The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach Author(s): Kathleen Sullivan Source: The American Mathematical Monthly, Vol. 83, No. 5 (May, 1976), pp. 370-375 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2318657 . Accessed: 10/03/2014 11:41 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 132.70.4.118 on Mon, 10 Mar 2014 11:41:19 AM All use subject to JSTOR Terms and Conditions MATHEMATICAL EDUCATION EDITED BY PAUL T. MIELKE AND SHIRLEY HILL Materialfor this Department should be sentto Paul T. Mielke,Department of Mathematics, Wabash College, Crawfordsville,IN 47933, or to ShirleyHill, Departmentof Mathematics,University of Missouri,Kansas City,MO 64110. PEAKS, RIDGE, PASSES, VALLEY AND PITS A Slide Study of f(x,y)=Ax2+By2 CLIFF LONG The adventof computer graphics is makingit possible to payheed to thesuggestion, "a pictureis wortha thousandwords." As studentsand teachers of mathematics we shouldbecome more aware of thevariational approach to certainmathematical concepts, and considerpresenting these concepts througha sequenceof computergenerated pictures. To illustratethis notion, consider the following. In thestudy of functionsof twovariables it is usuallyemphasized that a regularnon-planar point on a smoothsurface can be classifiedas one of threedistinct types: elliptic, parabolic, hyperbolic [1]. These maybe illustratedusing the functions f(x,y) = Ax2+ By2 withthe origin being: (a) ellipticif A *B > 0; (b) parabolicif A *B = 0, A,#B (planarif A = B = 0); (c) hyperbolicif A *B < 0. The slidesreproduced here (see page371) were made at BowlingGreen State University from the screenof an Owens-Illinoisplasma panel which is an outputdevice for a Data GeneralNova 800 mini-computer.Many slide sets and supereight movies have been produced by college mathematics teachersunder an NSF grantfor "Computer Graphics for Learning Mathematics." The institutewas heldat CarletonCollege in Northfield, Minnesota, 55057, during the summers of 1973 and 1974. It was underthe direction of Dr. RogerB. Kirchner,who, with no small amount of personal effort, has made theseslides and moviesavailable at minimumreproduction cost. It mustof course be keptin mindthat while one goodpicture may be wortha thousandwords, a thousandpoorly chosen ones maybe worthless. Reference 1. D. Hilbert and S. Cohn-Vossen, Geometryand the Imagination,Chelsea, New York, 1952. DEPARTMENT OF MATHEMATICS, BOWLING GREEN STATE UNIVERSITY, BOWLING GREEN, OH 43403. THE TEACHING OF ELEMENTARY CALCULUS USING THE NONSTANDARD ANALYSIS APPROACH KATHLEEN SULLIVAN In the 1960'sa mathematicallogician, Abraham Robinson, found a wayto makerigorous the intuitivelyattractive infinitesimal calculus of Newton and Leibniz, beginning a branch of mathematics callednonstandard analysis. When elementary calculus is developedfrom this nonstandard approach, thedefinitions of thebasic concepts become simpler and thearguments more intuitive (see Robinson [21or Keisler[1]). For example,the definition of the continuity of a functionf at a pointc is simply thatx infinitelyclose to c impliesthat f(x) is infinitelyclose to f(c). 370 This content downloaded from 132.70.4.118 on Mon, 10 Mar 2014 11:41:19 AM All use subject to JSTOR Terms and Conditions MATHEMATICAL EDUCATION 371 PWA PmPEWot 6y ZI -. y Z1 4 Z.i a z.i.v I z ixa-v 2.14.45 2144.6 v~~~~~~~ vS ,lv .-v -v I z-xea 2aa a-lt,v a a-12,t a a.lZ.y a a I Z.-1.y.1V2 I .-.S2.IV2 iI a 1 Z._,Z-. IZ -.aV I I Z.tM2.lVe I Z,#X2+1v2 I Z. 44.15 z..sa.as Z.6x415 It wouldseem that there ought to be considerablepedagogical payoff from this greater simplicity andcloseness to intuition.However, anyone considering using this approach will have questions that needto be answered.Will the students "buy" the idea ofinfinitely small? Will the instructor need to havea backgroundin nonstandardanalysis? Will the students acquire the basic calculus skills? Will theyreally understand the fundamental concepts any differently? How difficultwill it be forthem to makethe transition into standard analysis courses if they want to studymore mathematics? Is the nonstandardapproach only suitable for gifted mathematics students? In orderto get answersto some of thesequestions, an experimentwas carriedout, using an This content downloaded from 132.70.4.118 on Mon, 10 Mar 2014 11:41:19 AM All use subject to JSTOR Terms and Conditions 372 KATHLEEN SULLIVAN [May institutionalcyclic design (see Sullivan[3]). Fiveinstructors who were teaching a traditionalcalculus coursein schoolslocated in theChicago-Milwaukee area duringthe 1972-73 school year volunteered to use the nonstandardapproach during the 1973-74school year. The textused was H. Jerome Keisler'sElementary Calculus: An ApproachUsing Infinitesimals, published by Prindle,Weber & Schmidt,which is thefirst text book to adaptthe ideas of Robinson to a firstyear calculus course. It seemedimportant that the same teachers be involvedeach year,teaching basically the same student population.Otherwise, differences inattitude and performance might well be writtenoff to differences ininstructors orin studentpopulations. Only instructors who had taught calculus several times before wereasked to participate in the study so thatthey would be ina goodposition to makecomparisons. Fourof the five schools participating were small private colleges. The fifth was a publichigh school withover 2000 students in Glendale,an uppermiddle-class suburb of Milwaukee.At SaintXavier College,the control and experimental classes consisted of students planning to majorin mathematics; at Lake ForestCollege, of studentsin an honorsprogram; at NicoletHigh School, of a groupof acceleratedmathematics students in an advancedplacement class. Barat Collegeoffers only one calculuscourse, and thesame is trueof MountMary College. The assumptionthat the two groups would be comparablein abilitywas supportedby a checkof theSAT mathematicsability scores which were available, i.e., scoresfor 58 outof the 68 studentsin thecontrol group and for55 out of the68 studentsin theexperimental group. The distributionof scoresis shownin thetable below. TABLE 1: SAT Math AbilityScores Control Group ExperimentalGroup 700 + 13 13 600-699 29 29 500-599 14 11 400-499 2 2 The meanschosen to collectdata werethe following:a calculustest given to bothgroups of students;interviews with the instructorswho taughtthe controland experimentalclasses; and a questionnairefilled out by all thosewho had used Keisler'sbook withinthe past five years. The calculustest was a fifty-minuteexam which the instructors did notsee untilafter it had been givento bothgroups. The purposewas to explorewhether or notthere did seemto be differencesin performancebetween the two groups. The questionstested the ability of the students to definebasic concepts,compute limits, produce proofs, and applybasic concepts. The singlequestion which brought out the greatestdifferences between the two groupswas question3: Definef(x) by the rule f(x)=x2 forx 2, f(x)=0 forx=2. Prove,using the definition oflimit, that limx b2f(x) = 4. The responsesof thestudents are summarizedin Table 2. The factthat there were a greaternumber of students in theexperimental group willing to tryto solvethe problem was part of a pattern.Table 3 showsthe numbers in the two groups who attempted an answerto thecorresponding test questions. Therewas also a notabledifference between the two groups in the way in which they responded to question6, whichcalled for an explanationas to whya certainintegral represented the volume of a givensolid. In the controlgroup there were just 9 studentswho spoke of the integralas a sum, comparedwith 16 in the experimental group. And of these students, only 3 inthe first group, but 10 in thenonstandard group, saw it as a sumof smallvolumes rather than a sumof smallareas. This content downloaded from 132.70.4.118 on Mon, 10 Mar 2014 11:41:19 AM All use subject to JSTOR Terms and Conditions 1976] MATHEMATICAL EDUCATION 373 TABLE 2: StudentResponses to Question 3 Control Group ExperimentalGroup (68 students) (68 students) Did not attempt 22 4 Standard arguments satisfactoryproof 2 correctstatements falling short of proof (e.g., one is only concerned with x X 2) 15 14 incorrectarguments 29 23 Nonstandardarguments satisfactoryproof 25 incorrectarguments 2 TABLE 3: Number of StudentsAttempting a Solution Control Group ExperimentalGroup (68 students) (68 students) Definingbasic concepts 48 52 Computinglimits 49 68 Producingproofs 18 45 Applyingbasic concepts 60 60 One ofthe instructors mentioned during an interviewthat formerly his students had found the use ofthe symbol dx in theintegral very mysterious, but that the students in theexperimental class had seemedcompletely satisfied. This observation was supportedby the answers to question6. The first year,only 2 studentscommented on thesignificance of dx.The secondyear, 16 studentsbrought