The Mathematics Enthusiast
Volume 10
Number 3 Number 3
Article 9
7-2013
Mathematical Habits of Mind for Teaching: Using
Language in Algebra Classrooms
Ryota Matsuura
Sarah Sword
Mary Beth Piecham
Glenn Stevens
Al Cuoco
Follow this and additional works at: http://scholarworks.umt.edu/tme
Part of the Mathematics Commons
Recommended Citation
Matsuura, Ryota; Sword, Sarah; Piecham, Mary Beth; Stevens, Glenn; and Cuoco, Al (2013) "Mathematical Habits of Mind for
Teaching: Using Language in Algebra Classrooms," The Mathematics Enthusiast: Vol. 10: No. 3, Article 9.
Available at: http://scholarworks.umt.edu/tme/vol10/iss3/9
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TME, vol10, no.3, p. 735
MathematicalHabitsofMindforTeaching:
UsingLanguageinAlgebraClassrooms1
RyotaMatsuura2
St.OlafCollege
SarahSword
EducationDevelopmentCenter,Inc.
MaryBethPiecham
EducationDevelopmentCenter,Inc.
GlennStevens
BostonUniversity
AlCuoco
EducationDevelopmentCenter,Inc.
ABSTRACT:Thenotionofmathematicalknowledgeforteachinghasbeenstudiedbymany
researchers,especiallyattheelementarygrades.Ourunderstandingsofthisnotionparallel
muchofwhatwehavereadintheliterature,butarebasedonourparticularexperiences
overthepast20years,asmathematiciansengagedindoingmathematicswithsecondary
teachers.AspartoftheworkofFocusonMathematics,PhaseIIMSP,wearedeveloping,in
collaborationwithothersinthefield,aresearchprogramwiththeultimategoalof
understandingtheconnectionsbetweensecondaryteachers’mathematicalknowledgefor
teachingandsecondarystudents’mathematicalunderstandingandachievement.Wearein
theearlystagesofafocusedresearchstudyinvestigatingtheresearchquestion:Whatare
themathematicalhabitsofmindthathighschoolteachersuseintheirprofessionallivesand
howcanwemeasurethem?Themainfocusofthispaperisthediscussionofthehabitof
usingmathematicallanguage,andparticularlyhowthishabitplaysoutinaclassroom
setting.
Keywords:Mathematicalhabitsofmind,mathematicallanguage,algebra
1ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunder
GrantNo.0928735.Anyopinions,findings,andconclusionsorrecommendations
expressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflecttheviews
oftheNationalScienceFoundation.
[email protected]
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
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
x 2  6x 10 as (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
3
2
greaterthanthepreviousoutput.Makeatablethatmatchesthedescription.Canyou
makemorethanonetable?
Notethattheruleisincomplete,becauseitismissingthebasecase.Studentsexperiment
withthisrule,creatinginput/outputtablesandtryingtoderiveclosed‐formequations.
Matsuura et al.
Becauseoftheirdifferentchoicesofbasecases,theycomeupwithdifferentfunctions
definedbyexpressionsoftheform f (x) 
3
x  b .Studentsconcludethatthegraphsof
2
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 (n 1)  5. Tomakesenseofthisrecursiverule,Mr.Hartpointsoutthattheequation
f (n)  f (n 1)  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)  5n  3.Thereare
nowtwowaystodescribethefunctionathand,namelythe(stillincomplete)recursiverule
f (n)  f (n 1)  5 andtheclosedformrule f (n)  5n  3.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 (n 1)  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
 3
if n  0,
f (n)  
 f (n 1)  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 isasolutionto 6r  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’restatingthat 6r  2 willbeequalto 12  r. Andthey’reasking,“Is r  2 a
solution?”Soyougottotestitout,justasIaskedyoutotestoutthatonethatwe
justdid.So 6r  2  12  r. Substitutein r  2.So6times 2 plus2—doesthathave
thesamevalueas12plus 2 ?Andwehavetotest.Allright?We’reaskingourselves
thequestionof,doesthisequalthat?[Pointstoeachsideoftheequation.]OK?
ThenMs.Grahamleadstheclassthroughtheprocessofsubstituting r  2 intothe
equationandconcludingthatitisnotasolution,since r  2 yieldsunequalvaluesof 10 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
studentwhowrites 82  8  90  3  30  5  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.
Braunwritestheequation 3x  2 y  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  2 y  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’s 31 ?
Students(Ss):3.
T:
What’s 2  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: 2 1. 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
equation 3x  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
2  113 1
2
b) Graphthe (x, y) coordinatesthatsatisfytheequation x  y  5. [Gridsupplied.]
c) Whatshapeisthegraph?
and
Usetheequation 2x  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’s 3 3 ?
S:
9.
T:
What’s 2 1.5 ?
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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