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THE STUDENTEXPERIENCEOF LEARNING ADVANCED PLACEMENT CALCULUSAS IN A WEB-BASE DENVIRONMENT
by
o
DeanHollowayAthesis submittedtothe Schoolof GraduateStudies
in partialfulfilment of the requirementsfor thedegree of
Master of Education
FacultyofEducation Memorial University of Newfoundland
October 2002 51.John's, Newfoundland
Abstract
Theprimarypurposeof this study was toidentifyimpedimentsto learningcalculus conceptsin a web-based,Advanced Placement (AP)CalculusAB courseandto make recommendationsfortheimprovement oflearning inthat environment.The subjects of the study were 15,level IIIstudents from3high schoolsinNewfoundlandand Labrador, Canada who enrolled in Mathematics 4225,a provinciallyapproved,firstcalculuscourse whichprepared studentsto writeThe CollegeBoardAPCalculusA8examination for universitycredit.Inthis study,studentsandaninstructorin4 geographically distant sites in1 schooldistrictwerelinkedvia adigital Intranet to create avirtualclassroomwhere instructionoccurredsynchronously,using Microsoft NetMeeting andMeeting Point computersoftware,andasynchronously, through awebsite developed specifically for Mathematics4225 using webc'FandMicrosoftFrontPage 98.Studentsweresurveyed to determine their experienceoflearningcalculus in aweb-based environmentandto identifyfeatures of the course that needed tobeadded,modified orremoved.The course textbook was analyzedtodetermine the appropriatenessofthe epistemological messages conveyedtostudents.CalculusAB CollegeBoard examinationswerealsocompared in terms ofthe changesin structureand difficultylevelenabled by the recent authorized use of graphing calculators.On the basisof thegathereddata and the text andexamination analyses, recommendations are made for the improvement of learning.
Acknowled gements
Thenames of four very importantpeople cometomindasIwrite thispage.I wan tto thank my co-superv isors.Dr.KenStevensofMemorialUnivers ityof Newfoundland and Dr.DavidReid of AcadiaUniversity.The flex ib ility,guidan ce,and encourageme ntof these gentleme n have influe ncedmy growthasa personand as a teacherand made my experienceas a graduatestudentrewar dingand memor able.
Theunde rstand ingandsacrifices ofmy wifeTracy havenot gone unnoticed . She has offered unwave ring supportthrougha very demandingandtumul tuoustimeinour lives and allowedmetowork andstudy untethered.Iamvery fortunateto beableto sharemylife withsomeone so beautiful.
In themidstofmywork onthisthesis came a wo nde rfulch ildwhohasforever changed our lives.Ihad notrealizedthe depthoflove for achilduntilGrant becamethe first additiontoourfamily.He fills ourdays withjoyandlaughter andlove .Thisthesis is dedicated to him.
iii
Abstract...
Acknowledge ments...
List ofTables...
...ii
. iii
. vi
List of Figures xi
Listof Appe ndices ... . xii
INTRODUCTI ON ANDSTA TE MENTOFRESEAR CHQUESTI O N 1
Chapte r One :Introduction . LITERAT UR E REV IE W...
. 2
...7
Chapter Two:LearningTheory 9
Chapte r Three:Ca lculus Refonn... . . 12
Chapter Four:Advan ced Mathemat icalThinking.... . .32
Chapte rFive:Learnin gat aDistanc e .47
RESEAR C HPROCED URE S 69
Cha pter Six: Researc hProcedur es ... . 70
RESU LTS 88
ChapterSeven:Introdu ctio n and Description ofPartici pan ts 90
Cha pter Eight:Resultsof Part Oneof the Survey 101
ChapterNine:Result s ofPartTwo of the Survey 170
Chapter Ten : Comparis onof Text s and Examin ations 219
ANALYSIS 248
Chapter Eleven :Thc StudentExperie nceofLearning APCalculus AB 249
CONCLUSIONSAND RECOMMENDATIONS 253
Chapter Twelve :Con clus ions and Recommendati ons 254
References 274
Appendix A:StudentSurvey 289
iv
Appendix D;Student InformationSheet/RegistrationForm 342
AppendixC:SampleConsentForms.; . 344
ListofTables
Table I Description oflcons onthe"WelcomePage" . 74 Table 2 Interpretation ofResults from Part One oftheSurvey 101 Table 3 Accessto Internet-Ready Computers:Mean andStandard Deviation of
Survey Responses .. 102
Table 4 Dates for Classes:Mean and StandardDeviation of Survey Responses...104 Table5 Datesfor Evaluation:Mean and Standard DeviationofSurvey
Responses... . 105
Table 6 SpecialNotices:Mean andStandard Deviationof SurveyResponses 106 Table7 CalendarCompile Option:Mean andStandardDeviation of Survey
Responses... .. 106
Table8 MyRecord:MeanandStandardDeviationof SurveyResponses .107 Table9 Lessons:Mean and Standard Deviation of SurveyResponses 109 Table 10 Online Self-Evaluation:Mean and Standard Deviation of Survey
Responses... . 113
TableII Concept Map:MeanandStandard Deviationof SurveyResponses...116 Table12 Course Outline:Mean andStandardDeviationof Survey Responses...117 Table 13 OutcomeLinked toLessons:Mean andStandardDeviationof Survey
Responses... .. 118
Table14 CourseEvaluation:Mean andStandardDeviation of Survey
Responses 119
Table15 External Links:Mean and StandardDeviationof SurveyResponses...120 Table16 Communication:Mean and Standard Deviation of SurveyResponses...123 Table17 Password: Mean andStandardDeviation of SurveyResponses 125 Table18 AlternateAccesstoWebsite Features...
vi
. 129
Table19 Communication:Meanand StandardDeviationof SurveyResponses...132 Table 20 Interpretation ofResultsfromPart One of the Survey... . 137 Table21 RulesandFormulae:Mean and Standard Deviation of Survey
Responses 138
Table 22 Exercises andProblems: Mean and Standard Deviationof Survey
Responscs.; . 142
Table 23 ExercisesandProblems: MeanandStandardDeviation of Survey
Responses... . 144
Table24 Answers and Solutions:Mean and StandardDeviationof Survey
Responses . 146
Table25 Prerequisites Chapter: Mean and Standard Deviation of Survey
Responses... . 147
Table26 Communication:Mean and StandardDeviationof SurveyResponses...155 Table 27 SavingWhiteboardFiles:Meanand Standard DeviationofSurvey
Responscs. . 157
Table 28 StudentExpectationsvs.StudentPerceptions ofInstructor Activity
DuringOnlineClasses . 158
Table 29 Importance andUseofClasses:Meanand StandardDeviationof
SurveyResponses 159
Table30 Interpretation of ResultsfromPart Two ofthe Survey 170 Table 31 MathematicalModeling:Mean and StandardDeviationof Survey
Responses... . 170
Table 32 Class Size:Mean and Standard Deviation of Survey Responses 171 Table 33 Amount ofMaterial:Mean andStandardDeviation ofSurvey
Responses 171
Table 34 Word Problems:Mean andStandardDeviation of SurveyResponses...172 Table 35 Answers:MeanandStandardDeviationof SurveyResponses 173
vii
Table36 Memorization:Mean andStandardDeviation of SurveyResponses 174 Table 37 Symbolic Manipulation: MeanandStandard Deviation ofSurvey
Responses.... . 175
Table38 Student-Teacher Interaction: MeanandStandard Deviation ofSurvey
Responses... . 176
Table 39 Writing:MeanandStandardDeviationofSurveyResponses.. ...177 Table40 TopicsLinked to Careers: Meanand Standard Deviationof Survey
Responses.... . 178
Table41 Feedback:Meanand StandardDeviationof Survey Responses 179 Table 42 Pace:Mean and Standard Deviation of SurveyResponses 181 Table43 StandardsandExpectations:MeanandStandard Deviation ofSurvey
Rcsponses.; . 182
Table 44 Proof: Mean andStandardDeviation of SurveyResponses 184 Table 45 Numerical Methods:Mean and Standard Deviation of Survey
Responses.... ...184
Table 46 Class Presentations:Mean and Standard Deviation ofSurvey
Responses 185
Table 47 Realism:Meanand Standard Deviation of Survey Responses 185 Table48 Technology:Mean andStandard Deviation of Survey Responses 186 Table49 LinktoOtherDisciplines: MeanandStandard DeviationofSurvey
Responses... ...188
Table 50 Learning:Mean andStandardDeviationof SurveyResponses .189 Table51 Isolation:Mean and StandardDeviationof SurveyResponses 190 Table 52 Teacher Support:Mean andStandard Deviation of SurveyResponses..191 Table53 ConnectiontoPreviousKnowledge:Mean and Standard Deviation of
SurveyResponses 191
viii
Table 54 Discussion of Material:MeanandStandard DeviationofSurvey
Responses... . 192
Table 55 PreviousDistance EducationExperience:Mean andStandardDeviation
of SurveyResponses... . 192
Table 56 Deep VersusSurface Learning: Meanand Standard Deviation ofSurvey
Responses 193
Table57 AP Calculus ABExamination Formats:Mean and Standard Deviation of
Survey Responses 195
Table 58 ThinkingAbility:Mean andStandard Deviation of SurveyResponses..195 Table59 Applicationof Concepts:MeanandStandard Deviation of Survey
Responses... . 196
Table 60 Mathematical Language: MeanandStandard Deviation of Survey
Responses 196
Table61 On-Site Competition:Mean and Standard Deviationof Survey
Responses 197
Table62 Relevance and Interest ofProjectTopics:Mean andStandard Deviation
of Survey Responses . 197
Table63 ParticipationonOtherWeb-Based Courses: MeanandStandard
Deviationof Survey Responses 198
Table64 TopicsThat Should HaveReceivedMore Time 199 Table 65 Aspects ofthe CourseThatShould HaveReceived MoreEmphasis 199
Table 66 MeanstoImprove Leami ng 200
Table 67 Rules and Formulae:A Comparisonof Texts 221 Table6& Graphics,Tips,Notes,Remarks,&Summariesand Guidelines: A
Comparisonof Texts 222
Table69 Exercisesand Problems: A ComparisonofTexts 224
Table 70 Appendices:A ComparisonofTexts 225
ix
Table 71 Answers andSolutions:A Comparisonof Texts 226 Table72 PrerequisitesChapter: AComparison ofTexts 226
Listof Figures
FigureI Construct ionof objectsandprocesses .... Figure2 Facetsand layers ofaconcept ....
. .40
.. .44
.. 74
WelcomePage for Mathematics4225...
Amodel of self-regulatedlearning 66
Splash pagecontainingthe"Mathematics 4225"icon 73
Login screen forMathematics 4225 website 73
Frame linkedfrom the "Lessons"hyperlink 77
Links to Mathematics 4225CourseResources 79
Open ing pageofMicrosoftNetMeeting containin gthe IPaddressof the
MeetingPoint server andthe backupserver 80
Figure 10 Requestto joina conferenceinMicrosoftNetMeeting 81 Figure3
Figure 4 FigureS Figure6 Figure 7 FigureS Figure9
Figure II Selection of Mathematics 4225 conference 81
Figure 12 Passwordrequest screen... .. 82
xi
Listof Appendices
StudentInformationSheetIRegistrationForm 342 AppendixA
Appendix B AppendixC
StudentSurvey ..
Sample ConsentForms...
xii
... ...289
. .344
INTRODUCTIONANDSTATEMENT OF RESEARCHQUESTION
The StudentExperience2
Chapter One:Introductio n
Researchintothe learnin g ofcalculusis varied andexte nsive,especiallywith respecttothe learning that takes place inthetraditional classroomsetting.As well,much researchhas beencarriedout intolearnin gat a distance. Ho wever,there currently exists a lack of researchintostudent learn ing ingeneral, andinto the learni ngof ca lculusin particular,in a web-basedenviro nment.Thisthesis addressesthe needfor researchin this area by describingrecent research into thelearning that occursinsuchan environment.
The thesiswillfirstexaminethelearning experienceofstudents,located atthree geographically dista nt siteswithina schooldistrict,whoparticipatedina web-based, Advanced Placement (AP)Calculus ABcoursethat was delivered usin g both synchronous and asynchrono usinstruction.Thiswillbeaccomplishedthrough a stude nt survey derivedfrom areview ofrelevantliterature oncalculusreform,learning theory, advanced mathema tica lthinking,andlearningatadista nce .Itwill the ncompare the currenttextbook withanothertextwhich has received supportas amor e suitablefirst calculuscourse text in termsofthe appropria tenessof the epistemo logic al messages it contains. Finally,thethesiswillcompareofficialAPCalculusABexaminationsfrom 1988 and 1997 todetermi ne the changestothe examinat ionenabled byuse of graphing calcul ators .Itwillthenmake recommendations for improvementsinlearnin g in the generalcontextof a web-basedenvironment and in theparticular contextofanAP calculus course.
Thethesisconsistsof twelvechapters.The remainderof chapteronepresen ts the research question, which establishesthe overallfocus oft hethesis.Chapteronealso
The StudentExperience 3
ident ifiesthe significance and limitationsofthestudy.Theliterature review consists of fourchapters.Chaptertwoexamin estheconstructivisttheoryof learning.The student survey results areinterpretedin Iight of thistheory oflearni ng.Chapterthreeexamines calculusreform.This chapteridentifiestheproblemsassociatedwithcalculuscourses taughtinthe traditionalmanner,presents thevisionsandgoalsofanew,revitalized calculuscourse,andexaminestherole oftechnologyin calculu sreform.Chapterfour examinesadvanced mathem aticalthinkinginterm s of thesources ofobstaclestolearning calculusconcepts,those conceptswhichstudentstypicallyhavedifficultyin understand ing,andhow conceptual understandin gcan beattained. Chapterfiveexamines learn ing ata distanceand identifies thechallenges inlearning at a distance as wellashow distancelearning can beenhanced.Chaptersix(research procedures)describesthe contextin whichthe research takesplace,justifiesthe selection of a qualitati ve research methodology,and lists the actualmethodsofdatacollection. The results are contained in four chapters .Chaptersevencategorizesstudentsas completersornoncompletersand providesadescription of eachstudent ascompiledfrom the studentsurveys.Chapter eightpresentsthe resultsof part one of the studentsurvey.Thispartofthesurvey examinedthefeatures of the Mathematics4255 website,course text,onlineclasses, evaluation, collaboration andsocialinteractio n.Chapter ninepresents the resultsofpart two ofthe studentsurvey.Thispart ofthe surveyelicitedstudents'opinionson factors whichresearchliteratureindicatedwasimporta ntin learningcalculusconceptsandin learningatadistance.Thequantity ofresults presented in chapterseightand nine necessitated that thediscussion ofthe results foreachfeatureoccurimmediatelyafterthe
The StudentExperience4
result sthem selves were presented,instead ofbeingdelayed10alater chapter.Chapler ten containsa compariso nof the current textfor thecourse andan alternate textin terms of the features and epistemologicalmessages containedin each. This chap teralso compares College Board CalculusASexaminations from1988 and 1997in tcnn s of the changes enabledbythe use ofgraphingcalculators. As muchof the discuss ionof results occurredinearlierchapters,the analysischapter(chaptereleven) containsonly a succinc t general descriptio nof students'experiencein learningcalculusinthe contextof this study andidentifies someof the majorcontributorstothe obstaclesto learninginthe course.Chapter twelve containstheconclus ions andrecom mendations arisingfrom the study.
Statemen t oftheResearchQuestion
Thepurpose ofthis studyis to investigate,andto recommendstrategiesforthe improveme ntof, studentlearning of calculusin a web-basedenvironm ent. Hence the research question is;
Whatare the impedime nts tolearningcalculusconceptsin a web-based, Advanced Placement CalculusAScourse whichisdelivered via adistrict Intranet,and how can this learning be improved?
DefinitionofTenns
Student refers to fifteen,levelIIIhigh schoo lstudents inone schoo ldistrictin NewfoundlandandLabrador.
AdvancedPlacement Calculus AB Courserefers to afirst calculuscourse based on the topicaloutline(available;
The StudentExperience 5
hltp://ap cenlraI.collegeboard.cum/repusilory/apOl.cd_cal_4313.pdf}developedbyThe College Board,USA. The web-base version of this course referstothe onlinecourse devel opedin the Centre for TeleLeamingand Rural Education,Memorial Universityof Newfoundlandin 199&(available: http://www.stemnet.nfca/-vsddi/).
DistrictIntranet refers to the digitalconnectionof9 high schoolsin a schooldistrictfor thepurpose of deliveringAdvancedPlacement courses.
Significanceofthe Study
This studyissignificant because ofitsfocus onlearning early calculus in a web- based environment.Allaspects of learningcalculusin this contextshouldbeinvestigated, and any recommendationsforimpro vedlearning implemented,to ensurethat AP calculus distanceeducationisan attractiveand real alternative to traditiona lface-to-facecalculus instruction.The availabilityofviableweb-basedcourses is especiallyimportantin rural areas of'NewfoundlandandLabradorwherestudent enrolmentand teacher expertisemay prevent student participationin an AdvancedPlacementCalculuscoursetaughtin the traditionalmanner.Linkingschools electronically tocreatevirtualclassroomsprovides gifted mathematicsstudentsfrom ruralareas withthe opportunityto participatein AP coursesandhencecontributestothe equalityof educationalopportunityforstudentsin Newfo undland and Labrador, irrespective oftheir geographical location.As well, the further refinementof this web-basedmodelofcoursedeliverywillcreate a more powerfulleamingenvironmentforhigh schoolstudents who currentlyenrollin non-At' web-baseddistanceeducationcourseswhicharc deliveredusing theInternet.Thestudyis
The StudentExperience 6
also significantin thatitinvestigateslearning in an environmentwhichprovidesfor synchronousinstruction. The majorityofthe current researchinto web- basedlearn ing focusesprimarily onasynchro nousinstructio n. Hence, alongwithaddressingtheneed for research into learningin a web-basedenvironmentin general,thisstudyalsoaddresses the particularlack ofresearchinto thesynchronousaspectof web-based learning .
Limitations
I. Thefindings of this studyare specificto the contextof thisstudyandmaynot apply to other web-based,first calcul uscourses
2.Data gathe ringand participantobservationsoccu rredatthree AP sitesin oneschool districtof NewfoundlandandLabrador.However,two of these siteswerelarger high schoolsin the district and onewas a smaller,morerural all-gradeschool.Therefore, some ofthe findings arising fromgathered data maybe applicab leonlyto specific sites and not generally10allsitesin thestudy.
3.Theparticipantsin thisstudy were teenagersintheir finalyear of secondaryschool.
Therefore,findingsof this study maynotbe applica bletomore junior students or to the adultlearner .
LITERATURE REVIEW
The StudentExperience 8
Introd uction
Theliterature review for thisthesisconsistsoffour chapters.The purpose of chapterthree isto identifytheinadequaciesof calculuscoursestaught in the traditional mannerandto providean overallvisionof a revitalized ,first calculuscourse.This chapter oncalculus rcfonn will providethe benchmark foridentifying problemsand solutionsinthe web-based,AP calcuJuscourse understudy in thisthesis.Chapterstwo, four,andfive are moretheoretical in natureand provide a solidtheoretical basis for the conclusionsand recommendationsof chapter twelve.The chapteronconstruct ivist learningtheory examinesanalternativeto the behavioris tviewoflearn ing characteristic of the majorityof first calculuscourses.The chapteron advanced mathematicalthinking investigate scognitive theoriesofconceptualunderstanding.Thischapterconcentrateson Ed Dubinsky'sAction ProcessObject Schema(APOS)Theory andthe procept notionof David Tall.The final chapterof theliterature review examines distancelearningandin particularhow a deep approachtolearningcan be promotedin sucha context.
The StudentExperience 9
Chapter Two:Learning Theory
Skemp(1987)notesthat models ofleamingwhich see learni ng asapassive, reproductiveprocess have beenunsuccessfulinboth explainingand bringingabout"the higher formsof learning...of which mathematicsis a clear example"(p.134).Asa result, a theory of knowledge knownas constructivism has emerged incurrent mathematicalresearch. Thisviewoflearn ing has beensupportedinpublications by a varietyof influential educationa lorganizationsincluding theNationalResearchCoun cil (1989),the CommissiononStandardsfor SchoolMathematicsof theNationalCounc ilof Teachers ofMathematics (1989), the Commissionon Teaching Standards for Schoo l Mathematicsof the NationalCouncilofTeachersof Mathem atics(199 1)and the MathematicalSciencesEducationBoard(1990).
AccordingtoJeremy Kilpatrick,formereditorof the Journalfor Researchin MathematicsEducation,the constructiv istviewinvolvestwoprinciples:(I)knowledgeis active lyconstructed,not passivelyreceived,and(2) comingtoknow is an adaptive process of organizingone'sexperiencesanddocs notinvolve discoveringan independent, pre-existingworldoutsidethemindof theknower (cited in Selden&
Selden, 1990).Similar ly,vonGlasersfeld(1983) aSSl:I1Sthatmeaning isbuiltor constructedbytheindividualusinghis orher own experiencesandprevious knowledge as a guide and thus,conceptsreside withineachpersonand have subj ective representations. However,Seldenand Selden(1990)notethat this does notnecessarily implythataconceptcanmean whateverthelearner wishes.Itsmeaningmust be consistentwithexperience,the meaning of related conceptsand themeani ng constructed
The StudentExperience 10
by others. Accordingto thcauthors, this impliesthat theemphasis in mathematics teaching shouldshiftfrom ensuringthata studentcan correctly replicatewhat theteacher has demonstratedto helpingstudents organizeand modifytheir mentalschemata(i.e.
their internal conceptua lnetworks,hierarchiesandprocesses).Von Gtasersfeld contends thatmathematical knowledge is the productof consciousreflection onthepartof the student andthat themathematicsteacher should be a facilitatorof studentthought processes.Specifically,ateacher's roleis to suggest appropria tecritical examp lesor counterexamplesfor student reflection;the actualcognitiveconflict resolutionmustbe lefi to the student.
YoungandMarks-Maran (1998) describelearningin a constructivistframework.
Ratherthan describing learningas practic ing,performinga taskcorrectly,or as a passive endeavor,the authorscharacterizequalitylearningas (a) personalunderstandin g and meaning;(b) interpreta tive;(c) active;(d) learningtolearn;(e) constructive;(f) reviewing, redefiningandintegrative;and(g)inquiring, exploring andinq uisitive.
Constructivism itselfh as a variety offonns. The acceptanceof thefirst of Kilpatrick'sprinciples is knownas simple constructivism whileacceptanceofboth principles is referred to as radicalconstructivism.Whileconstructivis mitself maydefy a strictdefinition,Rcbler,Edwards and Havriluk(1997)have identifiedsomeinstruc tiona l characteristicsthat are commonly attributedto the influenceof constructivisttheories:
I. Proble m-oriented learning activities that are relevantto studen tinterests, requiresometime, a variety of skills, andseveralpeople workingtogether.
The StudentExperience II
2. Highlyvisualformat s such as those madepossiblewith videodiscsand multimediamaterials.
3 "Rich"learning environmen tsthat use avariety of resources, suchas computersand constructionkits,as opposed tominimal ist environments whichrelyprimarilyontheteacher,textbook,and prepared materia l.
4.Collaborative and cooperativegroup work.
5.Learning throughexploration,withanemphasison problem solvingrather than getting the right answer .
6. Authenticassessmentmethodswithqualitative(e.g.portfolios,teacher narratives, andpecfonnance measur es)rather than quantitative(e.g.objective paper-and-pencil tests) strategies.
This thesiswillassume a constructivistmodel in the analysisof learning inthe AP CalculusA8 course.Thismodelviews learningas a constructiveprocesswhere the person who islearningseeksandbuilds informationon the basisofpreviousknowledge andpastexperience.The assumptionunderlying this viewisthat human beingsact upon their environment;theydonot simplyrespondto it(Schuell,1986).Thischallenges the behaviorist viewthatthelearnerpassively receivesinformation Instead,the learneris seen as"a purposeful agent,extracting and imposing meaning"(Lauder,1996).
The StudentExper ience 12
ChapterThree: CalculusReform
Introd uction
The processof calculusreform had its officialbeginning at theTu lane Conference,aworkshop organizedby Ronald G.Douglas and heldinJanuary1986.
Subsequen tworkon reformled tothe publicat ionofCalculus for aNew Cemury :A Pump,Not a Filter,by the MathematicalAssoc iation of America(MAA)in 1988, are sult of a natio nalcolloquiumheld onOctober 28and 29,1987,in Wash ington,D.C..At this confe rence,over sixhundredmathematicians,scientists and educators presented 75 backgroundpapers,presentations,and responses ontheissue of calculusreform.These submissionshighlightedthe perceivedshortcomingsoftradi tional calculuscourses, offeredan overallvisionfora reformed calculus course,andincludedspecific suggestionsas tohow this vision mightbeachieved.
Problems withthe TraditionalCalculus Course
In herintroduction toCalculus for aNewCem ury:A Pump,Not a Filter,editor Lynn ArthurSteenstates that less thantwenty fivepercentofstudentswho studycalculus surviveto enterthe science and engineeringpipeline.Calculusthusbecomes a barrierto theseprofessi onaJcareersfor avastnumberof students.Aswell,manyof those who do survive calculus are 100 poorly motivated to fill graduateschools,toofewin number10 sustaintheneedsof busines s,academeandindustry,andtoo ill-suitedtomeetthe mathematical challengesofthe next century(Steen,1988a).
The StudentExperience 13
Problems in the traditionalcalculuscourse are numerous andserious.Current courses are highlystructured and automated (Moskowitz,1988).Thereislittle or no emphasis on conceptualization{Moskowitz.1988;Reed,1988),or modeling (Moskowitz, 1988).Classes aretoo large (Moskowitz,1988; Fulton, 1988),and courses containtoo muchmaterialand are tootechnica L(Reed,1988).There is alackof wordproblems (Reed, 1988),andthose that arc posed arc templateproblemswhichonly require straight forwardcalculations (llallett,1996;Steen,1988b; Carr,1992) and no justification of answers (Priestley,1988). Problems,dealingwith the theory ofcalculusandrequiring rigorouscalculus, have vanished(Hallett, 1996;Steen, 1988b).
Inthe contextof our present calculuscourses,the mathematicalpower and beauty of calculusthatmathematicians knowand relishcannot beillustrated to students(Curtis
&Northcutt,1988). Students seecalculus as a collectionof rulesand proble mswhich theymust memorize before beingallowedto proceed (Curtis&Northcutt,1988;
Lo velock&Newell,1988).Hence calculus is viewedas a filter,asa rite of passage (Curtis&Northcutt,1988;Malcom&Treisman, 1988). Students are seldomhelped to understandwhytheyneed calculus(White,1988)-that calculusin its prese ntand future forms willserveas a critical passagewayintomajorsthat leadto higherpaying technical andprofessional careers (Newman&Poiani,1988) or that calculusispart ofthe knowledge base oftheliberally educatedindividual.
Wenolo nger askstudentstounderstand,hut insteademphasizemanipulation (Cipra,1988; Kolata,1988,J.Ryan,1992).Alarge portion of the content of calculus coursesisdevoted to gainingskills in numericand algebraicoperations (Layton,1988).
TheStudentExperience 14
Unfortunately,skillssuchas differentiationandintegrationarequicklyforgotten and must be reviewed(Moskowitz, 1988). Most students achievebymem orizing algorithmic proceduresgiven inthetext orbyteachers. andworkingtextbook problems following thoseprocedure s (Anderson, 1988; Steen,1988c).Consequently,students·learnmuch of their mathematics bymemorizationandoftendo nothave muchexperiencewithbeing expectedtounderstand concepts(Layton, 1988). Thereis also the expectationon thepart of students that memorization ofalgorithmicprocedures willbetheprimaryfocus of the course(Ostebee, 1993).
Traditio nal textboo ks have also comeunderattackbythosewishing to transform calculus.Archerand Armstro ng(1988)state that current textbooksinclud e toomany topics, excessivehighlighting andsummar izingsections.toomany template problem s andtoomuch 'plugand chug'(p.61).NewmanandPoiani (1988),discussing the increasing percentage ofadultlearnerswhostudycalculus,claim thatthese adultshave anintellectual maturityand curiosity which demand s moredepthofunderstand ing about the relationshi pbetween mathematicsandthereal worldthanmosttexts can provide.
Kolata(1988) assertsthattextbook writers havemade calculuscoursessoinstructor - proofthatevenPh.D. mathematicians cannotintroduc eany concepts that mightreflect the beauty andexcitement of the field. Ostebee (1993)contends thatstandardtextbooks have a stro ngemphasisonsymbolicmanipulation.Lyo ns(1988)no tes thatevenwith the longhistoryofdissatisfaction withstandardtextbooks, there has beenno radicalchange in those that havebeenrecently published.The authornotesthat this is illustrated bythe fact thatmany popular texts arenowintheirthird.fourth,andfiftheditions. Anderson
The StudentExperience 15
(1988)states that textbooks that are adoptedare similarto pasttexts which have traditionalvalues.Texts withradical cha ngesseldo mgetadopted, thus they arenot ....-rittenandpublis hed. Kolata (1988),writingontextbook stagnatio n,asserts that traditionaltextbooks wouldbe allbut useless forareformedcalculuscourse but notes that developing newtexts would be goingagainstavery entrenc hed industry.Giventhis hurdle, Wilsonand Albers(1988)note thatit is generallyagreedthat calculuscourses are unlikely to changeunlessnew textbooksare produced.
Impeding the effortof ca lculusreform is thefactthatthere islill ie agreementas tothe basicpurposeofthereform.For example,Starr (1988)distinguishesproponentsof calculusas a servicecourse,as a course toprov idetechniqu es, methods and manip ulatio n, andas a coursewhich is a core part of the learningofaneducated individual. Kolata (1988)notes that variousclientdisciplin es wouldprefer emphasis alternatelyon approx imationtheory andnumerical solutionsto differe ntialequations, models and qualitative analysis,and breadthand problemsolv ing.
Disagreementalso arises on the issueof partitioni ngclassesby discipline.
Moskowi tz (1988)andCarr(1992)clai m thatpartitioning wouldallowthepossibility of focusi ng moreon applicatio ns that are relevant to students intheirrespectivediscipli nes.
However,Erdman and Malone (1988) notethatthereisastrong feeling thatthere sho uld notbe separa tesequencesand thatthe courseand educationalobjec tiveswouldbebetter served witha varietyof studentsinthe course.Stevenson(1988)warns that "princip lesof mathematicstaught inthe contextof specificapplication have the danger of not being recognizedfor their breadth of applicability" (p. 26).Bossert and Chinn(1988) also
TheStudent Experience 16
believe thatpartitioningis inappropriate,claiming thatstudentstaking afirstcalcul us course have notyet set career directions,thatthe valueofa mathematicscourseaspartof aliberaleducationwillbe compromised if itistaughtwitha narrowdisciplinary focus andthat "there is adangerthat specialization of a calculuscourse will obscure the generalityof themathematicalway ofthinkingthat separate disciplineshavecometo mathematics togain" (p.67),
Alsoretarding thereformeffortisthe significantgroup of scientistsandengineers whowishtomaintain the status quo. This groupisuninterested inchange, stating that the traditionalcalculuscourseis sufficient,andtheydonotwantto expend thetimeand effort neededto radicallychangeit(Kolata, 1988; Tucker&Leitzel,1995; see also Anderson,1988). This group alsoholdstheviewthatany studentwhocannotpassthe traditionalcalculuscourseshouldrevisetheirplanstobecomeascientist(Kolata,1988).
Vision/Goalsof aRevitalized Calculus Course Theoverallvisionofcalculusis thatofa pump whichcan fillourscientific pipeline.LynnArthur Steen, in herintroductiontoCalculus for a New Century:A Pump.
Nota Filter,states:
Tofillthis pipeline,wemust educateouryouthfor a mathematics ofthe future thai will function in symbiosiswith symbolic, graphical and scientific computation.Wemustinterestourstudents inthefascinationand power of mathematics-initsbeauty andinitsapplications, inits history and initsfuture.
[Wemustoffer]a vision ofthefutureofcalculus,afuture in whichstudents and facultyare togetherinvolvedinlearning, inwhich calculus isonceagain asubject
The StudentExperience 17
at thecuttingedge- challenging,stimulating.and immense lyattractive to inquisitive minds.(p. xii)
The purpose then,ofcalculusinstruction.is to helpstudents to succeed. not to weed them out toreduce the numbersin highlycompetitive programs (Malcom&Treisman, 1988).
More specifical ly,a vision ofa reformed calculusconsists,inpart, of a ubiqu itous emphasis on conceptualunderstan dingandmuchless emphasisonfacility insymbolic manipul ation (Davis, 1988;Levin,1988; Ralston, 1988; Ross,1996;Tucker&Leitzel, 1995),whichRalston believes,humans typicallydo poorly andcomputers do welland whosemasterydoes notaid the ability to apply calculusorto proceedtoadvanced subject matter.Salamon(1988)advocatesthatthe objectiveof instructionincalculus is to have students acquire the most thoroughandfunctional understanding of calculusthey can achieve.Davis (1988)supportsthe need forcalculuscoursesto emphasize clear thinkingand not merely symbolicmanipulation. Hodgson (1988)is certainthat, ina revitalizedcalculuscourse,a shiftwilltake placefrompurely computatio nalto more interpretativeskills-inotherwords,fromcalculationto meaning.O'Meara (1988) announcesthatmathematics educationatthe collegelevel will havetobe strippedof rote andconceptsemp hasized. Wilson and Albers(19 88)no tethe comm on agreementthat conceptbuilding is a fundamen talgoal.HainesandBoutilier (1988)callformuch more usc of theconceptualapproac h,whichlies between pure skillsandpure theory.Young andBlumenthal (19R8)statethatfundamentalunderstanding of underlyi ngconcepts is what is important for students.These views on theimportan ce of conceptual understandingarc also supportedby amyriad of otherauthors, including Anderson
The Student Experience 18
(1988),Armstrong,Gamer and Wynn (1994), Baileyand Chambers(1996),Buccino and Rosenstein(1988),Cipra (1988),Curtisand Northcutt (1988),Flashman,(1996),Fulton (1988), Gehrkeand Pengelley (1996),Harvey(1988), Kenelly (1996), Kenellyand Eslinger (1988),Kolata (1988),Lathrop (1988),Layton (1988), Levin (1988).Osrebee (1993), Schoenfeld (1995),Smith (1996), Starr(1988),Tucker and Leitzel(1995), Watnick(1993),and Zom and Viktora (1988).
Anotherpervasive goal is to have calculus contribute tothe broad aimsof undergraduate,liberaleducation (Alberts,1988;Buccino&Rosenstein.1988;O'Meara, 19118;Starr,1988;Steen,I988b)-that is to help studentsthinkclearly(Davis,1988;
Starr, 1988;Steen,1988b), to communicate effectively(Steen,I988b),both mathematicallyand scientifically (Fulton,1988),and to wrestlewith complexproblems (Hallett,1996; Steen,1988b;Tucker&Leitzel,1995).
Otheruniversal goals include cultivationof a fascinationwith mathematicsand of anappreciation(Hashman, 1996;Harvey,1988; Hodgson,1988;Levin,1988) for mathematics,incl udingits power(Anderson,1988;Buccino&Rosenstein,1988;Miller, 1995; Salamon,1988),beauty(Anderson,1988;Archer&Armstrong.1988;Barrett&
Teles,1988;Buccino&Rosenstein,1988;Salamon,1988),application(Cipra, 1988), andhistory (Barrett&Teles, 1988; International Commission onMathematical Instruction[IeM I],1998). Whatis needed is a calculus whichprovides a body of understandingthat contributesto the flexibilityand adaptabilityrequiredof scientists, engineersand managers(White,1988).Sucha calculus course must focus on intellectua l mastery of thesubj ect (Ralsto n,198 8),must developincreased analyticalability(Egerer
The StudentExperience 19
&Cannon, 1988 ;Prichett,1988;Tucker&Leitzel,1995),develop mathematicalmaturity
(Brito&Goldberg,1988;Haines&Boutilier,1988)reasoning andlogic (Davis,1988;
Harvey,1988;Prichett,1988),andemphasize(and illustrate)theway mathematiciansdo (andthink about)mathematics(Archer&Armstrong,1988;Lathrop, 1988)-inother words,develophigher orderthinking skills (Harvey,1988).Arevitalized calculuscourse shouldhave studentsand facultyinvolvedtogether in learning(Davis,1988),based on an apprenticeship system, inwhichstudentshavelaboratory experiences(A!arc6n&Stoudt 1997,Bressoud (1996),Dodge,1988;Ross, 1(96)and sustained,directcontact withreal mathematicsprofessors (Starr, 1988).Calculusmust becomemoredynamic (Davis, 1988)andintuitive (Archer&Armstrong,1988;ArmstrongetaI.,1994;Dodge,1988)in approach,but in doing so mustmaintain a balance between theteachingofnew skillsand thedevelopment ofmathematicalandintuitive reasoning(Stevenson,1988).Finally, it mustreturn a senseof discoveryandexcitementto calculus classrooms (Buccino&
Rosenstein,1988;Cipra,1988;Ferritor,1988)andleadto thedevelopmentofstudents whopose, and actively investigate, what ifquestions(Cipra, 1988;Moskowitz,1988).
AccordingtoLovelock and Newell(1988):
Whatisimportantis that we structure the curriculumso asto nurturethe students' abilitiesto develop clear thinking, do word problems andbewilling touse modem technology ....-ithconfidence. Former students who have some work experience are far more aptto saythat itistheir abilitytothink andlearnrather than theirprecise knowledgeofa given topic whichis importanttotheminlater life.(p.l 63)
The StudentExperience20
Hence it isclearthat calculus reform mustplace strong emphasisonreasoning, logic, investigation, adaptability,flexible andhigher order thinking skills,and real learningand understanding.
Another goal ofanew calculus is emphas ison ....'riting.Steen(1988b)notesthat even in calculus courseswhicharc very well done,itis probablythe casethat students canget a gradeofB,or maybe even an A,withouteverwritinga completesentence . Priestley(1988)states thatthemost overlookedshortcoming in theteaching of mathematics is the failureofteachers to insistthat students justifytheir answers-if not in completesentences, then at least with a few suggestive English phrases.Buccino and Rosenstein (1988) claimthat such ...riting enhancesstudents'higher-order thinkingskills, forces studentsto consider their results,and increasesstude nts'appreciationforthe problems of writing for the benefitof others.David Smith,ofDuke University,statesthat
"failureto read and analyzeinstructionsprevents studentsfrom gettingstartedon a problem,and their abilityto understanda solution processisrelated totheirabilityto explainin Englis hwhat theyhave done"(citedinCipra, 1988 , pp. 10 1.102).Prichett (1988)notes that theexpositoryskills of most collegestudentsare dismalandthatitis timeto embracewritingas a primaryinitiative incalcu tus.Harvey (1988) sees a re vitalizedintroductory calculuscourse as one in whichstudentsbecomemore precisein written presentations.Others authors,such asBressoud (1996),Carlsonand Roberts (1996),Fulto n (1988),Gehrkeand Pengelley (1996),Kenelly (1996),Layton(1988), Ostebee (1993),Ross (1996),Schoenfeld(1995),Smith(1996), andTuckerandLeitzel (1995),agree withthe importance of writingin calculus.
The StudentExperience 21
Another importantgoal is the provisionof meaningfu lapplicationsin calculus (Bossert&Chinn.1988; Carlson & Gulick,1988;Chrobak,1988; Cipra,1988;ICMI, 1998;Miller,1995;Malcom&Treisman,1988;Ralston,1988; Ross, 1996; Young&
Blumenthal,1988; Watkins, 1988),provided these applicationsare nonroutine, tro ublesome to students,and requireconsiderablediscourse withknowledgeableteaching staff (Fulton,1988).lIo wever,alack of totalagreement on theimportance of meaningful applicationsin a reformed calculus is illustratedby Boyce (1988)who states:
Iam notparticularlyenthusiast icabout 'realistic'applications .Theymust often be couched in terminology unfamiliarto manystudentsand requiretoo much time to describe the underlyingproblem.Itis better to use simpleproblemsandmodels (evenif 'unrealistic' ) so that everyone canunderstandthem. (p.42) Carr (1992) alsonotes that a difficultywithproblemsolvingisthat teacbers must rely on studentshaving a sufficientlygood backgroundin othernon-mathematicalsubjects.
Encompassedinthe goal of meaningfulapplicationsisthe goalof relevancein mathematics. Levin (1988)notes that biologists,who takecalculusas freshmen,have forgottenalltheylearned long beforecollege graduation.Levininsists that our goal in teachingmathematics shouldbe to demonstraterelevancethrough choiceofexamples andtopics,method ofpresentationand emphasison concepts.Otherauthorssuchas Bailey and Chambers(1996),GehrkeandPengelley (1996), KenellyandEslinger(1988), Prichett (1988),Ralston (19&8).Smith (19Q6).and Watnick(1993) support this view.
Levin alsoasserts that studentsmustbeabletodistinguishwhether what is learned is of
The StudentExperience 22
gene ral rel evan ce andapplicability,or whetheritis particular to the application presented .
Also relatedto thegoalofmeaningfulapplicationsisthe goal ofproblemsolving.
YoungandBlumenthal(1988)state that word problemsfromclientdisciplineswhich require studen ts to think aboutthemeaning oftheir solutionareimportant incalculus.
Barrett andTeles(19 88) assertthatstuden ts mustbetaught tothink aboutnew problems as wellas solvetempla teproblems.However,CurtisandNort hc utt (1988), and Tucker andLei tzel (1995)contendthat theseproblemssho uldbemorerea listic thanhas generallybeen thecase andthatthe parametersof atleast someofthese problemsnot be so carefully chos en. This wouldpreclude allsolutionsbeing givenin aclosedform.Zorn, (inCipra, 1988) suggests that stude ntssho uldbeeasedinto thepractice of problem solvingby givingprogressivelymore open-endedproblems.Ponzo (citedinCipra,1988, p.96) supportstheidea of'long winded' solutions ,allowingthe computertohandle ca lculations,leaving the studentto concentrate onproblem setupand the seque nceof fonn aloperationsthat willlead to a solution. Lathrop (1988)mainta insthatthecourse should contai nperiods inwhichexpertsare confrontedwithnew problemsand thatthese experts describe their thinkingas theyattempta solution.Lovelock andNewell(1988) assertthatitshouldbe emphasizedthatmathematicsis anexperimentalsubject andthat most realproblemsarenot solvedcorrectlythefirsttime .Theybelieveteachersshould emphasize that the answeraloneisnotgood enough-students need to explain their logic.and themeaningof theirsolution.They shouldalsobeencouragedtoexaminethe reasonablenessoftheir answers{Boyce,1988;Haines&Bo utilier,1988;Kola ta,1988;
TheStudentExpe rience 23
Lovelock&Newell, 1988 ).Problems with no solutionshouldalsobegiven since, according to Loveloc k and Newell and Alarcon and Stoudt(1997),most studen tshave the impressionthat allproblemshave solutions. Erdmanand Malone(1988),Anns trong et al.(1994) andWarnick(1993)also support pro blem solvingincalculus(see also Carr, 1992).
Thereis alsoconsiderable supportfor mathem aticalmodeling,especiallyamong biologists. BritoandGoldberg(1988) note a generalagreement that it isessential for studentstorepresent particularproblemsin thelanguage of mathematics. Egererand Canno n (1988 ) statethatamathematics courseshou ld provide studentswith the opportunity 10 construct,articulateand interpret formal modelsin their own discipline.
Levin (1988)assertsthatthemost importantmathematicalproblem sfacingbiologistsare conceptualand involve theproperformulationof amodel.He also contendsthat agoalof calculusshould be to provide biologistswitha better under standing ofmathematica l models. (seealsoBoyce,1988;Bossert&Chinn,1988;Bucchino&Rosenstein , 1988;
Carlson&Gulick,1988;ICMI,1998 ;Ross, 1996;Tucker&Leitzel,1995; Yo ung, 1988;
Young&Blumentha l,1988)
Homework , consistingof nonstandar dquestions,and regularand extensive feedbackconstitute two otherimportant goals ofa nontraditiona lcalculus course.Fulton (1988)statesthat thereshouldbeearly and frequent assessment ofeachstudent's grasp of conceptsas well as regular,oftenextensive,conferenceswithstudents.Strang (1988) notesthat, inmany calculus cour ses,students getno feedbacktohomework assignme nts andheasserts that we mayexpecttoomuch ofstudents to work well without recognition.
The Student Experien ce 24
Bucci noand Rosenstein (1988) claimthatgraded homeworkprovides val uable informationto thestudent abou tcours eobject ivesand teache rexpec tationsand provides theteacher with informa tio naboutclassandindividualprogress.Curtisand North cu tt (1988) contend that under standin gandcommuni cation ofideason thepartof the stude nt willnotbeachievedunlessthe stude nt recei ves high qualityfeedb ackfrom theinstruct or.
Brito and Go ldberg(1988),Bressoud (1996), Ko lata(1988),Schoe nfeld (1995),and Watkins(1988 )also suppo rt theneed forfrequen t,highquality feedback incalculus
Anoth er important goaJisthatof interac tive approachestoteaching andlearning incalcu lus.Kolata(1998) claim sthatstudents need thepersonalinvolvement ofa professor or teach ing assistantto dowellincalculus.Finkeland Monk claim that
"collective work isa keyingredientto intellectualgro wth"(cited inCipra,1988,p.101).
Douglasassertsthatcalculus cannotbetaught inan impersonalenvironment (in Kol ata, 1988).Salamon(1988) claimsthathelp sessionsand drop-inmathem atics labs wouldgo alo ng way to givingstudents controlover their learnin g,aswould20minutesfor questions beforeandaftereach lecture.Malco mand Treisma n(1988) report that a significa nt number ofChincscstudents achievedproficiencyin mathema ticsthrough work in informalstudygroups,which providedfeedbackaboutthequalityoftheirwork andaide d them in constructingconcepts. The twoauthors also statethat groupwo rk in whichtheinstructor entersintoconversation withstude ntsisimportantinthatit provide s impo rtan tinsightsinto thestudent'sthoughtprocesses . Alarco n andStoudt(1997), Armstro nget al.(1994), BaileyandChambers(1996), Carr (1992), Dodge (1988),Fulton
The StudentExpe rience25
(1988),Gehrke and Pcngcllcy (1996), Gillman(1988),Newmanand Poia ni(1988), Ross (1996),Schoenfeld(199 5),Smith(1996),TuckerandLeitzel (1995),andWatnick(1993) alsosuppo rtinteraction in calculus courses.
Timeisalso deemed to be important. Haines andBoutil ier(1988)statethat calculuscoursesmaybeimp roved by givingstude nts timeto lingeroverconcepts and considerthe consequences of their thoug htsandactions.Bradb urn (1988),commenting on howthe use of computersandsymbol ic manipu lationpack agescanfacilitate the teachi ng andlearni ng ofcalcu lus,asserts thatstudents willstill need time to think. 1.
Ryan (1992), whorecommends calculus teachers take timewithearly concepts, notes that allowingtimefor gradual developm entmay producelong-termdivide ndsin studen t performanceand satisfaction.Insupport ingBradbu rn's view,Ryan notesthattechnology canblur underlying ideasjustasmuch as traditionaldeliverytechniquesif timefor reflectionisnot builtintotheprocess. Gill man(19 88)andWatnick(1993)advocatethe inclusionofmore timeincalculusfor students to sitbackand reflect,to mulloverideas andexpressthem intheirownwords.
Itis alsopurported that high standards and expectationsshou ldcharac terizea calculuscourse.Boththe MathematicalAssociationofAmerica (MAA) andtheNational CouncilofTeac hers of Mathemat ics(NCTM)recommend that the calculuscourse offeredin the12'" gradebetreated as a collegelevel course .MalcomandTre isman claim thatwe shouldhaveexpectationsofcompetenceof calculusstudents,andnote that student performancein the autho rs'workshop program justifiesthisassumptionof competence.Sma ll(1988) assertsthat"o ffering a watered-downcollege level course with
The StudentExperience26
no expectationof studentsearning advancedplacement[is] not considered(0be[an]
acceptableoption" (p.225). Newmanand Poiani(1988) declare thai the maintenanceof standards of qualityandhighlevels of expectationsare essent ialfor allstudents in mathematicscourses.Bailey and Chambers(1996) support emphasison learningwhich requiresstudents10achieve at establishedproficiencylevelsrelatingto subjectmatter.
There issharp disagreementof the nature,and eventheinclusion,of proofina firstcalculus course.Itisarguedthai formal proofsare essentialto attract strongstudents (see Wilson&Albers,1988),whileothersthinkproofsshould be doneincalculususing eithcrintuitiveorformallyconstructed premises(Bradburn,1988).Hom (1988)asserts thatteachersmust"develop andemphasizeproofs, notfromfirst principles,but from agreed-on intuitive principles"(p. 21).However,Bradburn suggeststhat thecourse should include convincing examplesofcases whereintuition ledto incorrect conclusions.
Lathrop(1988)contendsthat proofshould bereserved forsecond courses in calculus,and eventhen shouldbe examinedin the context ofexamplesrelating thesignificance of the resultobtainedtopracticalapplications.There isagreementhowever,thatit is essent ial that statements ofproofs begiven inaclear,concise form andthat changesin the hypothesesbe exploredto helpmotivate theorems (Wilson&Albers,1988).
There isalso universalagreementforanincreased emphasison numerical methods.Redish,(1988)states that:
Practicalnumericalmethodsformtheheart of computerapproachesto real-world problems, yetthese areconsistentlyignoredin the traditionalintroductory sequence. The problemsof numerical integration and differentiation are
The StudentExperience 27
suppressedinfavorof extensivediscussionsofho wto differentiateandintegrate largenumbers of specialcases,despi tethe factthat the real-worl d problemsmost scientistsface will almostcertainlyhaveto be treated numericallyalarge fraction ofthet ime.(p.111)
Thereisalsosupportfornumerical methods fromBossertand Chinn(1988 ),Carlson and Gulick(1988),Horn (1988),Kolata (1988),Levin (19 88),Tucker andLe itzel(1995), Wilsonand Albers(1988), and Youngand Blumentha l (1988).
There are also numerous othergoals for arevi talized calculuscourse . These include:(a) theprovision ofexercisesrequiri nggood judge ment(Com pton,198 8;Davis, 1988;Hodgson,1988);(b)creatinglinksto otherdisciplines(Bailey&Cham bers,1996;
Bossert&Chinn,1988;Carlson&Roberts,1996;Carr 1992; Knight,1988;Malcom&
Treisman,19 88);(c)requiri ngestimat ion andapproximation (Lovelo ck&Newell,1988;
Salamon,1988; Watkins,1988);(d)error estimation(Boyce,19 88);(c) class presen tation s bystudents (Bressou d(1996),Fulton, 1988;Kenelly,1996;Ross,1996);(f) qualitativeanalysis(Bossert&Chinn,1988;Ko lata, 1988;Red ish ,19 88; Tucker&
Leitzel,1995;Wilson&Albers,1988 ); (g) readingcalc ulus (Buccino&Rosenstei n, 1988; Gehrke&Pen gelley,1996;Haine s&Boutilier,1988; Smith,1996);(h) smaller class size (Cipra,1988;Hughes,1988 ; Newman&Poian i,1988);(i)projects(Bresso ud, 1996;Gehrke&Pengelley,1996;Tucker&Leitzel,1995);(j)an appropriate mix of routinepractice, gradedhomework exercisesandexemplaryapplications(Hai nes&
Boutilier,1988); and (k)thedevelopm ent of intu ition(Bucci no&Rosenstein,19 88;
Cipra, 1988;J. Ryan,1992).There is also supportfor the introd uction ofelementsof
The Student Experience28
discretemathematics(Carlson&Gulick, 1988;Davis,1988; Erdman&Malone,1988;
Watkins,1988),and for student traininginthe detectionof anomalies (errors and invarian ts)duringthe process ingof informa tion(Lathrop,1988).
Layton(1988)states thatthe goalsoffirst-yearcalculusinclude: The ability to giveacoherentmathematicalargumentandthe abilitytobe able notonlyto give answersbutalso tojustify them.In addition,calculusshould teach studentshow to apply mathema tics in differentcontexts,to abstractand generalize,to analyzequantitativelyandqualitatively.Students should learn to read mathematicson their0\\011.Incalculus they mustalso learn mechanicalskills, bothbyhandandbymachine.
Asforthingsto know,students must understand the fundame nta lconcepts of calculus:changeandstasis,behavior at an instantandbehaviorinthe average, and approxi mat ionand error.Studentsmustknowthevocabulary ofcalculusused to describethese concepts,and theyshould feel comfortablewiththatvocabulary whenitis used in otherdisciplines(pp.150-151).
Accordingto Anderson(1988),what we seck in a revitalizedcalculuscourse is a balance between oldandnew.Toachieve thiswemustdesign oureducationalpracticesto conform tostudents' future needs inthe post-calculuslearning environment and inthe workplace.
The StudentExperience 29
Technology andCalculusReform
Areportfrom theNationalScience Foundation (1988)stalesthai" the explosion ofnew applicationsof mathematics andthe impactof computersrequiremajorchange in undergradu ate mathematics"(p.221).According10 Moskowitz(1988),thesechanges shouldincludea greaterfocus on calculusconcepts,modeling,andinterp retation and a relegationof computationto computers .Thesetechnologiesare importan tbecause they canenhance a student'sunderstan dingof the fundamentalideasof calculus (Dodge, 1988).Fulton(1988)claims thattechnologycanbeusedto instillnew calculusconcepts suchas approximation ,estimation, erroranalysis,asymptoticbehaviorand goodness of fit,inaddition tothe current conceptsof change,limit, andsummation.Zorn and Viktora (1988) describe how technology may affect calculuscurriculumandpedagogy.The authorsstate thateasyaccess10 technologymay (a)allowmore realisticapplications,(b) causesome topics andmethodsto becomeobsoleteandallow others to replacethem,(c) createamore active experimentalattitudeinstudents,(d)leadtodeeper, moreflexible understandings,(e) developstudentintuition,(f)supportqualitativereasoning,and(g) fostermoreeffective problem s solving.Harvey (1988)contendsthat theuse of technology (i.e.graphicscalculator)can permit expansionoftestinghigher-order thinking andproblem solvingskills. Gillman (1988),commentingon stude ntpracticeand feedback, writes that"a well-designedprogram,with thoughtfulconditionalbranching, willofferguidancewhile atthe sametime allowingstudentsto pick thetopics they need practice on.The instructoris[thenIfreedto devote full time to [he exchangeof ideas"(p.
218).Manyother authorssupportthe prudent usc oftechnologyincalculus, including
The Student Experience 30
Archerand Armstro ng(1988),Bail ey and Chambers(1996),BarrettandTclcs (1988), Buck(1988),Carlsonand Gulick(1988),Chrobak(1988),Cipra(1988),Curtisand Northcutt (1988),Davis (1988),Gillman (1988),Kola ta (1988),Layton(1988),Lovel ock andNewell(1988),NewmanandPo iani (1988),Ostc bcc(1993),Ralston(1988),Ross (1996 ),Steen (1988c)andWilsonandAlbers (1988).
Severalautho rsurge cautionin, andexpressconcernabout,the use ofteclmology incalculus.ZornandViktora(1988) note that itwouldbe a mistake to add morematerial to the alreadyovercrowdedsyllabusof a coursewhentime is saved through theuseof technology . Brad burn (1988) statesthatcaremusthetaken so thattechno logy doesnot take onthe role ofperforming"Mathernagic"(p.155). White(in Cipra, 1988) notes that, insteadof spendingthe timesavedbytheuse of graphicscalculatorson under stand ing underlyingconcepts, the time is sometimesused simplytoshow stude nts which butto ns to push.Hainesand Boutilier (1988),andNewmanandPoiani(19 88) advocatethatthe use oftechnology cannotbean add-on,itmustbeintegratedproperlyintothe course.
There isalso a fearofthe "blackbox syndrome";theuse of mechanic alsystems without understandingofthe underlyingconcepts. Boyce (1988)questio ns if we should permit studentstouse symbolicmani p ulationpackageswithoutunder stand ing what these packages are doing. Haines andBoutilier (1988)assert thatconcepts must he developed and understoodbeforebeingused ona calculatororcomp uter,whichcanthenbe usedto enhanceandexpand theunderstan ding of theseconcepts.Egererand Cannon(1988) questio n whether "black box" technologyisasuitablesubstituteforunderstand ing conceptsandprocedures.Th ere are those,however,whocontendthat suchsystems allow
The StudentExperience 31
studentsto concentrateon higher-levelreasoning,ins teadof on lowerlevel manipulative skills.
Giventhe cautio nsandfearsofsome authors,technologycan benefitcalculus learninginmany ways.Accordingto Zorn(1988),technologycan create better mathemat icallearn ing by (a) makingundergraduatemathematics more like real mathema tics,(b) better illustratingmathe maticalideas,(c)helping studentswork examples,(d) aidinginthe study of algorithms as opposed tothe performance of algorithms,(e) supportingmorevaried, realisticandilluminating applications,(f) improvi nggeometricintuition,(g) encou ragingexperiments,(h) facilitatinganalysis(i) teaching approximation, 0)prepari ng students to compute effective ly,butskeptical ly,(k) showingthemathema ticalsignificance ofthe computerrevol ut ion, and(I)making higher-level mathemat icsaccessib leto stude nts.Hencetechno logy hasthe potential to aidin attaining many ofthe goalsof a revitalized calculuscourse .
The StudentExperience 32
ChapterFour:AdvancedMarhematlca lThinking
Tall(1995)asserts thatadvanc edmathematica lthinkinginvolves using cognitive struct ures prod uced by a wide ran ge of mathemat icalactivities to constructnew ideas that build on and extendan ever-grow ingsyste mof establishedtheorems.Dreyf us (1991) statesthatadvanced mathe maticalthinkingis extremely complex, requiring the intricate interactionof alargenum ber of compone ntprocesses. He notes no sharp distinction betweenmanyof the processes ofelementary and advanced mathema ticaJ thinking.
Processes suchas abstrac tion,representation,analysis andvisualizatio nare presentin elemen tary mathematical thinking aswell as in advancedthinking.However , he identifiesthe manage mentof complexconcepts as a distinguishingfeatureofadvanced mathematica lthinking.Tall (1995)writ es that "elementary mathematical thinking becomes advanced mathematicalthinkingwhenthe conceptimage s in the cognitive structureare reformulatedas conceptdefinitions[themathematically accepteddefi nition] andusedto construc t formal concepts that are part ofa systematic body ofshared mathematicalknowled ge" (p.61).Talldefinesconceptimageas theterm used todescribe the"tota lcogn itivestruct ure that is associatedwith the concept,which includesall the mentalpictures and associated properties and processes" (p. 7).Tall(1991,1992) also identifiesthepossibility offormaldefinitionand deduction as another factor which distinguishes adva nced mathema ticalthinking. The move fromelemen tary to advance d mathema ticalthinking involves a significant transit ion :that from describing to defining andfrom convincingtoprov ingina logicalmannerbased onthose definitio ns.
The Stude nt Experience 33
SourcesofObstacles inLearningMathematics Tall(1991) asserts thatitis often the incongruitybetween concept imageand conceptdefinit ionwhich is thesource of cognitiveobstacles forlearne rs. Learn ersform a personalconcept image whichmayor may notbeequivalenttothe formalconcept definition.Itisthe conceptimage instead ofthe concept definitio nwhichreceives emphasis(Vinner,1992), andthis mayresult ina lackof conceptual understand ing onthe part of the student.According toTall (1992),themove tomore adva nced mathematical think inginvol ves adifficulttransitio nfrom thedevelopm ent of concepts on an intuitive basis,foundedon experience, to one whe reconceptsare specifiedby formaldefinitions andthei r propert ies reco nstructedthrou ghlogicaldedu ctions.The growingbodyof deductiveknowledgewillexistsimultaneo uslywithearlierexperiencesandpro pert ies and this can produce cognitiveconflictswhich actas obstacles tolearn ing.
Students may alsonever experience the fullrange of thought proce sse s commonly used by mathe maticiansandthismay betheresult oftheinstructio nthat stude nts receive.
Skemp statesthatstuden tsaretypicallytau ght "theproducts of mathematicalthought insteadof the processof mathematica l thinking"(citedin Tall,1991,p. 3).Instruct ors tend topresent mathematics inits final,polishedform,following the standardsequence oftheorem-proo f -application,eventhoug hthey areawarethatmathema ticsiscreated through processes such asintuit ion,trial anderror, conjecturing, andte sting. Such presentationof mathematical products enables a well-plannedandsched uledcourse presentationbutlacksflexibility in itsadaptability10 students.As well,this logical sequenceof instruc tion may notbeappropriate for thelearner's cognitive dev elop ment.
The StudentExperience 34
Theresult of this approach is that most students learnto carry outlarge numbersof standard procedu res,cast in precisely definedformalisms,for obtaininganswers to clearlydelimitedclasses of exercises.Many"successful" studentsendup with considerablemathematicalknowledgebutwithout theabilityto usc theirknowle dgein a flexible marmerand transfer ittounknown problemtypes;that is, theylackthe methodology of the workingmathematician(Dreyfus,1991).
Language may also be a barrierto understanding in mathematics. For example, the connotations of the word "limit"in everyday life are often atodds with the mathematicalideaof limitandmay leadtounavoidable misconceptionsofthelimit concept.As well,terminologysuchas "gets close to" and"approaches"is often used in the developmentof thelimit conceptin mathematicsandcarriesthe implication that an expression cannever equalthe limit value.Wells (1999)has publisheda 200 page Handbook ofMathematical Discourse whichis a compilationof mathematicalusage with afocus onusage whichcauses problemformathematics students -anexamplebeingthe logical errors which are a resultofthe use of everyday Englishlanguage andits meaning in a mathematical contextThe authornotes that mathematicalEnglishlanguagemay be foreign tostudents as it uses familiarwords and grammatica lconstructio nswhichmay havemeanings which arevery differentfromthose to whichstude ntsare accustomed.
Concepts WhichCauseProblems inCalculus
Researchhasidentifiedavariety of calculusconceptswhichcauseproblemsfor students . Russian andVinner (1997),in a studyof high schoolstudents,found thatthe
"specialcase"approach to odd and even functionscausedtheformationof concept
The StudentExperience 35
images which createdserious difficulties in theformulation andthe applicationof concept definitions. Davydov(inBinns, 1994)notes that this inductiveorspecial to generalapproachhas failedin Soviet schools. Bills (1996)andMcel (1997)offeran explanation, referring topsychological evidence produced by Tall and Vinner(1981) and Tall(1989),which indicates thatirrelevant attributes of examples from whicha concept has been abstractedare not forgotten once thc concept is formed. These attributesare retained aspart of the conceptimage and canform the basis for obstaclestothe understandingof mathematicalconcepts.
Ursini and Trigueros (1997),in a study of starting college students,foundthat the majority of studentswere restrictedto the action oonceptof variableand that this conceptionof variableprecludessome proceduresfrom beinginteriorized.Aprocedureis interiorized whenthe learner nolo nger has to perform the operationin ordertothink aboutthe procedure.White and Mitchelmore (1996) notedthat such studentsare stillat the condensationphase of theirdevelopment oftheconcept of variable and concluded thatan abstract-generalconcept of variableat or near the pointof reification isnecessary forthe successfuJstudyof calculus.Condensationrefers to the stage where a complicated processis condensedinto a fonn that becomes easierto cognitively manage. Reification isthe stagewhereconceptsare conceivedas objects (Foran expanded discussion of interiorization,condensationand reification,seethe next section).Mamona-Dow ns (IQQ7),in a study offirst year universitystudent s,assertedthat an understa ndingof variable is necessary forthe objectificationofthe concept of function,which in tum is crucialfor the understandingof calculusconcepts.Trigucros, Ursini andReyes (1996)
The StudentExperience 36
studiedstarting colle ge student'sand foundthatmostdidnot havetheunderstanding of variable required foradvanced mathematicalthinking.
Crawford, Gordon,Nicolas and Prosser (1994), studyingfirst year mathematics students,foundthat studentsexhibitedawide rangeofawarenes sof the conceptof functionbut that mostfocused onthe representation of functionrather thanits essentia l meaning.DeMaro isandTall(1996), DreyfusandEisenberg(1992 ), Eisenberg(1991), Hollar(1996),RassianandVinner(1998), Vinner(1983) andVinnerandDreyfus (1989) have identi fiedother problemswith functions.
Researchhasalso revealed student problems with theconceptsof:(a)proof (Finlow-Bates,1994),(b) infinity(Shama&Movshovitz-Hadar,1994; Tirosh,199 1;
Tsamir, 1992),(c)rate of change(Martin,1996; Orton,1984;Porzio.1997), (d)average rate andaveragevalue(Bezuidenhout,Human&Olivier,1998),(e)tangent (Vinner, 1982;Tall,1987),(f) differentialequations[Esquinca ,1996;Mochon, 1996; Rasmussen, 1998),(g)the chainrole (Vidokovic&Czamocha,1996),(h) derivati ve (Ferrini-Mundy
&Graham,1994;Porzio,1997;Risnes,1997;Villarreal&Borba,1998)and (i) limits
(Azcarate&Espinoza,1995;Cornu, 199 1). Foragenera ldiscussionofstudent difficulties andmisconceptions see Thomas and Hong (1996) andNaidoo(1998).
ConceptualUnderstanding
Dubinsky.Czarnocha,Prabhuand Vidakovie(1999)identify five learni ng theoriesfoundin researchliterature.TheseincludeSfard's operational/structural characterizatio n (Sfard,1991),the concept image/conceptdefinitiondichotomyof Tall andVinner(Tall&Vinner,1981), the didacticalengineerin g view(Artig ue,1991), the
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proceptnotionof Gray and Tall(1991, 1994)and Dubinsky'sown Action Process Object Schema(APOS) Theory (Dubinsky,1991,1992;Dubinsky et aI.,1999).The sections whichfollowfocusmainly on APOStheory,althoughatleast partialdescriptionsare includedofmostof the other four theories.The sectionendswith a critiqueofAPOS theorygiven byDavid Tall,in whichheacknowledges the usefulnessand successof APOS theory in the development of undergraduate mathematics curricula,but questions theuniversalapplicabilit yofthetheory,especiallyin thelearning ofgeometry.
Given thatthe conceptimageis oftenat oddswiththe formalconceptdefinition, and that cognitiveobstacles are frequentlytheresult of thisincongrue ncy,how arethe conceptualentities, that are the essence ofadvanced mathematics,formed?David Ta ll,in thepreface ofthe bookAdvancedMathematica lThinking,asserts thatconceptualentities are formedthrough theprocess of reflectiveabstract ion,which,accordingtoDubinsky (1991)is aPiagetian conceptused to describe the constructionoflogico-mathernatical strueturesbyan individualduring thecourse of cognitivedevelopment.Dubinsky(1999) identifies andbrieflydescribes the mainmental constructions thatneedto bemade in the learning ofmathematics.He describesan actionas a transformationwhichis a reactionto stimuliwhichthe subjectperceivesas external.Theauthornotesthat altho ughtheaction conception islimited,itisan importantbasis forunderstandinga concept.An action becomes aprocesswhentheindividualreflects on theactionand interiorizesit. The individual can thenestablishcontrolover it. When the individ ual reflectson the operationsapplied to a process,becomes aware oftheprocessinitstotality,realizesthat the process canbe transformed and is ableto constructthetransformations,the processis
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thoughtofas anobject.Finally, Dubinsky definesaschemaas amoreorless coherent collectionof processes,objectsand otherschema thatisinvoked to deal withane w mathematicalproblem situation. liemaintains thatlearninginvolvesapplyingreflective abstraction to existingschemainorder to constructnew schemafor understanding concepts.
Dubinsky(1991), whoemphasizesits constructive aspects,assertsthat reflective abstraction isthe constructionof mentalobjectsand ofmental actionson these objects.
lie identifies five kindsofconstructionmechanisms whichare importantforadvanced mathematicalthinking;interiorization,composition or coordination,encapsulation, generalizatio nandreversal.
Goodson-Espy(1998)describes Sfard's three stages of conceptdevelopment, whichare similar tothat of Dubinsky's.He describes stage one(interiorization)as the stage where thelearner performsoperationsonlower level mathematical objects.
Eventually,the learner becomesfamiliar with theseprocessesandcan thinkabout them without actuallyhavingto carry outtheprocess.Whenthis occurs,theprocessis saidto have beeninteriorized.Forexample,alearner mayinitiallyfind the sum of 5 and3by beginning withfivefingersandcountingthreemore fingersinsuccessiontoarrive atthe sum of8.Eventually,thelearner canperformthis operationwithouthavingto carryout the fingercounting processand 'Nil! arrive atthe answerof8.Atthis pointthe process of additionofnaturalnumbershas been interiorized.
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Newprocesses in a schema canbeconstructedbythe coordinationoftwoor more processes (Dubinsky,1992).Forexample,the process of finding(x+3)2canbeviewed as the coordinationof the process of adding3to a numberandtheprocess ofsquaring.
Dubinsky (1991,1992) definesencapsulationasthe conversionofa dynamic process intoa staticobject.For exam ple,the expression7x+5 canbe viewed asthe process of multiplyinganumberby 7 and then adding5 or asthe binomial7x+5 (an object).When7x+5canbeconceived of as an objectas opposedto a process, encapsulationis saidto have occurred.This construct ionis viewed as the mostimportant, andthemostdifficult,forstudentsof malhernatics (see alsoGoo dson- Espy,1998).The abilitytoflexibly view a conceptas process or objectis what makes mathematical thinking powerfu l(Tall,1992).Thesymbol7x+5bccom es a vproceptrwhen 7x+5cornes to be viewedflexiblyas eitheraproc essora concept(binomial)(Gray&
Tall,1994,p.121).
Generalizationoccurs whenthe learnercanapplyan existingschemato a wider collectio nof phen ome nathan was previouslythecase.Forexample,when the schema for additionis appliedto obtainmultiplication,genera lizationof the schema for addition has occurred . Tall(1991)makes a distinctio nbetweenthedifferent types of generalizatio nin accordancewith the cognitiveactivitiesinvolved.Anexpansive generaliza tionis one whichextends the leamer'sexistingcognitive struct urewithoutrequiringchanges in curren tideas. A reconstructive generalizationrequiresreconstructi onof the existing cognitivestructure.
