RidingtheDoubleFerrisWheel:Students’InterpretationandUnderstandingof

TrigonometricFunctionsinRealisticSettings

GeorgeSweeneySanDiegoStateUniversityfiresnake77@hotmail.com

Thispaperexamineshowtwostudentsunderstoodthenotionsofangularvelocity

andmovementandhowthatunderstandingcontributedtotheirconstructionofa

trigonometricfunctionthatdetailedtheheightofarideronadoubleFerriswheel

overtime.Theanalysisiscarriedoutfromasituatedperspectivewithparticular

emphasisonwhenstudentsreasoningtookplaceandhowthatfacilitatedtheir

understanding.Thispapercontributesbydetailinghowthestudents’conceptions

ofangularvelocityandmovementchangeastheyconstructgraphsandfunctions.

Furthermore,itillustrateshowthestudentsutilizedthoseconceptionstocontribute

totheirabilitytoreasonaboutquantitiesco‐varyingintheproblemandalsowhatit

meanttocomposetheangularmovementwiththefunctionfortheheightverses

changeinradians.

Thispaperisdividedintothreemainsections.Thefirstsectionisa

backgroundsectioninwhichIgiveabriefsynopsisoftheliteratureon

trigonometry,detailthetheoreticalperspectivethatwillbeusedinthepaperand

discussthetaskandmethodsutilized.Thesecondsectionanalyzesthestudents’

engagementinthetaskandalsohowtheyconstructedtheirmathematical

understandingofthephenomenon.Finally,thelastsectionwilldiscusshowthis

examinationcanshedlightontherolethatfunctioncompositionandco‐variation

playinmodelingperiodicphenomenausingtrigonometricfunctions.

Background

LiteratureReview

Theliteratureanalyzingtrigonometry,particularlystudentunderstandingof

trigonometryissparseandreflectsadisparatenumberofdifferentperspectives.

Weber(2005)hasexaminedstudentsrelationshipsbetweenthetriangle,unitcircle

andthesineandcosinefunctions.Hedemonstratedhownon‐traditionalclassroom

instructionmadeanimpactinthestudents’abilitytoreasonabouttrigonometric

functions.Ozimek,Engelhardt,Bennett,&Rebello(2004)foundthatstudentsin

theirstudyhadsignificantlygreaterdifficultydealingwiththeunitcirclethanthe

functionandtrianglerepresentationsofsinefunctions.Theyalsofoundthatthere

wasnosignificanttransferfromtrigonometrytophysics.Shama(1998)suggested

thatstudents’understandingofperiodicitywasrelatedtoGestaltstructures,where

periodicphenomenaare“understoodasawholeprocesswithunifiedlaws(273)”.

AndGerson(2008)analyzedhowstudentsunderstooddifferentrepresentationsof

periodicfunctions.Furthermore,sheshowedthatstudentsdevelopedtheirown

conceptimagesofthesefunctionsdespitethefactthattheteacherfocusedonlyon

proceduralfluency.

Someauthorshavealsoexaminedhowtheteachingofanglemeasure,radian

andarclengthhavepossiblyledtoalackofcoherenceinthetrigonometry

curriculum(Thompson,Carlson&Silverman,2008;Thompson2008).These

authorscontendthatthetrigonometrycurriculumdoesnotallowenoughtimeor

emphasisonstudentscreatingmeaningaboutanglemeasureandarclength.

Furthermore,theyalsonotethatteachersoftrigonometryneedtohaveprofessional

developmentthatencouragescoherencebetweentheirunderstandingofangle

measure,arclengthandtrigonometricfunctions.

TheoreticalPerspective

Thesituatedperspectivetakesasfirstprinciplethatindividuals’mathematical

activitycannotbeseparatedfromthesituationinwhichitarises(Lave,Murtaugh,&

DeLaRocha,1984;Nunes,Schliemann,&Caraher,1993;Säljö&Wyndham,1993;

Brown,Collins&Duguid,1989).Theuseofthisperspectiverequiresthatwere‐

examinehowstudentscometoknowandthepatternsofactionthattheyuseto

undertaketasks.Greeno(1998)definestheemergentproblemspaceasoneinwhich

theproblemsthatareindisputeduringanyinteractionareaproductofthecontext

inwhichanyproblemarises.Consequently,salientcharacteristicsofanyproblem

arenotimpliedintheoriginalproblem,butratherariseintheinteractionofthe

personorpersons.Thisproblematizescharacterizingcharacteristicfeaturesofa

problemassurfaceorrelevantbecausethestatusofthefeaturedependsonthe

personsolvingtheproblemandhowtheyareinterpretingthesituationina

moment‐to‐momentfashion.Thus,whenexaminingAndrewandOscar’sattempts

atdealingwiththetasksoftheexperiment,Iconsiderwhatfeaturesbecomesalient

andwhatquestionsOscarandAndrewaskatanygivenmomenttounderstandthe

natureoftheproblem.Aswell,Iexaminedhowthenatureofthetaskandthe

patternsoftheiractivitycontributedtotheirunderstandingsofthegraphsand

functionofSandra’sheightversustimeonthedoubleFerriswheel.

Methods

Asmallgroupteachingexperiment(Confrey,2000)wasconductedoveraperiodof

twoweeks.Threegroupsoftwostudentsattendedtwo,1½‐2½hourworking

sessionsinwhichtheywereaskedtoconsideraseriesoftasksrelatingtothepath

travelledbyarideronadouble‐Ferriswheel.Thestudentsweregivenanapplet,

whichmimickedthemovementofthedouble‐FerrisWheel(figure1)thattheycould

controlandmoveastheywished.Furthermore,laterintheinterview,thestudents

weregivennumericalvaluesforthesizeofthewheelsandperiodsofthewheels

movement.Theteachingexperimentcenteredaroundfourbasictasks:

• ConstructarepresentationoftheSandra’sride.

• ConstructaqualitativelycorrectheightversustimegraphofSandra’sride.

• ConstructamathematicallyaccurategraphoftheSandra’sheightversustime

ontheDouble‐Ferriswheel..(Eg.Addscaleandimportantpointstothe

graphandtheircorrespondingvalues.)

• CreateafunctionforSandra’sheightversustime.

Thestudentswereallowedtouseasmuchtimeastheywishedtocompletethe

tasksandtheywerefrequentlyaskedquestionsabouttheparticularworktheywere

doingorhowtheywerethinkingatanyonemoment.Furthermore,therewas

considerationthatthestudentsmaynotbeabletomoveontolaterpartsofthetask

withoutsomeintervention.Asmyintentionwastocharacterizethestudents’

activityastheyinteractedwiththewheel,Idonotseethisasbeingproblematic.

Anytimemyownactionsservetofurtherstudentunderstandingofthetask,Iwill

accountforitintheanalysis.

Figure11 Thetwostudentsanalyzedinthefollowingsectionsofthepaperwereboth

enrolledinaclassonthinkingabouthighschoolmathematicsfromanadvanced

perspective.Theywerebothengineeringmajorsandwereontheirwaytoearninga

minorinmathematics.Theybothconsideredteachinghighschoolmathematicsas

anoptiontoworkinginengineering.Furthermore,bothstudentshadextensive

1Keymath.(2009).[AppletforthedoubleFerriswheel].DiscoveringAdvanced

AlgebraResourcesatKeymath.Com.Retrievedfromhttp://www.keymath.com/x3361.xml.

backgroundinmodeling,asevidencedbytheirreferencestomodelingthe

phenomenonandtalkingabouthowtheywouldmodelcircularmotion.

AnalysisInitialImpressions

AndrewandOscarfirstdrewaparametricgraphofSandra’srideonthe

doubleFerriswheel.Bothwereconvincedthatsuchagraphwassinusoidalin

nature.Andrewreasonedthatthiswasbecausethegraph,onceitreacheditslowest

pointwouldrepeat.ForAndrew,thisrepetitionmeantthatthegraphwasperiodic

andhencewouldbesinusoidal.Oscarechoedthesesentiments.Whenaskedwhy

theybelievedthat,theystatedthattheyhadseensimilargraphsintheirmodeling

classesandtheyalwaysendedupbeingsinusoidal.Itisfollowingthisinteraction

thatthefirstmentionofangularvelocityarises.Inthiscase,forAndrew,angular

velocitywasnecessaryastheargumentforasinusoidalfunction.Whenqueriedas

towhythatwasthecase,hesaidthatitwasbecausetheywerewheelsandthe

motionofwheelsneedstobeexpressedintermsofangularvelocity.Herelatedthe

periodicnatureofthemovement(thewheelreturnstothesamepointaftera

certainnumberofrevolutionsofthecomponentwheels)tothefunctionsrootsin

trigonometry.However,angularvelocityissoontobeproblematicasAndrewand

Oscarengagefurtherinthetasks.

Separatingthetwowheels

AngularvelocityaroseagainasAndrewandOscarattemptedtoaddscaleto

theheightversustimegraph(Figure2).Thestudentshaddrawnaqualitatively

correctgraphoftheheightversustimeofSandraonthewheel.Duringthat

discussion,Iaskedthemtocharacterizehowmanytimesthesmallwheelhad

turnedverseshowmanytimesthebigwheelhadturned.Therewassome

disagreementbetweenthetwostudentsastohowfarthesmallwheelhadactually

travelled.Oscarcontendedthatthesmallwheel2movedthreerevolutionsforevery

tworevolutionsofthebigwheel.However,Andrewcontendedthatthesmallwheel

onlymovedonerevolutionforeverytworevolutionsofthebigwheel.Heascribed

theappearanceofthesmallwheelrevolvingthreetimestohiscountbeing“with

referencetothepositionofthebigwheel.”Consequently,whenheaddedthe

positionofthebigwheelplusthepositionofthesmallwheelwith“referenceto”the

bigwheel,hearrivedatthepositionofthesmallwheelwithreferencetoit’sstarting

position.Forexample,ifthebigwheelhasmoved¼andthesmallwheelhas

moved1/8with“referencetothebigwheel,”thenthesmallwheel3/8withregard

tothestartingpoint.Thus,whenAndrewandOscarlookedatthevelocityofthe

wheel,Andrewcontendedthatthesmallwheelmovedmoreslowlythanthelarge

wheel.WhenAndrewwasmadeawareofthetimethateachwheeltooktomakea

revolution,hehadproblemscompromisinghowhecalculatedthespeedofthe

wheelwiththedatathathewasgiven.Consequently,heclaimedthattheonlyway2Inthisteachingexperiment,Andrew,OscarandIrefertothewheelthatSandraisdirectlyconnectedtoasthesmallwheel.Theleverarm,inblueontheapplet,isreferredtoasthebigwheel.TheotherredwheelwithoutSandraisneverreferredtoandneverbecomesasalientissueforthepair.

thatthatcouldoccurwouldbeforthebigwheel’svelocitytobeaddedtothesmall

wheel’svelocity,astatementthatOscarclearlydisagreedwith.Consequently,atthe

endoftheinteractionbothAndrewandOscaragreedtodisagree.

It’simportanttonotethatwhileOscarandAndrewcontinuedtoworkon

thisgraphandtoconstructitsscale,thisparticularpointofcontentionwasnot

addressed.Infact,whileworkingontheheightversustimegraph,dealingwith

angularvelocityisnotanexplicitissue.Therelationshipbetweenhowfastthe

wheelsareturningandtheheightversustimegraphisimplicit.Thespeedatwhich

eachwheelmovesorhowtheyworkinconjunctiontocreatethemotionneednotbe

addressedbecausepointscanbeconsideredinisolation.Theonlyindicationthat

angularvelocityisaconcernwouldbetherelationshipbetweenthescaleofthe

graphandtheshapeofthegraph.However,becauseAndrewandOscaronlydrewa

singleperiodofSandra’sheightversustimethatrelationshipisnotexplicitly

addressed.Consequently,whenconstructingtheheightversustimegraphand

plottingthemaximumsandminimums,AndrewandOscarrarelyhadtodealwith

angularvelocitydirectly.

RateofChangeinHeightVersusTimeandtheAngularVelocity

Theroleofangularvelocityreturnedasanexplicitcontentionwhenthey

wereaskedtogivetheirintuitionsaboutthefunctionforheightversustimethat

couldbedrawnfromtheircompletedgraph(Figure2).AndrewandOscarboth

agreedthatthefunctionforthisgraphwouldmostlikelybesinusoidal.They

mentionedthattheyhadseengraphslikethisinphysicsandtheyhadsinusoidal

functions.Furthermore,fluctuatingoftheheightversustimegraphupanddown

alsoindicatedthatthismightbesinusoidal.Andrewreturnedtotheappletandhe

onceagainsaidthathewastryingtofigureouttheangularvelocity.Oscarvocally

expressednotwantingtodiscussangularvelocityagainwithhim.Infact,although

bothstudentshadbeenconfidentthatthegraphtheydrewrepresentedSandra’s

heightversustimeassherodetheride,Oscaropenlydoubtedifthegraphthatthey

hadconstructedactuallywastherightgraphandthatitwouldbebetterforthemif

theyconstructedthefunctionusingonlytheappletandwhattheyknewaboutthe

dimensionsofthewheelanditassociatedvalues.

Figure3

Hereatensionaroseregardingtherelationshipbetweentheheightversus

timegraphandthemovementofthetwowheels.SomuchsothatOscardecidedto

doubtthegraphaltogetherandrelateonlytothephenomenon.Thisisfurther

complicatedbythefactthatAndrewandOscarhavebeenaskedtounderstandthe

functionviatheheightversustimegraph.Theirfirstrepresentation(Figure2)was

asimpletranslationfromtheappletontothepage.Theyevenconstructeditby

placingatransparencyovertheappletanddrawingupontheapplet.Inthecaseof

theheightversustimegraph,thestudentswerereducingthephenomenontofewer

salientfeatures.Furthermore,withrespecttoangularvelocity,inthefirst

representationtheangularvelocitycanbethoughtofasmovingaroundthewheel

andsoangularvelocityismoreexplicitlyimportantinconstructingthefunction.

Butaswasmentionedearlier,theangularvelocityisonlyimplicitintheheight

versestimegraph.

Andrewexaminedhisnewlynotatedgraphandnotedtheminimumsand

maximumsasbeingplaceswherethevelocitywaszero.Furthermore,therewere

placeswherethechangeinheightversusthetimeisincreasinganddecreasing

whichindicatedthatthevelocitywasnotinfactconstantattheseplaces.

Oscar:Weknowthevelocitiesofthecirclesareconstant.Sothe

velocityincreaseordecreasedependsontheheight.Ifyou’resaying

thattheydochange.Andthenyousaid…

Andrew:Butthisisum…

Oscar:Butyousaid,um…thevelocityincreasesasyoumoveaway

fromthecenter.

Andrew:That’sangularvelocityright.Isn’tangularvelocitytheone

thatincreasesaswemove?

Andrewthengesturedwithhisfingerbymovingthefingerinaclockwisedirection

infrontofhim.Asheisreferringtoangularvelocity,Andrewthenstates,“

Fast…alright,let’sthinkaboutit.”Andrewthenplacedhisthumbandforefingeron

thedrawngraphinamannerthatheusedearliertodescribethechangesinvelocity

thatheobservedonthenewlynotatedgraph.Hethenpointedtothecenterofthe

graph.AsOscarnotes,Andrewhasobservedthattherateofchangeintheheight

decreasesasSandra’spositiononthedoubleFerriswheelnearsthecenteraxisof

thewheel.ThenAndrewstates,“Ifthisisthecenter….Idon’tknow.Ithinkthe

relationshipthathasherbeclosertothecentergiveshersmallervelocity.“Hethen

pointedhispenatthepaper.“…thathastodowithangularvelocity,”indicatinga

localmaximumontheheightversustimegraph.

Thisseriesofutterancesandgestureshelpedtoestablishsomeofthe

problemsthatAndrewwashavingunderstandingtherelationshipbetweenthe

angularvelocityofthetwowheelsandthechangeinheightversesthechangein

time.Andrewattemptedtounderstandhowitwasthattherateofcircularmotion

playedintounderstandingthefunction.Secondly,thelocalmaximum,atwhichthe

twoparticipantsagreedthatthevelocitysloweddown,wasseenasespecially

problematicbecauseatthispointthederivativeoftheheightversestimefunction

wasobviouslynotconstantandeventuallyhitzero.However,Oscarwasstill

unabletoconsciouslyiteratethatrelationship,asevidencedbyhisstatement,“I

guessthere’sarelationshipbetweentheangularvelocity,”hepointeddownatthe

paperwiththeheightversustimegraphandtraceshispenacrossthecurve,”and

thisstuffrighthere.”

EventhoughOscarwasunabletoiteratewhattherelationshipwasbetween

theheightversestimegraphandtheangularvelocitiesofthetwowheels,hedoes,

atleasttentatively,convinceOscarthatangularvelocityisimportant.

Oscar:(Hemoveshishandupwardstowardshischestandturnshis

finger)SoSandra’smoving(hemoveshisfingerupanddown)in

height(hemoveshisfullpalmupanddown).Herheightischanging.

Andthenthere’sangularvelocity(hemoveshisfingerinacircular

motion)….that’srelated….Thatcorrespondstothecircle….the

circles….Andthen,there’sthederivative(traceshisfingeralongthe

graph)…Let’ssee(heplaceshisthumbandforefingeroveraportionof

theheightversestimegraph)it’sthederivativeoftheheight,soI

guessI’mtryingtomakesenseofthat…It’sthevelocitythatshehas

(picksuphispen)ontheheightaxis.Sothisis…(drawsanx‐y

Cartesiancoordinateaxis)aCartesiancoordinatesystem,wherethis

istheheighthere,she’llhavesomevelocity(hedrawsanupanddown

arrowonthepaper),eithergoingupordown.Whichwouldbethe

derivativeofthis…(pointsattheheightversestimegraph)Iwould

see…Whichisnotthesameasangularvelocity….

OscarrecapitulatedAndrew’sgesturefortheangularvelocity,butaddedtothat

gesturetheupwardanddownwardmovementsinpositionwithrelationtothe

ground.Furthermore,hedifferentiatedbetweenthetwomotionsbyswitchingfrom

thefingermotionusedintheangularvelocitygesturetothefullhandforthechange

inheightgesture.Hisgesturesdifferentiatedbetweenthetwodifferenttypesof

movement,circularandupanddown.Hisnextgesture,whichmovedalongthe

graphoftheheightversustimeisaccompaniedbyhisdiscussionofthederivative.

Hedrewaplanewithanup‐downarrow.Atthispoint,Andrewseemedconvinced

thattheangularvelocityandthederivativeoftheheightversustimegraphwerenot

thesamething.

ThisisthefirsttimethatAndrewandOscarexplicitlydifferentiatedbetween

therateofchangeforthewheelsandtherateofchangeforSandra’sheight.This

parsingiscruciallyimportantforthestudentsandalsoforthepurposesofthe

mathematics.ForAndrewandOscaritallowedthemtoconsiderseparatedifferent

aspectsoftheappletforindividualconsiderationandmathematization.Froma

mathematicalstandpoint,theseparationofthetworateshelpedthestudentto

distinguishthedifferentfunctionsthatwillbeconstructedandcomposedforthe

finalheightversustimefunction.

TheCovariationoftheSmallWheel,theBigWheelandtheHeightvs.TimeGraph

AndrewandOscar’srealizationthatthechangeintheheightvs.timeandthe

angularvelocityweredifferentdidprovideanothersteptowardstheirconstruction

ofthefunction,butitalsoestablishedanewdillema.Theyneededtobetter

understandhowthegraphthattheydrewillustratedthemovementofthetwo

wheels.

Thisrealizationledthepairtoreturntotheapplettoseeiftheycould

somehowdealwiththeirapparentproblem.Andrewmovedhischairnexttothe

appletandfollowedthemovementofthedoubleFerriswheel.

Andrew:TheonlythingthatIcanseeisthatthevelocityofthebig

circlecancelswiththevelocityofthesmallcircle.Likethebigcircleis

going(hemoveshisfingerovertheappletasSandramovesalongthe

wheel.Hethentakeshiswholearmandusesthearmtotracethe

movementofthewheel.)Andatonepointthesmallcircleisgoingthe

oppositeway(hepointstotheintersectionofthesmallwheelandthe

bigwheel)[Interviewer:Okay.]Andthat’stheonlywaythatIcansee

thatitwouldcancelout,butitdoesn’thappen…

Int:Cancellingout?Soexplainwhatyoumeanbycancellingout?

Oscar:She’llhavezerovelocity.That’swhathemeans.

Int:So,zerovelocity.Explainhowdoyouknowwherethereisgoing

tobezerovelocity?

Andrew:I’mtryingtorelateitbutthishappens(hemovesfromthe

tablebacktotheappletandstartstheappletagain.Hestopsthe

applet)Sothishappensthatinbetweenonequarterand…(looksat

thescreen)somewhereinthereright….(hepointstothehighpoint

thatSandrareaches.)

Oscar:Thelargeleverarmisgoinggoingup(hegestureswithhis

handoutstretched.Hemoveshishandupwardpivotingfromthe

elbow)…butshestartscomingdown…

Themovebacktotheappletseemedtofacilitateabetterunderstandingofhowthe

movementofthetwowheelsledtowhatappearedtobeapointofzerovelocity.

AndrewandOscarreferredtothisactionascancellingout.InbothOscarand

Andrew’scase,themotionofonewheelappearedtobeinoppositiontothemotion

oftheotherwheel.AndrewpointstothespotontheappletwhereSandra’sheight

isatitsmaximum.OscarmimicsAndrew’smovementwithhisarm.Hethensays

thatshestartscomingdown.

Thisinteractionmarksashiftintheirparticipationandinbothindividuals’

reasoning.Insteadofdecidingifangularvelocitywasimportantandrelatedtothe

function,thepairattemptedtoexplainhowthemovementofthetwowheelsleadto

whattheyobservedintheheightversustimegraph.Thegraphandthequestionof

whetherornottheangularvelocitywasnecessarytoconstructafunctionofthe

heightversustimeledtoaproblemthatthepairneededtosolve.Imaginingthe

arm’smovinginoppositedirectionwashighlyproblematicforbothAndrewand

Oscar.Themovementoftheirgesturesforboththelargewheel(fullarmmotions)

andthesmallwheel(singlefingermotions)moveinthesamecounter‐clockwise

direction.Oscarhesitatesashedescribeshowthetwowheelscouldbe“cancelling”

eachotherout.

Andrew:Thisisthemaxremember(hepointsatapointonthe

applet,whichcorrespondstothelocalmaximumontheheightversus

timegraph).Atthispoint,thiswheelstartsrotating(moveshisfinger

aroundasthesmallwheelontheappletmoves)downwardwhilethe

otherone(moveshiswholearm)startsmoving….(longpause)No,

thatdoesn’tmakeanysensedoesit?

Atthispoint,AndrewrealizedthatwhileSandraismovingdownward,thepoint

wherethewheelthatSandraisonandthebigwheelismovingupward,butboth

wheelsaremovinginthesamedirection.

Thiseliminatedthetwomovinginoppositedirectionsasanexplanationfor

the“cancellingout”thattheyobservedintheheightversustimegraph.Oscar

iteratedasmuch,“They’realwaysrotatingthesametime,but…they’realways

(rotateshisfingerinacircularmotion)…they’rerotatinginthesamedirection.”

Oscar’sgesturesandhisutterancesindicatesthatheunderstoodthattheywere

movinginthesamedirectionandmovingwiththeirseparateconstantvelocities.He

thenofferedanexplanationforhowtheconstantvelocityofthetwowheelscanbe

thecaseandyetthegraphoftheheightversustimecanhaveapointthatindicates

zerovelocity.

Oscar:They’rebothrotatinginthesamedirection(circleshisfingerin

theairincounterclockwisedirection).Butsincetheyhavedifferent

angularvelocities,atsomepoi:::nt(hepauses)Likewhen,asfaras

heightisconcerned.Thelargeleverarmisgoingway(hemoveshis

forearminacounterclockwisemotion)andthis(hemoveshisfinger

towardshisarminacounterclockwisemotion)iscomingdown

already.Andthat’swhereyousay(pointstothemiddleoftheheight

versustimegraph)thatshehaszerovelocity.

Atthispoint,Oscar’sdemeanorchanged.Hehasfoundanexplanationthatwas

consistentwiththeapplet,thegraphandhisunderstandingofthesituation.Itis

clearthattheexplanationalsoresonateswithAndrew,“Yeah,shedoesn’treally

havezerovelocity,likeshe’salwaysmovingright?[Int:Right]Itsalwaysrelative

betweentheangularvelocityofthebigwheelwiththesmallwheel.”

AndrewandOscarcreatedanexplanationoftherelationshipbetweenthe

heightversustimegraphandthemovementofthetwowheelsontheapplet.From

apersonalstandpointithelpstoestablishtherolethatangularvelocityplaysin

theirunderstanding.Elaboratingtherelationshipbetweenthetwowheelsand

Sandra’sheightontheFerriswheelallowedAndrewandOscartoestablishthe

independenceoftherotationofthetwowheels.Thisiskeytodevelopingafunction

fortheheightversustime.Thestudentsneedtobeawarethattherearemultiple

independentbutinterrelatedquantitiesthatcomeintoplay.Theimportanceofthis

knowledgeiscontextualizedviahowthesequantitieschangeandhowitisthat

thesechangescontributetoSandra’sheight.Although,AndrewandOscardidnot

saysoexplicitly,theirgesturesandtheirutterancesillustratethattheyare

coordinatingfirstthechangeinheightofthesmallwheelversusthechangein

heightofthesmallwheel.

Finally,theycoordinatehowthedifferencesintheangularvelocitiesofthe

twowheelsallowedforbigwheeltobemovingawayfromthegroundwhilethe

Sandra’spositiononthesmallwheelmovestowardstheground.Thisisa

complicatedcovariantrelationshipinwhichthemovementofonewheelcombined

withthemovementofthesecondwheelleadstooverallchangeintheheightversus

timegraph.Inactuality,thestudentsneededtodealwith6distinctco‐varying

quantities,thetime,theangularmovementofthesmallandbigwheel,theheight

versusradiansfunctionofthesmallandbigwheel,andtheheightversustime

functionforSandra’srideonthedoubleFerrisWheel.AndrewandOscarutilized

thegraph,theapplet,andtheirgesturestoworkoutthiscomplexrelationship.

Theydemonstratedhowunderstandingoftheroleofangularvelocityinthis

situationandtrigonometricsituationsingeneralrequiresworkingoutnotonlythe

angularvelocityitself,buthowthatvelocityrelatestotheotherquantitiesinthis

situationandthemovementinthephenomenon.Consequently,AndrewandOscar

finallyconcludedthattheangularvelocitiesofbothwheelsplayalargerolein

constructingthefunctionandwhatexactlythatrolewas.

PuttingitAllTogether:ConstructingtheFunction

OnceAndrewandOscarhavedistinguishedthedifferentquantitiesinvolved

intheheightversustimefunctionandalsodistinguishedhowthewheelsinteracted,

thestudentscouldworkondevelopingthefunctionbyanalyzinghowthequantities

combinedanddependedoneachother.Atfirst,AndrewandOscarwantedtofind

anangularvelocityforthemovementofbothwheels.Theyarguedthattheangular

velocitiesofbothwheelscouldbecombinedinsuchawayastocreateasingle

angularvelocityforthedoubleFerriswheel.

Oscar:Sothatwhatyou(Andrew)aretalkingaboutisequaltothe

angularvelocityofthesmallcircleandthebigcircle.Sothat’swhat

weneedtodo.

Theangularvelocitiesofbothwheelsarenotwhatneedtobecombined.Butthe

studentsareclearthatsomethingneedstobecombined.Thishighlightsthe

difficultyofestablishinghowthedifferentquantitiesinvolvedinthedoubleFerris

wheelcombine.Notonlyinthisproblemdidthestudentsneedtocombinetwo

heightversustimefunctions,theyalsoneededtokeeptheangularmovementof

bothwheelsseparateandcomposethosefunctionswiththefunctionfortheheight

versustimeofeachwheel.

AtthispointIintervenedintheconversationandhadthestudentsconstruct

theheightversustimeofSandraifshewasonlyridingthesmallwheel.Whenthe

timecametoconstructafunctionforSandra’sheightversustime,thestudentshad

differentwaysofinterpretinghowtheywouldfindthefunction.Andrewdecidedto

explainhowtocombinethesinewavesforeachwheeltogenerateasinglewavefor

theheightversustime(Figure3)ofSandra’srideonthedoubleFerriswheel.Oscar,

ontheotherhand,explainedhowhecouldimagineaddingtheheightoftheone

wheeltotheheightofthesecondwheelatanygiventimeandthatwouldgiveyou

Sandra’sheightoverallatanygiventime.

Figure3 Forbothofthesestudents,thiswasthefinalinsightnecessaryforcombining

themultiplequantitiesintoasinglefunction.Bothstudentshadtodealwiththe

relationshipbetweentheangularvelocityandtheheightversustimefunctions.

Combiningthetwoheightversustimefunctionsrequiresthestudentstocoordinate

theheightsofthetwowheelsastheymovethroughtime.Andrew’scoordination

usingthetwowavesexemplifiesconsideringthemovementofthetwowheelsin

conjunctionwitheachother.Thescaleofthegraph,whichwasonlysomewhat

importantforthepurposesofdrawingtheheightversustimegraph,isvery

importantinthisinstance.Andrewneedsfortheheightsofthewavestobe

coordinatedproperlyinorderforthevaluesfortheheightversustimefunctionto

becorrect.Thus,Andrewneededtofactorin,atleastimplicitly,theangularvelocity

foreachofthewheels.AsforOscar,hisexplanationexemplifiesadifferent,butno

lesscomplexunderstanding.Addingtheheightsforagiventimecanbeinterpreted

asaddingtheheightfunctionsforeachwheel,implicitlycoordinatingtheother

covariantquantitiesinthetask.Thetwoexplanationshighlightthedifferent

understandingsforSandra’sheightversustimethatthetwohaveelaboratedasthey

constructthefunction.

Conclusions

TheconnectionthatOscarandAndrewmadebetweentheangularvelocityof

bothwheelsandtheirconstructionoffunctionsandgraphsforthemovementofthe

rideronthewheelwasakeycomponentintheireventualabilitytomodelthe

doubleFerriswheelsituation.Theirinitialintuitionsaboutthefunctionbeing

sinusoidalandsorequiringangularvelocityfortheargumentofthefunctionproved

tobecorrect.However,simplyrecognizingthatangularvelocitywasnecessarywas

insufficienttoultimatelycreateandinterpretthefunctionofsandra’sheightversus

time.

FourkeyinsightscontributedsignificantlytoAndrewandOscarestablishing

angularvelocitiesroleinmathematizingSandra’srideonthedoubleFerriswheel.

First,AndrewandOscarrecognizedtheindependenceofthetwowheels.They

concludedthatthevelocityofoneofthewheelswasnotbeingaddedtotheother.

Second,theyestablishedthattherateofchangeinthegraphoftheheightverses

timewasdifferentfromtheangularvelocity.Third,theydistinguishedthe

relationshipbetweenthemotionofthetwowheelsandthegraphoftheheight

versustime.Thisallowedthemtoconsiderhowitwasthattheycouldhavepoints

ofzerovelocity(rateofchangeinheightversestime)butstillcouldhaveconstant

motioninbothwheels.Finally,theyparsedoutofthedifferencebetweenthe

angularvelocityandtherateofchangeinheightversestimeallowingthestudents

toconsiderthefunctionofheightversestimedistinctfromthefunctionforthe

angularmovementwhichconstitutedtheargumentfortheheightversestime

function.

AndrewandOscar’sworkwiththedoubleFerriswheelhighlightsthe

situatednatureofstudent’sunderstandingsofcomplexphenomena.Whatis

importantinanyproblemsituationandwhyitisimportantarecrucialquestionsfor

anyproblemsolver,butthisparticularepisodealsoemphasizesthecrucialrolethat

whenplays.Atanygivenmoment,forAndrewandOscardifferentquestions

becomeimportantandsodifferentfeaturesoftheproblembecomeimportantas

well.

Thequestionofwhatisimportantinthissituationwasnottrivialeven

thoughthestudentshadanimmediateintuitionaboutthesinusoidalnatureofthe

graphandthatangularvelocitywouldbeimportant.However,whyandinwhat

waystheywereimportantaroseastheydealtwiththedoubleFerriswheel.Inthis

situation,angularvelocityoperatedasawayforthestudentstounderstandthe

circularmotionofthewheel.InAndrewandOscar’scase,theyneededtocontend

withtwoseparatefunctions,oneforeachwheel,whoseadditionconstitutesthe

functionforSandra’sheightversustime.Twootherfunctionsarealsoatplayinthis

situation,thefunctionsfortheangularmovementofthewheel.Thesetwofunctions

needtobecomposedwiththefunctionsofeachwheelinordertocomeupwitha

functionthatmodeledtheheightversustime.Thisparticularsituationrequiresthe

students’attentiontofourco‐varyingquantities,butissuesofco‐variationarealso

apparentindealingwithsituationsthatonlyhavesinglemotion.Studentsinthat

casestillmustconstructfunctionstomodelthesinusoidalnatureofthe

phenomenon,andtheywillalsohavetoconstructfunctionsthatmodeltherateof

changeofthosefunctions.Furthermore,theyneedtobeabletocomposethosetwo

functionstoadequatelymodelthephenomenon.

Asforwhenandwhereissuesofangularvelocitywereimportant,

conversationsaboutangularvelocityaroseasstudentsengagedwiththeirfirst

inscriptionoftheride,theirconstructionoftheheightversustimegraphand

constructionofthefunction.Indealingwiththeinitialinscription,angularvelocity

wasanotionthatwasbelievedtohaveimportance,buthowitwasimportantwas

stilluncertain.Asthestudentsconstructedtheheightversustimegraphtheangular

velocitywasimplicitintheirconstructionandinterpretationofscale,butasforthe

qualityandshapeofthegraph,itsrolewasyettobeworkedout.Thatworkingout

tookplaceduringtheconstructionofthefunctionfortheheightversustime.This

makessomesenseifweconsiderwheretheissueoftheangleandthechangeinthe

angleariseintrigonometry.Therelationshipbetweenthechangeintheangleand

thechangeintheheightisexplicitontheunitcircle.IfIimaginemovingfasteror

sloweralongtheedgeofthecircleorchangingmyangleinrelationshiptothecenter

ofthecircleatafasterorslowerrate,thenIcanalsoimaginethattherateatwhichI

gethigherorlowergoesatafasterorslowerrateaswell.However,changesin

radianmeasurearestaticpointsonaheightversestimegraphandinthiscasewere

usedtofitpointsontothegraph.Thestudentsdidnotneedtoconsidertherateat

whicheachwheelwaschangingtogetasenseofwhatthegraphwouldlooklike.

Onlyaftertheyfinishedthegraphdidtheyneedtoworkouthowtheangular

velocityrelatedtoshapeofthegraph.Thishighlightstheneedtomakeexplicit

relationshipsbetweenrateofchangeontheunitcircleinscriptionsandtherateof

changeontheheightversustimegraph.Understandingthatshiftsinthesteepness

ofthecurvesofthegraphrepresentincreasesintherateofchangeoftheangleon

theunitcircleallowsstudentstonegotiatebetweenthetwoinscriptionsand

possiblyrelatethemtophenomena.

Finally,inconstructingthefunctionforSandra’sheightversustime,Andrew

andOscarneededtoconsiderangularmovementasafunction.Bythetimethey

neededtoconstructthefunction,thestudentshadalreadylaidthegroundworkfor

theirunderstanding.Buttheirworkhighlightstheneedforconstructingmeaning

forthatfunction.Understandingthefunctionentailsnotjustbeingabletoconstruct

itfromagraphoreventhephenomena,butitalsomeansunderstandinghowthe

angularmovementfunctionyouareconstructingcontributestothechangeinthe

heightversusthechangeinthetime.

AndrewandOscar’smathematizingofthedoubleFerriswheelillustratesthe

rolethatfunctioncompositionandcovariationplayinmodelingsituationswith

periodicmotion.Italsohighlightsthecomplexrelationshipsthatareimplicitand

explicitintrigonometricfunctionsandinscriptions.Whenmodelingthesekindsof

situations,teachersneedtoaidstudentsinparsingout,understandingand

mathematizingthemultiplerelationshipsthatareimplicitorexplicitinany

situation.Imentionmathematizingwithparsingandunderstandingbecausefor

OscarandAndrewtheactofmathematizingthephenomenoncontributed

significantlytotheirunderstandingofangularvelocity.Engagementwith

constructingthefunctionservedasthegroundfortheirinsightsintotheco‐

variationoftheratesofthetwowheelsandtheirunderstandingofthecomposition

ofthetwofunctions.Furthermore,togainafullunderstandingoftheangular

velocityinthissituation,thestudentsneededtocontendwitheachoftheir

representations.Thisunderscorestherolethatsituatingthemathematicsplaysin

studentsunderstandingofcertainmathematicalconcepts.

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