bothcases some genuinesuccess can be discerned.But much of Moore'soriginal program was considerablymuted. In generalthe NCMRtook a farmore defensive posture, aware of the attackson mathematicseducation that had eruptedover the previous20 years.The

October2001] E. H. MOORE'S REFORM PROGRAM 693 high hopes for the "laboratorymethod" seem to have largelydissipated, nor was there any mentionof the desirabilityof aligningmathematics education with engineering. Belatedlyrealizing that few of the studentssurging into the high schoolswere going to become scientistsand engineers,the NCMRmodified Moore's original call to justify mathematicsas a tool of science,and instead tried to reinvigoratethe mentaldiscipline thesis. But for all its efforts the NCMR did not succeed in vanquishingthose attack- ing mathematicsin the schools. High schools increasinglydropped mathematics as a graduationrequirement. Even requiredninth-grade mathematics, a point at which the NCMRhad chosento makea stand,was underchallenge. Percentages of students takingalgebra and geometryin gradesnine throughtwelve of the public schools de- clined steadilyinto the 1930s [26, p. 195]. The period 1915-1940 has been called a "twenty-fiveyear depression" in school mathematics[5, p. 24]. 5. CONCLUSIONS. Whatwent wrongwith Moore's program? Many reasons might be cited, fromthe deathof Moore'sstrong supporter, University of Chicagopresident WilliamRainey Harperin 1906, to Moore's increasingpreoccupation with his own puremathematical research program in whathe called GeneralAnalysis [11, pp. 134- 135]. Certainlythe decentralizedAmerican educational system provided a formidable barrierto even the most enthusiasticand resourceful of reformers. Some criticswere distressedby Moore'sapparent abandonment of rigorand formal drill. WilliamFogg Osgood (1864-1943) of Harvard,speaking as the retiringAMS presidentin 1906, implicitlycriticized Moore's approach:

Let us not for a moment fail to recognize the fact that, whateverchanges it may be desirable to make in the suggestive instructionof the course, the process by which the youth actually acquiresthe ideas of the calculus is to a large extent and essentially throughformal work of substantialcharacter [17, pp. 449-450].

Therewas also an unavoidableincongruity in the spectacleof a mathematiciansuch as Moorearguing on behalfof intuitionand applications. Moore was the purestof pure mathematicians;it is very difficultto construeany of his researchas being applied mathematics.Even as he was preachingthe value of developinggeometric intuition for instructionalpurposes, the explorationof the axiomaticfoundations of geometry was being characterizedby a fellow Americanmathematician as follows: "Geometric intuitionhas no place in this orderof ideas which regardsgeometry as a meredivision of pure logic." This observernamed the internationalleaders of this anti-intuitional approach;in Americait was E. H. Moore [19, p. 158]. But two otherfactors deserve emphasis as well. Moore'spedagogical program was in conflictboth with the environmentenveloping the secondaryschools and with the professionalizationproject of Americancollege anduniversity mathematicians. Withregard to the school environment,Moore, like many othereducators, largely failed to foreseethe consequencesof changingdemographics. In the face of the surge of studentsinto the schools, calls for educationalefficiency that had emergedduring the last half of the nineteenthcentury became much more insistent and attractive. The efficiencyadvocates claimed to offermeans to controlthe floodof studentsby carefully circumscribingrequirements in termsof time and effort.In contrastMoore's "mathematicallaboratory", which called for suchextravagances as performingall demonstra- tionsin two differentways andfor blurringof subject-matterboundaries, could well be seen as a prescriptionfor waste andconfusion. Moreover, at the very time thatMoore was proposingto justify mathematicseducation primarily as an aid to science anden- gineering,the populationof high school studentswas explodingwith students,most of

694 ? THE MATHEMATICALASSOCIATION OF AMERICA [Monthly108 whomwere not aimingto becomescientists or engineers.Why shouldsuch students be requiredto take substantialamounts of mathematics?This was an awkwardquestion for Moore,who had almostentirely jettisoned mental discipline and related concepts from his vocabulary.In the dawningera when school subjectswere being judged on "socialefficiency", mathematics was in a precariousposition for whichMoore and his allies were unprepared[10, pp. 278-280]. Secondly,the communityof researchmathematicians, which Moore'sown efforts did so muchto invigorate,was increasinglyreluctant to involveitself with educational matters.This was evident immediatelyafter his 1902 address.The researchershad been shocked to find their presidentstooping to discuss education.They were dis- appointed,as one observerput it, "thata man who stood amongthe few recognized leadersin highermathematics in this countryshould lose the opportunityoffered to considerthe greatproblems of the scienceper se" [25, p. 135]. Insteadof following Moore'scall to seek closer engagementwith the schools, the AMS soughtto protect the sanctityof its researchmission by relegatingdiscussion of educationalissues to the MAA. Centralto Moore's basic argumentfor enticingresearch mathematicians into the educationalarena was the contentionthat the healthof the researchcommunity was ultimatelydependent on the healthof mathematicsin the schools. But considerhow difficultit wouldhave been to defendthis positionduring his lifetime.As noted,school mathematicsexperienced hard times during the firstdecades of the 20thcentury. Mean- while the AMS was growing,college mathematicsdepartments were expanding,and Moore'sdoctoral students in particularwere colonizingdepartments of mathematics across the country.In short,the connectionargued for by Moore, that the research communitydepended on school mathematics,was not obviousin the early twentieth century.I leave as an open questionwhether it has become moreobvious since. A strongclaim can be made for Moore as a primaryinitiator of the use of graph paperin the Americanmathematical curriculum, although nomography never came to have the importancethat Moore envisioned. It is also likely thathe pavedthe way for makingthe functionconcept more central to instruction,although widespread adoption wouldprove slow. But Moore'sbroader aims for reorientingmathematical instruction were not achieved,and with the luxuryof hindsightit seems fairto say thatthey were underminedby his failure to analyze sufficientlyboth the internalworkings of his own professionand the importof wider social developments.Today's mathematical educationreformers might do well to take heed of the difficultiesexperienced even by such a one as "E. H. Moore mit seinem aggressivenEnthusiasmus und seinem Organisationstalent".

REFERENCES

1. AMS Council Minutes, 2 January 1915, Records of the American Mathematical Society, Box 12, Folder 47, Brown University ManuscriptsDivision. 2. James K. Bidwell and Robert G. Clason, eds., Readings in the History of Mathematics Education, Na- tional Council of Teachersof Mathematics,Washington, 1970. 3. Oskar Bolza, Aus Meinem Leben, E. Reinhardt,Muinchen, 1936. 4. CharlesH. Butler, The reorganizationreport of 1923, Mathematics Teacher44 (1951) 90-92, 96. 5. W. L. Duren, Jr., CUPM, The history of an idea, Amer Math. Monthly 74 [Fiftieth AnniversaryIssue] (1967) 23-37. 6. Phillip S. Jones, ed., A History of Mathematics Education in the United States and Canada, National Council of Teachers of Mathematics Thirty-Second Yearbook, National Council of Teachers of Mathe- matics, Washington, 1970. 7. David StarrJordan, The making of a Darwin, Nature 85 (1911) 354-359. 8. Roland S. Keyser, The readjustmentof the school curriculum,School Review 1 (1893) 131-140. 9. Jeremy Kilpatrick,Confronting reform, Amer. Math. Monthly 104 (1997) 955-962.

October2001] E. H. MOORE'S REFORM PROGRAM 695 10. EdwardA. Krug, The Shaping of the AmericanHigh School 1880-1920, Harper& Row, New York, 1964. 11. Saunders Mac Lane, Mathematics at the University of Chicago: A brief history, in A Century of Math- ematics in America-Part II, ed. Peter Duren et al., American MathematicalSociety, Providence, 1989, pp. 127-154. 12. Kenneth 0. May, ed., The Mathematical Association of America: The First Fifty Years, Mathematical Association of America, Washington, 1972. 13. E. H. Moore to William Rainey Harper,29 July 1902, University of Chicago Presidents' Papers, 1889- 1925, Box 17, Folder 2, University of Chicago Special Collections. 14. Eliakim Hastings Moore, On the foundations of mathematics,Science 17 (1903) 401-416. 15. , The cross-section paper as a mathematicalinstrument, School Review 14 (1906) 317-338. 16. William Mueller, Reform now, before it's too late!, Amer Math. Monthly 108 (2001) 126-143. 17. Wm. F. Osgood, Calculus in our colleges and technical schools, Bull. Amer.Math. Soc. 13 (1907) 449- 467. 18. Karen Hunger Parshall and David E. Rowe, The Emergence of the American Mathematical Research Community,1876-1900: J. J. Sylvester,Felix Klein, and E. H. Moore, American Mathematical Society, Providence, 1994. 19. James Pierpont, The history of mathematics in the nineteenth century,Bull. Amer Math. Soc. 11 (1904) 136-159. 20. The Reorganization of Mathematics in Secondary Education, Mathematical Association of America, [Oberlin], 1923. 21. Report to the AMS, 2 January1915, Records of the American MathematicalSociety, Box 12, Folder 47, Brown University ManuscriptsDivision. 22. R. G. D. Richardson, The Ph.D. degree and mathematical research, in A Century of Mathematics in America-Part II, ed. Peter Duren et al., American Mathematical Society, Providence, 1989, pp. 360- 376. 23. David Lindsay Roberts, Mathematics and Pedagogy: Professional Mathematicians and American Edu- cational Reform, 1893-1923, Ph.D. dissertation,The Johns Hopkins University, Baltimore, 1998 (UMI number 821188). 24. Raleigh Schorling, ed., The First Yearbookof the National Council of Teachersof Mathematics,National Council of Teachers of Mathematics, [New York], 1926. 25. David Eugene Smith, Movements in mathematicalteaching, School Science and Mathematics 5 (1905) 135-39. 26. George M. A. Stanic, The growing crisis in mathematicseducation in the early twentieth century,Journal for Research in MathematicsEducation 17 (1986) 190-205. 27. Laurence R. Veysey, The Emergence of the American University, University of Chicago Press, Chicago, 1965. 28. G. M. Wilson, A Surveyof the Social and Business Usage ofArithmetic, Bureau of Publications, Teachers College, New York, 1919. 29. Henry Aaron Yeomans, Abbott Lawrence Lowell, HarvardUniversity Press, Cambridge,MA, 1948.

DAVID LINDSAY ROBERTS is an independent scholar, especially interested in the history of mathematics education. He earned an A.B. in mathematicsfrom Kenyon College in 1973, and then proceeded to the Univer- sity of Wisconsin-Madison, where he obtained an M.A. in mathematics and an M.S. in industrialengineering. After eleven years as an operationsresearch analyst, he entered Johns Hopkins University, completing a Ph.D. in the history of science in 1998. Since then, with the aid of the Smithsonian, the Educational Advancement Foundation,and the National Academy of Education, he has been exploring the "New Math"of the 1950s and 1960s. 12226 ValerieLane, Laurel, MD 20708 [email protected]

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