TRUObservationGuideforMathAlpha
FifthCompleteDraft
November25,2016
TheTeachingforRobustUnderstanding
(TRU)ObservationGuideforMathematics
AToolforTeachers,Coaches,Administrators,
andProfessionalLearningCommunities
ThisTeachingforRobustUnderstandingObservationGuideforMathematicsisdesignedtosupport
teachers,coaches,administrators,andprofessionallearningcommunitiesinplanning,conducting,and
reflectingonobservationsinmathematicsclassrooms.ItisbasedontheTeachingforRobust
Understanding(TRU)Framework(see,e.g.,Schoenfeld,2013,2014;Schoenfeld&theTeachingfor
RobustUnderstandingProject,2016).Thekeyideabehindtheframeworkisthatthefivedimensionsof
classroomactivitydescribedinfigure1arecentralindeterminingthedegreetowhichstudentswill
emergefromtheclassroombeingproficientmathematicalthinkersandproblemsolvers.
The Five Dimensions of Powerful Mathematics Classrooms
TheMathematics
Theextenttowhich
classroom activity
structuresprovide
opportunitiesfor
studentstobecome
knowledgeable,
flexible,and
resourceful
mathematical
thinkers.Discussions
arefocusedand
coherent,providing
opportunitiesto
learnmathematical
ideas, techniques,
andperspectives,
makeconnections,
anddevelop
productive
mathematicalhabits
ofmind.
Cognitive
Demand
EquitableAccess
toMathematics
Theextenttowhich
studentshave
opportunitiesto
grapplewithand
makesenseof
important
mathematicalideas
andtheiruse.
Studentslearnbest
whentheyare
challengedinways
thatprovideroom
andsupportfor
growth,withtask
difficulty ranging
frommoderateto
demanding.The
levelofchallenge
should beconducive
towhathasbeen
called“productive
struggle.”
Theextenttowhich
classroom activity
structuresinviteand
supporttheactive
engagementofall
ofthestudentsin
theclassroomwith
thecore
mathematical
contentbeing
addressedbythe
class.Classrooms in
whichasmall
numberofstudents
getmostofthe“air
time”arenot
equitable,no
matterhowrichthe
content:allstudents
needtobeinvolved
inmeaningful ways.
Agency,
Ownership,and
Identity
Theextenttowhich
studentsareprovided
opportunitiesto“walk
thewalkandtalkthe
talk”– tocontributeto
conversationsabout
mathematicalideas,
tobuildonothers’
ideasandhaveothers
buildontheirs– in
waysthatcontribute
totheirdevelopment
ofagency(the
willingness to
engage),their
ownershipoverthe
content,andthe
developmentof
positiveidentitiesas
thinkersandlearners.
Formative
Assessment
Theextenttowhich
classroom activities
elicitstudent
thinking and
subsequent
interactionsrespond
tothoseideas,
building on
productive
beginnings and
addressing emerging
misunderstandings.
Powerfulinstruction
“meetsstudents
wheretheyare”and
givesthem
opportunitiesto
deepentheir
understandings.
Figure1.Thefivedimensionsofpowerfulmathematicsclassrooms
TRUObservationGuideforMathAlpha
FifthCompleteDraft
November25,2016
ThisObservationGuideforMathematicsispartofasupportsystemforcollaborativepartnerships
betweenteachersandobservers1.Optimally,eachobservationisoneofaseriesofclassroomvisits
contributingtoteachergrowth.Thereshouldbeampletimetoplanobservations,toobservelessons,
andtodiscusstheobservations,overthecourseofatermorayear.
Priortoanobservation,itisusefulfortheteacherandobservertodiscussthelessonplananddecide
onthemainpointsoffocusfortheobservation.Theobservationmightbegeneral;itispossiblefora
practicedobservertotakenotesonalldimensions.Alternatively,theteacherandobservermight
agreetofocusononeortwoareastheteacherwantstoaddressindetail.Eitherway,reflecting
beforehandongoalsforthelessonandfortheobservationisagoodwaytomakethemostofthe
observation.AusefultoolforplanninganddebriefingistheTeachingforRobustUnderstanding
ConversationGuide(Baldinger,Louie,andtheAlgebraTeachingStudyandMathematicsAssessment
Project,2016).TheConversationGuidelaysoutaseriesofquestionsforeachdimensionthatteacher
andobservercanuseinplanningthelesson,andinreflectingonitaswell.
Whenplanningobservations,itisusefultothinkofwhattheclassroomexperiencelooksandfeelslike
fromtheperspectiveofastudent–students,afterall,aretheonesexperiencingtheinstruction!The
questionsinFigure2provideanorientationthathelpsinseeingthelessonfromthestudent
perspective.
Observethelessonthroughastudent’seyes
The
Mathematics
•
•
What’sthebigideainthislesson?
HowdoesitconnecttowhatIalreadyknow?
Cognitive
Demand
•
•
•
HowlongamIgiventothink, andtomakesenseofthings?
Whathappens whenIgetstuck?
AmIinvitedtoexplainthings, orjustgiveanswers?
EquitableAccess •
toMathematics •
DoIgettoparticipateinmeaningful mathematicallearning?
CanIhideorbeignored?
Agency,
Ownership,and
Identity
•
•
DoIgettoexplain,topresentmyideas?Aretheybuilton?
AmIrecognizedasbeingcapableandabletocontribute in
meaningful ways?
•
•
Doclassroomdiscussions includemythinking?
Doesinstruction respond tomythinking andhelpmethinkmore
deeply?
Formative
Assessment
Figure2.Observingamathematicslessonfromthestudentperspective
Theformoftheobservationguideanditsusearestraightforward.Eachobservationsheetfocuseson
onedimensionoftheframework,andisonepagelong.Eachobservationsheetlookslikethis:
1AdditionaltoolscanbefoundattheMathematicsAssessmentProjectandAlgebraTeachingStudywebsites(at
<http://map.mathshell.org/>and<http://ats.berkeley.edu/tools.html>respectively.Amoreextensivesetoftoolswill
behousedat<http://TRU.berkeley.edu/>.
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TRUObservationGuideforMathAlpha
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November25,2016
Thenameofthedimension
Adescriptionofit
Alistofsomestudentbehaviors
tolookfor…
Alistofsomeactionsbyteachersthat
couldsupportsuchstudentbehaviors…
Spaceforfocalpointsforthisparticularobservation,ifdesired
Spacefornote-taking
Abriefdescriptionofkeygoalsrelatedtothisdimension
Figure3.Thestructureofanobservationsheet2
Thetopandbottompartsofeachobservationsheetprovideconcisedescriptionsoftherelevant
dimensionandgoalsforit.Beneaththedescriptionofthedimensionaresomeexamplesof“lookfors”-
actionsonthepartofstudentsandteacherthatareoftenindicatorsthatthingsaregoingwell.They
arethingstoaimforingeneral,andovertime–theyareNOTalistofthingstobecheckedoffinany
particularlesson.Weimagineteacherandobserverdiscussingthesepriortoalessonanddeciding
which,ifany,mightbethingstofocusonintheupcomingobservation.Thelistisnotmeanttobe
comprehensive;teacherandobservermaydecideonanotherfocusandwriteitinthespaceprovided.
Thecenteroftheobservationsheetprovidesspaceforwritingdownobservations.
Therearemanypossiblegoalsforclassroomobservations.Teacherandobservermaydecidetofocus
ononeortwoissues,ortheymayagreethattheobserverwillprovideasystematicrun-throughofall
thedimensions.Itisuseful,andtypicallymostcomfortable,forthepost-lessonconversationtostart
withthemainfocalpoints–onagreed-uponfoci,alongwitheventsinthelessonthatwereparticularly
interestingandsalient.But,evenifparticularfocihavebeenchosenfortheobservation,itisvaluable
torunbrieflythroughallofthedimensions–theTeachingforRobustUnderstandingFrameworkis
intendedasawayofseeingandtalkingaboutinstruction,anditprovidesalanguageforthinkingabout
it.Afterafewobservations,itbecomesanaturalwayforteachers,coaches,administrators,and
professionallearningcommunitiestotalkaboutteaching.
2WeareindebtedtotheSanFranciscoUnifiedSchoolDistrictforitsdevelopmentof“ObservationalToolforLEAD,”
whichprovidedtheinspirationforthedesignformatofthisGuide.
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TRUObservationGuideforMathAlpha
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HereareafewimportantpointsabouttheTRUframeworkanditsuse.TheTeachingforRobust
Understandingframeworkhighlightsfivedimensionsofclassroommathematicalactivity.Theyare
describedseparatelybecauseeachcanbetheobjectofcoherentfocus,aspartofongoingprofessional
development.Intheclassroom,however,theyarealldeeplyinterrelated.Inparticular:
•
Issuesrelatedtomathematicalcontentpermeateallfivedimensions–andallclassroomactivities.
Dimension1focusesonthequalityofthecontentperse.Ifthemathematicsisn’trich,thereis
nothingmeaningfulforthestudentstolearn.Butwhatmattersinadditionisthesetof
opportunitiesthateachstudenthastoengagewithandmakesenseofthemathematics.Thus
Dimension2,cognitivedemand,shouldbeconceivedofasopportunitiesforproductivestruggle
withcoremathematicalconceptsandpractices.Issuesofaccess(Dimension3)andopportunitiesto
developagency,ownership,andidentity(Dimension4)concernthewaysinwhicheverystudent
relatestothebigideasofthediscipline.And,ofcourse,thepurposeofformativeassessment
(Dimension5)istofacilitateaccesstothemathematics.
•
Similarly,issuesofequityalsopermeateallfivedimensionsandshouldbecentralatalltimes.The
keypointisthateverystudentshouldbesupportedindevelopingapositivemathematicalidentity
(Dimension4)throughmeaningfulaccessandparticipation(Dimension3)torichmathematical
content(Dimension1).Thatparticipationcanonlybemeaningfulforastudentifthelevelof
cognitivedemandisrightforsensemaking(Dimension2),somethingachievedbyformative
assessment(Dimension5).Atthesametime,Dimension3,EquitableAccesstoMathematics,does
needspecific,focusedattention:teachinginwaysthatprovidemeaningfulopportunitiesforall
studentstoengagewithcentralmathematicalcontent,andtobuildproductivemathematical
identities,isextremelychallenging.
•
Asnotedabove,eachobservationsheethasroomforspecificobservationalgoalsestablishedby
theteacherandobserver.OneplacewherethiswillbeessentialisforDimension1,thecontent.
The“lookfors”onthefirstobservationsheetaregeneralacrossmathematics,andtheyshouldbe
supplementedbyspecificsforthelessonbeingobserved.
ThisObservationGuideforMathematicsisaproductofTheTeachingforRobustUnderstandingproject
attheUniversityofCalifornia,Berkeley,withsupportfromtheNationalScienceFoundation(Grant
1503454,“TRUmathandLessonStudy:SupportingFundamentalandSustainableImprovementinHigh
SchoolMathematicsTeaching”)andtheBillandMelindaGatesFoundation(GrantOPP1115160,
“NetworkedImprovementCommunitytoSupportCommonCoreStateStandards”).Itbuildsonwork
bytheAlgebraTeachingStudy(U.C.BerkeleyandMichiganStateUniversity:see
<http://ats.berkeley.edu/>)andtheMathematicsAssessmentProject(U.C.BerkeleyandtheUniversity
ofNottingham:see<http://map.mathshell.org/>).
Suggestedcitation:
Schoenfeld,A.H.,andtheTeachingforRobustUnderstandingProject.(2016).TheTeachingforRobust
Understanding(TRU)observationguideformathematics:Atoolforteachers,coaches,administrators,
andprofessionallearningcommunities.Berkeley,CA:GraduateSchoolofEducation,Universityof
California,Berkeley.Retrievedfrom:<http://TRU.berkeley.edu>or<http://map.mathshell.org/>or
<http://ats.berkeley.edu/>.
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TRUObservationGuideforMathAlpha
FifthCompleteDraft
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Thismaterialmaybereproducedanddistributed,withoutmodification,fornon-commercialpurposes.All
otherrightsreserved.
References
Baldinger,E.Louie,N.,andtheAlgebraTeachingStudyandMathematicsAssessmentProject.
(2014).TRUMathconversationguide:Atoolforteacherlearningandgrowth.Berkeley,CA&E.
Lansing,MI:GraduateSchoolofEducation,UniversityofCalifornia,Berkeley&Collegeof
Education,MichiganStateUniversity.Retrievedfrom:http://ats.berkeley.edu/tools.htmland/or
http://map.mathshell.org/materials/pd.php.
Schoenfeld,A.H.(2013).Classroomobservationsintheoryandpractice.ZDM:TheInternational
JournalonMathematicsEducation,45(4),607-621,DOI10.1007/s11858-012-0483-1.
Schoenfeld,A.H.(2014).Whatmakesforpowerfulclassrooms,andhowcanwesupportteachers
increatingthem?Astoryofresearchandpractice,productivelyintertwined.Educational
Researcher,43(8),404-412.
Schoenfeld,A.H.,&theTeachingforRobustUnderstandingProject.(2016).AnIntroductiontothe
TeachingforRobustUnderstanding(TRU)Framework.Berkeley,CA:GraduateSchoolofEducation.
Retrievedfromhttp://map.mathshell.org/trumath.phporhttp://ats.berkeley.edu/tools.htmlor
http://tru.berkeley.edu.
Theobservationsheetsfollow.
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TRUObservationGuideforMathAlpha
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November25,2016
THEMATHEMATICS
Theextenttowhichcentralmathematicalcontentandpractices,asrepresentedbyStateorhe
CommonCoreStateStandards,arepresentandembodiedininstruction.Everystudentshouldhave
opportunitiestograpplemeaningfullywithkeyideasand,indoingso,tobecomeaknowledgeable,
flexible,andresourcefulmathematicalthinkerandproblemsolver.Teachersshouldhave
opportunitiestoconsideranddiscusshoweachlesson’sactivitiesconnecttotheconcepts,
practices,andhabitsofmindtheywantstudentstodevelopovertime.
EachStudent…
Teachers…
•
•
•
•
•
Engageswithgradelevelmathematics
inwaysthathighlightimportant
concepts,procedures,problemsolving
strategies,andapplications
Hasopportunitiestodevelop
productivemathematicalhabitsofmind
Hasopportunitiesformathematical
reasoning,orallyandinwriting,using
appropriatemathematicallanguage
Explainstheirreasoningprocessesas
wellastheiranswers.
•
•
•
•
Highlightimportantideasandprovide
opportunitiesforstudentstoengagewiththem
Usematerialsorassignmentsthatcenteronkey
ideas,connections,andapplications
Explicitlyconnectthelesson’sbigideastowhat
hascomebeforeandwillbedoneinthefuture
Supportthepurposefuluseofacademic
languageandofrepresentations(e.g.,graphs,
tables,symbols)centraltomathematics
Supportstudentsinseeingmathematicsasbeing
coherent,connected,andcomprehensible
•Otherfocalpointsforobservation:
Whatarethebigideasinthislesson?Howdotheyconnecttowhathascomebefore,and/or
establishabaseforfuturework?Howdothewaysstudentsengagewiththematerialsupportthe
developmentofconceptualunderstandingandthedevelopmentofmathematicalhabitsofmind?
Goal:Allstudentsworkoncoremathematicalissuesinwaysthatenablethemtodevelop
conceptualunderstandings,developreasoningandproblemsolvingskills,andusemathematical
concepts,tools,methodsandrepresentationsinrelevantcontexts.
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COGNITIVEDEMAND
Theextenttowhichstudentshaveopportunitiestograpplewithandmakesenseofimportant
mathematicalideasandtheiruse.Studentslearnbestwhentheyarechallengedinwaysthat
provideroomandsupportforgrowth,withtaskdifficultyrangingfrommoderatetodemanding.
Thelevelofchallengeshouldbeconducivetowhathasbeencalled“productivestruggle.”
Eachstudent…
Teachers…
•
•
•
•
•
•
•
Engagesindividuallyandcollaboratively
withchallengingideas
Activelyseekstoexplorethelimitsof
theircurrentunderstandings
Iscomfortablesharingpartialor
incorrectworkaspartofalarger
conversation
Reasonsandtestsideasinwaysthat
connecttoandbuildonwhattheyknow
Explainswhattheyhavedonesofar
beforeaskingforhelp
Continuestowrestlewithanideaafter
theteacherleaves
•
•
•
•
Positionstudentsassensemakerswhocan
makesenseofkeyconceptualideas.
Useoradaptmaterialsandactivitiestooffer
challengesthatstudentscanuse,individuallyor
collectively,todeepenunderstandings
Buildandmaintainclassroomnormsthat
supporteverystudent’sengagementwiththose
materialsandactivities
Monitorstudentchallenge,adjustingtasks,
activities,anddiscussionssothatallstudents
areengagedinproductivestruggle
Supportsstudentswithoutremovingthe
challengefromtheworktheyareengagedin
•Otherfocalpointsforobservation:
Whatopportunitiesdostudentshavetomakesenseofmathematicalcontentandpractices?How
aretheysupportedinsensemakingsothattheyarenotlost–yetrealchallengehasbeen
maintained,sothattheyhaveopportunitiestograpplewithimportantideas?
Goal:Allstudentshaveopportunitiestomaketheirownsenseofimportantmathematicalideas,
developingdeeperunderstandings,connections,andapplicationsbybuildingonwhattheyknow.
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EQUITABLEACCESSTOMATHEMATICS
Theextenttowhichclassroomactivitiesinviteandsupportthemeaningfulengagementwithcore
mathematicalcontentandpracticesbyallstudents.Findingwaystosupportthediverserangeof
learnersinengagingmeaningfullyisthekeytoanequitableclassroom.
Eachstudent…
Teachers…
•
•
•
•
•
•
•
Contributestocollectivesense
makinginanyofanumberof
differentways(e.g.,proposing
ideas,askingquestions,creating
diagrams…)
Activelylistenstootherstudents
andbuildsontheirideas
Supportsotherstudents’
developingunderstandings
Explains,interprets,appliesand
reflectsonimportant
mathematicalideas
Participatesmeaningfullyinthe
mathematicalworkoftheclass
•
•
•
•
•
Createsafeenvironments
Usetasksandactivitiesthatprovidemultipleentrypoints
andsupportmultipleapproachestothemathematics
Provideopportunitiesforstudentstoseethemselves,and
theirpersonalandcommunityinterests,reflectedinthe
curriculum
Validatedifferentwaysofmakingcontributions
Buildandmaintainnormsthatsupporteverystudent’s
participationingroupworkandwholeclassactivities
Supportparticularneeds,suchasthoseoflanguage
learners,forfullparticipation
Expectandsupportmeaningfulmathematicalengagement
fromallstudents,helpingthemcontributeandbuildon
contributionsfromothers
•Otherfocalpointsforobservation:
Inwhatwaysdoeseachstudentengageintheworkoftheclass?Howcanmoreopportunitiesfor
everystudenttoparticipateinmeaningfulwaysbecreated?
Goal:Allstudentsaresupportedinaccesstocentralmathematicalcontent,andparticipateactively
intheworkoftheclass.Diversestrengthsandneedsarebuiltonthroughtheuseofvarious
strategies,resources,andtechnologiesthatenableallstudentstoparticipatemeaningfully.
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AGENCY,OWNERSHIP,ANDIDENTITY
Theextenttowhicheverystudenthasopportunitiestoexplore,conjecture,reason,explain,andbuild
onemergingideas,contributingtothedevelopmentofagency(thewillingnesstoengage
academically)andownershipoverthecontent,resultinginpositivemathematicalidentities.
Eachstudent…
Teachers…
•
•
•
•
•
Takesownershipofthelearningprocessin
planning,monitoring,andreflectingon
individualand/orcollectivework
Asksquestionsandmakessuggestionsthat
supportanalyzing,evaluating,applying
andsynthesizingmathematicalideas
Buildsonthecontributionsofothersand
helpothersseeormakeconnections
Holdsclassmatesandthemselves
accountableforjustifyingtheirpositions,
throughtheuseofevidenceand/or
elaboratingontheirreasoning
•
•
•
•
Providetimeforstudentstodevelopand
expressmathematicalideasandreasoning
Worktomakesureallstudentshave
opportunitiestohavetheirvoicesheard
Encouragestudent-to-studentdiscussionsand
promoteproductiveexchanges
Assigntasksandposequestionsthatcallfor
mathematicaljustification,andforstudentsto
explaintheirreasoning
Employarangeoftechniquesthatattribute
ideastostudents,tobuildstudentownership
andidentity
•Otherfocalpointsforobservation:
Whatopportunitiesdoallstudentshavetoseethemselvesandothersasproficientmathematical
thinkers,tograpplewithchallengesandconstructnewunderstandings,tobuildonothers’ideas,
anddemonstratetheirunderstandings?Howcanmoreoftheseopportunitiesbecreated?
Goal:Allstudentsbuildproductivemathematicalidentitiesthroughtakingadvantageof
opportunitiestoengagemeaningfullywiththedisciplineandshareandrefinetheirdevelopingideas.
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FORMATIVEASSESSMENT
Theextenttowhichclassroomactivitieselicitallstudents’thinkingandsubsequentinteractions
respondtothatthinking,bybuildingonproductivebeginningsorbyaddressingemerging
misunderstandings.Highqualityinstruction“meetsstudentswheretheyare”andgivesthem
opportunitiestodevelopdeeperunderstandings,bothasshapedbytheteacherandinstudent-tostudentinteractions.
Eachstudent…
Teachers…
•
•
Explainstheirthinking,evenif
somewhatpreliminary
Seeserrorsasopportunitiesfor
newlearning
Consistentlyreflectsontheir
workandtheworkofpeers
Seesfellowstudentsas
resourcesfortheirownlearning
Providesspecificandaccurate
feedbacktofellowstudents
Makesuseoffeedbackin
revisingtheirwork
•
•
•
•
•
•
•
•
•
Createsafeclimatesinwhichstudentsfeelfreetoexpress
theirideasandunderstandings
Usematerialsthatelicitmultiplestrategies,andhave
studentsexplaintheirreasoning,inordertogain
informationaboutstudent’emergingunderstandings
Flexiblyadjustcontentandprocess,providingstudents
opportunitiesforre-engagementandrevision
Providetimelyandspecificfeedbacktostudents,aspart
ofclassroomroutinesthatpromptstudentstomake
activeuseoffeedbacktofurthertheirlearning
Createopportunitiesforstudents’individualand
collaborativereflectionontheirknowledgeandlearning
•Otherfocalpointsforobservation:
Whatopportunitiesexistforallstudentstodemonstratetheirunderstandings?Whatopportunities
existtobuildonthethinkingthatisrevealed?Howdoteachersand/orotherstudentstakeupthese
opportunities?Wherecanmorebecreated?
Goal:Everystudent’slearningiscontinuallyenhancedbytheongoingstrategicandflexibleuseof
techniquesandactivitiesthatallowstudentstorevealtheiremergingunderstandings,andthat
provideopportunitiesbothtorethinkmisunderstandingstobuildonproductiveideas.
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