TME, vol10, no.3, p. 735

The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, no.3, pp.735-776 2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana

MathematicalHabitsofMindforTeaching:UsingLanguageinAlgebraClassrooms1

RyotaMatsuura2St.OlafCollege

SarahSword

EducationDevelopmentCenter,Inc.

MaryBethPiechamEducationDevelopmentCenter,Inc.

GlennStevens

BostonUniversity

AlCuocoEducationDevelopmentCenter,Inc.

ABSTRACT:Thenotionofmathematicalknowledgeforteachinghasbeenstudiedbymanyresearchers,especiallyattheelementarygrades.Ourunderstandingsofthisnotionparallelmuchofwhatwehavereadintheliterature,butarebasedonourparticularexperiencesoverthepast20years,asmathematiciansengagedindoingmathematicswithsecondaryteachers.AspartoftheworkofFocusonMathematics,PhaseIIMSP,wearedeveloping,incollaborationwithothersinthefield,aresearchprogramwiththeultimategoalofunderstandingtheconnectionsbetweensecondaryteachers’mathematicalknowledgeforteachingandsecondarystudents’mathematicalunderstandingandachievement.Weareintheearlystagesofafocusedresearchstudyinvestigatingtheresearchquestion:Whatarethemathematicalhabitsofmindthathighschoolteachersuseintheirprofessionallivesandhowcanwemeasurethem?Themainfocusofthispaperisthediscussionofthehabitofusingmathematicallanguage,andparticularlyhowthishabitplaysoutinaclassroomsetting.Keywords:Mathematicalhabitsofmind,mathematicallanguage,algebra

1ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderGrantNo.0928735.Anyopinions,findings,andconclusionsorrecommendationsexpressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheNationalScienceFoundation.2matsuura@stolaf.edu

Matsuura et al.

OurPhilosophyandApproach

Buildingontwodecadesofpriorwork,theFocusonMathematics(FoM)MathandScience

Partnershipprogram(MSP)has,overthelastdecade,developedandrefinedadistinctive

frameworkforamathematics‐centeredapproachtodevelopingteacherleaders,andithas

builtamathematicalcommunitybasedonthatframework.TheFoMapproachinvolves

teachers,mathematicians,andeducatorsworkingtogetherinprofessionaldevelopment

activities.Thecommonthreadrunningthroughthistightlyconnectedsetofactivitiesisan

explicitfocusonmathematicalhabitsofmind.

Wedefinemathematicalhabitsofmind(MHoM)tobethewebofspecializedwaysof

approachingmathematicalproblemsandthinkingaboutmathematicalconceptsthat

resemblethewaysemployedbymathematicians(Cuoco,Goldenberg,&Mark,1997,2010;

Goldenberg,Mark,&Cuoco,2010;Mark,Cuoco,Goldenberg,&Sword,2010).Thesehabits

arenotaboutparticulardefinitions,theorems,oralgorithmsthatonemightfindina

textbook;instead,theyareaboutthethinking,mentalhabits,andresearchtechniquesthat

mathematiciansemploytodevelopsuchdefinitions,theorems,oralgorithms.Some

examplesofMHoMfollow:

Discoveringthestructurethatisnotapparentatfirstbyexperimentingandseeking

regularityand/orcoherence.

Choosingausefulrepresentation—orpurposefullytogglingamongvarious

representations—ofamathematicalconceptorobject.

Purposefullytransformingand/orinterpretingalgebraicexpressions(e.g.,rewriting

x2 6x10as (x 3)2 1torevealitsminimumvalue).

TME, vol10, no.3, p. 737

Usingmathematicallanguagetoexpressideas,assumptions,observations,

definitions,orconjectures.

OurworkoverthepastdecadehasconvincedusoftheimportanceofMHoMfor

studentsandforteachersofmathematics,particularlyatthesecondarylevel.Thesehabits

fosterthedevelopmentanduseofgeneralpurposetoolsthatmakeconnectionsamong

varioustopicsandtechniquesofsecondaryschoolmathematicscontent;theycanbring

parsimony,focus,andcoherencetoteachers’mathematicalthinkingand,inturn,totheir

workwithstudents.Inthissense,weenvisionMHoMasacriticalcomponentof

mathematicalknowledgeforteaching(Hill,Rowan&Ball,2005)atthesecondarylevel(i.e.,

theknowledgenecessarytocarryouttheworkofteachingmathematics).

Webeginthispaperbydescribingthemathematicalcommunitythatwehavebuilt

andtheimpactthatithashadonourteachers,inparticular,theimpactonteachers’

mathematicalunderstandingandinstructionalpractices.Thenwediscusstheresearchthat

grewoutofourdesiretostudyscientificallyhowMHoMmightbeanindicatorofteacher

effectiveness.Lastly,weshedlightononehabitthatemergedprominentlyinour

research—usingmathematicallanguage.Weexaminehowateachermightusethishabitin

aclassroom,possibleimplicationsforstudentlearning,andhowuseofthehabitrelatesto

teachers’useofothermathematicalhabitsintheclassroom.

Weendthissectionwithafewremarks.Althoughwedescribeourresearchon

MHoM,theemphasisofthispaperisnotonourstudy,onitsparticularoutcomes,oronthe

measurementinstrumentsindevelopment.Instead,weintendtoillustrate,usingexamples,

ourmotivationforwhywethinkthesemathematicalhabitsareimportant.Hence,themain

focusofthepaperisthediscussionofthehabitofusingmathematicallanguage,and

Matsuura et al.

particularlyhowthishabitplaysoutinaclassroomsetting.Weincludeadetailed

discussionoftheFoMMSP,partlytosituateourworkwithintheMSPcontextinthisspecial

issueofTheMathematicsEnthusiast.Wealsowanttoprovidebackgroundfortheresearch

thatemergedfromandismotivatedbyourongoingMSPworkwithsecondaryteachers.

Indeed,ourstudyofteachers’MHoMandcorrespondinginstrumentdevelopmentarose

fromourdesiretomeasureprogressinandcontinuetoimproveourworkwithourown

FoMteachers.

FocusonMathematics

FocusonMathematics(NSFDUE0314692)isatargetedMSPfundedbytheNational

ScienceFoundationsince2003.Ourpartnershipisdevotedtoimprovingstudent

achievementinmathematicsthroughprogramsthatprovideteacherswithsolidcontent‐

basedprofessionaldevelopmentsustainedbymathematicallearningcommunitiesinwhich

mathematicians,educators,administrators,andteachersworktogethertoputmathematics

atthecoreofsecondarymathematicseducation.

TheoriginalFoMdistrictpartnersincludetheMassachusettsschoolsystemsof

Arlington,Chelsea,Lawrence,Waltham,andWatertown.Thesesystemsrangefrom

suburbantourban,withmiddleandhighschoolstudentpopulationsfrom1,300to6,000.

Overtheyears,FoMhasofferedavarietyofprofessionalopportunitiesforteachers,

including:(a)apubliccolloquiumseriesdevotedtomathematicsandeducation;(b)

partnership‐widemathematicsseminars;(c)week‐longsummerinstitutesforteachers;

(d)onlineproblem‐solvingcourses;and(e)anewMathematicsforTeachingMasters

ProgramatBostonUniversity.Twoactivitiesdeservespecialmention.

TME, vol10, no.3, p. 739

PROMYSforTeacherssummerinstitute,asix‐weekintensiveimmersionin

mathematics,engagesparticipantsinexperiencingmathematicsasmathematicians

do,solvingproblemsandpursuingresearchprojectsappropriateforthem.Each

summer,theinstitutecombinesteachersfrommultipledistricts,Grades5–12.

Academic‐yearstudygroupsaredistrict‐based—oftenbuilding‐based—groupsthat

meetbiweeklyfortwotothreehoursoverthecourseofayear.Thoughfocusedon

doingmathematics(ratherthanbeingtaughtitsresultsorhowtoteachit)—again,

experiencingmathematicsasamathematicianwould—thesetradetheintensityand

immersionofthesummerinstituteforlong‐term,ongoingstudy.

Thesemathematicallearningcommunitieswithcoreinvolvementof

mathematiciansaredesignedtohelpteachersdevelopthemathematicalhabitsofmind

thatarecentraltothedisciplineofmathematics.Ourteachershaveresponded

enthusiastically,withcommentssuchas:

“[Thestudygroup]isthebest‘professionaldevelopment’thatIhavebeeninvolved

inthroughoutmy35‐yearteachingcareer.Iguessthebesttestamentforthesuccess

ofFocusonMathematicscomesfromthecontinuedattendanceofsomanyteachers.

Wecontinuetotalkaboutthetopicsdiscussedatourstudygroupslongafterthe

weeklysessionisover”(Cuoco,Harvey,Kerins,Matsuura,&Stevens,2011).

“The[Masters]programhasexpandedmyknowledgeofmathematicsanddeepened

myunderstandingofhowchildrenlearnmathematics,but—moreimportantly—I

amnowconnectedtopeoplewhoareaspassionateaboutchildrenlearningand

doingmathematicsasIam”(Cuoco,Harvey,Kerins,Matsuura,&Stevens,2011).

Matsuura et al.

TostudytheimpactofFoM’sprofessionaldevelopmentprogramsonteachers’

professionallives,theProgramEvaluationResearchGroupatLesleyUniversity(FoM’s

evaluators)collectedandanalyzedteacherandstudentdataoverfiveyears(Lee,

Baldassari,Leblang,&Osche,2009)andconductedcasestudiesofteachers(Baldassari,

Lee,&Torres,2009).Belowarethosefindingsmoststronglyinformingourcurrentwork:

Teacherbeliefsandattitudesaboutthenatureofmathematics:Ininterviews,

teachersreportedunderstandingthestructureofmathematicsingreaterdepth—

howtopicsandideasareconnectedandhowtheyaredevelopedthroughthegrade

levels.Teachersreferredtodevelopingamorecompletepictureorunderstandingof

mathematicsasasystemandunderstandingtheconnectionsbetweendifferent

threadswithinit(Lee,Baldassari,&Leblang,2006;Lee,Baldassari,Leblang,Osche,

&Hoyer‐Winfield,2007).

Teacherchangesininstructionalpractice:Manyoftheinstructionalchangesteachers

reportedstemfromthewaysinwhichtheyexperiencedlearningthroughFoM(Lee

etal.,2006).Whenteachersdevelopedadeeperunderstandingofmathematics,

theirconfidenceoftenincreasedandtheydevelopedmoreflexibilityintheir

teachingandtheabilitytoadjustlessonsbasedonstudentresponses.

ThroughourworkinFoM,wehaveseenthatMHoMisindeedacollectionofhabits

teacherscanacquire,ratherthansomestaticyou‐have‐it‐or‐you‐don’twayofthinking.And

teachersreporttousthatdevelopingthesehabitshashadatremendouseffectontheir

teaching.Wehavecollectedampleanecdotalevidence,butrecognizetheneedfor

scientifically‐basedevidencetoestablishthattheseteachershaveindeedlearnedMHoM

TME, vol10, no.3, p. 741

andthatthesehabitshavehadapositiveimpactontheirteachingpractices.Wealso

recognizetheneedtostudystudentoutcomesaffectedbyteachers’usesofMHoM.

MathematicalHabitsofMindforTeachingResearchStudy

FocusonMathematics,PhaseII:LearningCulturesforHighStudentAchievement(NSF

DUE0928735)isanMSPprojectthatbeganin2009.InFoM‐II,wecontinuedtorefineour

mathematicallearningcommunitiesandbegananexploratoryresearchstudyfocusedon

teachers’mathematicalhabitsofmind.

Asabasisforbeginningtheresearchstudy,weusedthetheoreticalframeworks

developedbyClarkeandHollingsworth(2002)fortheir“InterconnectedModelofTeacher

ProfessionalGrowth,”whichischaracterizedbynetworksof“growthpathways”among

four“changedomains”inteachers’professionallives—theexternaldomain(E),the

personaldomain(K)(ofknowledge,beliefsandattitudes),andthedomainsofpractice(P)

andsalientoutcomes(S).Significant,fromourpointofview,istheClarke‐Hollingsworth

theoryofprofessionalgrowth(asdistinctfromsimplechange),whichtheyrepresentas“an

inevitableandcontinuingprocessoflearning”(p.947).Theyaptlydistinguishtheir

frameworkfromothers:“Thekeyshiftisoneofagency:fromprogramsthatchange

teacherstoteachersasactivelearnersshapingtheirprofessionalgrowththroughreflective

participationinprofessionaldevelopmentprogramsandinpractice”(Clarke&

Hollingsworth,2002,p.948).Theagencyofteachersintheirownprofessionalgrowth

characterizesvirtuallyallFoMprograms,soweseetheClarke‐Hollingsworthmodelof

professionalgrowthaswellsuitedforourpurposes.

WeillustrateouruseoftheClarke‐Hollingsworthframeworkwithanexample.

ShowninFigure1isachangeenvironmentdiagramfor“Ms.Crew,”amiddleschool

Matsuura et al.

teacherandactivememberoftheFoMlearningcommunity.Thediagramrepresentsthe

changedomainsasfourboxes,labeledE,K,P,andS,asexplainedabove.Thesolidarrows

refertogrowthsduetoenactment,whilethedashedarrowsdepictthoseduetoreflection.

TheloopontheboxEreferstointeractionbetweenstudygroupsandtheimmersion.

Figure1.SchematicdiagramofMs.Crew’schangeenvironment

ThisparticulardiagramdepictsactivityrelatedtoMs.Crew’sresearchon

PythagoreanTriplesandshowshowthisactivityledtohergrowth,bothmathematically

andasateacher.EacharrowrepresentsagrowthinMs.Crewthatoccurredasaresultofa

changeinherprofessionallife.Forexample,arrow6depictshowherincreasedbelief

aboutherself(achangeinboxK,thepersonaldomain)leadstoMs.Crewencouragingher

studentstoperformmoreexplorations(achangeinboxP,thedomainsofpractice).

Moreover,arrow6issolid,becausethechangeinherclassroomisdueanenactment,i.e.,a

particularcourseofactionthatshetookasateacher.Thearrowsarenumberedin

chronologicalorder,soarrow1denotesagrowthinMs.Crewthatoccurredbeforethat

depictedbyarrow2,andsoon.ThedashedarrowfromboxEtoKhasmultiplenumbers

TME, vol10, no.3, p. 743

(asdoesthesolidarrowfromKtoE).Here,thedashedarrowmaybeinterpretedasthree

separatearrows(arrow1,arrow3,andarrow5)—wesimplycondensedthemintoone

arrowtosavespaceinthediagram.

Ms.CrewfirstencounteredtheconceptofPythagoreanTripleswhilestudying

Gaussianintegersduringhersummerimmersionexperience.Thetopicleftsuchan

impressiononher(reflectivearrow1)thatshepursuedit(enactivearrow2)asaresearch

projectundertheguidanceofanFoMmathematician.Throughmonthsofhardwork—

familiarizingherselfwithPythagoreanTriplesthroughdozensofexamples,makingcareful

datarecordingandanalysis,discoveringbeautifulpatterns,comingupwithinteresting

conjectures(someweretrue,somewerefalse),andfinallywritingdownclearandconcise

propositionsandprovingthem—shecametounderstand(reflectivearrow3)featuresof

Pythagoreantriplesthatwouldhavebeenbeyondherconceptionbeforethisexperience.

Ms.Crewproducedanindependentresearchpaperandaone‐hourmathematicstalkfor

herpeers(enactivearrow4).

Neitherthesummerimmersionexperiencenortheindependentresearchproject

waseasyforMs.Crew,whocameintoourprogramwitharatherweakmathematics

background.Butcompletingthisprojecthadasignificanteffectonhermathematicalself‐

confidence(reflectivearrow5).TheloopsofthisupwardspiralbetweendomainsKandE

repeatedmanytimes.Amongstherpeers,Ms.Crewbecameoneoftheleadersinherstudy

group(4).Inhercurriculumplanning,shenowhasmorebeliefinherdecision‐making

abilities(5).Andinherclassroom,sheengagesherstudentsinperformingmathematical

exploration(6).Thisnewclassroomatmosphere,aswellashernewattitudetowards

mathematics,ledtomorecuriosityandquestionsfromherstudents(7,8).Andwhileshe

Matsuura et al.

maynotbeabletoanswerallofthemonthespot,shenowwelcomesmathematicaldialogs

anduncertaintyinherclassroom(9,10).Allofthisrepresentssignificantprofessional

growthandMs.Crew’schangediagramenablesustoseetheelementsofthatgrowthata

glance.

LookingatMs.Crew’schangediagram,onecannotfailtonoticetheintenseactivity

takingplacearoundthenodeK,whichincludesgrowthinMs.Crew’sknowledgeof

mathematics.Butitseemstousthatmoreisinvolvedthansimplyknowingmathematicsas

abodyofknowledge.Ms.Crewislearningmathematicsinacertainway.Herbeliefsabout

thenatureofmathematicsarechanging.Sheisacquiringcertainmathematicalhabitsof

mindandsheisfindingthesehabitsusefulforherworkintheclassroomandalsofor

leadershiprolesintheschool.

Applyingthisframeworkofteacherchange,webegantobuildforourselvesa

theoreticalunderstandingofhowMHoMplaysaroleintheworkofteaching.Recognizing

theneedforascientificapproachtotestthetheory,andindeedinvestigatethewaysin

whichMHoMisanindicatorofteachereffectiveness,weconductedanexploratorystudy

titledMathematicalHabitsofMindforTeachingthatcentersonthefollowingquestion:

Whatarethemathematicalhabitsofmindthatsecondaryteachersuseintheir

professionandhowcanwemeasurethem?

Toinvestigatethisquestion,wedevelopedadetaileddefinitionofMHoMandhavebeen

buildingthefollowingtwoinstruments:

Apaperandpencil(P&P)assessmentthatmeasureshowteachersengageMHoM

whendoingmathematicsforthemselves.

TME, vol10, no.3, p. 745

Anobservationprotocolmeasuringthenatureanddegreeofteachers’usesofMHoM

intheirteachingpractice.

Weemphasizethatbothinstrumentsareneeded,becauseinourworkwithteachers,we

haveseenthosewhohaveverystrongMHoMforthemselvesbutdonotnecessarilyemploy

thesamemathematicalhabitsintheirteachingpractices.

Ourcurrentworkfitsintoalargerresearchagendathatwearedevelopingin

collaborationwithleadersinthefield,withtheultimategoalofunderstandingthe

connectionsbetweensecondaryteachers’mathematicalknowledgeforteachingand

secondarystudents’mathematicalunderstandingandachievement.

OperationalizingMHoM

TooperationalizetheMHoMconcept,wereliedonourownexperiencesas

mathematiciansdoingmathematicswithsecondaryteachers(Stevens,2001).Wealso

studiedexistingliterature—inparticular,Dewey’s(1916)andDeweyandSmall’s(1897)

earliertreatmentsofhabitsandhabitsofmind,theStudyofInstructionalImprovement

(SII)andtheLearningMathematicsforTeaching(LMT)projectstodevelopmeasuresof

mathematicalknowledgeforteaching(MKT)forelementaryteachers(Ball&Bass,2000;

Ball,Hill,&Bass,2005;Hill,Schilling,&Ball,2004;Hill,Ball,&Schilling,2008),andthe

descriptionbyCuocoetal.ofmathematicalhabitsofmind(1997,2010).Andweconsulted

thenationalstandards,i.e.,theNCTMPrinciplesandStandardsforSchoolMathematics

(NationalCouncilofTeachersofMathematics[NCTM],2000)andtheCommonCore

StandardsforMathematicalPractice(NationalGovernorsAssociationCenterforBest

PracticesandtheCouncilofChiefStateSchoolOfficers[NGACenter&CCSSO],2010).But

aboveall,wewentintotheclassroomsofFoMteachers,whereweobservedabroad

Matsuura et al.

samplingofMHoMstrengths.Someteachersexhibitedpreciseuseoflanguageandcareful

reasoningskills;othershadstrongexplorationskills,weregoodatdesigningmathematical

experiments,orshowedspecialstrengthatgeneralizingfromconcreteexamples.

Fromthesevarioussources,webegantocompilealistofhabitsthatconstitute

MHoM.Asthelistgrew,weidentifiedfourbroadandoverlappingcategoriesintowhichour

mathematicalhabitsnaturallyfell:

● Seeking,using,anddescribingmathematicalstructure

● Usingmathematicallanguage

● Performingpurposefulexperiments

● Applyingmathematicalreasoning

Indeed,thesearecategoriesofmathematicalpracticesthatareubiquitousinthediscipline.

Andinordertoconductafine‐grainedstudyofthesecategories,weteasedapartmultiple

habitswithineachcategorythatwewantedtomeasure,someofwhichwereidentified

earlier.Thatbeingsaid,weprimarilyenvisionMHoMasbeingcomprisedofthefour

categories,withthelistofhabitswithineachcategoryprovidingmoredetailandtextureto

thesefour.Bynomeansisourlistfinal.Infact,weconsideritanevolvingdocumentthat

wewillcontinuetoreviseasweobtainmoredatausingourinstruments.Fromourdata,

wewilllearnwhichhabitsaremoreprominentlyusedbysecondaryteachers,bothwhen

doingandteachingmathematics.

PaperandPencil(P&P)Assessment

WedevelopedapilotP&Passessmentthatmeasureshowsecondaryteachersuse

MHoMwhiledoingmathematics.Thisassessmentcontainssevenopen‐endedproblems

andisdesignedtobecompletedinonehour.Inparticular,wedevelopedproblemsthat

TME, vol10, no.3, p. 747

mostteachershavetherequisiteknowledgetosolve,oratleastbegintosolve.Andwhat

weareassessingishowtheygoaboutsolvingit.Itisthechoiceoftheirapproachthatwe

areinterestedin,asopposedtowhetherornottheyhavethenecessaryknowledge/skills

tosolveit.Eachitemisdesignedtorevealwhathabitsandtoolsteacherschoosetousein

familiarcontexts.Todate,wehavegonethroughseveralroundsofdesign,pilot‐test,data

analysis,andrevisionofthisinstrument.Forourlatestpilot‐testinthesummerof2011,we

administeredtheP&Passessmentto43secondarymathematicsteachersparticipatingin

theNSF‐fundedstudyChangingCurriculum,ChangingPractice(NSFDRL1019945).Wewill

carryoutanotherfieldtestwithapproximately50teachersinthesummerof2012.

Togatherinitialdataontherolethatteachers’approachtosolvingmathematics

problemsplaysintheirapproachtomathematicsinstruction,weaskedafollow‐up

questiontosomeofourP&Passessmentproblems:Whatstrategieswouldyouwantyour

studentstodevelopforaproblemlikethis?Our43respondentsalmostunanimously

reportedthattheywanttheirstudentstoapproachtheproblemsexactlyastheydid

themselves.(Note:Afewteacherswantedtheirstudentstoappreciateavarietyof

approaches.)Thisfindingprovidesinitialevidencethatteachers’ownmathematicalwork

maybeindicativeofhowtheychoosetoexplain/formulatethesubjectmatterfortheir

students.Recognizingtheneedforfurtherstudyofthishypothesis,webegantocreatean

observationprotocol.

ObservationProtocol

Weareintheprocessofdesigninganobservationprotocolandcodingschemethat

measurethenatureanddegreeofteachers’usesofMHoMintheirclassroominstruction.

Todeveloptheinstrument,weconductedliveandvideotapedobservationsoftwotothree

Matsuura et al.

consecutivemathematicslessonscollectedfromatotalof30secondaryteacherstoidentify

teacherbehaviorsthatreflecttheusesofaparticularmathematicalhabit.Inaddition,we

developedasimpleprotocolforpre‐andpost‐interviewswithteacherswevideotape.We

alsocollectedclassroomartifacts(lessonplans,in‐classworksheets,homework,and

assignments)fromeachclassroomweobserved.

Animportantfeatureofourobservationprotocolisthatitmeasureshowteachers

useMHoMintheirinstruction.Thusteachersarecodednotforpossessingcertain

mathematicalhabitsintheabstract,butforchoosingtobringthemtobearinaclassroom

setting.Todevelopsuchaninstrument,wearecurrentlystudyingourvideosandslicing

theselessonsintosmallepisodes—i.e.,shortinstructionalsegmentslasting30secondsto4

minutes.Ineachepisode,wedeterminewhethertherewerebehavioralindicatorsthat

reflectedteachers’usesofMHoM,andwecreatecodesthatgeneralizeandcharacterize

theseteacherclassroombehaviors.Weemphasizethatourcurrentfocusisonteacher

behaviorsandusesofMHoMintheclassroom.Wearestillastepawayfromconnecting

teachingpracticescenteredonMHoMtostudents’developmentofMHoMandtostudent

achievement—partlybecausewedonotyethavetheinstrumentstoassessthesehabitsin

students—butimpactingstudents,ofcourse,isourultimategoal.

Later,wedescribethreeteachersfromwhomwegatheredvideodataforour

observationprotocoldevelopment.Specifically,wewilldiscusshowtheyapplythehabitof

usingmathematicallanguageintheirclassroominstruction.Wewillalsoconsiderhow

teacheruseofthisparticularhabitmayaffectstudentunderstanding.

TME, vol10, no.3, p. 749

RelevantLiteratureandRelatedWork

ThetheoryofmathematicalhabitsofmindisphilosophicallygroundedinDewey’s

(1916)andDeweyandSmall’s(1897)earliertreatmentsofhabitsandhabitsofmind.

Theirseminalworkhassinceencouragededucators(Duckworth,1996;Meier,1995)and

educationresearchers(Kuhn,2005;Resnick,1987;Tishman,Perkins,&Jay,1995)to

furtheroperationalizetheconceptofhabitsofmind—thatis,torespondtothegeneral

question:Whatdohabitsofmindlooklikeinthecontextoflearning?Notasevidentinthe

literaturearethehabitsofmindthatpromotesuccessfullearninginspecificdisciplines.In

thecaseofmathematics,thequestionthathasgainedresearchattentionwithinthelast

decadeis:Whatdohabitsofmindlooklikeinthecontextoflearninganddoingmathematics?

Whileaddressingthisquestionisnotanunfamiliartask(Hardy,1940;Polya,1954a,1954b,

1962),whatislessfamiliaristhetaskofgatheringevidenceofmathematicalhabitsofmind

fromteachersofmathematics.WebeganthisworkinourFoM‐IIstudy;weareinthelong‐

termprocessofdevelopingvalidandreliableinstrumentsthatwillallowustomore

rigorouslyinvestigatetherelationshipbetweenteachers’ownMHoM,theirusesofMHoM

intheirteachingpractice,andstudentachievement.

Asmentionedearlier,weenvisionMHoMasanintegralcomponentofMKTatthe

secondarylevel.ThenotionofMKThasbeenstudiedbymanyresearchers(Ball,1991;Ball,

Thames,&Phelps,2008;Heid,2008;Heid&Zembat,2008;Heid,Lunt,Portnoy,&Zembat,

2006;Hilletal.,2008;Kilpatrick,Blume,&Allen,2006;Leinhardt&Smith,1985;Ma,1999;

Stylianides&Ball,2008).Ourunderstandingsofthisnotionparallelmuchofwhatwehave

readintheliterature,butarebasedonourparticularexperiencesoverthepast20years,as

mathematiciansengagedindoingmathematicswithsecondaryteachers.

Matsuura et al.

Asmathematiciansworkinginschoolsandprofessionaldevelopment,wehavecome

tounderstandsomeofthewaysinwhichteachersknowandunderstandmathematics.

Thesefitintofourlargeandoverlappingcategories:

(1) Teachersknowmathematicsasascholar:Theyhaveasolidgroundinginclassical

mathematics,includingitsmajorresults,itshistoryofideas,anditsconnectionsto

precollegemathematics.

(2) Teachersknowmathematicsasaneducator:Theyunderstandthethinkingthat

underliesmajorbranchesofmathematicsandhowthisthinkingdevelopsin

learners.

(3) Teachersknowmathematicsasamathematician:Theyhaveexperiencedasustained

immersioninmathematicsthatincludesperformingexperimentsandgrappling

withproblems,buildingabstractionsfromtheexperiments,anddevelopingtheories

thatbringcoherencetotheabstractions.

(4) Teachersknowmathematicsasateacher:Theyareexpertinusesofmathematics

thatarespecifictotheprofession,includingtheabilityto“thinkdeeplyofsimple

things”(Jackson,2001,p.696),thecraftoftaskdesign,andthe“mining”ofstudent

ideas.

Thefirsttwoofthesewaysofknowingmathematicsarecommontomostpre‐serviceand

in‐serviceprofessionaldevelopmentprograms.FoMhaspaidparticularattentiontothe

lasttwo,whichtypicallyreceivelessemphasis.Wehavebecomeconvincedthat(3)greatly

enrichesandenhancestheotherwaysofknowingmathematicsandthatmanyteachers

whogothroughsuchanexperiencedevelopthehabitsofmindusedbymany

mathematicians.Furthermore,wehaveseenthatparticipationinamathematicallearning

TME, vol10, no.3, p. 751

communityhelpssuchteachers“bringithome”inthesensethattheycreatestrategiesfor

helpingtheirstudentsdevelopthemathematicalhabitsthattheythemselveshavefoundso

transformative.

Otherresearchersaredevelopinginstrumentstoassesssecondaryteachers’content

knowledgeanduseofmathematicsintheirclassrooms(Bushetal.,2005;Ferrini‐Mundy,

Senk,McCrory,&Schmidt,2005;HorizonResearch,Inc.,2000;MeasuresofEffective

TeachingProject,2010;Piburn&Sawada,2000;Reinholzetal.,2011;Shechtman,

Roschelle,Haertel,Knudsen,&Vahey,2006;Thompson,Carlson,Teuscher,&Wilson,n.d.).

Indevelopingourowninstruments,wehavedrawninsightfromalloftheseprojects.But

wehavemostcloselyfollowedthemodeldevelopedbyBallandHill—specifically,their

MKTassessmentandMathematicalQualityofInstruction(MQI)protocolfordocumenting

MKTinelementaryteachers(Hilletal.,2005;LearningMathematicsforTeaching,2006).

Theirinstrumentsmeasure“specialized”mathematicalknowledge,thatis,knowledgethat

teachersuse,asdistinctfromthemathematicalknowledgeheldbythegeneralpublicor

usedinotherprofessions,whosecomponentsincluderepresentationofmathematical

ideas,carefuluseofreasoningandexplanation,andunderstandinguniquesolution

approaches.Theseskillsresemblethekindsofmathematicalhabitsthatweareinterested

instudyingatthesecondarylevel.

ThecollectiveeffortsofthefieldwillallcontributetowhatweknowaboutMKT,but

thereareimportantdifferencesbetweenourinstrumentsandthoseofothers.The

differencesarelistedbelow.

AfocusonMHoM—themethodsandwaysofthinkingthroughwhichmathematics

iscreated—ratherthanonspecificresults(Cuocoetal.,1997).Itisimpossible,even

Matsuura et al.

inthreeorfouryearsofhighschoolmathematicsalignedwiththeCommonCore,to

equipstudentswithallofthefactstheywillneedforcollegeandcareerreadiness.

Butlearningtothinkincharacteristicallymathematicalwaysisatickettosuccessin

fieldsrangingfrombusiness,finance,STEM‐relateddisciplines,andevenbuilding

trades.

Thecoreinvolvement,ateverylevel,ofmathematicianswhohavethoughtdeeply

abouttheimplicationsoftheirownhabitsofmindforprecollegemathematics

curricula,teaching,andlearning(Bass,2011;Schmidt,Huang,&Cogan,2002).

Ourinstrumentsare,therefore,aimedatdiscerningtheextenttowhichsecondary

classroomsarecenteredonthepracticeofdoingmathematicsratherthanonthespecial‐

purposemethodsthatoftenplaguesecondarycurricula(Cuoco,2008).Inourworkwith

teachers,wehaveseenhowexpertteachersusecoremathematicalhabitsofmindintheir

profession—inclass,inlessonplanning,andincurricularsequencing.And,astheCommon

Corebecomesthenationallyaccepteddefinitionofschoolmathematics,teacherswillbe

expectedtomakethedevelopmentofmathematicalhabitsanexplicitpartoftheirteaching

andlearningagenda.Ourwork,therefore,makesauniquecontributiontothefield’s

increasinglevelofattentiontosecondarymathematicsteaching.

UsingMathematicalLanguage

Inthissection,wewillfocusonaspecificmathematicalhabit—usingmathematical

language—andexaminehowteachersusethiscorehabitintheirinstructionalpractice.We

willalsoconsideritspotentialimplicationsforstudentlearning,andhowthishabitmay

workinconjunctionwithothermathematicalhabitsintheclassroom.

TME, vol10, no.3, p. 753

Inparticular,wewilldiscussexamplesofthreeteacherswhoseAlgebra1

classroomsweobservedinourresearchstudy.WewillbeginwithMr.Hart,whouses

mathematicallanguagetoencapsulatetheexperiences,observations,anddiscoveriesofhis

students.Second,wewilllookatMs.Graham,whousespreciseandoperationalizable

languageasawayofpromotingconceptualunderstandingandeaseofproblem‐solving.

Andthird,wewilldescribeanexampleofateacher,Mr.Braun,whosechoiceoflanguage

caninterferewithstudents’engagementinactivitiesdesignedtopromoteotherMHoM.

AllthreeoftheseteachershaveshownevidenceofstrongMHoMintheirowndoing

ofmathematics.Mr.Harthasheldformalandinformalleadershiprolesinanumberof

FoM’smathematicallearningcommunities;andinthoseroles,hehasexhibitedstrong

MHoM.TheothertwoteachersperformedwellonourP&Passessment.Thenamesofthese

teachershavebeenalteredtoprotecttheiridentities.

Mr.Hart

WeconsiderMr.Hart,anAlgebra1teacherwhousesmathematicallanguageto

encapsulatetheunderlyingstructurethatstudentsdiscoveredthroughexperimentation.

Themathematicaltopicofthedayisrecursiverules.Theclassbeginswithstudents

workingonthefollowingwarm‐upproblem.

Afunctionfollows[thisrule]forintegervaluedinputs:Theoutputforagiveninputis 32

greaterthanthepreviousoutput.Makeatablethatmatchesthedescription.Canyou

makemorethanonetable?

Notethattheruleisincomplete,becauseitismissingthebasecase.Studentsexperiment

withthisrule,creatinginput/outputtablesandtryingtoderiveclosed‐formequations.

Matsuura et al.

Becauseoftheirdifferentchoicesofbasecases,theycomeupwithdifferentfunctions

definedbyexpressionsoftheform f (x) 3

2xb .Studentsconcludethatthegraphsof

thesefunctionsareparallellineswithdifferenty‐intercepts.Mr.Hartalsoasks,“Sowhat’s

thepartwhereyougettobecreativeinmakingthesetables?”Hethenexplains,“Soyouget

topickonenumber,andtheneverythingelseisdecidedbythepartthatIgaveyou[inthe

warm‐up].Butthere’sstillanawfullotofdifferentnumbers.”Here,heisforeshadowingthe

needtofixthebasecase.

ThenMr.Hartformallyintroducesthenotionsofrecursiveruleandbasecaseto

summarizestudents’experiencesandtocapturetheunderlyingstructuretheyobserved

whenworkingonthewarm‐upproblem.Hesays,

Arecursiverule,that’sjustthedescriptionthattellsushowtogetfromanoutput—

toanoutputfromthepreviousones.Sobasically,whatweweredoing.Nowasyou

saw,there’sanotherpiecethat’snotreallyenoughinformation.It’sjustmetelling

youhowtogetfromone,tothenext,tothenext.Tohaveacompleterule,wealso

needtoknowwheretostart.Becauseotherwise,wewon’tknowifwehavetherule

that—thefirstrule,thesecondrule,thethirdrule,orsomeotherrulecompletely.

(Videotranscript,February14,2011.)

Next,theclassstudiesthefunctiondescribedbythefollowingtable:

n f (n)

0 3

1 8

2 13

3 18

TME, vol10, no.3, p. 755

4 23

5 28

6 33

Inthistableofdata,studentsrecognizethe+5pattern,i.e.,“Youadd5totheoutput.”

Throughdiscussion,Mr.Hartguidesthemtoarticulatetherelationshipmoreprecisely:

f (5) f (4)5. Usingthisconcreteexample,studentsareabletoderiveageneralequation:

f (n) f (n1)5.

Tomakesenseofthisrecursiverule,Mr.Hartpointsoutthattheequation

f (n) f (n1)5“letsusrelateanyoutputtoapreviousone.”Inessence,itisthesymbolic

representationofwhathetoldstudentsinthewarm‐upproblem.Thenhedescribesthe

needforthebasecase,saying,“Butthatwasn’tquiteenoughbecauselotsofyouwrote

downdifferentrules.And[Student1]hadone,[Student2]hadadifferentone,[Student3]

hadadifferentoneprobably,andsoon.Soweneedsomethingelsetosortoffixitinplace.”

Here,astudentinterruptsandproposesaclosed‐formrule: f (n) 5n3.Thereare

nowtwowaystodescribethefunctionathand,namelythe(stillincomplete)recursiverule

f (n) f (n1)5andtheclosedformrule f (n) 5n3.Hesays,“[Therecursiverule]tells

ushowtoworkourwaydownthetable.IfIknowonevalue,Iknow23,Icanfindthenext

onereallyeasily.Nowthisone’s[pointstotheclosed‐formrule]nicetoobecauseitletsme

workacrossthetable.IfIknowtheinput,Icansaytheoutputreallyquickly.”Inthisshort

episode,Mr.Hartusesthesymbolicrepresentationofeachruletodiscussitsunderlying

structure.

Mr.Hartreturnstotheequationwrittenontheboard(i.e., f (n) f (n1)5)and

says,“Butstill,this—thisrulealmosttellsmethewholetable,butitdoesn’tquitebecause

Matsuura et al.

I’mmissingonecriticalpieceofinformation.”Astudentchimesin,“Well,youdon’tknow

whatyoustartedwith.”Mr.Hartrespondswith,“That’sagoodpoint.Yeah,solike

[Student]’ssayingthis3inthetable,that’swherewe’restarting.Sowekindofneedto

knowthat.Sotheway(pause)agoodwaythatwecansortofkeeptrackofthisandwrite

ourrule...”Almost20minutesintothelesson,Mr.Hartfinallyintroducesthecomplete

notation

f (n) 3 if n 0,

f (n1)5 if n 0.

Heexplainsthisnewequationbysaying,“Sothisformulacapturesexactlywhatwedid.The

keypartistherecursivepartthatwehadwrittendownalready.Andthisjustaddsthatlast

bit,thebasecase,sowecansummarizeitintoonecompactrule.”

Insteadofbeingastartingpoint,thisnotationistheculminationofthestructures

thatstudentsdiscoveredthroughtheirexperimentationandthefollow‐updiscussion.

Studentsreadilymakesenseofthenewnotationandtheaccompanyingideasthatit

encapsulates,becausetheexperiencegainedthroughtheir“struggles”allowsthemto

connectthenewlanguagetoalready‐establishedideas.

Mr.Hartusesthestructurethatstudentsfoundthroughtheirexperimentstomotivate

thelanguageneededtodescribetheirobservedresults.Forinstance,students’experiments

withthewarm‐upproblem,inwhichtheyproposedifferentfunctionsthatallsatisfythe

givenrule,maketheneedforthebasecasecomealiveforthem.Indeed,hismathematical

habitsofmindallowMr.Harttocreatealearningenvironmentwherestudentsbuildnew

knowledgefromtheirexperiences(NCTM,2000).

TME, vol10, no.3, p. 757

Ms.Graham

ThroughMs.Graham,welookathowanAlgebra1teacherusespreciseand

operationalizablelanguageasawayofpromotingeaseofproblem‐solving.More

specifically,shehelpsstudentsmakesenseoftheobjectiveofthegivenproblemand,

subsequently,providesinsightintohowtoproceed.

Inthisepisode,astudentasksaboutthefollowingquestion:

Determineif r 2 isasolutionto6r 2 12 r.

Ms.Grahamasks,“Didwenotunderstandwhattheywereasking?”Thestudentconfirms,

“Yeah,obviouslythere’saneasierwaytodoit,butIjustdidn’tknowhow.”Thenthe

followingdialogueoccurs,inwhichMs.Grahampressesforthemeaningoftheword

“solution”:

Teacher(T): Allright.Whenweusetheword“solution,”allright,we’vetalkedalotabout

whatasolutionis.Whatdoes“solution”mean?

Student(S): Like,does—it—whenitworks.

T: Whenyousaid“itworks,”whatdoyoumean?BecauseIthinkyou’reontheright

track.

S: Like,doesitmakesense?

T: Bealittlemorespecific.

S: Idon’tknowhow,like…

T: Whatdoes“solution”mean,anyoneknow?Allright.

Newstudent(SN): Theanswer?

T: “Theanswer.”Wetalkedaboutthisalot.What’sasolutiontoanequation?

SN: Somethingthatcangointomakeanequationwork.

Matsuura et al.

T: Somethingthatmakestheequationtrue,OK?

AswewillseelaterinMr.Braun’sexample,“works”isoftenusedbystudentsand

teacherstodescribewhatitmeansforanumbertobeasolutiontoanequation.Ms.

Grahamdoesnotsettleforthisnorotheroft‐usedphrasessuchas“itmakessense”and

“theanswer.”Thelanguageusedbystudentsdoesnothelpthemunraveltheproblemto

understandwhattheyarebeingaskedtodo.Onlyaftertheoperationaldefinitionof

“solution”hasbeengivencanMs.Grahamcontinuewithanexplanationofhowtoproceed.

T: We’restatingthat6r 2 willbeequalto12 r.Andthey’reasking,“Is r 2 a

solution?”Soyougottotestitout,justasIaskedyoutotestoutthatonethatwe

justdid.So6r 2 12 r. Substitutein r 2.So6times2 plus2—doesthathave

thesamevalueas12plus2?Andwehavetotest.Allright?We’reaskingourselves

thequestionof,doesthisequalthat?[Pointstoeachsideoftheequation.]OK?

ThenMs.Grahamleadstheclassthroughtheprocessofsubstituting r 2 intothe

equationandconcludingthatitisnotasolution,since r 2 yieldsunequalvaluesof10

and10forthetwosidesoftheequation.Thestudentwhooriginallyinquiredaboutthis

questionsays,“Ok.NowIgetit.”Thedefinitionof“solution”providedbyMs.Graham—

namely,“somethingthatmakestheequationtrue”isoperational(i.e.,studentscanusethis

definitiontounderstandandaccomplishthetaskposedbythegivenquestion).Indeed,

oncethedefinitionhasbeengiven,substituting r 2 andcheckingifitmakesthe

equationtrueisanaturalnextstep.

Ms.Grahamconcludesthisepisodebyforeshadowingwhatstudentswillbelearning

next,byprovidingthemwithanotherdefinition:

TME, vol10, no.3, p. 759

T: We’regettingtothepointwherewe’regoingtoaskyou,“Whatisthevalueofrthat

makestheequationtrue?”Andthat’scalledsolvingtheequation.

Throughoutthelesson,Ms.Grahamconsistentlyuseslanguagecarefully.Shecorrectsa

studentwhowrites828 903 305 25,callingita“run‐onsentenceinmath.”When

astudentdescribestwosidesofanequationbysaying,“It’sequals,”Ms.Graham

immediatelyresponds,“They’reequaltoeachother.”Sherepeatedlytellsstudentstocheck

theiransweraftersolvinganequation,remindingthemwhat“solution”means.Sheisalso

preciseinherinstructions(e.g.,askingthestudentsto“writeanexpressionfortheright

sideoftheequation,sothatyou’vegotanequationthatworksandistruewhen x 3”).

Mr.Braun

Oneoftheissueswehaveencounteredinthedevelopmentofourobservation

protocolis,“Whatcountsasevidenceofnon‐useofMHoM?”Inthecaseofthehabitofusing

mathematicallanguage,wedoseemomentsinwhichteacherschooselesscarefullanguage.

Forexample,ateachermightchoosetouseinformallanguage.Sometimesthereisevidence

thattheteacherismakingthischoicebecausetheinformallanguageseemsmoreaccessible

tostudents.Butsuchchoices—ifnotmadecarefully—canleadtostudentconfusion.

Inthefollowingexample,Mr.Braunissettingupaninvestigationthataimstolay

thefoundationthatthegraphofanequationisarepresentationofthesolutionsetofthe

equation(EducationDevelopmentCenter,Inc.,2009b).Tolaunchtheinvestigation,Mr.

Braunwritestheequation3x 2y 12 ontheoverheadprojectorandasksstudents,

“What’stheanswer?”Hethendescribessomeofthesolutionsstudentsofferas“that

works”or“thatdoesn’twork.”Thefollowingisanexcerptfromthelaunchofthe

investigation.Therearetwothingstonote.First,Mr.Braunismodelinghowstudents

Matsuura et al.

mightexperimentwithnumbersasawayofmakingsenseoftherelationshipbetween

graphsandequations.Second,observehowfrequentlyheusestheword“works.”

T: 3x 2y 12 .What’stheanswer?

SN: It’scomplicated.

T: Oh,no.Whatdoyouthink?

SN: 1and2?

T: YouthinkIcanuse1and2?

S: xis1andyis2.

T: xis1andyis2.HowwouldIfindoutif[name]isright?Icouldputinthenumbers

thathegaveme,soI’mgoingtoputin1forxandI’mgoingtoputin2fory,anddoI

get12,likeI’msupposedto?What’s31?

Students(Ss):3.

T: What’s2 2 ?

Ss: 4.

T: What’s3+4?

Ss: 7.

T: DidIget12?

Ss: No.

T: Man,[name],that’sabummer.OK,so—

SN: Oh,Iknowit.

T: —thatwassomethingthatdidn’twork.It’snotbadtofindoutthingsthatdon’t

work.Sometimes,you’regoingtobeaskedintheseinvestigationstofindthingsthat

don’twork,sorememberhowwedidthat.

TME, vol10, no.3, p. 761

Atthispoint,theteachercontinuestotakestudentguessesforxandy.Students

makeguessesandonestudentsuggests x 2 and y 3.Mr.Brauntriesthatsuggestion,

andseesthatindeed,3(2) 2(3) 12.

T: OK,sowefoundoutthat1and2didnotwork;wefoundoutthat2and3didwork.

Doyouthinkthereareanymorethingsthatdon’twork?

SN: Yes.

T: Alotmorethingsthatdon’twork.OK,doyouthinkthereareanymorethingsthat

dowork?

S: Yes.

T: Canyouthinkofanotherthingthatdoeswork?[...]

SN: 3(3)…

T: OK,ifIputathreethere,OK.

S: Andthen,the2yis2,2(1).

T: 21. OK,thisis9,right?Plus2,makes11insteadof12.So,wefoundanotherthing

thatdoesn’twork.So,I—[name],youmusthavebeenright,thereweremorethings

thatdonotwork.Canyoufindanythingelsethatdoeswork?

SN: 4and1.

T: Youthink4and1works?WheredoIputmy4,forxorfory?

S: Forx,yeah.

T: OK,soIputin3(4)+2(1),thatgivesme12+2=14.Wefoundanotherthingthat

doesn’twork.

S: Actually,put3fory,plus1.5.

T: […]2(1.5),whatarewegoingtoget?

Matsuura et al.

Ss: It’s3.

T: 3,andwehad9.Is3+9=12?

Ss: Yes.

T: Hey,lookatthat.Allright,now,that’sthekindofthingIwantyoutodo.You’rejust

goingtotrysomethings.Someofthemwillwork;someofthemwon’twork.

Mr.Braunhasmodeledadetailedinvestigationoflookingforpointsthatsatisfythe

equation3x 2y 12, usingtheword“works”asasubstitutefor“satisfiestheequation.”

Heusesthephrases“works”and“doesn’twork”repeatedly.Hethenhandsoutaworksheet

forinvestigationthatincludestheproblems:

Eachpointinthefollowingtablesatisfiestheequation x y 5.

a) Completethetable.

x y (x,y)

1 4 (1,4)

2

3

0

12

2

113

b) Graphthe (x, y) coordinatesthatsatisfytheequation x y 5. [Gridsupplied.]

c) Whatshapeisthegraph?

and

Usetheequation2x 3y 12.

TME, vol10, no.3, p. 763

a) Findfivepointsthatsatisfytheequation.

b) Findfivepointsthatdonotsatisfytheequation.

Studentsbegintheinvestigation.Somedonotknowwhatitmeansforapointto

“satisfyanequation.”Mr.Braunhadcreatedtheworksheetbasedonproblemsinan

Algebra1textbook—inthebook,studentsareremindedthat“Ifapoint’scoordinatesmake

anequationtrue,thepoint‘satisfiestheequation’”(EducationDevelopmentCenter,Inc.,

2009a,p.251).Mr.Braunhadleftthatreminderoffofhisworksheet,andsomeofthe

studentsgetstuck.Forexample:

S: …Please!

T: Youjusttoldme,though.[Laughter]Whatarewetryingtodo?What’sitaskingyou

todo?

S: Findthispoint…

T: OK,whatdoes“satisfy”mean?That’sthesameequationweplayedwithatthe

beginningofclass,right?

S: Idon’tknow.

T: Itis,right?Wedidn’tsay“satisfy”and“notsatisfy”;whatwerethewordsthatwe

used?

S: Idon’tknow.Idon’tknow.

T: When[name]gaveus3and1.5,whatdidwesay?

S: Decimal?

T: Well,wesaidtheyweredecimals,wesighedat[name],butbesidethat,whatelse

didwesay?Whatdoesthissideequal?

S: x?y?What?

Matsuura et al.

T: What’s33?

S: 9.

T: What’s21.5?

S: 3.

T: What’s9+3?

S: 12.

T: So,whatdidwesay?“[Name]’ssolution...”

S: Works?

T: Works!“Works”isanotherwordfor“satisfies.”Ifyouwanttosoundsmart,yousay,

“Itsatisfiestheequation.”OK?Allright.

Similarly,anotherstudentasks:

S: Idon’tunderstandwhatit’saskingus![Laughter]

T: Allright,fairenough.Itsays,“Sketchagraphofallthe(x,y)coordinatesthat

satisfy”—work—“inthisequation,”andhere’smyequation.

Ononehand,thisisnotabigdeal.Theteachercantravelfromgrouptogroup,

remindingthemwhat“satisfiestheequation”means,butheusuallysimplysaysthat“it

means‘works.’”However,“works”asadescriptionisnotoperational.Whenstudentsare

solvingproblems,theyrepeatedlyaskaboutthephrase“satisfiestheequation.”Rather

thanoffertheoperationalizabledefinition:“ifapoint’scoordinatesmakeanequationtrue,

thepointsatisfiestheequation,”Mr.Braunreturnstothephrase“works.”

Itisworthnotingthatthefollowingday,Mr.Braunposesawarm‐upquestiontohis

class:“Whatdoesitmeantobeasolution?”Althoughhedoesnotspecificallyaddressthe

TME, vol10, no.3, p. 765

definitionofapointsatisfyinganequation(andtheissuecontinuestopersistforstudents),

hedoesstartworkingonunpackingthatlanguageforstudents.

CommonThemesintheExamples

Severalobservationsandquestionsemergeforusintheseexamples.First,what

strikesusagainandagainisthecomplexityofteachers’usesofMHoM.Thesehabitscannot

bedeployedindependentlyintheclassroomanymorethantheycanbewhenteachers(and

mathematicians)domathematicsforthemselves.Infact,wesawthatthehabitofusing

mathematicallanguagecaneithercomplementorgetinthewayofstudent

experimentationandinquiry,dependingonhowtheteacherusesthehabit.InMr.Hart’s

class,theprecisedefinitionofrecursivefunctionismotivatedbythestructurethathis

studentsdiscoveredthroughexperimentation.And,inturn,Mr.Hartplanstousethis

functionnotationasaninvestigativetooltoexplorefurthertopics(e.g.,theconnection

betweenlinearandexponentialfunctions).Mr.Braunalsobringsexperimentationintohis

classroom.Indeed,hisstudentsconductaninvestigationtoexploretherelationship

betweenanequationanditsgraph.However,somestudentshavedifficultybeginningthe

investigation,becausetheydonotunderstandthelanguagetheyencounterinthetask.

Here,anoperationaldefinitionofthephrase“satisfiestheequation”mayhaveledthemto

understandtheproblemstatementsandgiventheminsightintohowtoproceed.

Throughouttheseexamples,wealsosawhowtheuseofmathematicallanguagecan

supportstudents’understanding.InMs.Graham’sclass,weseehowshepushesher

studentstoclearlystatethemeaningoftheword“solution.”Anditsdefinitionbecomesa

vehiclethatfacilitatestheproblem‐solvingprocess.Incontrast,weseeMr.Braunwhose

studentsencounterthephrase,“satisfytheequation.”Insteadofprovidingausable

Matsuura et al.

definition,heoffersanalternative,namely“works.”WebelieveMr.Brauniswell‐

intentionedhere.Specifically,thereisevidencethatheistryingtomakethelanguageless

intimidatingforstudentsbyofferingamoreinformalphrase.Indeed,hesays,“‘Works’is

anotherwordfor‘satisfies.’Ifyouwanttosoundsmart,yousay,‘Itsatisfiestheequation.’”

Butasdiscussedearlier,“works”isaphrasethatisdifficulttooperationalize.Itleadsto

confusionforhisstudents,becausetheydonotknowhowtouseit.Oneofthemathematical

practicesadvocatedbytheCommonCoreisattendingtoprecision.TheCommonCore

statesthat,“Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.

Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning”

(NGACenter&CCSSO,2010,p.7).That“usability”oflanguageisanimportantpartof

communicatingprecisely,andonethatseemsespeciallyimportantforteachers.

Inparticular,thecarefuluseofmathematicallanguagenotonlyhelpsclarifyideas

forstudents,asitdidinMs.Graham’sclass,butithelpsthemunderstandthemathematics

itselfinadeeperway.WeseethisinMr.Hart’slesson,wheretherecursiveformulafor

f (n) capturesthepropertiesofthefunctionthatstudentsfoundthroughtheir

investigations.Indeed,thisformulaisbothaproductandareflectionoftheirexperiences.

InourworkwithFoMteachers,wehavefoundthatencapsulatingvariousinsightsinto

preciselanguage—aswesawinMr.Hart’sclass—helpsonebetterunderstandtheideas

themselves.

Mr.Hartalsorecognizesthepowerofpreciselanguagetodrivefurther

investigations.Laterintheschoolyear,thesestudentswillusefunctionnotationtostudy

transformationsoffunctions(e.g.,stretches,shrinks,andtranslations).Headds,“Ithink

TME, vol10, no.3, p. 767

thatwillbeaplacewherestudentswillreallyappreciatethefunctionnotationin

representingthosetransformationsmoreeasily.”

Mr.Hartconcludesthepost‐interviewbydescribinghowtoday’slessonispartofa

biggerunitandhowitsetsthefoundationforlaterlessons.Heplanstousetheserecursive

rulesasavehicleforbetterunderstandingtheirclosed‐formcounterparts.Inafuture

lesson,studentswillinvestigatetheconnectionbetweenlinearandexponentialfunctions.

“Iwantmystudentstoseethatrecursively,exponentialfunctionsarevery,verysimilarin

theirrepresentationtolinearfunctions.Ithinkthatwillprovideanicefoundationfor

studyingexponents,”hesays.Here,Mr.Hartisusingthelanguageofrecursivefunctionsto

shedlightontheconnectionsbetweentheircorrespondingclosed‐formrepresentations.

Ourowngoalsinwatchingthesevideoshavebeentobetterunderstandteachers’

usesofMHoM,andtolearnabouthowwemightmeasurethatuse.Partofourdesireto

measuretheusestemsfromourdesiretounderstand(eventually)thelinkbetween

teachers’usesofMHoMandlearningoutcomesforstudents,particularlyifwecanmeasure

students’usesofMHoMorstudents’facilitywithCommonCore’sMathematicalPractices,

whichincludesignificantoverlapwithMHoM.Withinthecontextoftheexamplesinthis

paper,mightteachers’useoflanguagehaveanimpactonstudentachievement?Evento

begintoanswersuchaquestion,wemusthavesomeobjectivewayofdecidingwhetheror

notagiventeacherisusingclear,usable,andpreciselanguage.This,too,iscomplex.

Establishingwhatcountsas“clear,usable,andprecise”languagedependsverymuchonthe

classroomcontext.Mr.Braunusestheword“works”soconsistentlyinhisclassroom

discussion,thatifitdidnotcauseconfusion,surelywewouldwantto“rate”thatastotally

acceptablelanguage,takenassharedbythewholeclassroom.

Matsuura et al.

ImpactandNextSteps

Webeganourresearchworkpartlybecausewewantedtoassesstheeffectsofour

ownMSPprofessionaldevelopmentprogramsusingtoolsthatwereconsistentwiththe

goalsofourMSP,andpartlybecausewewantedtounderstandtheMHoMofsecondary

teachersbetter.Wedidnotfindinstrumentsthatmeasuredteachers’MHoM—eitherwhen

doingmathematicsforthemselvesorteachingmathematicsintheirclassrooms—in

existenceinthefield,sowebegantocreateourown.Althoughweexpectedtolearnfrom

thedatagatheredusingourinstruments,wedidnotanticipatetheimmediateimplications

thatourresearchwouldhaveontheprofessionaldevelopmentprogramsinourMSP.For

example,basedonwhatwehadlearnedfromourresearch,wepilotedtheMathematical

HabitsofMindShadowSeminarinthesummerof2011,gearedtowardteacherparticipants

returningtoPROMYSforTeachers(oursummerimmersionprogram)forasecond

summer.Throughdiscussions,readings,curriculumanalyses,andlessondesigns,thegoal

ofthisseminarwastoexplore(a)thewaysinwhichsecondaryteachersknowanduse

MHoMintheirprofession,and(b)theeffectsthatalearningenvironmentthatstresses

MHoMmighthaveonsecondarystudents.Wewillcontinuetoofferandrefinethiscourse

aspartofoursummerimmersionprogramforteachers.

Wealsodidnotanticipatethepotentialforimpactonthefield.Whiledevelopment

andvalidationoftrulyreliabletoolsisbeyondthescopeofthecurrentFoM‐IIstudy,we

havebeenlayingthegroundworkforourMHoMinstruments—theP&Passessmentandthe

observationprotocol—overthelastfewyears.Thisexploratoryphaseofinstrument

developmentalsocoincidedwiththeemergenceoftheCommonCoreStateStandardsand

itsadoptionby45states(NGACenter&CCSSO,2010).OurMHoMconstructisclosely

TME, vol10, no.3, p. 769

alignedwiththeCommonCore,especiallyitsStandardsforMathematicalPractice,and

thereisconsiderableoverlapinthetwo.Forexample,bothplaceimportanceonseeking

andusingmathematicalstructure,usesofprecision,andtheactofabstractingregularity

fromrepeatedactions.Aswepresentedourpreliminaryfindingsatnationalconferences

(Matsuura,Cuoco,Stevens,&Sword,2011;Matsuura,Sword,Cuoco,Stevens,&Faux,

2011),wereceivedseveralrequeststouseourinstruments,eventhoughtheywereinthe

pilotphaseofdevelopment.Onedistrictleaderwantedtodiagnosethepreparednessofher

teacherstoteachfromacurriculumbasedontheCommonCore.Otherswantedtousethe

instrumentsaspre‐andpost‐measuresforevaluatingprofessionaldevelopmentprograms

alignedtotheCommonCore.Wehavebecomeabundantlyawareofthenationalneedfor

validandreliableinstrumentstomeasureteachers’knowledgeanduseof

MHoM/MathematicalPractices,aswellasguidelinesforacceptableuseofsuch

instruments.Thus,inthenextphaseofourresearch,weplantosubjectourpilot

instrumentstorigorousscientifictesting.Theexamplesinthispaperareexemplarsof

thosethatprovideboththecontentbasisfortheP&Passessmentandthebehavioral

indicatorsfortheobservationprotocol.

Matsuura et al.

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