THE IMPACT OF TEACHER QUESTIONING AND … · The purpose of this inquiry project was to investigate teacher questioning in the ... support students’ metacognitive and communicative

THEJOURNALOFTEACHERACTIONRESEARCH 68

JournalofTeacherActionResearch- Volume4,Issue3,2018,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights

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



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



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?



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.



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



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



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



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.



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).



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



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.”



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



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



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.



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.



Thefindingsledtodevelopaplanofactiontofurthersupporttheuseofopen-ended

problemsandqualityteacherquestioningatthiscampus.

AbouttheAuthors

CinthiaRodriguez,MS.Ed.isanelementarymathematicsspecialistatNorthside

IndependentSchoolDistrictandadoctoralcandidateatTheUniversityofTexasasSan

Antonio.Herresearchfocusesonmathematicseducationintheelementaryschoolcontext

andincludeseffectiveteachingpracticesfordiversepopulationsoflearners.Email:

[email protected]

EmilyP.Bonner,Ph.D.isanAssociateProfessorofCurriculumandInstructionatthe

UniversityofTexasatSanAntonio.Shespecializesinmathematicseducation,andher

researchinterestsincludeequityinschools,actionresearch,andcommunity-basedproblem

solvingmodels.Email:[email protected]



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AppendixA:ExamplesofThirdGradeWorkasSeeninFigure1

GroupActivity

IndividualActivity



AppendixB:TeacherandStudentSolveSamplesasSeeninFigure2

5thGradeTeacher-September2013

5thGradeStudent-March2014



AppendixC:Samplesofa4thGrader’sWorkOverTime

October2013

January2014



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.

Documents

THE IMPACT OF TEACHER QUESTIONING AND … · The purpose of this inquiry project was to investigate teacher questioning in the ... support students’ metacognitive and communicative