Graphing formulas: unraveling experts’ recognition processes

Peter M.G.M. Kop

, Fred J.J.M. Janssen

, Paul H.M. Drijvers

, Jan H. van Driel

aIclon,LeidenUniversity,TheNetherlands

bFreudenthalInstitute,UtrechtUniversity,TheNetherlands

cMelbourneGraduateSchoolofEducation,TheUniversityofMelbourne,Australia

a rt i c l e i n f o

Articlehistory:

Received10June2016

Receivedinrevisedform12January2017 Accepted25January2017

Availableonline9February2017

Keywords:

Graphingformulas Experts’recognition Functionfamilies Prototypesandattributes

a b s t ra c t

Aninstantlygraphableformula(IGF)isaformulathatapersoncaninstantlyvisualize usingagraph.TheseIGFsarepersonalandserveasbuildingblocksforgraphingformulas byhand.Thequestionsaddressedinthispaperarewhatexperts’repertoiresofIGFsareand whatexpertsattendtowhilerecognizingtheseformulas.Threetasksweredesignedand administeredtofiveexperts.Thedataanalysis,whichwasbasedonBarsalouandSchwarz andHershkowitz,showedthatexperts’repertoiresofIGFscouldbedescribedusingfunction familiesthatreflectthebasicfunctionsinsecondaryschoolcurriculaandrevealedthat experts’recognitioncouldbedescribedintermsofprototype,attribute,andpart-whole reasoning.Wegivesuggestionsforteachinggraphingformulastostudents.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Algebraicconcepts,likefunctions,canbeexploredmoredeeplythroughlinkingdifferentrepresentations(Duval,2006;

Heid,Thomas,&Zbiek,2012).Graphsandalgebraicformulasareimportantrepresentationsoffunctions.Graphsseemtobe moreaccessiblethanformulas(Leinhardt,Zaslavsky,&Stein,1990;Moschkovich,Schoenfeld,&Arcavi,1993).Inaddition, graphsgivemoredirectinformationoncovariation,thatis,howthedependentvariablechangesasaresultofchangesof theindependentvariable(Carlson,Jacobs,Coe,Larsen,&Hsu,2002).Agraphshowsfeaturessuchassymmetry,intervals ofincreaseordecrease,turningpoints,andinfinitybehavior.Inthisway,itvisualizesthe“story”thatanalgebraicformula tells.Thereforegraphsareimportantinlearningalgebra,inparticularinlearningtoreadalgebraicformulas(Eisenberg&

Dreyfus,1994;Kieran,2006;Kilpatrick&Izsak,2008;NCTM,2000;Sfard&Linchevski,1994).

Studentshavedifficultiesinseeingafunctionbothasaninput-outputmachineandasanobject(Ayalon,Watson,&

Lerman,2015;Gray&Tall,1994;Oehrtman,Carlson,&Thompson,2008;Sfard,1991).Graphsappealtoagestalt-producing ability,andinthiswaycanhelptoconsolidatethefunctionalrelationshipintoagraphicalentity(Kieran,2006;Moschkovich etal.,1993).Graphsarealsoconsideredimportantinproblemsolving.Graphsareusedforunderstandingtheproblem situation,recordinginformation,exploring,andmonitoringandevaluatingresults(Polya,1945;Stylianou&Silver,2004).

So,theabilitytoswitchbetweenrepresentations,representationversatility,inparticularconversionsfromalgebraic formulastographs,isimportantinunderstandingalgebraandinproblemsolving(Duval,2006;NCTM,2000;Stylianou, 2011;Thomas,Wilson,Corballis,Lim,&Yoon,2010).

∗ Correspondingauthor.

E-mailaddress:koppmgm@iclon.leidenuniv.nl(P.M.G.M.Kop).

http://dx.doi.org/10.1016/j.jmathb.2017.01.002 0732-3123/©2017ElsevierInc.Allrightsreserved.

Inapreviousstudyaframeworkwasdevelopedtodescribestrategiesforgraphingformulaswithoutusingtechnology (Kop,Janssen,Drijvers,Veenman,&VanDriel,2015).Intheframework,itisindicatedhowrecognitionguidesheuristic search.Whenonehastographaformulatherearedifferentpossiblelevelsofrecognition:fromcompleterecognition(one immediatelyknowsthegraph)tonorecognitionatall(onedoesnotknowanythingaboutthegraph).Foreverylevelof recognitiontheframeworkprovidesstrongtoweakheuristics.

Forthetwohighestlevelsofrecognitionthegraphiscompletelyrecognizedortheformulaisrecognizedasamemberof afunctionfamilywhosegraphcharacteristicsareknown.Forinstance,atthehighestlevelofrecognitionthegraphofy=x² isinstantlyrecognizedasaparabolawithminimum(0,0).Atthesecondlevelofrecognition,y=4·0.75^x+3isrecognizedas amemberofthefamilyofdecreasingexponentialfunctions,andsothehorizontalasymptoteisreadfromtheformula.In thiswaythegraphcanbeinstantlyvisualized.Anotherexampleatthislevel:y=−x⁴+6x²isrecognizedasapolynomial functionofdegree4;becauseofthenegativeheadcoefficientitsgraphhasanM-shapeoran-shape;ashortinvestigation of,forinstance,thezeroeswillinstantlygivethegraph.

Atthesetwohighestlevelsofrecognitionintheframework,formulascanbeinstantlylinkedtographs.Therefore,these formulasaredefinedasinstantlygraphableformulas(IGF).AlargesetofIGFsisbeneficialtoproficiencyingraphingformulas.

Thecurrentstudywasfocusedonexperts’recognitionprocesseswhendealingwithIGFs.Forthisstudywedefinedanexpert asapersonwithatleastamaster’sdegreeinmathematicsandatleast10yearsofexperienceteachingatthesecondary orcollegelevel,withexperienceingraphingformulasbyhand.Althoughtheseexpertsareexpectedtobeabletoinstantly linkmanyformulastographs,theirrepertoiresofIGFsremainunknown.Inaddition,weinvestigatedwhatexpertsattend towhenrecognizingIGFs.ThisinformationmightgivesuggestionsforarepertoireofIGFsforstudentsandforafocusin teachingstudentsIGFs.

2. Theory

2.1. Cognitiveunitsasbuildingblocks

IGFscanbeseenasbuildingblocksinthinkingandreasoningwithandaboutformulasandgraphs.BarnardandTall(1997) introducedtheconceptof“cognitiveunit”,anelementofcognitiveknowledgethatcanbethefocusofattentionaltogether atonetime.Forexperts,well-connectedcognitiveunitscanbecompressedintoanewsinglecognitiveunitwhichcan beusedasjustonestepinathinkingprocess(Crowley&Tall,1999).Inthiswayexperts’knowledgeiswellorganizedin hierarchicalmentalnetworkswithcomplexcognitiveunits,whichcanbeenlistedwhennecessary(Campitelli&Gobet, 2010;Chi,Feltovich,&Glaser,1981;Chi,2011).

AsIGFsarecognitiveunitsingraphingformulas,theycanbecombined(addition,multiplication,chaining,etc.)andcan formnew,morecomplexIGFs.Forinstance,whendealingwithy=−x⁴+6x²,novicesmayrecognizetheIGFsy=−x⁴and y=6x²andhavetocombinethesetwoIGFstodrawagraph,whereasy=−x⁴+6x²isanIGFforexperts,whorecognizea 4thdegreepolynomialfunction.Forexperts,aformulalikey=x²−6x+5cantriggerothercognitiveunits,like“itsgraph isaparabolawithaminimumvalue”,andtheequivalentformulasy= (x−1)(x−5) andy= (x−3)²−4,whichcangive informationaboutthezeroesandtheminimumvalue,etc.Expertsareexpectedtohavemore,andmorecomplex,IGFsthan novices,whichgenerallyenablethemtographformulaswithfewerdemandsontheworkingmemory(Sweller,1994).

Thecurrentstudywasfocusedonrecognition:inparticular,whichformulasand/orfunctionfamilieswereinstantly recognizedbyexpertsandhowtherecognitionprocessescanbedescribed.

2.2. RecognitiondescribedusingBarsalou’smodelwithprototype,attribute,andpart-wholereasoning

Barsalou(1992)showedhowhumanknowledgeisorganizedincategoriesorconcepts.Peopleconstructthesecategories basedonattributes.Whenataskrequiresadistinctiontobedrawnbetweenexemplarsofacategory,peopleconstructnew attributesandinthiswaynewcategories(Barsalou,1992).Forinstance,fortheconceptbird,attributes(variables)likesize, color,andbeak,withseveralvalues,canbeusedtodistinguishdifferentexemplars.Categoriescanhavealargediversityof exemplars,buthaveagradedstructure(Eysenck&Keane,2000;Barsalou,2008).Someexemplarsinacategoryaremore centraltothatcategorythanothers;thesearecalledprototypes.Forinstance,arobinisconsideredamoretypicalexample ofabirdthan,forinstance,achickenorapenguin.Whendealingwithexemplarsofacategory,peopletendtoassociate prototypicalfeatureswiththeseexemplars(Barsalou,2008;Schwarz&Hershkowitz,1999).Thetendencytoreasonfrom prototypescanposeproblems.Sinceconceptformationisnotnecessarilydoneusingpuredefinitions,WatsonandMason (2005)emphasizedtheneedtogobeyondprototypesandtosearchfortheboundariesofaconcept.Inthiswayonebecomes awareofthedimensionsofpossiblevariationandineachdimensionoftherangeofpermissiblechange(Billsetal.,2006;

Sandefur,Mason,Stylianides,&Watson,2013;Watson&Mason,2005).Thepersonalexamplespace,thecollectionof examplesandtheinterconnectionbetweentheexamplesapersonhasathis/herdisposal(theaccessibleexamplespace), playamajorroleinhowapersonmakessenseofthetaskshe/sheisconfrontedwith(Watson&Mason,2005;Goldenberg

&Mason,2008).VinnerandDreyfus(1989)usedconceptimagetoemphasizethepersonalcharacterofpeople’smental networks.Theseconceptimagesdeterminewhataperson“sees”whendealingwithconceptsorcategories,andareusedin rapididentification.

theexemplar(s)withthesetofhighestfrequencyofattributevaluesinthecategoryorwiththehighestcorrelationwith otherexemplarsinthecategory(Barsalou,1992).Prototypesaretheexamplesthatareacquiredfirstandareusuallythe examplesthathavethelongestlistofattributes:thecriticalattributesofthecategoryandtheself-attributes(non-critical attributes)oftheexemplar(Schwarz&Hershkowitz,1999).Prototypesareusedasareferencepointforjudgingmembership ofthecategory:anexemplarisjudgedtobeamemberofacategoryifthereisagoodmatchbetweenitsattributesandthose ofthecategoryprototype(Barsalou,2008;Eysenck&Keane,2000).Whenaskedforaprototypeofacategory,itisexpected thatapersonwillnotuseadefinitionofprototypebutwilluseageneralideaaboutwhatprototypesare:namely,the mostcentralexemplar(s)ofacategoryfromhis/herpersonalperspective.Asaconsequence,whendealingwithacategory, theprototypesarethefirstexamplesthatcometoone’smindandarethenaturalexamplesthatareusedwithoutany explanation.Examplesinthedomainofgraphingformulasincludeprototypicalformulaslikey=x²andy=x³,withtheir prototypicalgraphs.Inthisstudyweusedthetermprototypereasoninginthisway.

Attributeunderstandingcanbedefinedastheabilitytorecognizetheattributesofafunctionacrossrepresentations (Schwarz&Hershkowitz,1999).Forinstance,fromtheformulay= (x−1)(x−5),itisconcludedthatitsgraphisaparabola, ithaszeroesatx=1andatx=5andasymmetryaxisatx=3.Theseattributesorpropertiesofthisfunctioncanberecognized inthegraphical,tabular,andalgebraicrepresentations.

Inhisproperty-orientedviewoffunctions,Slavit(1997)usedproperties(orattributes)likesymmetry,monotonicity, horizontalandslantasymptotes,intercepts(zeroes),extrema,andpointsofinflection.

Dependingonthetask,peopleconstructattributestobeabletodistinguishexemplars:inthisstudy,formulasandgraphs (Barsalou,1992).Todistinguishdifferentgraphsof4thdegreepolynomialfunctionsinFig.1,onecanuseattributeslike symmetry,infinitybehavior,numberofturningpoints,numberofzeroes,andlocationofzeroesrelativetothey-axis.When relatingformulasandgraphs,asingraphingformulas,onechoosesorcreatesattributestofocusonfeaturesofformulasand graphs.Wecallthisreasoningaboutattributesandtheirvaluesattributereasoning.

Part-wholereasoningreferstotheabilitytorecognizethatdifferentformulasordifferentgraphsrelatetothesameentity:

inthiscase,tothesamefunction.Inthegraphicalrepresentation,differentscalingcanresultindifferentpicturesofgraphs belongingtothesamefunction.Inthealgebraicrepresentation,formulamanipulationcanresultindifferentformulasofthe samefunction:forinstance,y=x²−4x,y=(x−2)²−4,andy=x(x−4).Fromthesedifferentformulasdifferentattributesof thegraphcanberead.Therefore,part-wholereasoningisimportantintherecognitionofIGFs.

Forattributereasoningandpart-wholereasoningonehastograspthestructureofaformula.Intheliteraturethisiscalled symbolsense(Arcavi,1994).Symbolsenseisaverygeneralnotionof“whenandhow”tousesymbolsandhasseveralaspects, suchastheabilitytoreadthroughalgebraicexpressions,toseetheexpressionasawholeratherthanaconcatenationof letters,andtorecognizeitsglobalcharacteristics(Arcavi,1994).PierceandStacey(2001)usedalgebraicinsighttocapture thesymbolsenseintransformationalactivitiesinthe“solving”phaseofproblemsolving(PierceandStacey,2001).The algebraicinsightisdividedintwoparts:algebraicexpectationandtheabilitytolinkrepresentations.Algebraicexpectation hastodowithrecognitionandidentificationofobjects,forms,keyfeatures,dominantterms,andmeaningsofsymbols (Kenney,2008;Pierce&Stacey,2001).Algebraicinsightisshownwhenapersonhasexpectationsaboutgraphsthatare linkedtofeaturesofthesymbolicrepresentationandwhenequivalentalgebraicexpressionsarerecognized(Ball,Stacey,&

Pierce,2003;PierceandStacey,2001,2004).

Thethreeaspectsprototype,attribute,andpart-wholereasoningfromSchwarzandHershkowitzcanbeusedtodescribe therecognitionprocessingraphingformulas.ABarsaloumodelforrecognizingIGFsisformulatedinFig.2.Inthecase ofgraphingformulas,itisdifficulttomentionallpossiblevalues.Forinstance,theattribute“zeroes”canhavevalueslike 0,1,2,3,toindicatethenumberofzeroes,butalsothelocationcanbeusedasvaluesofanattribute(forinstance,azeroat x=5).Forthesakeofreadability,thevaluesbelongingtotheattributesareomittedinFig.2.

TheBarsaloumodelinFig.2showshowfunctionfamiliesareconstructedbyusingvaluesetsonasetofattributesand allowsadetaileddescriptionofhowformulascanbelinkedtographs,andsooftherecognitionofIGFs.Startingwitha formula(ontherightsideofFig.2),thereareseveralpossibilities:theformulacanbemanipulated(part-wholereasoning) intoanotherformula,theformulacanberecognizedasamemberofafunctionfamily,ortheformulacanberecognizedasa

Fig.2. IGFsintheformofaBarsaloumodelbasedonSchwarzandHershkowitz(1999).

prototypeofafunctionfamily.Itisthenpossiblethatthegraphisdirectlyknown,orthat,usingattributereasoning,agraph canbevisualized.

Someexamplescanillustratethisrecognitionprocess.InIGFy=4·3^x+2theprototype3^xcanberecognized(prototype reasoning),andviaatranslation(attributereasoning)thegraphcanbevisualized.InIGFy=−2x(x−3)(x−6),theprototype x³canberecognized,−x³asareversion(attributereasoning),andviazeroesatx=0,x=3,x=6(attributereasoning)the graphcanbevisualized.However,wheny=−2x(x−3)(x−6)isnotrecognizedasamemberofafunctionfamilyorprototype ofafunctionfamily,theformulaisnotanIGF(Kopetal.,2015).Inthiscasethegraphhastobeconstructedby,forinstance, reasoningaboutattributeslikeinfinitybehaviorandzeroes.If,whengraphingy=4x⁻²,theformulacanberewrittento y=4/x²(part-wholereasoning)andrecognizedasa1/x²(prototypereasoning),theformulaisanIGF.Butwhenfromthe formulay=4/x²itisreadthatithasaverticalasymptoteatx=0,andthatalloutcomesarepositiveandwhenx→±∞then y=0(infinitybehavior),thenwesaythatthegraphisconstructedthroughqualitativereasoning(Kopetal.,2015),andso theformulaisnotanIGF.

2.3. Globalandlocalperspectives

Covariationalreasoningisessentialforgraphingformulas.Incovariationalreasoning,oneisabletoimaginerunning throughallinput-outputpairssimultaneouslyandsotoreasonabouthowafunctionisactingonanentireintervalofinput values(Carlsonetal.,2002).InrecognizingIGFsonehastohaveapictureofthefunctionasanentity.Intheliteraturethis perspectiveofthefunction,seeingthefunctionasawhole,isalsoaddressedastheobjectorglobalperspective(Confrey

&Smith,1995;Even,1998;Gray&Tall,1994;Oehrtmanetal.,2008;Sfard,1991).Thereisalsoanotherperspectiveofthe function,namely,toseeafunctionasaninput-outputmachine.Thisperspectivehastodowiththefundamentalviewon functions(whatitmeansthatacertainy-valuebelongstoagivenx-value),andisaddressedasthepointwise,process, orcorrespondenceperspective.Switchingbetweenbothkindsofperspectiveisnecessaryforreasoningaboutfunctions.

Slavit(1997)spokeaboutthelocalandglobalnatureoffunctionalgrowthpropertiesinaddressingbothkindsofperspective (Slavit,1997).Theglobalgrowthpropertiesconcernattributeslikesymmetry,monotonicity,horizontalandslantasymptotes, integrability,andinvertibility,whereasthelocalpropertiesareaboutextrema,intercepts,cusps,andpointsofinflection.In anin-betweenclass,Slavitalsomentionedcontinuity,sign,differentiability,domain,andrange.Graphscanbedescribed

Fig.3.Anumberofthecardsusedintask1.

usingthesepropertiesorattributes.Beforethecurrentresearch,itwasunknownwhichattributesexpertsuseinrecognizing IGFs.

2.4. Researchquestions

Inthecurrentstudywefocusedonexperts’repertoiresofformulasthatcanbeinstantlyvisualizedusingagraph(IGFs) andontheirconceptimagesofIGFs,withattributes,prototypes,andpart-wholereasoning.Weexpectedthatexpertswould havelargerepertoiresofIGFsthatarestructuredincategories.However,wedidnotyetknowwhatanexpertrepertoireof IGFswouldbe.

Weexpectedexpertstobeabletomanipulatealgebraicformulas(part-wholereasoning),tousesymbolsenseandin particularalgebraicinsight,andtousesetsofattributeswithvaluesetstodistinguishdifferentgraphs.However,wedidnot knowwhichprototype,attribute,andpart-wholereasoningtheywoulduseinlinkingformulasandgraphsofIGFs.

Thisleadtothefollowingresearchquestions:

Canwedescribeexperts’repertoiresofinstantgraphableformulas(IGFs)usingcategoriesoffunctionfamilies?

WhatdoexpertsattendtowhenlinkingformulasandgraphsofIGFs,describedintermsofprototype,attribute,and part-wholereasoning?

3. Method

Thecurrentstudycanbecharacterizedasanexploratorystudy,inwhichweinvestigated“snapshots”ofexperts’concept imagesoffunctionfamilieswiththeiralgebraicformulasandgraphs.

3.1. Tasks

Threedifferenttasksweredevelopedtoelicittheexperts’repertoiresofIGFsandtoexploretheexperts’prototype, attribute,andpart-wholereasoning:acard-sortingtask,amatchingtask,andamultiplechoicetask.

Card-sortingtasksareoftenusedinelicitingstructuredknowledge(Chietal.,1981;Jonassen,Beissner,&Yacci,1993;de Jong&Ferguson-Hessler,1986;Goldenberg&Mason,2008;Sandefuretal.,2013).

Intask1,60formulasweregivenandtheparticipantswereaskedtocategorizethemaccordingtotheirgraph.After this,theywereaskedtogiveanameandaprototypicalformulaforeachoftheircategories.Westructuredthistaskby addinggraphstothecardsshowingtheformulas.Whensuchtasksaregivenwithoutstructuringbeforehand,gettinga completepictureorcomparingtheresultscanposeproblems,becauseofthedifferentcriteriathatcanbeusedtosortthe cards(Ruiz-Primo&Shavelson,1996).Becauseweaddfourgraphstothe60cardswithformulas,theparticipantswere explicitlycompelledtofocusonthegraphsoftheformulas.Wedidnotindicatewhetheraparticipantshoulddiscriminate betweenparabolaswithamaximumorminimumbecausethelevelofdetailcanbeanindicatorofexpertise.Fig.3shows 20cardsfromtask1.Mostoftheseformulas,butnotall,arerelatedtooneofthebasicfunctionfamilies,whicharestudied ingrades10–12:y=xⁿ,y=a^x,y=log₂(x),y=1/x,y=√

x,y=ln(x),y=e^x.Since,weusedthebasicfunctionsfromsecondary schoolcurricula,weexpectedthatmanyformulas,butnotall,wouldbeIGFsfortheexperts.Thiscategorizationtaskgave informationaboutdimensionsofvariationandtherangeofpermissiblechangeexpertsusedindiscriminatinggraphs.The namesgivenforthedifferentcategorieswiththeprototypesgaveinsightintothegraphfamiliesandthusintheattribute andvaluesetsexpertsused.

Fig.4.Somealternativesoftask2.

InTask2,thematchingtask,alistof40formulaswasgivenandtheparticipantswereaskedtoselectthecorrectalternative outof21alternatives:20graphsandonealternativestating“noneofthese”.Thislastalternativewasprovidedtodiscourage guessing.Inthistaskthefocuswasoninstantlinkingofformulastotheglobalshapeofgraphs.Thereforeastricttime limitwasusedtoencouragerecognitionandtodiscourageconstructionofagraph.Wechoseamatchingtaskwithmany alternativesratherthanagraphingtasktoindicatethelevelofdetailthatwasneeded:theexpertshadtorecognizethe globalshapeofthegraphofthegivenformula.

The formulas used in this task resembled the formulas used in the first task. The following are some exam- ples:y=2x(x−2)(x+4);y=6x²−2x⁴; y=e^2x+1;y=x−4/x;y=4−2x+x/4; y=4/2^x; y=√

x−6+2;y=ln(e²·x); y=2x⁻⁴; y=2(x−1)⁴−4;y=



8−x²;y=ln(4/x);y=9x/√³

x;y=ln(e²·x).EightofthealternativegraphsareshowninFig.4.

Task2wasalsodevelopedtoelicitparticipants’repertoiresofIGFs.Thereforesomefunctionswereaddedthatdonot belongtothefunctionfamiliesofbasicfunctions,forinstance,y=



8−x²,y=30/(x²−16),y=x−4/x,becausewewanted toinvestigatetheboundariesoftheexperts’repertoiresofIGFs.Becausetheformulasusedweresimilartothoseintask1, thistaskwasusedtovalidatetheresultsoftask1.When,forinstance,intask1nodistinctionwasmadebetweenincreasing anddecreasingparabola,butintask2thisdistinctionwasmade,itwasconcludedthattheparticipantcouldindeedmake suchadistinction.

Tasks3Aand3B,thinkingaloudmultiplechoicetasks,weredevelopedtoelicittheparticipants’prototype,attribute,and part-wholereasoningandinthiswaytogetmoredetailedknowledgeoftheparticipants’conceptimages.Theparticipants wereaskedtochoosethecorrectalternativeoutoffouralternatives.AsimilartaskwasusedbySchwarzandHershkowitz (1999)intheirstudyofconceptimagesoffunctions.Bothtasksconsistedofsixitems.Intask3Baformulawasgivenandthe expertshadtofindthecorrectgraph.Intask3Aagraphwasgivenandtheexpertshadtoprovideaformula.Ingeneral,tasks like3Aareconsideredtobemorechallenging.Butthisisnotclearwhendealingwiththefunctionfamiliesofwell-known basicfunctions.Inthiswaywegotmoredetailedinformationabouttheexperts’conceptimagesofIGFs.Threeexamplesof thistaskareshowninFig.5.

Theformulaswereagainchosenfromthesamesetoffunctionsasintasks1and2.Participantshadtoconsiderall alternativesbecausemorethanonealternativecouldbecorrect.

Intasks1and2thefocuswasonsketchesofgraphs;inthistask,moredetailedanswerswereneeded.Forinstance,in tasks1and2itwasnotnecessarytodistinguishy=−2x(x−2)(x−4)andy=−2x(x+3)(x+6),butintask3thisdistinction hadtobemade(seetask3A-4inFig.5).

3.2. Participants

Fivemathematicalexpertswereinvitedtoparticipateinthisstudy.WeassignedthelettersP,Q,R,S,andTtoourfive experts.Theexpertshaddifferentbackgrounds:twomathematicianswhohadbeenteachingcalculusandanalysistofirst- yearstudentsatuniversity(Q,R),oneauthorofamathematicstextbookseries,whohadbeenateacherinsecondaryschool (T),onemathteacherwhowasinvolvedintheNationalMathExamsandhadbeenasecondaryschoolteacher(S),andone mathteachereducatorinuniversity(P).AllhadaMaster’sdegreeinmathematicsandtwohadaPhDinmathematics(Q, R).Allofthemhadbeenworkingasateacheratuniversityorinsecondaryeducationformorethan20yearsandhadbeen graphingmanyformulaswithouttechnologyduringtheireducationandduringtheirwholeteachingcareer.Therefore,we consideredthemexpertsingraphingformulas.

Fig.5.Someexamplesoftask3:task3A-4,3A-6,3B-3.

3.3. Dataanddataanalysis

3.3.1. Datacollectionprocedure

Writteninstructionswerehandedoutforeverytask,togetherwithanindicationofthetimeneededtoperformthetask.

Fortask1,atimeindicationofmaximum40minwasgiven;fortasks2and3,20min.Foralltasks,thetimeneededwas recorded,asthetimerequiredtoperformataskcanbeanindicationofexpertise.Duringthetasksthefirstauthoronly emphasizedtheneedtokeeponthinkingaloudwhentheexpertsstoppedtalking.Aftereachtask,thefirstauthoraskedthe expertstolookbackandtodescribethestrategies,theyhadusedinthetask.Theinterviewswerevideotaped.

Intask1,thecard-sortingtask,60cardswerelaidonatableandtheparticipantscouldphysicallygrouptheformulas intodifferentcategories.Afterwards,thecategoriesweregluedonalargesheet.Theparticipantsthenwrotethecategory namesandtheprototypicalformulasforeachcategory.Intask2andtask3,theparticipantsfilledintheanswersonaform.

Duringtasks1and3theparticipantswereaskedtothinkaloud;thiswasvideotaped.Thinkingaloudisconsideredto givereliableinformationabouttheproblem-solvingactivitieswithoutdisturbingthethinkingprocess(Ericsson,2006).For task3thethinking-aloudprotocolsweretranscribedinordertoanalyzetheprototype,attribute,andpart-wholereasoning.

3.3.2. Dataanalysis

3.3.2.1. Task1.Theaimoftask1wastogatherinformationonwhichcategoriesexpertsuseintheirrepertoiresofIGFs.It wasexpectedthatexpertswouldusesalient,globalpropertiesofgraphs,likesymmetry,in/decreasing,verticalasymptotes, infinitybehavior,andnumberofturningpoints,tocategorizetheirIGFs.Basedonthesesalientproperties,thefirstauthor madeatheoretical,hypotheticalexperts’categorizationbeforethestartofthisstudy.Thecategorizationsofthefiveexperts werecomparedwitheachotherandwiththefirstauthor’scategorization.Basedonthesefindingsacommoncategorization wasconstructed.Thiswasdoneinseveralsteps.First,commonelementsinthecategoriesandprototypesintheexperts’

categorizationsandthefirstauthor’scategorizationweredetermined.Fromthesefindingsapreliminarycommonexpert categorizationwasformulated.Inthesecondstep,thelevelofdetailwasconsidered.Ahigherlevelofdetailmeantthat subcategorieswereused.Ifoneormoreexpertsusedahigherlevelofdetail,thenthislevelofdetailwasusedinthe(final) commonexpertcategorization.Inthelaststep,thedistancesbetweenindividualcategorizationsandthecommonexpert categorizationwerecalculated.Weconsideredwhethersmalladjustmentsinthecommonexpertcategorizationwould resultinalowerminimumofthetotalofalldistances.Whennoprogressioncouldbemade,thefinalcommonexpert categorizationwasfound.

Todeterminethedistancebetweenanindividualcategorizationandacommoncategorization,thefollowingprotocol wasused:

-Iftheindividualcategorizationhadthe“same”categorybutaformulawasnotmentionedordidnotbelongtothatcategory, thenthedistanceincreasedby+1

-Ifnosubcategoriesweremadeintheindividualcategorizationandthecommonexpertcategorizationmadeadistinction betweenincreasinganddecreasing,thenthedistanceincreasedby+2(forinstance,nosubcategoriesbetweenparabolas withmaximumandparabolaswithminimumgaveanincreaseofthedistanceby+2ifthecommonexpertcategorization madethisdistinction)

-Iftwocategoriesofthecommonexpertcategorizationweremergedintheindividualcategorization(otherthanthe distinctionbetweenincreasinganddecreasing),thenthedistanceincreasedby+4(forinstance,3rdand4thgradefunctions wereputtogetherinonecategory)

-Ifa completely newcategory, differentfromthecommon expertcategorization, wasformulated,then thedistance increasedby+6.

3.3.2.2. Task2.Inthistaskthenumbersofmistakesperexpertwerecounted.Themistakeswereindicatedinatablein ordertoseewhethertheyweremadeinparticularfunctionfamilies.

3.3.2.3. Task3.Toanalyzetheresultsoftask3,thetranscriptswerecutintofragmentswhichcontainedcrucialstepsof explanations:ideaunits.Ideaunitsareprimitiveelementsinthejustificationsofparticipants(Schwarz&Hershkowitz, 1999).TheseideaunitswereencodedusingtheelementsfromFig.2:prototype,attribute,orpart-wholereasoning.

Sinceprototypesarethenaturalexamplesofcategoriesthatcanbeusedwithoutanyexplanation,graphsandformulas thataparticipantusedasthestartofareasoningprocesswereconsideredprototypesfortheexpert.Ifnoprototypereasoning wasusedorfunctionfamilywasmentioned,wesaidthattheformulawasnotanIGF,andthatthegraphwasconstructed.

Thefragmentsoftheprotocolswereencodedasfollows:

-pr(prototypereasoning):onlyaprototypicalexemplarwasmentioned;forinstance,“itlookslikealog”,“itisanxinthe power6”,“itisanexpo”,“itisanoscillation”.Ifafunctionfamilywasmentioned,likein“itisanexponentialfunction”or

“4thdegreepolynomial”,thiswasconsideredprototypereasoning

-att(attributereasoning):anattributewasmentioned;forinstance,“thisonehasaverticalasymptoteatx=0”,“itisalways positive”,“itgoestominusinfinity”.

-pw(part-wholereasoning):theformulawasmanipulatedtoanequivalentformula,forinstance,y=4x⁻²toy=4/x² -con(construction):nofunctionfamilyorprototypewasmentioned,theformulawasnotanIGF:thegraphwasconstructed

through,forinstance,attributereasoningorcalculatingpoints.

WegivetwoexamplesoftheencodinginFig.6.

4. Results

4.1. Resultsoftask1

Theexperts’andauthors’categorizationsareshowninAppendixA.Theexpertsshowedagreatdealofagreementintheir choicesofcategories,namesofthesecategories,andprototypesofthecategories.OnlyexpertSusedadifferentapproachin hiscategorizationofpolynomialfunctions.Hebasedhiscategorizationonthenumberofturningpoints.Theotherexperts allusedthedegreeofpolynomialfunctions.The4thdegreepolynomialfunctionsweredividedintographswithaW-form, aM-form,andaV-form(orastheexpertsmentioned,“increasingordecreasing”).Nolargedifferenceswerefoundon exponentialfunctionsandlogarithmicfunctions,althoughsomeexperts(PandR)madenodistinctionbetween“normal”

and“reversed”graphs(forinstance,y=e^xversusy=e^−xandy=ln(x)versusy=−ln(x)).Allexpertsagreedonlinearbroken functionsandsquare-rootfunctions.Moredifferenceswerefoundinthecategoriesofpowerfunctions,whereonlyexpert Qmadedistinctionsbasedondomainand/oronconcavity.

Intheconstructionofthecommonexpertcategorization,thedistancesbetweenindividualcategorizationsandthe commonexpertcategorizationwerecalculated.ThefinalcommonexpertcategorizationisshowninTable1.

Thefollowingdistancesfromthefinalcategorizationwerefound:11,3,19,20,and15(forP,Q,R,S,andT,respectively).

Theexpertsneededanaverageof18min:23,20,11,14,and21min(forP,Q,R,S,andT,respectively).

Forthistasktheexpertsusedalotofpart-wholereasoninginordertocategorize,forinstance,thefollowingformulas correctly:ln(e^2x),(1−x)(2+x)+x²,ln(1/x),x(x−1)/(x+1)(x−1),(2x¹³)

5

.

Fromtheinterviewsandobservationsweknowthattheexpertsfirstmadeaglobalcategorization.Latertheylookedin greaterdetailandusedmoreattributestodiscriminatebetweentheformulas.Theexpertsdescribedtheirstrategyas“from simpletomorecomplex”(expertP),“ImadeapreliminarycategorizationbasedonthefunctionfamilieswithwhichIwas broughtup:withpolynomial,exponential,logarithmic,power,broken,androotfunctionsandonlyafterthisIdidfocus onthegraphs.”(expertQ),and“someIseeatfirstsight,othersonlywithsecondthoughts,likex−4/x”(expertR).Most oftheformulasinthistaskcouldbeconsideredIGFsandtheexpertsdidnotconsiderthistaskdifficult:“notadailytask andnicetodo,butnotdifficult”(expertT).Someexpertsmentionedthe“things”theycouldinstantlyseefromtheformula, likedefinitiondomain,asymptotes,singularities,even/oddfunctions,infinitybehavior.Someexpertsindicatedthatamore

Fig.6.Examplesoftheencodingoffragmentsofprotocolsoftask3.

Table1

Commonexpertcategorization.

Categories:

Linear

x+5(4−x),ln(e^2x),(1−x)(2+x)+x² Parabola

parabolawithmax:x(9−x),−(x−3)²+2,(x+5)(3−x),2x−3(x+2)(x−2),−(x−1)²+2(x−1)+6;

parabolawithmin:x²−7(x−5),(6−x)²,x²+(−x+1)² 3rddegreeoscillation

(x²−7)(x−5),2(x−3)²(x+3),2x³+4x²−16x,x³−9x,e^3ln(x) 4thdegree

W-shape:x⁴−16x²+28,(x²−7)²;(6^edegreeW-shape:3(x⁴−6)(x²−8));

M-shape:−3(x²−4)(x²−6),2x³(6−x);V-shape:(x+3)⁴−9,x²(9+x²) Exponential

increasing:4^−3+x,2.(√

2)^x;decreasing:18·0.3^x,2^6−x,8e^−x,10^−2x+5,8/3^x; reversedexponential:6−2^x;100−e^x

Logarithmic

inceasing:ln(e²·x),1+²log(x),ln(x)+ln(2);decreasing:−ln(x),ln(1/x) distractor:1/ln(x)

Hyperbola

hyperbola:x(x−1)/(x+1)(x−1),(4x+2)/x,1−5/(x+1)

powerfunctionswithnegativeoddpower:8x⁻³;withnegativeevenpower:2/x⁴,3x⁻² slantasymptote:x−4/x;twoverticalasymptotes(x²−1)⁻¹,2/x−3/(x−1)

‘Roots’

increasing‘√ x-like’:3√

x+6,2√

x−6;(100x)¹²;decreasing‘√ x-like’:4√

10−x,(2−x)¹²+2 halfacircle:



8−x²;V-shape:



8+x² powerfunctions

exponent‘¹₃-like’<1:2√³ x,8√³

x⁴/2x;exponent‘¹₃-like’>1:(2x¹³)

5

;exponent‘1¹₂-like’:2x√ x

Table2 Resultsoftask2.

Participant Numbermistakes Mistakes

P 3 (4x+1)/(x+2);7x√

x;10/x³

Q 1 (4x+1)/(x+2)

R 1 (4x+1)/(x+2)

S 0

T 5 6x²−2x⁴;2x⁻⁴;4−2x+x/4;(4x+1)/(x+2);5x⁷

Table3

Timeneededfortask3Aand3B,thetotalnumberofmistakes,andthenumberofIGFsandconstructions.

Participants Time3A Time3B Numberofmistakes NumberofIGFs Numberofconstructions

P 4:54min 7:44min 0 10 2

Q 4:16min 2:13min 0 11 1

R 3:56min 6:27min 0 9 3

S 4:02min 2:44min 0 6 6

T 6:16min 2:46min 0 9 3

detailedcategorizationwouldbepossible,butnotwithoutcalculations:“InthenextstepIwouldhavetomakecalculations;

Iwouldnottrustmyselftosaymoreaboutthiscategorizationoffthetopofmyhead”(expertQ).

4.2. Resultsoftask2

Theresultsoftask2(seeTable2)showedthatthreeoutofthefiveexpertsmadenomistakesoronlyonemistake.Most mistakesweremadewiththeformulay=(4x+1)/(x+2).Fourofourexpertsselectedthealternativewiththeincreasing hyperbola.Fromtheotheralternativesitcouldhavebeenconcludedthatadistinctionhadtobemadebetweenanincreasing andadecreasinghyperbola.Sinceastricttimelimitofonly30sforoneformulawasusedandallexpertsfinishedthistask easilywithinthistimelimit,itwasconcludedthatalltheformulasthatdidnotbelongtothealternative“noneofthese”

couldbeconsideredIGFsfortheexperts.

Fromtheobservationsandinterviewswelearnedthatallexpertsfirstexaminedthe20graphalternativesandhada globalviewoftheformulastogetanimpressionofwhichaspectswouldplayaroleinthistaskandwhattheyhadtofocus on.Allexpertsreadalmostallgraphsbymentioningafunctionfamilythatfittedthegraph.Whenperformingthistask,they usedpart-wholereasoningifnecessary,recognizedafunctionfamilyandusedattributereasoningtodiscriminatebetween differentoptionsofthesamefunctionfamily.Forinstance,y=4^x−5y=3e^−0,5x+4−4,y=4/2^xwereallrecognizedasmembers oftheexponentialfunctionfamily,attributereasoning,likeinfinitybehaviorandreversingaprototypicalgraphwasused tochoosethecorrectalternative.

4.3. Resultsoftask3

Intask3theprotocolswereanalyzedusingprototype,attribute,andpart-wholereasoning.Fromtheencodedprotocols, wefoundthatexpertsoftenstartedwithprototypesoffunctionfamilies,followedbyattributereasoning.

Wegivefourexamples(pr=prototype;att=attributereasoning;pw=part-wholereasoning):

Example1(:expertQintask3A-4(3rddegreepolynomialinFig.5)). Somethingwithahigherdegree(pr),decreasing(att), let’ssee;thisissomethingthatincreases(att),zeroesindeedat0,2,and4(att),thatlooksreliable;andthisat0,3,and6, andthatwillbepossible(att);andthisoneincreases,oh,noitdecreasestoo(att);wouldbeapossiblealternative;andthis onenot,ithasitszeroesonthewrongside(att).

Example2(:expertQintask3A-6(4thdegreepolynomialinFig.5)). Let’ssee,4thdegree(pr),downwards(att);A.thisone hasnooscillations,andisonlytranslated(att);B.ispossible,wherearethezeroes?,factorizinggivesme−x²+9(pw),so zeroesatx=3andx=−3(att);C.isnotpossible,becausewhenIdividedbyx²(pw)thennoextrazeroes;d.whenIdivided byx³(pw),itgavemeonlyonemorezero;soithastobeB

Example3(:expertSintask3B-3(exponentialfunctioninFig.5)). 100−50·0.75^xisanexponential;function(pr)withy=100 asahorizontalasymptote(att);thatleavesB.andC.;itis100minus....,soitcomesfrombeneaththeasymptote(att),soit hastobeC.

Example4(:expertTintask3B-2(Indicatewhichgraph(s)canfity=−x(x−2)(x−4))). Thisisapolynomialfunctionofdegree 3(pr)andthosegraphsalllookofdegree3(pr);itis−x³(att),sothatmeansthesealternativesarenotpossible(indicatedA.

andB.);thesetwoarepossiblebutitisonlythisone(C.)becauseD.hasnotthecorrectzeroes(att).

Expertsmadenomistakesinthistaskandworkedfast:seeTable3.However,notallformulascouldbeconsideredIGFs fortheexperts,assomegraphshadtobeconstructedbyreasoningaboutattributes.Inparticular,thegraphofthelogistic functiony=500/(2+3·0,75^x)(task3B-4)hadtobeconstructedbyallourexpertsandtheformulay=6x⁻²(task3B-5)was

ofapolynomialfunctionofdegree3respectivelydegree4.InFig.7itisindicatedwhichexpertsstartedintask3Atheir thinkingaloudwithmentioningaprototypeofafunctionfamily.Theotherexpertsworkedfromthealternativeformulas tothegraph.

Fromtheprotocolsweseethattheyexpertsusedprototypicalformulasandprototypicalgraphsofbasicfunctions.

Theyusedprototypesofexponential,logarithmic,even,andpolynomialofdegree2,3and4functions.Also,y=



a−x² (half acircle)wasconsidereda functionfamily.OnlyexpertQusedy=1/x² asafunctionfamily. Attributesthat were usedtodiscriminatebetweendifferentalternativeswere:increasing/decreasingofgraphlinkedtopositive/negativehead coefficient,infinitybehaviorandhorizontalasymptote,translations,verticalasymptote,numberofzeroesandlocationof zeroes,reversingagraph,positive/negativeoutcomes,domain,andpointofinflection.

ExpertSseemedtousealotofconstructions,perhapsbecauseaprototypeorfunctionfamilywasnotmentioned.From theprotocolsandresultsoftheothertasksitwasconcludedthatthesefunctionfamilieswereimplicitlyusedbythisexpert.

Fromobservationsandinterviewswelearntthattheexpertsthoughtthefunctionsusedintask3Awere“easier”than thoseusedintask3Bbecausetheyonlyrequiredsimpletransformations.Anotherreasonforthedifferencesbetweentask 3Aand3Bwastheamountofvisualinformationintask3B:“fourformulasandonegraphiseasiertodealwiththanfour graphsandoneformula”(expertQ).ExpertPmentionedthatingeneral“itismoredifficulttothinkfromthegraphthan tothinktothegraph”.Nevertheless,allexpertsindicatedthatbothtasksrequiredthesameknowledgeelements:namely, linkingvisualfeaturesofthegraphsandfeaturesoftheformulas.

5. Conclusionsanddiscussion

5.1. Conclusions

Thefirstaimofthecurrentresearchwastodescribeexperts’repertoiresofIGFs.Wehypothesizedthatexpertswould usecategoriestoorganizetheirknowledgeofgraphsandformulas.Theexperts’resultsintask1showedthatthecategories theyconstructedwereverysimilarandalsothatthecategorydescriptionsweresimilar.Thesedescriptionswereclosely relatedtothefunctionfamiliesofbasicfunctionsthataretaughtinsecondaryschool:linearfunctions,polynomialfunctions, exponentialandlogarithmicfunctions,brokenfunctions,andpowerfunctions.OnlyexpertSuseddescriptionscontaining numbersofturningpointsforthepolynomialfunctions.Thereforeacommoncategorizationcouldbeconstructed.Thedis- tancesbetweentheindividualcategorizationsandthefinalcategorizationvariedfrom3to20.Manyofthesedifferences couldbeexplainedbytheabsenceofsubcategories.Forinstance,someoftheexpertsdidnotdistinguishbetweenincreas- inganddecreasingexponentialgraphsorbetweenparabolawithamaximumorwithaminimum.However,theexperts’

performancesintask2confirmedthattheycouldrecognizethesedifferencesbetweensubcategoriesastheymadealmost nomistakesinthistask.

Thetimetheexpertsneededtoperformthiscategorizationtaskvariedfrom11to23min.Whentakingabout20min tocategorize60cards,theexpertsneededonly20spercardtoread,torecognize,tocomparewithothers,andtogroup formulaswithsimilargraphs.Thismeantthattherewasalmostnotimefortheconstructionofnew,unknowngraphs.

Someoftheformulas,likey=1/ln(x),y=(x²−1)⁻¹,y=x−4/x,y=2/x−3/(x−1)werecategorizedinacategorywithasingle formula,oftenwithamentionofsomeattributes,butwithoutagraph.Therefore,itwasconcludedthattheexpertsused thefunctionfamiliesofthebasicfunctionsfromsecondaryschooltoorganizetheircategoriesofIGFs:linearfunctions;2nd, 3rdand4thdegreepolynomial,exponential,logarithmicandrootfunctionswith,ineveryfunctionfamily,adistinction betweenincreasinganddecreasing;brokenlinearfunction;powerfunctionsxⁿ,withnodd/even,andn=p/qwithp>q, p<q.Thisshouldcomeasnosurprise,sinceweusedpredominantlyformulasofbasicfunctionsfromthesecondaryschool curricula.Theexpertswerebroughtupwiththesecategories,astheyindicatedintheinterviews.Theyshowedthroughtheir highproficiencythattheyhadtrulyinternalizedthiscategorizationofbasicfunctions.Theformulasseemedtobecomplex enoughtocapturetheproficiencyoftheexperts,assomeformulascouldnotbeinstantlyvisualizedorwerenotcorrectly categorized.

ThesecondaimofthecurrentstudywastodescribewhatexpertsattendtowhenlinkingformulastographsofIGFs.The recognitionprocesswhenworkingfromformulastographscanbewelldescribedusingtheBarsaloumodelofFig.2.Itis showninTable3thatinrecognizingIGFs,theexpertsoftenstartedwithprototypes.Thisprototypereasoningwas,when necessary,followedbyattributereasoning.Forinstance,y=−2x(x−3)(x−6)isrecognizedasaprototypical“x³”,whichis

“reversed”andhaszeroesat0,3,and6;y=log₂(x+3)asalogtranslatedtotheleft;y=−(x+2)⁴+16asan“x⁴”,reversedand translated;y=



6−x²as“half-a-circle”.TheseexampleswereinlinewiththefindingsofSchwarzandHershkowitz(1999), whofoundthatproficientstudentsusedprototypesasleversforhandlingotherexamplesandshowedgreaterunderstanding of(critical)attributes.

Theexpertsalsorecognizedprototypicalgraphsforwell-knownfunctionfamilies,astheyshowedintask2andtask3A.

Forwell-knownfunctionfamiliesthereseemedtobelittledifferencebetweenworkingfromformulatographandworking fromgraphtoformula.WhenworkingwithIGFs,theexperts’conceptimagesthatweretriggeredbythegivenformulaor givengraph,seemedtocontainequivalentformula(s),graph(s),attributesofgraphsandofformulas,functionfamilywith prototypes,formulasofotherfunctionsinthisfunctionfamily.

Inordertoelicitexperts’attributereasoning,allattributestheexpertsusedintask3weregathered:translationtothe right/leftandabove/below,stretchinghorizontalorvertical,reversion(oftenindicatedbyreasoningaboutnegativehead coefficient),infinitybehavior(withhorizontalasymptotes),increasing/decreasing,numberandlocationofturningpoints, locationandnumberofzeroes,positive/negative,domain,pointofinflection,andverticalasymptotes.

Particularattributesseemedtobelinkedtoparticularfunctionfamilies.Asshownintask3,theseconnectionscould workbothways:fromfunctionfamiliestosalientattributesofgraphsandfromgraphswithsalientattributestofunction families.Thesesalientattributesofafunctionfamilyarecharacteristicofthemembersandprototypesofthefunctionfamily.

Forinstance,averticalasymptotewasdirectlylinkedtologarithmicfunctionsorbrokenfunctions.And,whenconfronted withpowerfunctionswithn=p/q,someinstantlystartedwithafocusondomainandconcavity.Forthedifferentfunction familiesinourresearch,theexpertsusedsalientattributes:limiteddomainwaslinkedtorootfunctions,powerfunctionsand logarithmicfunctions;verticalasymptoteswerelinkedtologarithmicfunctionsandbrokenfunctions;horizontalasymptotes werelinkedtoexponentialfunctionsandbrokenfunctions;symmetrywaslinkedtoevenpolynomialfunctions.

Expertsusedattributesappropriatetothetasks.Forinstance,whentheyhadtolinkformulastoglobalgraphs(tasks1 and2),theypaidnoattentiontothefactor7iny=7x√

x,ortothefactor3andterm1iny=4^−x+3+1.Butwhenparameters influencedtheglobalshapeofthegraph,theseparametersweregivenampleattention.Forinstance,theminussignsof theheadcoefficientiny=−x⁴+9x²andiny=2√

8−xwhichreversedtheprototypicalgraphsweredirectlynoticedand mentioned.Whenmoredetailedgraphswererequested,asintask3,theexpertsagainonlyusedthoseattributesthatwere neededforthetask.Forinstance,theydidnotmentionanythingaboutthefactor0.1iny=0.1·x²orabouttheterm12inthe formulay=x⁶+12,becausethesewerepositivenumbers.Butwhenthetaskdemandedit,theexpertsquicklynoticedthe attributesandvaluesneededtographtheformulas.Forinstance,theexpertsinstantlyrecognizedthedifferentlocationsof thezeroesiny=x(3−x)(x−6)andy=−2x(x+3)(x+6).Thesefindingsshowthattheexpertsworkedefficientlyanddidnot payattentionto“whatisnormal”(Chietal.,1981;Chi,2011).

Theexpertssometimeshadtoshowtheirabilitiesinalgebraicmanipulation.Asexpected,theyhadnoproblemswiththis aspectofpart-wholereasoning:thiswasshownin,forinstance,y=6x⁻²(intask3B),andy=8x⁻³,y=x(x−1)/((x+1)(x−1)), andy=(1−x)(2+x)+x²(intask1).

Theseresultsshowthatexperts’processesof recognitionof IGFscanbedescribed usingthemodelin Fig.2:with prototypes,supportedbyattributeandpart-wholereasoning.

5.2. Discussionandimplications

5.2.1. ABarsaloumodelforrecognitionofIGFs

Thecurrentfindingshighlightthetwohighestlevelsofrecognitionoftheframeworkforstrategiesingraphingformulas (Kopetal.,2015).WedefinedformulasattheselevelsasIGFs,instantlygraphableformulas.Wedescribedtheexperts’

repertoiresofIGFsanddescribedwhatexpertsattendedtoinrecognizingIGFs.Weshowedthattheexpertsusedprototypes andattributereasoninginrecognizingIGFsandfoundhowparticularattributeandvaluesetswerelinkedtoparticular functionfamilies.Forinstance,givenalogarithmicformulasuchasy=log3(2x+4)−3,aprototypey=log3(x)ory=log(x) wasinstantlyidentifiedandattributereasoning(translation,domainx>−2,and/orverticalasymptoteatx=−2)resultedin agraph.Wealsofoundthatforfunctionfamiliesofbasicfunctions,theexpertscouldeasilyworkfromgraphtoaformula.

Givenagraph,theyinstantlyrecognizedafunctionfamilythatfittedthegraph.Forinstance,agraphwithattributeslike domainx>a,averticalasymptoteatx=aandconcavedownwasinstantlyidentifiedasalogarithmicfunction.Thisimplies thattheBarsaloumodelbasedonSchwarzandHershkowitzinFig.2canbeexpandedwithlinkagesbetweenattributeand valuesets,prototypesandfunctionfamiliesandwithlinkagesfromgraphtoattributes,prototypes,andfunctionfamilies.In Fig.8,forsomeofthefunctionfamiliesintheexperts’categorizations(logarithmic,polynomialwithdegree2,exponential, andbrokenfunctions),aprototypeisdescribedusingattributesandvalues;forotherexamplarsofthefunctionfamilysalient attributesareindicated.

Fig.8. ABarsaloumodelbasedonSchwarzandHershkowitzwithfunctionfamiliesandtheirsalientattributes.

5.2.2. Globalpropertiesingraphingformulas

Theexpertsinourstudyfocusedonattributesandvaluesthatinfluencedtheglobalshapeofthegraph.Forinstance, aparameterthatreversedtheprototypicalgraphofy=x⁴wasgivenampleattention,like−2iny=−2x⁴,butaparameter thatresultedinonlyasmallchangeoftheprototypicalgraphwasnotmentioned,suchas0.1iny=0.1x⁴.Intheirattribute reasoning,theexpertsfocusedonattributesandvaluesthatgaveagreatdealofinformationaboutthewholegraph.These attributescanbeconsideredtheglobalgrowthpropertiesofSlavit’sclassificationoffunctionproperties(Slavit,1997).

Startingwiththeseglobalpropertiesisconsideredtobemoreefficientingraphingformulasthanusinglocalproperties (Even,1998;Slavit,1997).

InthecurrentstudywefoundthattheexpertsusedasetofattributesandvaluesthatdifferfromSlavit’sglobalproperties, partlybecauseSlavit’sfocuswasmoreonthefunctionconcept,whereasourfocuswasontherelationbetweenformulaand graph.SeveralglobalpropertiesSlavitused,suchasintegrabilityandinvertibility,werenotmentionedatallbyourexperts.

Basedonthecurrentresults,wesuggestthatrelevantglobalpropertiesforrecognizingIGFsmaybesymmetry,infinity behavior(includinghorizontalandslantasymptotes),verticalasymptotes,domain,increasing/decreasingonintervals,sign (reverse),andconcavity.Forlocalproperties,wesuggestzeroes,turningpoints,pointsofinflection,andindividualpoints.

5.2.3. Suggestionsforfurtherresearchandteaching

Indiscussingtheirideasaboutgraphsandformulas,theexpertsusedanamplerepertoireofdescriptions:“avalley”,“it goesintherightdirection‘,“ithastogodownwards’,”itrunsflat”,“tailsgotominusinfinity”,“thisonehasnooscillations”,

“areversed....”,“itgoestoinfinity”,“itgoesup”,“itcomesfrombelow”,“ininfinityitis...”,“thisoneisonlypositive”,“log totheright”,“anoscillationdownwards”,“a−x³”.

Thesedescriptionsshowthattheexpertsoftendidnotusetheformalmathattribute/propertyconcepts,butusedboth picturesofthewholegraphandactionlanguagesuchas“it(thegraph)runs...”.Peopletalkubiquitouslyaboutabstract conceptsusingconcretemetaphors(Barsalou,2008).Metonymiesandmetaphorsarenecessaryforefficientcommunication andinthelearningofmathematicalconcepts(Presmeg,1998;Zandieh&Knapp,2006).Furtherresearchisnecessarytofind outhowtheseexperts’metonymiesandmetaphorscanbehelpfulintheefficientteachingofgraphingformulas.

ArepertoireofIGFsisnecessaryforgraphingformulas.EisenbergandDreyfus(1994)wroteabouttheneedforarepertoire ofbasicfunctionsandknowledgeofthecharacteristicsoftherepresentationsofthesefunctions.Slavit(1997)speaksabout

“propertynoticing”,theabilitytorecognizeandanalyzefunctionsbyidentifyingthepresenceorabsenceoftheseproperties andtheneedfora“library”offunctionalproperties.Ourfindingsshowhowexpertsusedprototypeandattributereasoning forgraphingformulasandsogiveanimpressionofanexpert“library”ofproperties.Fig.8,aBarsaloumodelbasedonSchwarz andHershkowitz,showshowforIFGsthesefunctionfamilies,prototypes,attributes,andpart-wholereasoningareintegrated intheexperts’conceptimages.Ourfindingsmaybehelpfultofurtherdescribetherecognitionandidentificationofobjects, forms,keyfeatures,anddominanttermsusedinPierceandStacey’salgebraicinsight(PierceandStacey,2001;Kenney, 2008).NotonlyingraphingformulasbutalsowhenusingCASorgraphicalcalculatorsoneneedsthisalgebraicinsight.For instance,Heidetal.(2012)showedhowsolvingtheequationln(x)=5sin(x)requiredknowledgeofthecharacteristicsof functionfamiliesofbothformulasy=ln(x)andy=5sin(x)andtheabilitytolinkthegraphimagestotheformulas(Zbiek&

Heid,2011).

Theresultsofthisstudycanberelevantforteachingalgebraandinparticularfunctions.Studentscontinuetoexperience difficultieswithseeingtherelationshipbetweenalgebraicandgraphicalrepresentations,althoughgraphingtechnologycan supportstudents’understandinginlinkingrepresentationsoffunctions(Kieran,2006;Ruthven,Deaney,&Hennessy,2009).

Inordertofurtherimproveeducation,wefirstneedadomain-specificknowledgebase(DeCorte,2010).Expertiseresearch canprovidesuchaknowledgebase(DeCorte,2010;Campitelli&Gobet,2010;Stylianou&Silver,2004).Thecurrentfindings showwhatknowledgeexpertsusedinrecognizingIGFs:theyusedthebasicfunctionstoorganizethefunctionfamilies,used prototypestohandleotherexemplarsoffunctionfamilies,andusedprototypesandattributestolinkgraphsandformulas offunctionfamilies.Insecondaryschoolcurriculamuch attentionispaidtobasicfunctions,inparticulartolinearand quadraticfunctions.Ourstudysuggeststhatonlylearningandpracticingbasicfunctionsisnotenoughtobecomeproficient inlinkingtheformulasandgraphsoffunctions.Studentsneedtoknowhowtohandleparametersinformulasandneed opportunitiestointegratetheirknowledgeofprototypesandattributesoffunctionfamiliesintowell-connectedhierarchical mentalnetworks.Besidessuchaknowledge-baseforrecognition,studentsneedheuristicmethods,likesplittingformulas andqualitativereasoning,whenrecognitionfallsshort(Kopetal.,2015).

Forgraphingformulasonehastobeableto“read”algebraicformulas.Furtherresearchisnecessarytoinvestigatewhether graphingformulasindeedimprovesymbolsense,inparticularalgebraicinsightandhowgraphingformulascanbeeffectively andefficientlytaughttostudents.

AppendixA.

Fiveexperts’categorizationsandtheresearcher’scategorizationwithcategorynamesandprototypes.

P.23:22min Q.20:28min R.11:25min S.14:25min T.21:00min Researcher’s

categorization Linear:ax+b Straightlines:ax+b Linear:ax+b Linear

y=x

Linearfunctions Linear

Increasing/decreasing Degree2:ax²+bx+c Parabolawithmax:

−x²

Parabolawithmin:x²

Degree2:ax²+bx+c 1turningpoint y=x²

Degree2 Parabolawithmaxand withmin

Polynomials:



Degree3(odd):x³ Degree3:

ax³+bx²+cx+d

2turningpoints:

y=x³−3x

Degree3 Degree3increasing

Degree4,decreasing:

−x²(x²−1) Degree4,increasing:

x²(x²−1)

Degree4:ax⁴+bx³..etc 3turningpoints:

y=(x²−1)² 5turningpoints y=x²(x²−1)(x²−2)

Degree4,with W-shape Degree4without W-shape

Degree4with W-shape M-shape V-shape

Degree6withW-shape (ax+b)^k/(cx+d)ⁿ Hyperbola:1/x

Quotientfunctionswith morethan1vertical asymptote:

y=1/(x²−1)

Brokenfunctions Verticalandhorizontal asymptotes

y=1/x

Verticaleandslant asymptotesy=x−4/x 2verticalasymptotes:

y=1/(x²−1)

Linearbroken:

y=(ax+b)/(cx+d)

Hyperbolaand Powerfunctionwith highernegativeodd exponent

2verticalasymptotes Slantasymptote