The Teaching of Elementary Using the 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

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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 , 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, , found a wayto makerigorous the intuitivelyattractive 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

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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 , 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 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: 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 it intotheir explanations. As forsuccess in givingdefinitions, there was little difference between the two groups, except that thenonstandard group gave nonstandarddefinitions. (Since both standard and nonstandarddefini- tionsare presentedin Keisler'sbook, a choicewas involved.)The twogroups also did aboutequally wellin computinglimits. The onlystriking difference was that 23 studentsin thestandard group took derivativesin computinglimits without there being any justification for doing so (includingsome studentsfrom each of the 5 schools).Only one student in the experimental group made this mistake. Seekingto determinewhether or notstudents really do perceivethe basic concepts any differently is notsimply a matterof tabulating how many students can formulate proper mathematical definitions. Mostteachers would probably agree that this would be a veryimperfect instrument for measuring understandingin.a college freshman. But further light on thisand other questions can be soughtin the commentsof the instructors. The instructor questionnaire was responded to byall ofthe 12 instructors who had taughta courseusing the textElementary Calculus: An ApproachUsing Infinitesimals withinthe past three years. The responsesare tabulatedin Table 4, withthe numbers indicating how manyinstructors gave each response. Notethat the group as a wholeresponded in a wayfavorable to theexperimental method on every itemin PartOne (agreeingwith the advantages cited, disagreeing with the objections proposed). Note also thatin PartTwo therewas almostcomplete agreement that the proofs of are easierto explainand closerto intuition.This seems quite remarkable since the instructors would be muchmore familiar with standard proofs. The experimental group was also given the edge regarding theease withwhich they were able to learn the basic concepts. One instructorcommented that, "When mymost recent class were presented with the epsilon-delta definition oflimit, they were outraged by its obscuritycompared to whatthey had learned." In the individualinterviews with the 5 teacherswho had taughtthe controland experimental classes,3 ofthem said theyfelt that the students in theexperimental class had a muchbetter feeling forlimits. One ofthem found the explanation in the fact that a studentcan move forward in thought (if

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 374 KATHLEEN SULLIVAN [May

TABLE 4: InstructorQuestionnaire:Part One

(-2 stronglydisagree, - 1 disagree,but not strongly,0 neitheragree nor disagree, 1 agree, but nor strongly,2 stronglyagree)

The responses below referto the experimentalclasses.

-2 -1 0 1 2

1. The studentshad a problem with accepting for the hyperrealnumbers. 3 4 1 4 2. The studentsseemed to find"infinitely small" a naturalconcept. 1 2 4 5 3. The time that mustbe used for nonstandard materialmakes it difficultto cover topics that ought to be included in a 1st year course. 8 2 1 1 4. The course seemed to give the studentsa betterfeeling for the historicaldevelopment of mathematicsthan a standardcourse. 1 3 3 4 1 5. I thinkthat a studentwho has had two semesters of nonstandardcalculus will be at a disadvantage in a standard3rd semestercalculus course. 6 4 1 1 6. I enjoyed teachingcalculus using this approach. 2 2 8 7. I probablyshould not have tried to teach the course withouta betterbackground in nonstandard analysis. 9 1 1 1 8. I feel that the experienceof teachingcalculus fromthe infinitesimalapproach will enrichmy futureteaching of calculus. 1 5 6 9. I am afraid that the introductionof infinites- imals leftthe studentsconfused about the real numbers. 4 6 2 10. I would preferto use the nonstandardapproach the next time I teach calculus. 2 1 4 5

InstructorQuestionnaire: Part Two

(S Standard,NS Nonstandard,ND No Difference)

The instructorsindicated which approach seemed to them to have the advantage or that neitherapproach seemed to have an advantage over the other.

S NS ND

1. The studentslearn the basic concepts of calculus more easily. 8 4 2. The studentsseem to be more "turned on." 5 7 3. The proofsare easier to explain and closer to intuition. 1 10 1 4. The studentsfind it easier to formulate theirquestions. 2 9 5. The studentsend up with a betterunderstand- ing of the basic concepts of calculus. 5 7

I picka pointx whichis infinitelyclose to c,f(x) willbe infinitelyclose to L) ratherthan backward (if I wantf(x) to be withinepsilon of L, I mustpick x to be withindelta of c). It wasalso pointed out that the nonstandardapproach makes it easier to illustrateconcepts like the derivativeon a static blackboardsince one is dealingwith a singlenumber - a representativeinfinitesimal - and applying operationsto it ratherthan letting something approach zero.

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Two instructorsexpressed the opinionthat the nonstandardmethod of learningcalculus has specialmerit for students planning to majorin engineeringor physics,fields in whichinfinitesimals havealways been considereda usefultool. On theother hand, some uncertainty was voicedon the questionof howwell students who want to studymore analysis will be able to makethe transition froman experimentalclass to a traditionalcourse. Conversations with students at theUniversity of Wisconsin,who had been in nonstandardcalculus classes, suggest that the attitude of the instructor in the standardclass maybe the crucialfactor. The studentswill have the necessarymathematical backgroundto makethe transition, but perhaps they ought to be preparedfor the fact that their future instructorsmay or maynot be convincedof thisfact. As G. R. Blackleyremarked in a letterto Prindle,Weber & Schmidt,concerning Elementary Calculus: An ApproachUsing Infinitesimals, "Such problemsas mightarise with the book willbe political.It is revolutionary.Revolutions are seldomwelcomed by theestablished party, although revolutionaries often are." Certainlythere is no claimto havefound an instantsolution to theproblem of teaching calculus. Thisis not"Calculus made easy." Hard work and practice will still be requiredof the student, and a poorbackground in algebrawill still make it difficultto learncalculus. On theother hand, there does seemto be considerableevidence to supportthe thesis that this is indeed a viable alternateapproach to teachingcalculus. Any fearson the partof a would-be experimenterthat students who learn calculus by way of infinitesimalswill achieve less masteryof basicskills have surely been allayed. And iteven appears highly probable that using the infinitesimal approachwill make the calculus course a lotmore fun both for the teachers and for the students.

References

1. H. J. Keisler,Elementary Calculus: An Approach Using Infinitesimals,Prindle, Weber & Schmidt,Boston, 1971. 2. A. Robinson, NonstandardAnalysis, North Holland, Amsterdam,1970. 3. K. A. Sullivan,Doctoral Dissertation,University of Wisconsin,Madison, 1974.

DEPARTMENT OF MATHEMATICS, BARAT COLLEGE, LAKE FOREST, IL 60045.

USE OF CANNED COMPUTER PROGRAMS IN FRESHMAN CALCULUS

MARGARET S. MENZIN

1. Introduction.Many articles and newsletters([1], [2], [3], [4], and theirreferences) have been publishedon thesubject of "computer calculus," which means a calculuscourse in which the students writeand runcomputer programs. It is thepurpose of thisarticle to describea freshmancalculus coursethat uses cannedcomputer programs, to argue thatcanned programs are an excellent alternativeto user-writtenprograms, and to encourageinstructors who have not used any computer programsin freshmancalculus to tryusing canned ones. 2. Generaldescription. The followingdescription refers to a courseof 100 math,science and economicsmajors taught at SimmonsCollege. Students were enrolled in 4 sections,each of which met 3 hoursa week.The textwas Thomas' Calculus and AnalyticGeometry ([5]). Studentshad periodic accessto twoteletype terminals hooked into a HewlettPackard H P-2000C.The terminalusage was a requiredpart of the course; computer exercises were collected, but not graded. Students, working in pairs,ran 6 programseach semester,for a totalexpenditure of 12 hoursor $7.50 per studentper semester. 3. The logisticsof terminalusage. One-thirdof the way through the fall semester, the calculus class was shownhow to use the terminal.Students did the actualtyping of responsesto thefirst

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