THE IMPACT OF TEACHER QUESTIONING AND … · The purpose of this inquiry project was to investigate...
Transcript of THE IMPACT OF TEACHER QUESTIONING AND … · The purpose of this inquiry project was to investigate...
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THEIMPACTOFTEACHERQUESTIONINGANDOPEN-ENDEDPROBLEMSONMATHEMATICALCOMMUNICATIONCinthiaRodriguez
NorthsideIndependentSchoolDistrict
EmilyP.Bonner
TheUniversityofTexasatSanAntonio
AbstractThispaperreportsonanactionresearchprojectthatinvestigatedthewaysinwhichteacherpracticeimpactedstudents’mathematicalcommunication,particularlyintermsofteacherquestioningwiththeuseof
open-endedproblems.GradelevelteamsinaTitleIschoolwereengagedinaprofessionaldevelopment
modelthatfocusedonintegratingproblem-basedlessonsthatwouldelicitproductivemathematicaldiscussion
amongstudents.Resultsshowedthattheuseofopen-endedproblemsrefinedteachers’questioningskillsand
producedmoreproductivestudentdialogue.Teachersandstudentsalsodemonstratedmoreeffective
communicationingeneral,andteachersspecificallyweremorereflectiveintheirplanningandteaching.
Keywords:teacheractionresearch,questioning,open-endedproblems,mathcommunication
Introduction
Recentreformeffortsaretransforminghowmathematicsistaughtinelementaryschools.
Traditionalmodelsforteachingmathematicsarebeingreplacedwithconstructivist,
community-basedteachingclassrooms,increasedstudentexpectationsaroundconceptual
understanding,andmorerigorousstandardizedachievementmeasures(McConney&Perry,
2011).Onesuchchangeincludestheexplicitemphasisontheroleofquestioningand
communicationinmathematicsand,morespecifically,engagingstudentstorepresent
mathematicalideasinmultipleways(NCTM,2014)togenerateproductivediscussion.As
such,thereisaneedfortask-basedmathematicsandinstructionalpracticesthatproduce
purposefulmathematicaldiscussionsamongstudentsinwholeandsmallgroupsettings.
Throughthesepractices,teacherscanmorereadilysupportstudents’conceptual
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understandingsofcomplexmathematicalideasandtheconnectionsbetweenthem(Yackel,
Cobb,&Wood,1991).
Thepurposeofthisinquiryprojectwastoinvestigateteacherquestioninginthecontextof
anopen-endedproblem-solvingenvironment,andtheimpactoftask-basedlessonson
studentmathematicalcommunication.Theprojectfollowedtheimplementationofa
problem-solvingplanatanelementaryschool,inwhichcampusmathematicsspecialists
incorporatedteachertrainingcoveringquestioning,theproblem-solvingprocess,andthe
useofopen-endedmathematicswordproblems.Teacherfeedback,studentartifacts,and
personalobservationswereusedtogaininsightsontheutilizationofquestioningstrategies
withpre-selectedwordproblemsandtheirimpactonstudentmathematical
communication.
LiteratureReview
Thenationalandstatemathematicsstandardsdrawpredominantlyfromsourcessuchas
theProfessionalStandardsforTeachingMathematics(1991),PrinciplesandStandardsforSchoolMathematics(2000),andPrinciplestoActions:EnsuringMathematicsSuccessforAll(2014).Theseresourcesemphasizetheimportanceofmathematicalcommunicationinthe
classroomandtheteacher’simpactonstudentresponses.Forexample,theteacherisseen
asonewhonavigatesdialoguethroughtheuseofquestioningstrategiesthatprobedeep
studentthinking.Studentshaveauthorityandautonomytoquestion,justify,andengagein
productiveargumentsaswellasprovideevidenceofthinkingthroughvariousformsof
communicationsuchasoral,written,andsymbolictext(NCTM,1991,2000,2014).Thus,
mathematicalcommunicationisnotdefinedasaone-waydiscoursefromteacherto
student.Instead,thestandardsunearththeimportanceoftheinterrelationshipbetween
studentandteachertouseacomplexmathematicallanguagethatsupporttheconnection,
analyzation,andexpressionofaccuratemathematicalideas.Thus,thestandards
encompassashifttowardsdifferentresearch-basedcriteriafortherolesofteacherand
student.
Mathematicalstandardsidentifytheteacher’sroleasoneoforchestrator,facilitator,
monitor,andprovokerofstudentexplanations,justifications,andarguments.Thisdiffers
fromatraditionalmodel,whichcustomarilybeginswithteachermodelingofproblemsand
algorithms.Typically,theteacherthenguidesstudentsthroughaseriesofapplication
questionsthatrequirestudentstoreproducestepsinsteadofgeneratingsolutions(Cazden,
1988;Barnes,1976).Thisinitiation-response-feedbackmodel(I-R-F),isstillapracticed
method,butisnolongersufficientinmeetingthecurrentmathematicsstandardsrelatedto
communication(Kyriacou&Issitt,2007).Teachers“mustrefinetheirlisteningskills,
questioning,andparaphrasingtechniques,bothtodirecttheflowofmathematicallearning
andtoprovidemodelsforstudentdialogue”(NCTM,2000,p.197).Moreover,teachers
mustprovidestudentswithopportunitiestosharetheirthinkingandlearnfromthethinking
ofothers.Forexample,studentsneedopportunitiestosharemathematicalideasinvarious
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ways,suchasspeaking,writing,listeninganddrawing(Gojak,2011).Therefore,using
strategiesthatprovideopportunitiesforstudentstoengageinmathematicalthinkingand
communicationareanecessityintheelementaryclassroom.
Studieshaveshownthatpurposeful,high-level,problem-basedquestionshelpteachers
extendstudents’mathematicallanguage(DiTeodoroetal.,2011;McConney&Perry,2011;
Strom,2001;Webb,2009;Webb,2014).Thesestudiescollectivelyimplythatusingopen
questioning,wheremorethanonecorrectresponseispossible,aswellasaskingquestions
thatconnectstudentideasandprobeforfurtherexplanation(e.g.“whydidyou…?”“how
couldyouboth…?”and“whatif…?”),havebeenfoundtoincreasemathematical
communication.Inthisregard,newareasofcurriculumdevelopmentandtrainingsupport
teachers’questioningstrategiesbyprovidingresearchbasedtoolsandtechniquesthat
supportstudents’metacognitiveandcommunicativeskills(Walsh&Sattes,2011).This
includesusingopen-endedquestions,connectingstudentideas,andprobingstudent
thinking.
Additionally,usingopen-endedmathematicsproblemsiseffectiveinthepromotionof
mathematicalcommunication.Oftenheuristicinnature,open-endedproblemsprovide
studentsthechoicetoselectvariousstrategies,arriveatmultipleanswers,andperform
multi-stepoperationsindifferentcombinations(Clarke,Sullivan,&Spandel,1992).These
diverseoptionsallowstudentstoexpresstheirmathematicalthinkinginnumerousforms
andengageinvaluabledialoguewiththeirteachersandpeers.Studentswhosolveopen-
endedproblemsarepronetoactivelyparticipate,expresstheirideasmorefrequently,and
discusstheirsolutionswithotherstudents.Moreso,utilizingopen-endedproblems
providesanapproachtoevaluatehigher-order-thinkingskillsandimproveteachingand
learning(Becker&Shimada,1997).Usingacombinationofeffectivequestionswith
thoughtfullyconstructed,multifacetedmathematicswordproblemsmayprovidean
effectivewayforteacherstoincreasemathematicalcommunicationintheirclassrooms.
Thus,thisactionresearchinquiryprojectaimedtoexplorethefollowingquestions:
1. Howdoestheuseofquestioningstrategieswithopen-endedwordproblems
impactstudents’mathematicalcommunicationatanelementaryschoolcampus?
2. Whatclassroominstructionalpracticesonthiscampusneedtobemodified,
basedonthestudy’sfindings,toimprovemathematicalcommunicationamong
students?
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Methodology
SchoolDescriptionandSample.ThisprojectwasconductedataTitleIelementaryschoolin
alarge,urbandistrictintheSouthernU.S.Atthetimeofthestudy,theschoolhouseda
totalof542studentswiththefollowingdemographicbreakdown:81%Hispanic,7%White,
6%Black,4%Asian,and2%other.Ofthepopulation,85%ofstudentsqualifiedforfreeand
reducedlunchand75%wereconsideredtobeoflowsocio-economicstatus.Theschool
hadfourkindergarten,fourfirst-grade,threesecond-grade,fourthird-grade,fourfourth-
grade,andthreefifth-gradeteachers.Ofthe22teachers,10werenewstaffmembersat
thecampus.All22teachersandtheirstudentsparticipatedintheproblem-solvingprogram
thatwasimplementedthroughthisproject.Typicalcasesamplingwasusedtoselectone
first-grade,onesecond-grade,twothird-grade,andtwofourth-gradeteachersandtheir
studentsasparticipantsoftheprojectfordatacollectionpurposes(Creswell,2012).Wewill
provideanoverviewoftheprogram,followedbyanoverviewofthedatacollectionand
analysisprocess.
CampusGoalsandTraining.Incollaborationwithadministrationandteachers,thetwo
campusmathematicsspecialistsimplementedaseriesofinitiativestoimprove
mathematicalinstruction.Oneparticularareaoffocuswasclassroommathematical
communication.Thisincluded,butwasnotlimitedtothefollowinggoals:
• UseeffectivequestioningstrategiesthatsupporttheuseoftheTexas
EssentialKnowledgeandSkills(TEKS)ProcessStandards
• Supportstudents’useofthedistrict’sproblem-solvingmodel
• Providestudentsandteacherswithmathematically-richwordproblems
• Supportstudentmathematicalcommunication,definedastheuseof
discussions,drawings,text,andmanipulativestodemonstratedevelopment
inmathematicalknowledge
Thestaffdevelopmentprogram,Problem-SolvingwithaPurpose,wasassembledand
presentedtoteachersatthebeginningoftheschoolyeartoaddresstheseobjectivesand
provideteacherswithtraininginthreeareas:thedistrict’sproblem-solvingmodel,
questioningstrategiesembeddedwiththestate’sprocessstandards,andtheuseofopen-
endedwordproblems.Animportantobjectiveofthetrainingwastoemphasizetheneed
forteacherstosupportstudentsinbecomingproblem-solversthatcommunicate,connect,
prove,andreasonmathematicalideas(Gojak,2011).Specifically,throughtheuseofquality
questionsandopen-endedproblems,teachersweretrainedtoguidestudentsthroughthe
problem-solvingprocessandengagelearnersinthesevariousformsofmathematical
communication.Resourcescreatedbytheschooldistrictaswellasothersupplemental
materialswereusedtodevelopthetraining.
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First,informationaboutthedistricts’problemsolvingmodel,Facts-Action-Solve-Think
(FAST),waspresentedtotheentirestaff.Althoughtheschoolhadusedthismodelfor
severalyears,foundationaltrainingwasnecessaryduetoahighnumberofnewteachersat
thecampusaswellasend-of-yearfeedbackfromseniorstaff.ThecomponentsofFacts(gatheringnecessaryfacts),Action(selectinganappropriatestrategy),Solve(findingasolution),andThink(explainyourthinkinginwords),weremodeledandexplained.
Secondly,conceptsofteacherquestioningwerepresentedusingWalshandSattes’(2011)
ThinkingThroughQualityQuestioning:DeepeningStudentEngagementandQualityQuestioning:Research-BasedPracticestoEngageEveryLearner.
TheconceptsofqualityquestioningwerethenconnectedtotheTEKSProcessStandards.
TheProcessStandardsemphasizethatstudentsshould“usemultiplerepresentations,
includingsymbols,diagrams,graphs,andlanguagetodisplay,explain,andjustify
mathematicalideasinvariousways”(TEA,2012).Thus,studentsneedtobeengagedin
multipleformsofmathematicalcommunicationandusingquestionsthataligntotheTEKS
ProcessStandardsmayprovebeneficialtoreachthisobjective.TeachersanalyzedtheTEKS
ProcessStandardsandparticipatedingrade-leveldiscussionstosharewaysofincorporating
standard-basedquestioningthroughoutthevarioussectionsoftheschool’sproblem-solving
model.Forexample,oneprocessstandardstatesthatstudentsshould“analyze
mathematicalrelationshipstoconnectandcommunicatemathematicalideas”(TEA,2012).
Thus,questionssuchas“canyourelatethisproblemtoanotherproblemyouhavesolved”
and“canyouthinkofamathematicalequationtomatchthestory?”wereconsideredand
discussedforapplicationintheclassroom.Furthermore,grade-levelwordproblemsand
studentsampleswerepresentedwhileteacherspracticedselectingandcreatingquestions
thatpromotedmathematicalthinkinganddialoguethroughouttheFASTprocess.Examples
intheSolvestageoftheFASTproblemsolvingprocessincludedquestionssuchas“howcan
wedrawamodelthatrepresentsthisproblem?”and“canyouconvinceyourpartnerthat
yoursolutionmakessense?”Thisteachertrainingactivityledtoacompiledcollectionof
questionsthatalignedtothefirstgrade-levelopen-endedproblemthatteacherswoulduse
intheirclassrooms.
Lastly,characteristicsandexamplesofopen-endedproblemswerehighlightedand
discussed.Thissectionofthetrainingfocusedontheimportanceofselectingandusing
mathematicallyrichwordproblemsthatprovideteacherswithopportunitiestoask
meaningfulquestionsandengagestudentsinvariousformsofmathematical
communicationthroughouttheproblem-solvingprocess.Thisincludedwaystoengage
studentsinpurposefuldiscussionsthatprovidestudentswithopportunitiestoreason,
connect,explainandjustifythinking(NCTM,2014).Additionally,theschool’smathematics
supplementalresources,oftenunderutilized,wereshowcased.Theteachersthengenerated
open-endedproblems,basedontheconceptualsupportthatthesematerialscanprovide,
touseintheirclassrooms.Thetrainingconcludedwithanoverviewoftheschool’s
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problem-solvingplan,termedProblemsoftheMonth,whichincorporatedtheconceptsdiscussedinthetraining.
Problems-of-the-MonthCampusPlan.TheProblemsoftheMonthinitiativeincludedasequenceofopen-endedproblemsingradesK-5forteacherstousebi-monthlyintheir
classroomsalongwithadistrictrubrictoassessmathematicalunderstanding.Atotalof18
problemswereselectedfromtheresources,Exemplars’(1999)ExemplarsDifferentiatedProblemSolvingandPearson’s(2012)ReadyFreddy:DailyProblemSolving(seeAppendixD).Thefirstproblemforeachgradelevelalongwiththeinitialquestionswascompleted
duringthetraining.TheteacherswerechallengedtopresenttheProblemsoftheMonth,engagestudentsintheproblem-solvingprocess,andusestandard-basedquestioningto
promotemultipleformsofmathematicalcommunication.
TeachersimplementedtheProblemsoftheMonthintheirclassroomswereencouragedto
usethestrategiesprovidedinthetraining.Teachersmodeledtheprocessforthefirsttwo
monthsproducingteachersamplesthatwereturnedintothemathematicsspecialists.
Subsequently,teachershadthechoicetocompletetheproblemsinavarietyofways
includinginwhole,partner,andsmallgroups.Asstudentsworkedthrougheachproblem,
teacherswouldselectonestudentworksamplefromtheproblemsolvingsessiontodisplay
ontheschool’smathematicsbulletinboard.Theresultwasamonthlyboardshowcasinga
collectionofcompletedproblemssolvedinavarietyofwaysfromkindergartento5thgrade
thatdemonstratedstudentmathematicalrepresentationsinavarietyofways.Throughout
theyear,teachersreflectedontheirquestioningskillsandobservedtheirstudents’
mathematicalprogress.Teachersinformallydiscussedtheirexperiencesinmonthly
meetingsfacilitatedbythecampusmathematicsspecialists.
DataCollection.Threetypesofdatawerecollectedduringtheseven-monthproject.The
firstartifactwasacollectionofstudentworksamples.Theartifactsweretakenfromthe
mathematicsbulletinboardwhereteachersdisplayedtheirselectedstudentpieces.The
secondartifactwasasetoffiveteachersurveysrangingfromfirstthroughfourthgrades.
Thesurveysweregivenattheendoftheschoolyear,andfocusedonteacherbeliefsabout
questioningstrategiesaswellastheirperceptionsoftheirstudents’mathematical
communicationduringProblemsoftheMonth.Halfofthesurveyquestionswereinopen-endedformat.Theremainingquestionsincludeda5-pointLikertscale;theseweresortedby
questionandquantified(seeTable1).Thelastartifactwasanassemblyoffieldnotesbased
onobservationsduringthemonthlyreflectivemeetingswithteachers.
DataAnalysis.Data,whichincludedstudentwork,teachersurveys,andfieldnotes,wereanalyzedusingthematicanalysis.Afterexaminingeachartifact,weselectedourunitsof
analysisforeachcomponentofthedata.Thestudentworksamplesweregroupedby
teacherandplacedinchronologicalorder.Wesearchedforpatternsthatemergedinthe
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data;includingbothpositiveandnegativeevidenceofstudents’mathematical
representationinwritten,modeled,andothertextform.Morespecifically,wecategorized
thewaysthatstudentswererepresentingtheirthinking,andthestrategiesthattheywere
usingintheproblemsolvingsessions.Wedidnotuseapredeterminedsetofcriteriafor
thisanalysis;rather,thestrategiesandrepresentationsemergedfromtheworkitself.We
lookedforsophisticationinrepresentationovertime,abstractions,andstrategy
development,especiallyastheserelatedtocommunicationofmathematicalideas.For
example,wefoundthatopen-endedproblemsencouragedmoreanddiversemethodsof
communicationovertime,andelicitedmultiplerepresentations.Teachersurveyswere
transcribedandre-organizedbyquestiontocodewithinafocusedtopic.Weparaphrased
theresponsesseparatelyandthenmettoconfirmourresults.Thus,thefindingswere
double-codedforconsistency.Themeswereemergent,butwealsoutilizedconstant
comparisonthroughoutthedataanalysisprocess.Fieldnoteswereanalyzedsimilarlyand
providedtriangulationforemergentthemes.
Results
Amajorthemethatappearedconsistentlyintheanalysiswasalignedtothenotionof
opportunity.Ourobservations,teachersurveys,andstudentworksamplesdemonstrated
thatProblemsoftheMonthprovidedteachersnotonlymoretime,butinstructional
venturestoexplorein-depththeopen-endedproblems,askmeaningfulquestions,and
engagestudentsinvariousformsofmathematicaldiscourse.Theselearningopportunities
impactedbothteachersandstudents’abilitiestocommunicatemathematicsinamultitude
ofways.Inthissection,thefindingsarepresentedbythematictopicwithsupportfrom
worksamplesandtheresultsoftheend-of-studyteachersurveyshownbelow(Table1).
Table1.TeacherBeliefsaboutQuestioningandStudentCommunication
Question StronglyAgree
Agree Disagree StronglyDisagree
DidNotUse/Uncertain
1.ThePOMsupportedmathematicalclass
discussions
0 5 0 0 0
2.ThePOMprovidedopportunitiesto
probestudentthinking
2 3 0 0 0
3.ThePOMprovidedopportunitiesto
connectstudentmathematicalideas
1 4 0 0 0
4.ThePOMprovidedopportunitiestoask
open-endedquestions
2 3 0 0 0
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7.ThePOMincreasedmathematical
communicationinmyclassroom
1 4 0 0 0
ThePOMimprovedmathematical
communicationinmyclassroom
1 4 0 0 0
TeacherQuestioning.Akeyfindingthatemergedfromthedatashowedthatusingopen-
endedproblemswithstandards-basedquestioningrefinedteachers’questioningskills.
Teachershadtothinkcriticallyabouttheirselectionofquestionssincethenatureofthe
problemsencompassedmultiplesolutionroutes.Themajorityofteacherswhowere
surveyedfeltthattheProblemsoftheMonthpositionedthemtoaskbetterquestions.One
particularteacherstated,“manyoftheproblemsweremulti-step,andthatchallengedme
toscaffoldtheirlearningandunderstandingateverystepatwhichtheyhaddifficulty.”
Anotherteacherrevealedthatshefeltherquestioningskillsimprovedthroughouttheyear
becauseshewas“abletoaskmanyopen-endedquestionstodiscussthedifferentwaysto
workouttheproblems.”Furthermore,ourfieldnoteobservationsrevealedthatsincethe
mathematicsproblemswerenoteasilysolvable,teachershadtothink,plan,andbe
selectiveofthequestionstheyasked;theseexperienceshelpedtoimprovetheirown
inquiryskills.
Secondly,thedatarevealedthatusingopen-endedproblemsallowedteacherstoask
diverseandspecifictypesofstandard-basedquestionstosupportmathematical
communication.Outofthefiveteacherssurveyed,twostronglyagreedandthreeagreed
thatProblemsoftheMonthprovidedopportunitiestoaskquestionsforspecificpurposes.Oneteachermentioned,“Iwasabletoaskmanyopen-endedquestionsandevaluation
questions.Sincethereweredifferentwaystosolvetheproblems,itwaseasytoaskquality
questions.”Thisteachermadeadistinctionaboutquestiontypes,notingdifferencesamong
theintentionofthequestion,inthiscase,toevaluatestudentknowledge.Thisideais
furthersupportedbyanotherteachers’reflection.“Ithinktheproblemsofthemonth
improvedmyquestioningskillsbecauseitallowedmeopportunitiestoaskmoreprobing
questions,whichallowsmetoobservehowmystudentscanandcan’tsupporttheir
thinking.”Thisteacheralsocategorizedatypeofquestion,probing,foraspecificpurpose,
tolookatthestrengthsandchallengesofstudents’abilitytosupporttheirmathideas.
However,notallteachersfeltthattheproblemsofthemonthsharpenedtheirquestioning
skills.Forexample,ateacherparticipantnotedthattheproblemsofthemonthdidnot
greatlyimpactherquestioningskills,butitdidallowformoreopportunitiestohave
enrichingmathconversationswithstudents.Inall,identifyingandusingquestiontypes
purposefullyhelpedteachersanalyzeandevaluatestudentthinkingthroughoutthe
problem-solvingprocess.
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StudentCommunication.Teacherswhousedacombinationofqualityquestioningwith
open-endedproblemscreatedalearningspacetoengagestudentsinvariousformsof
mathematicalcommunication.Thefindingsshowedthatthemultiplechoiceofstrategies
andsolutionswerecriticalfactorsinstudentdiscussion,symbolicmodelcreation,and
writtenmathematicalcommunication.
MathematicalDiscussions.Oneformofcommunicationthatwaspositivelyimpactedwas
studentdialoguewithteachersandpeers.ThedatashowedthatduringProblemsoftheMonth,studentsusedmathematicslanguagetodiscussdifferentsolutions,strategies,and
generatenewideas.Theteacherparticipantsnotedthattheyusedthemultiplesolution
pathsoftheopen-endedproblemstoengagestudentsinrichdiscussion.Thesurvey
demonstratedthat4outof5teachersagreedthatthewordproblemsprovided
opportunitiesforstudentstoconnectmathematicsideas.Mostteachershadsimilar
responses,acknowledgingthatthechoicesthemathematicsproblemsofferedwerea
contributingfactortostudentdiscussion.Oneteacher’sobservationreinforcesthisnotion:
“Theproblemsofthemonthenabledmystudentstohavediscussionsaboutwhy
theydidcertainoperations.Theywouldengagemoreactivelywhentheyweretrying
tosupporttheiranswers.Ithinkthediscussionsledthemtoseehowothers
approachedthesametypesofproblems.Theyrealizedthattherewasmorethan
onewaytosolvingaproblem.Itwasveryenlighteningobservingtheirdiscussions.”
Alongwiththeteachersurveyresults,thefieldnotesfrominformaldiscussionsrevealed
thatoverall,teachersfeltthattheProblemsoftheMonthallowedformultipleopportunities
formathematicaldiscussionsbefore,during,andaftersolvingtheopen-endedproblems.A
teachercommentsthattheproblemslendthemselvesto“discussingdifferentproblem-
solvingandplanningstrategies,”whileanotherteacheraddsthatsincethereweremany
waystosolvetheproblems,it“helpedgeneratenewideasandapproachestosolving.”
Moreso,teachersfeltthatstudentshadatimeandspacetoparticipateindiscussionsand
sharetheirthinkingwithothers.
MultipleRepresentations.Students’increasedproductionofmathematicalmodelsand
symbolstocommunicatemathematicalthinkingwasalsoseeninthedataastheyearwent
on.Teacherswhoaskedquestionsthatencouragedmultipleformsofsymbolic
representation(e.g.“canyoushowmeadifferentway?”,“whocandrawadifferent
model?”)impactedstudentresponsesinanassortedofways.Twoexamplesareseenin
Figure1(enlargedfigurescanbefoundinAppendixA).
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Figure1.Samplesof3rdgradework.
3rdGrade,GroupActivity
3rdGrade,IndividualActivity
Thisideawasalsovisibleintheteachersurvey,wereeveryteacherfeltthattheProblemsoftheMonthprovidedauniqueopportunitytoengageandexposestudentsincreatingmultiplemathematicsmodels.Ateacheremphasizesthisnotionbyclaimingthefollowing:
“ThePOMwasanexcellenttoolforproblemsolvingwithpicturesanddrawings.
Beforestudentsdrewthepicture--theproblemwasveryabstract.Inoticedthatfor
mystudentswhodrewpictures,itwasmucheasierforthemtosolvethePOM’s.
ThisisanexcellentstrategythatIreinforcedaily.”
Furthermore,thestudentexamplesdiscussedabovearerepresentativeofstudentwork
wheretheteachernotonlyspenttimeguidingstudentsthroughtheprocessofdeveloping
multiplemathematicsmodels,butalsosettingexpectationsforstudentstoshowmorethan
onerepresentationoftheirsolutions.Ontheotherhand,thedatashowedthatteachers
whodidnotmodelthecreationofmultiplerepresentationnorsettheexpectationto
producethemhadlessintricatestudentsamplesinregardstomultipleformsof
mathematicalrepresentations,asshownbelow.
InFigure1,theteacheraskedherthirdgradestudentstoshareideasandcreatevarious
waystorepresentsolutionstotheproblem.Thestudentsworkedingroups,compared
strategiesandselecteddifferentwaystoshowtheirmathematicsthinking.Throughteacher
scaffoldinganduseofquestionsthatconnectedstudentideas,thelearnerswereableto
discussandcreatemodels,tables,andnumbersentencesthatrepresentedtheirsolutions.
Thesecondexampledisplaysanindividualstudent’sworkwithsimilarresults.Thisthird
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gradestudentusedatable,model,andnumberlinetocommunicatehismethodand
solution.
Figure2.TeacherandStudentSolveSamples
5thgradeTeacher,September2013
5thgradestudent,March2014
Thisselectionshowsafifthgradeteachers’modeledProblemoftheMonthatthebeginningoftheyearcomparedtoanendofyearstudentsamplefromthesameclass.Inparticular,
theSolvesectionoftheteacher-modeledproblemshowsonlyanalgorithmicsolution,with
littletonodetailoftextualrepresentationoftheproblemorsolutionstrategy.Thestudent
sampleisstrikinglysimilarinregardstousingonlytheoperationtoarriveatthesolution
withminimalemphasisoncommunicatingmathematicalideasusingmultiplemodelsand
strategies.
MathematicalWriting.Studentmathematicscommunicationintheformofwritingwasalso
positivelyimpactedthroughtheuseoftheProblemsoftheMonthandteacherquestioning.
Teacherswhoprobedstudentthinkingandencouragedstudentstowritetheirprocedures
andjustificationincreasedthequantityandqualityoftheirstudentswrittenexplanations.
WhenaskedhowtheProblemsoftheMonthimpactedstudentwriting,teachersresponded
positively.Oneteachermadethefollowingstatements.
“Theproblemsofthemonthpositivelyimpactedmystudentswritingskillsbecause
theyhadtoutilizemathvocabularytosupporttheiranswersandtheirwayof
thinking.Theyhadtobeveryspecificontheirstepsandthusmadethemthinkmore
carefullyaboutthestepstheytookinsolvingproblems.”
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Anotherteacheradded,“writingindetailabouttheirproblemsolvingnotonlyhelpedtheir
writingskills,buthelpedstudentsrememberandthinkaboutstepstheytooktosolve
problems.”Accordingly,studentsnotjustsolvedproblems,butreflectedandexplained
theirthinkinginwrittenform.Thefindingsrevealedthatteacherswhousedopen-ended
problemsandprobedthinkingthroughquestioningincreasedthequalityoftheirstudents
writtenexplanationsregardingtheproblemsolvingprocess.Anexampleisseenthrougha
fourthgradestudent’sworkovertime(AppendixC).
Here,afourthgrader’ssamplesaresequencedchronologically.Theartifactshowsthatboth
thestudent’squantityandqualityoftheirwrittenexplanationsincreasedthroughouttime.
Thisstudentnotonlywrotemoreastimeprogressed,butprovidedmoredetailinthe
methodtakentosolveproblems,theoperationsandstrategiesthatwerechosen,and
demonstratedmoreelaborateexplanationsregardingtheirmathematicaljustifications.This
student’steacherfurtheraddedthatstudentsinherclasshadthetimeduringProblemsoftheMonthtoexplaintheirthinkingregardingmathematicalconcepts,whichhelpedwith
theiroverallacademicwriting.
Theresultsofthestudyweremultifacetedwithimplicationsformathematicsclassrooms,
districts,teachers,andadministrators.Overall,inthecontextofproblemsolving,student
communicationaboutmathematicsoccurredatahigherlevel.Moreover,teacher
communications,suchasquestioningtechniques,abilitytoengagestudentsinmeaningful
discussionandconnectstudentstrategies,wereimprovedinfrequencyofspecificquestion
types.Moreso,theresultsdemonstratedthemediationofcertainteachermovesrelatedto
choicesaroundstudentgroupingandquestioningtechniques.
Discussion
OverallthefindingsshowedthatclassroomteacherswhousedtheProblemsoftheMonthwithdiversequestioningtoengagestudentsindiscussion,model-making,andwrittenform,
hadpositivefeelingstowardstheuseofopen-endedproblems.Consequently,their
studentsshowedincreasedimprovementinallthreeformsofmathematical
communication.Thishasimplicationsforclassroomteachersinthatitsupportsthenotion
thatteachingmathematicsthroughopen-endedproblemsolvingsessions,asopposedto
traditionallectureandworksheetdriveninstruction,increasesnotonlystudent
mathematicalunderstandingbutalsoteacherpractice.Further,whenteaching
mathematicsthroughproblemsolving,teachersareinherentlyandcontinuouslyassessing
studentsbycirculating,listening,andaskingquestionsaboutthinking.
Theimplicationsatthedistrictlevelaresimilar–mathematicscurriculashouldbewritten
withaproblemsolvingfocusinordertosupportstudentunderstandingsandpedagogical
development.Teachersshouldbetrainedinthistypeofinstruction,andshouldbeableto
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elicitmathematicalideasfromstudents.Thistypeofteachingismoreequitablebecause
thestudentsaredoingthecognitivework,andthereforemaintainagreaterlevelofpower
intheclassroom.Ratherthantheteacher“holding”theknowledge,itisco-constructed
throughdiscussion.
Teachereducationprogramshaveadoptedthistypeofpedagogymorereadily,butshould
beconscioustoplacestudentteachersinclassroomswheremathematicsistaughtthrough
problemsolving.Further,moreresearchisneededtounderstandthedevelopmentof
teachersasproblemsolvers.
LimitationsandChallenges
Thoughtheresultsofthisstudyoverwhelminglysupportthepracticeofteaching
mathematicsthroughproblemsolvingattheelementaryschoollevel,thereweresome
challengesthataroseandshouldbeaddressed.Manyofthesewereatthecampuslevel,
butdospeaktothefactthatsuccessinimplementationisbasedonmanyfactors,someof
whichareoutofateacher’simmediatecontrol.
CampusChallenges.Althoughtheuseofopen-endedproblemswithstandard-based
questioningprovedtohavepositiveresults,classroomsshoweddifferentlevelsofqualityin
theirwork.Mathematicsspecialists’fieldnotesandexaminationofallstudentworkhelped
formabetterunderstandingofthevarieddegreesofmathematicalcommunication.
Teacherexpectations,studentgrouping,andyearsofexperiencewerefactorsthatimpacted
thequalityandquantityofstudentmathematicalcommunication.Onesuchfinding
revealedthatteacherswhodidnotsethighexpectationsatthebeginningoftheyear
throughtheirmodeledsampleshadlowermathematicalcommunicationthanteacherswho
sethighexpectations.Teacherswhotooktimetoengagestudentsindiscussion,used
questioningtoconnectideas,andmodeledcorrectformsofdetailedrepresentationshad
superiorresults.Additionally,theresearchshowedthatteachersgroupedstudentsin
differentways;studentsworkedinpairs,smallgroups,orindividually.Thisledtovaried
degreesofmathematicaldiscussion.Studentswhoworkedinpairsorsmallgroupsengaged
inmorepeerdialoguethanstudentswhoworkedontheactivitiesindividually.The
groupingofstudentsalsoledtoexposuretovariedformsofmathematicalcommunication
fromotherpeers.
Furthermore,thefindingsrevealedthatfirst-yearteachershadlowerstudentmathematical
communicationcomparedtotheclassroomsofmoreexperiencedteachers.Student
samplesfromtwofirst-yearteachersexposedtheirmisunderstandingofthedistrict’s
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problemsolvingmodelaswellaslessdetailedwork,whichinturn,negativelyaffectedtheir
students’mathematicalcommunication.
AnotherareaofconcerndealtwiththeorganizationalaspectsoftheProblemsoftheMonth,mainlyseenwithalignmentandfrequencyissues.Theteachersurveyrevealedthat
althoughtheProblemsoftheMonthwereseenasbeneficialforstudentmathematical
communication,thesequenceoftheplandidnotalwaysalignwiththedistricttimeline.
Thiscausedsomeissuesforteachers,sinceattimestheyweresolvingchallengingproblems
thatrequiredskillsthatwerenotyettaught.Althoughitpushedstudentstosolveproblems
usinginnovatedways,teachersexpressedconcernduetothetimeandchallengesit
created.TeacherfeedbackalsorevealedthatcompletingtheProblemsoftheMonthbi-weeklycausedsomesetbacksinkeepingupwiththedistricttimelinesandassessments.
Sincetheopen-endedproblemsrequiredampletime,teachersofteneithershortenedor
condensedthedailydistrictmathematicslessons.
CampusChanges.Theimplementationoftheproblem-solvingplanprovedtoenhance
teacherquestioningskillsandstudentmathematicalcommunication;however,thisinquiry
alsoexposedcampusissuesthatrequirefurtheraction.Thus,modificationstothecampus
problem-solvingplanandstafftrainingsopportunitieswerecreatedtorespondtothe
researchfindings.
First,theselectedProblemsoftheMonthwerere-evaluatedandmodifiedtoalignwiththe
districttimeline.Thischangehashelpedteacherspresentrelevantmathematicsproblems
tostudentsaftertheyhaveacquiredsomebackgroundknowledgeandskilledpractice.
Additionally,thedatabaseofproblemsisavailabletoteachersformodificationstotheplan,
thusthegoalisforteacherstoconsidertheselectedproblems,butalsoallowforteacher
autonomy.Secondly,theProblemsoftheMonthchangedfromabi-weeklytoamonthly
activity.Althoughtheresearchexposedthebenefitsofstandard-basedquestionsandopen-
endedproblems,theProblemsoftheMontharenottheonlyavenuetoaccomplishthis
positiveimpactonmathematicalcommunication.Acampusgoalfornextyearisto
encourageteacherstousetheProblemsoftheMonthasopportunitiestoengagestudentsindeepmathematicalunderstandingandcommunication,buttheexpectationistoapply
theresearch-basedstrategiesacrossthemathematicscurriculum.Furthermore,the
mathematicsspecialistshavedevelopedaplantoaddressnewteachermisconceptionsby
modelingeffectivestrategiesandprovidingsupportfornoviceteachersthroughoutthe
year.
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Lastly,afocusfornextyear’sstaffdevelopmentwillincorporatetrainingrelevanttothe
researchfindings.Thisincludessharinganonymousexamplesofqualitystudentworkand
teachermodelswithallstafftoexposeanddiscusseffectivestrategiestofurtherenhance
mathematicalquestioningandstudentcommunicationatthecampus.
Conclusion
Theproceduresandfindingsoftheactionresearchprojectaddtotheeducationalliterature
byexposingvaluableconsiderationsforadministrators,mathspecialists,andteachersto
developandsupportmathematicalcommunicationattheelementaryschoollevel.First,
theneedforadministratorsandmathspecialiststocultivatecampus-widegoalsand
supportstaffintheimplementationofreform-basedmathematicalinstructionisimportant.
Specifically,theresultsshowedthatthemathspecialists’roleofsettinginitiatives,
conductingstaffdevelopment,andprovidingmathematicalresourceswerefactorsthat
supportedteachersintheimplementationofeffectiveinstructionalstrategies.Thus,math
specialistsneedtostaycurrentwithmathematicalpracticesandcollaboratewith
administratorstodispensemathematicalknowledgetoclassroomteachers.
Secondly,theprojectdisclosedinstructionaltechniquesthatbenefiteducatorswhowork
withelementaryschoolstudents.Theuseofopen-endedproblemsincombinationwith
meaningfulquestioningprovedtoincreasethequalityofteachers’questioningskillsand
reflectiveplanning.Hence,creatingspacesforteacherstodiscussandcollaboratewith
othereducatorsisacentralcomponenttoenhanceinstruction.Likewise,challengingand
encouragingteacherstoimplementteachingtechniquesinvariouscombinationsimproves
instructionandlearning.Furthermore,practicingthisstrategywasalsofoundtoincrease
students’mathematicaldialogue,writtenexplanations,andsymbolicrepresentation.Thus,
theresultsexposethebenefitsofusingthisreform-basedstrategytohelpstudentsexplain
andjustifytheirmathematicalreasoningthroughmultipleavenues.
Overall,theprojectrevealedthattheintegrateduseofstandard-basedquestioningwith
open-endedproblemspositivelyimpactedthecampus’mathematicalcommunication.
Teachersenhancedtheirquestioningskillsandengagedstudentsinmathematical
discussions,model-making,andwrittenexplanations.Moreover,theProblemsoftheMonthprovidedteachersthetimetoaskdiversesetsofquestionsandguidestudents
throughcomplexproblemsolving.Studentsweregivenopportunitiestoengagewith
teachersandpeersindialogicinteractionsthatledtotheco-constructionofstrategiesand
solutionsinmultipleforms.Theresearchfurthershowedthateachclassroomvariedin
degreeofmathematicalproductivity.Factorssuchasteachingexperience,grouping,and
classroomexpectationsimpactedthequalityandquantityofmathematicalcommunication.
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Thefindingsledtodevelopaplanofactiontofurthersupporttheuseofopen-ended
problemsandqualityteacherquestioningatthiscampus.
AbouttheAuthors
CinthiaRodriguez,MS.Ed.isanelementarymathematicsspecialistatNorthside
IndependentSchoolDistrictandadoctoralcandidateatTheUniversityofTexasasSan
Antonio.Herresearchfocusesonmathematicseducationintheelementaryschoolcontext
andincludeseffectiveteachingpracticesfordiversepopulationsoflearners.Email:
EmilyP.Bonner,Ph.D.isanAssociateProfessorofCurriculumandInstructionatthe
UniversityofTexasatSanAntonio.Shespecializesinmathematicseducation,andher
researchinterestsincludeequityinschools,actionresearch,andcommunity-basedproblem
solvingmodels.Email:[email protected]
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References
Barnes,D.R.(1976).Fromcommunicationtocurriculum.NewYork,NY:PenguinEducation.
Becker,J.P.,&Shimada,S.(1997).Theopen-endedapproach:Anewproposalforteachingmathematics. Reston,VA:NationalCouncilofTeachersofMathematics.
Cazden,C.(1988).Classroomdiscourse:Thelanguageofteachingandlearning.Portsmouth,NH:Heinemann.
Clarke,D.J.,Sullivan,P.,&Spandel,U.(1992).Studentresponsecharacteristicstoopen-endedtasksin mathematicalandotheracademiccontexts.TheCentre.
Creswell,J.W.(2012).Qualitativeinquiryandresearchdesign:Choosingamongfiveapproaches.Thousand Oaks,CA:Sage.
DiTeodoro,S.,Donders,S.,Kemp-Davidson,J.,Robertson,P.,&Schuyler,L.(2011).Askinggoodquestions:
Promotinggreaterunderstandingofmathematicsthroughpurposefulteacherandstudent
questioning.CanadianJournalofActionResearch,12(2),18-29.
Exemplars.(1999).K-8DifferentiatedBestofMathExemplarsI,II,andIII.[CDROM]Underhill,VA.
Ferrini-Mundy,J.,&Martin,W.G.(2000).Principlesandstandardsforschoolmathematics.Reston,VA:NationalCouncilofTeachersofMathematics.
Gojak,Linda.(2011).What’syourmathproblem?Gettingtotheheartofteachingproblemsolving.Huntington Beach,CA:ShellEducation.
Kyriacou,C.,&Issitt,J.(2007).Teacher-pupildialogueinmathematicslessons.BSRLMProceedings,61-65.
McConney,M.,&Perry,M.(2011).Achangeinquestioningtactics:Promptingstudentautonomy.
InvestigationsinMathematicsLearning,3(3),26-45.
NationalCouncilofTeachersofMathematics.CommissiononTeachingStandardsforSchoolMathematics.
(1991).Professionalstandardsforteachingmathematics.Reston,VA:NationalCouncilofTeachersofMathematics.
NationalCouncilofTeachersofMathematics.(2000).Principlesandstandardsforschoolmathematics(Vol.1).
Reston,VA:NationalCouncilofTeachersofMathematics.
NationalCouncilofTeachersofMathematics.(2014).Principlestoactions:Ensuringmathematicalsuccessforforall.Reston,VA:NationalCouncilofTeacherofMathematics.
PearsonLearningSolutions(2012).ReadyFreddy:DailyProblemSolving.Boston,MA:Pearson.
Strom,D.,Kemeny,V.,Lehrer,R.,&Forman,E.(2001).Visualizingtheemergentstructureofchildren's
mathematicalargument.CognitiveScience,25(5),733-773.doi:10.1207/s15516709cog2505_6
TexasEducationAgency(2012).Chapter111.Texasessentialknowledgeandskillsformathematics. SubchapterA.Elementaryschool.RetrievedOctober30,2017. fromhttp://ritter.tea.state.tx.us/rules/tac/chapter111/ch111a.html
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Walsh,J.A.,&Sattes,B.D.(2011).Thinkingthroughqualityquestioning:Deepeningstudentengagement.ThousandOaks,CA:CorwinPress.
Webb,N.M.,Franke,M.L.,Wong,J.,Fernandez,C.H.,Shin,N.,&Turrou,A.C.(2014).Engagingwithothers’
mathematicalideas:Interrelationshipsamongstudentparticipation,teachers’instructionalpractices,
andlearning.InternationalJournalofEducationalResearch,63,79-93.
Webb,NoreenM.FrankeMeganL.DeTondraChanAngelaG.FreundDeannaSheinPatMelkonianDorisK.
(2009).'Explaintoyourpartner':Teachers'instructionalpracticesandstudents'dialogueinsmall
groups.CambridgeJournalofEducation,39(1),49-70.doi:10.1080/03057640802701986
Yackel,E.,Cobb,P.,&Wood,T.(1991).Small-groupinteractionsasasourceoflearningopportunitiesin
second-grademathematics.Journalforresearchinmathematicseducation,390-408.
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AppendixA:ExamplesofThirdGradeWorkasSeeninFigure1
GroupActivity
IndividualActivity
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AppendixB:TeacherandStudentSolveSamplesasSeeninFigure2
5thGradeTeacher-September2013
5thGradeStudent-March2014
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AppendixC:Samplesofa4thGrader’sWorkOverTime
October2013
January2014
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AppendixD:SamplesofProblemoftheMonth
(AdaptedfromExemplars,1999andPearsonLearningSolutions,2012)
K-2ndGradeProblems
1.ClassPetsFASTFreddy’sclasshas7goldfish.HelpFASTFreddyputtheminto3bowls.
• Eachbowlmusthaveatleast1goldfish
• Nobowlmayhavemorethan3goldfish
Howmanyfishwouldyouputintoeachbowl?
2.CoinsYouandyourfriendareonyourwaytothestoretobuysomemilk.Whenyougetthereyourfriendrealizes
thatsheis40centsshortsofwhatsheneedsandasksifshecanborrowsomemoneyfromyou.Youhave
pennies,nickels,dimesandquarters.Whataredifferentwaysyoucancombinethesecoinstoloanyourfriend
40cents?
3rd-5thGradeProblems1.LuggingWaterJustinandAnnawerecampingwiththeirfamily.Theyjoinedtheirdadatthecampwaterpumpwherehehad
partiallyfilled6containers.Thecontainershadnohandles.Ashefilledeachone,helabeledthefractional
amounttowhicheachcontainerwasfilled.Theamountsareshownbelow.
JustinandAnnaeachhadacontainerthatwasthesamesizeastheonestheirdadfilled,buttheirshad
handles.Theirtaskwastopourthewaterfromthe6containersintotheir2containerssotheycouldeasily
carrythewaterbacktocamp.WhichcontainersshouldJustinandAnnapourintoeachoftheircontainersso
togethertheycantransportthewaterinonetrip?Showyourmaththinking.
2.FishDilemmaThereare3boats.Thereare4peoplefishingoneachboat.Eachpersonmaycatchupto3fish.Howmanyfish
couldbecaught?
Besuretoexplainyourreasoningusingwords,numbers,diagramsand/orcharts.