RidingtheDoubleFerrisWheel:Students’InterpretationandUnderstandingof
TrigonometricFunctionsinRealisticSettings
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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