Teaching geometry to students with learning disabilities and emotional disturbance using graduated and peer-mediated in...

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Teaching geometry to students with learning disabilities and emotional disturbance using graduated and peer-mediated instruction
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Dobbins, A., Gagnon, J. C., & Ulrich, T. (2013). Teaching geometry to students with learning disabilities and emotional disturbance using graduated and peer-mediated instruction. Preventing School Failure: Alternative Education for Children and Youth, 58, 17-25.
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Geometry is a course that is increasingly required for students to graduate from high school. However, geometric concepts can pose a great challenge to high school youth with math difficulties. Although research on teaching geometry to students with mathematics difficulties is limited, teachers are challenged daily with providing support for youth who do not make adequate progress through high-quality core mathematics instruction. This article provides practical and promising practices for providing additional small group support (i.e., Tier II interventions) to youth that promote understanding of finding the area of a trapezoid. Specifically, the authors focus on the use of a graduated instructional sequence and peer-mediated instruction within an explicit instruction model. Recommendations and sample lessons are provided.
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This article was downloaded by: [joseph Gagnon]On: 13 December 2013, At: 12:00Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK Preventing School Failure: Alternative Education forChildren and YouthPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/vpsf20 Teaching Geometry to Students With Math DifficultiesUsing Graduated and Peer-Mediated Instruction in aResponse-to-Intervention ModelAngela Dobbins a Joseph Calvin Gagnon b & Tracy Ulrich ba University of Virginia Charlottesville VA USAb University of Florida Gainesville FL USAPublished online: 11 Dec 2013. To cite this article: Angela Dobbins Joseph Calvin Gagnon & Tracy Ulrich (2014) Teaching Geometry to Students With MathDifficulties Using Graduated and Peer-Mediated Instruction in a Response-to-Intervention Model, Preventing School Failure:Alternative Education for Children and Youth, 58:1, 17-25, DOI: 10.1080/1045988X.2012.743454 To link to this article: http://dx.doi.org/10.1080/1045988X.2012.743454 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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PreventingSchoolFailure ,58(1),17–25,2014 CopyrightCTaylor&FrancisGroup,LLC ISSN:1045-988Xprint/1940-4387online DOI:10.1080/1045988X.2012.743454 TeachingGeometrytoStudentsWithMathDifculties UsingGraduatedandPeer-MediatedInstructionina Response-to-InterventionModelANGELADOBBINS1,JOSEPHCALVINGAGNON2,andTRACYULRICH2 1UniversityofVirginia,Charlottesville,VA,USA2UniversityofFlorida,Gainesville,FL,USA Geometryisacoursethatisincreasinglyrequiredforstudentstograduatefromhighschool.However,geometricconceptscanpose agreatchallengetohighschoolyouthwithmathdifculties.Althoughresearchonteachinggeometrytostudentswithmathematics difcultiesislimited,teachersarechallengeddailywithprovidingsupportforyouthwhodonotmakeadequateprogressthrough high-qualitycoremathematicsinstruction.Thisarticleprovidespracticalandpromisingpracticesforprovidingadditionalsmall groupsupport(i.e.,TierIIinterventions)toyouththatpromoteunderstandingofndingtheareaofatrapezoid.Specically,the authorsfocusontheuseofagraduatedinstructionalsequenceandpeer-mediatedinstructionwithinanexplicitinstructionmodel. Recommendationsandsamplelessonsareprovided. Keywords: geometryinstruction,graduatedinstruction,mathematicsdifculties,peer-mediatedinstruction,responsetointervention, TierIIinstructionIntroductionIsn’tageometrycourseoneofthoseextracoursesthatisnot reallynecessaryforhighschoolstudents?Moreandmore, theanswerisaresounding,“No!”Changesinmathematics expectationshavebeenfueledbythepoorperformanceof Americanhighschoolstudents.ResultsfromtheProgram forInternationalAssessment(OrganizationforEconomicCooperationandDevelopment,2010)indicatethat17countries outperformedAmericanstudents.Alsodisconcertingisthat only26%oftwelfthgradestudentsperformedatorabovethe procientlevelontheNationalAssessmentofEducational Progress(NationalCenterforEducationStatistics,2010). Inadditiontopoorperformanceonnationalandinternationalassessments,theNationalCouncilofTeachersof Mathematics(NCTM)standardshaveinuencedmathematicseducationacrosstheUnitedStates.Allbutonestatehas revisedtheirmathematicscurriculumandcontentstandards asaresultoftheNCTMstandards(Woodward,2004).Forexample,NCTM(2011)supports4yearsofmathematicsinhigh school.By2012,atleast23stateswillrequire3yearsofhigh schoolmathematicsand11stateswillrequire4years(Reys, Dingman,Nevels,&Teuscher,2007).Geometryisamong thecoursesthatareincreasinglyrequiredforahighschool diploma.Themostrecentevidencesuggeststhatatleast20 stateshave“course-basedlearningexpectations”(although AddresscorrespondencetoAngelaDobbins,P.O.Box117050, Gainesville,FL32611,USA.E-mail:dobbinad1@u.edunotspeciccourserequirements)thatincludegeometryfor highschoolgraduation(Reysetal.,2007,p.2). Geometryisoneofthecurricularareascoveredbythe NCTMcontentstandardsandisdenedasa“branchofmathematicsthatdealswiththemeasurement,properties,andrelationshipsofpoints,lines,angles,surfaces,andsolids;broadly: thestudyofpropertiesofgivenelementsthatremaininvariant underspeciedtransformations”(NCTM,2000).AsidentiedintheNCTMstandards(2000),thegoalsofgeometryare forstudentstobeabletoaccomplishthefollowing: € analyzecharacteristicsandpropertiesoftwoandthreedimensionalgeometricshapesanddevelopmathematicalargumentsaboutgeometricrelationships € specifylocationsanddescribespatialrelationshipsusing coordinategeometryandotherrepresentationalsystems € applytransformationsandusesymmetrytoanalyzemathematicalsituations € usevisualization,spatialreasoning,andgeometricmodelingtosolveproblems Inlightofincreasingexpectationsonstudentswithregard togeometry,thereareconcernsthatsomestudentswillhave seriousdifcultiesmasteringgeometricconcepts.Youthclassiedashavingalearningdisabilityareatparticularriskfor failureinmoreadvancedmathematics,suchasgeometry.For example,only6%ofyouthwithlearningdisabilitiesin12th gradewithinpublicschoolsscoredatorabovetheprocient levelontheNationalAssessmentofEducationalProgress(NationalCenterforEducationStatistics,2010).

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18 Dobbins,Gagnon,andUlrichYouthwithMathDifcultiesandGeometry Althoughyouthwithmathematicslearningdisabilitiesstruggle,thereexistsabroadercategoryofyouthwithmathematics difcultieswhoarealsoatriskforfailureinmathematics. Youthwithmathematicsdifcultiesareidentiedasranging fromlowaverageperformancetowellbelowaverageperformance.Morespecically,identicationofmathematicsdifcultiesiscommonlydenedasastandardizedmathematicaltestscorethatfallsbelowthe35thpercentile(Mazzocco, 2007).Maccini,Strickland,Gagnon,andMalmgren(2008) summarizedcharacteristicscommontoyouthstrugglingin mathematics:(a)memoryproblemsthatincluderememberingandusingmultiplestepstosolveamathematicsproblem; (b)receptive(e.g.,comprehendingmathematicsproblems)and expressive(e.g.,justifyingananswer)mathematicsvocabulary difculties;and(c)cognitivedecitsthatmayinhibitprocessingofmathematicsconcepts,proceduralstrategies,andrules. Thesecharacteristicsmaynegativelyaffectsecondary-level studentperformanceingeometry,particularlywhencomputingtheareaofgeometricgures. Geometryconceptssuchasndingtheareaofgeometric shapesrequiretherecallandrecognitionoftheappropriate areaformulas.Forexample,whenaskedtondtheareaof atrapezoid,youthmustrecall(a)geometriccomponents ofatrapezoidand(b)theformulaforndingtheareaofa trapezoid.Thisinstructionalconceptrequiresthatyouthuse workingandlong-termmemory;areasinwhichstudentswith mathematicsdifcultiesoftenexhibitdecits(Montague& Jitendra,2006).Also,geometryrequirestheuseofhigher ordercognitiveskills(i.e.,metacognitiveskills).Themetacognitiveskillsinvolvedincomputingtheareaofageometric shapeincludeprediction,planning,monitoring,andevaluationofmathematicalinformationpresentedineacharea problem(Carr,Alexander,&Folds-Bennett,1994;Lucangeli &Cornoldi,1997).Morespecically,themetacognitive strategiesincomputingareainvolve(a)readingthearea problem,(b)developingaplanforsolvingtheproblem,(c) monitoringunderstandingoftheproblemandcompletionof stepstosolveforthearea,(d)evaluatingtheaccuracyofthe stepsandsolution.MontagueandJitendra(2006)connected thedevelopmentofsuchcognitivestrategiestoone’sability tounderstandandintegrateprocedural,declarative,and conceptualknowledge.Toeffectivelycomputetheareaofa geometricshape,youthmusthaveastrongunderstanding ofthedeclarative(i.e.,factualinformation),procedural (i.e.,stepsandprocedures),andconceptual(i.e.,logical relationships)knowledge.Youthwithmathematicsdifculties oftenstrugglewiththeintegrationandapplicationofthese threeareasofknowledge,whichcansignicantlyaffect achievementinsecondary-levelgeometryclasses. Acknowledgingthatstudentswithmathematicsdifculties mayhavesignicantdifcultieswithlearningmathematical conceptssuchastheareaofgeometricshapesisimportant indevelopinginstructionalstrategiestomeettheneedsofa diversegroupoflearners.SlavinandLake(2008)foundthat themosteffectiveeducationalreformmethodtoaddressing theneedsofadiversegroupoflearnersinvolveschangesto classroominstructionalpractices.Furthermore,youthwith mathematicsdifcultiesbenetfromexplicitinstruction,combinedwithstrategyinstructiontoassistintheirabilitytosolve complexmathematicsproblems(Montague&Jitendra,2006). Onewaytoassistyouthwithmathematicsdifcultieswithin thegeneraleducationenvironmentistoprovideadditional smallgroupsupport(Burns&Gibbons,2008).Theideaof supplementaryinstructionisalignedwithTier2withinthe ResponsetoIntervention(RTI)model.Teachersmayndthat additionalsmallgroupsupport,consistentwithTier2inRTI, maysufcientlysupportyouthandallowthemtosucceedin geometry. MeetingtheChallenge Ourgoalistoprovidepromisingandpracticalinstructional approachestoteachersthatpromotestudentlearningofgeometryconcepts.Specically,wefocusonexplicitinstructionthroughsmallgroup,strategy-focusedinterventions(e.g., peer-mediatedlearning,useofagraduatedinstructionalsequence),consistentwithTier2intheRTImodel,whichcan beusedforteachingtheconceptof area .First,weprovide backgroundonRTIandTierIIinterventions,thenweaddressthekeycomponentsnecessaryforeffectivelyusinggraduatedinstructionandpeer-mediatedlearningforgeometry instruction.Next,wesupplyideasforcombiningthetwoapproachesforgeometryinstructionandprovideaconcreteexampleforteachersonteachingstudentstondtheareaof trapezoids.RTIandTierIIInterventionsTheessentialdesignofaresponse-to-Intervention(RTI) modelformathematicsconsistsofthree(orfour)tiersofinstructionalsupport,withlevelsofintensityandindividualizationincreasingateachtier.(Bryant&Bryant,2008;Bryant, Bryant,Gersten,Scammacca,&Chavez,2008)Tier1,thepreventativephase,ensuresthatallstudentsreceivehigh-quality coremathematicsinstruction.Studentachievementisassessed throughuniversalscreeningconductedperiodicallythroughouttheschoolyear.Coreinstructionalchangesaremadewhen screeningdatarevealsagroupdecitinmathematicsskills. StudentsthatareunabletomakeadequategainswithoutadditionalinstructionalsupportaremovedintoTier2andreceive supplementalsmall-groupmathematicsinstruction.Thethird tierofanRTImodelconsistsofone-on-oneorsmall-group instructionforstudentswhoarenotmakingsufcientgrowth towardgrade-levelmathematicsbenchmarkswithTier2supports.Insomemodels,specialeducationservicesareincluded withinTier3,whereasothershaveanadditionaltierspecicallyforthoseidentiedasneedingspecialeducationsupports (Gersten&Beckman,2009).Throughouteachofthetiers,studentprogresstowardgrade-levelmathematicsbenchmarksis continuouslymonitored,withthefrequencyofmonitoring increasingateachtier.Thelevelandformofinstructional supportwithinanRTImodelisheavilyinuencedbyastudent’sabilitytomakeadequategainsinskillareas(Fuchs& Fuchs,2006;Kashima,Schleich,&Spradlin,2009).Asnoted,

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TeachingGeometrytoStudentsWithMathDifculties 19studentsinthegeneraleducationclassroomthatarenotmakingadequateprogresstowardgrade-levelexpectationswith Tier1levelofsupportareprovidedadditionalintervention supportthroughTier2oftheRTIservicedeliverymodel. EmpiricalresearchontheeffectofTierIIhasgenerallyshown thatwhengroupsof4–6studentsareprovidedatleast30minutesofinterventioninadditiontocoreinstructionfor3–5days oftheweek,therearesignicantincreasesinstudentperformanceinmathematics(Fuchs,Fuchs,&Hollenbeck,2007). Atthesecondarylevel,TierIIinterventionsforstudentswith mathematicsdifcultieshaveshowntosignicantlyimprove mathachievementwhenprovidedatleast2timesperweek, for30mineachsession(Calhoon&Fuchs,2003).Inaddition, Fuchsandcolleagues(2008)postulatethattherearekeycomponentstoeffectiveTierIIinterventionsforsecondarystudentsexperiencingdifcultiesinmathematics.TierIIinstructionmustprovideaconnectionbacktothecoreinstructional curriculumbyinsuringthattheinstructionalcontentofTierII alignswiththatofTierI.TierIIinstructionshouldincreasethe frequencyandintensityofinstructionreceivedatTierI,which isaccomplishedthroughprovidingstudentswithincreasedexposuretomathematicsmaterialandadditionalopportunities topracticelearnedskills.Inaddition,TierIIinstructionincludesexplicitandsystematicinstruction,guidedandindependentpractice,andcumulativereviewofpreviouslylearned material(Fuchs,2011).Byincorporatingthesecomponents intoTierIIinterventionsstudentswillhaveopportunitiesto increaseconceptualknowledgeofthesubjectarea,whichmay furthertheirabilitytomaintainandtransferlearnedconcepts andskills(Witzel,Riccomini,&Schneider,2008). Althoughthereissupportforbroadessentialcomponents ofTierIImathematicsinterventions,limitedinformationis availablethatdetailseffectiveinterventionsthatcanbeincorporatedintosecondarymathematicsinstructioninanRTI model.Oftheresearchthathasfocusedonsecondary-level mathematics,twoseparateinstructionalpracticeshavebeen identiedaseffectiveinteachingmathematicsskills.Graduatedinstructionandpeer-mediatedinstructionaresmallgroup interventionsthatalignwiththeNCTMPrinciplesandStandards(NCTM,2000)forgeometryinstruction.Bothinstructionalpracticeshaveindividuallinesofempiricalsupportfor applicationwithinmathematics(Allsopp,1997;Calhoon& Fuchs,2003;Mercer&Miller,1992;Witzel,2005).While limitedresearchstudieshaveexaminedtheeffectivenessof graduatedinstructionandpeer-mediatedinstructioninrelationtogeometryskills(Cass,Cates,Smith,&Jackson,2003), researchershavenotedthepotentialofcombiningtheseinstructionalapproachestosupportstrugglinglearnersduring geometryinstruction(Mulcahy&Gagnon,2007). Moreover,weacknowledgetheimportanceofproviding recommendationforteacherscurrentlyintheeldbasedon presentunderstandingandavailableliterature.Assuch,the combinationofgraduatedinstructionandpeer-mediation instruction,twocompatibleandvalidatedinstructionalapproacheswithinanexplicitinstructionmodelwillbethebasis ofoursuggestionsandlessonexample.Inthisarticle,wediscusseachinstructionalapproachandthenrecommendations forcombiningtheapproaches. GraduatedInstruction Youthwithmathematicsdifcultiescouldbenetfromunderstanding,“thefundamentalmathematicsconceptsprior toadvancingtogeneralizationofrules,facts,oralgorithms” (Maccini,Gagnon,Mulcahy,&Leone,2006,p.210).TheNationalCouncilofTeachersofMathematicsPrinciplesand StandardsforSchoolMathematicsemphasizestheimportanceofconceptualunderstandingofmathematicsasthis approachenhancesstudents’learningandtheirpostschool opportunities(NCTM,2000).Usingagraduatedsequence thatincludeshandsonmanipulativestoteachdifcultmathematicalconceptsinaconcreteandprogressivelymoreabstract mannerallowsstudentstounderstandabstractconceptsmore easily(Devlin,2000).Inaddition,graduatedinstructionalsequenceshavebeenfoundtobeaneffectivemethodforteaching studentstheproceduralstepsassociatedwithamathematics problem,aswellastheconceptualconnections(Witzeletal., 2008). Oneexampleofthistypeofinstructioninmathematics isknownastheConcrete–Representational–Abstract(CRA) technique(Steedly,Dragoo,Arafeh,&Lake,2008).During theCRAsequence,instructionfollowsathree-partsequence beginningwiththeconcretestage(seeTable1).Thisstage involvesstudentsphysicallymanipulatingmaterialssuchas geometricshapes(i.e.,parallelogramsortrapezoids)todisplayandsolvegeometryareaproblems(Miller&Mercer, 1993).Studentswhouseconcretematerialsdevelopmorecomprehensivementalrepresentations,betterapplymathematics tolife,oftenshowmoremotivationandon-taskbehavior, andexhibitadeeperunderstandingofmathematicalconcepts (Clements,1999;Moyer,2001;Sowell,1989).Therepresentationalorsecondstage,movestheinstructionfrommanipulativestopictorialrepresentations,whichcouldincludedrawingsorstampsofgeometricshapes(Maccini&Gagnon,2000). Thenalteachingstage,knownasabstract,isenactedwhen studentsarereadytomovetosymbols,numbers,andnotationstorepresenttheareaproblems(Harris,Miller,&Mercer, 1995).Eachofthesethreephasesbuildononeanotherand studentsmustunderstandwhathasbeentaughtateachstep beforeteachersprovideopportunitiesforlearningatthenext stage(NCTM,2000). TheCRAmodelhasbeeneffectiveinteachingavariety ofmathematicalconceptsincludingbasicmathfacts,coin sums,multiplication,placevalue,perimeter,fractions,andTable1. GraduatedInstruction Phase Instructional practiceExamples ConcreteHands-onmaterialsGeometricshapesand gures RepresentationalPicture representation Picturesofgeometric shapesandgures onaworksheet AbstractAbstractnotationsUseofareaformula foraparallelogram

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20 Dobbins,Gagnon,andUlrichalgebra(Butler,Miller,Crehan,Babbitt,&Pierce,2003;Cass, Cates,Smith,&Jackson,2003;Mercer&Miller,1992;Miller, Harris,Strawser,Jones,&Mercer,1998;Miller,Mercer,& Dillon,1992;Peterson,Mercer,&O’Shea,1988;Witzel, 2005).Itisalsoeffectivewithstudentsfrommultiplegrade levels(Maccini&Gagnon,2000),forthosewithdiverselevels ofmathematicsachievement(Mercer&Miller,1992;Witzel, 2005),andinavarietyofinstructionalgroupingsincluding wholegroupinstruction(Witzel,Mercer,&Miller,2003), smallgroupinstruction,andone-on-oneinstructionwitha student(Miller,Mercer,&Dillon,1992).Last,CRAhasbeen implicatedinincreasedlevelsofretentionandtransferof skills(Butleretal.,2003;Cassetal.,2003). Specictogeometry,researchers(Cass,Cates,Smith,& Jackson,2003)reportedtheefcacyoftheCRAtechniquefor teachingperimeterandareaproblemsolvingtomiddleand highschoolstudents.IntheCassandcolleagues’(2003)study, threehighschoolstudentswhowereclassiedwithalearning disabilitywereselectedtoreceiveacombinationofCRAinstructioninconjunctionwithmodeling,guidedpractice,and independentpractice.Infewerthan7days,theinstructionresultedinacquisitionandmaintenanceoftheareaandperimeterskillsforallthreestudents.Cassandcolleagues(2003)also notedthatstudentsmaintainedgains. TheresultsoftheCassandcolleagues(2003)study,as wellasnumerousotherstudiesshowingbenetsoftheCRA (aswellastheRAsequence)inotherareasofsecondary mathematics(seeButler,Miller,Crehan,Babbitt,&Pierce, 2003;Maccini&Ruhl,2000;Witzeletal.,2003),indicatethe promiseofCRAwithteachinggeometry.Givenourcurrent knowledge,itisrecommendedthatCRAbeused,particularly whenteachingareatoyouthwithmathematicsdifculties,as amethodforsupportingtheunderstandingandintegrationof conceptualknowledgewithproceduralanddeclarativeknowledge.TheAppendixcontainsapracticalapplicationexample ofCRAinstructionforgeometry.Thisexampledemonstrates thegraduatedsequenceofCRAinstructioninrelationtondingtheareaofgeometricshapes.TheclassroomteacherfacilitatesthisCRAsequencebyincorporatingexplicitinstruction andmodelingofappropriategeometricstrategiestosolvearea problems. Peer-MediatedInstruction AccordingtotheNCTMstandards,mathematicsinstruction shouldbeactive,social,andinteractive(NCTM,2000).Effectivemathematicsinstructionatthesecondarylevelinvolves (a)theuseofsmall,interactivegroups;(b)theuseofdirected questioningandresponses;and(c)usingextendedpractice withfeedback;allofwhicharecomponentsofpeer-mediated instruction(Swanson&Hoskyn,1998).Specically,peermediatedinstructioninvolvespairsofstudentsworkingina collaborativemanneronstructured,individualizedactivities (Kunsch,Jitendra,&Sood,2007). Aformofpeer-mediatedinstructionthathasreceived substantialsupportisPeerAssistedLearningStrategies (PALS;Fuchs,Fuchs,Mathes,&Martinez,2002).Originally developedinalignmentwithTennesseestatemathematicsTable2. PALSImplementation PALSImplementationProcedures Step1Engageinmediatedverbalrehearsal Step2Providestep-by-stepfeedback Step3Engageinfrequentverbalandwritteninteraction Step4Reciprocityofrolesastutorandtutee curriculumstandards,PALSusespeer-tutoringdyadsand asystematicapproachtopracticinginstructionalcontent. PALSistypicallyused2–3daysperweekasasupplementto coremathematicsinstructionandconsistsofcoachingand independentpractice(seeTable2).Studentdyadsareconguredbasedonlevelofability,withhigherandlowerability studentspairedtogether.Eachstudenthasanopportunityto actasthetutorandtuteeduringeachPALSsession.Tutors engageinteachingtheirpeerthroughmodelingprocedural stepsforproblems,askingstructuredquestionstopromote conceptualknowledge,andprovidingimmediatefeedback. IndependentpracticeisusedwithinPALSasamethodof reinforcingnewlylearnedmaterialandpracticingpreviously taughtinformation. CalhoonandFuchs(2003)examinedtheapplicabilityof peer-mediatedinstructionalstrategiessuchasPALSatthe secondarylevelwithstudentsidentiedashavinglearning difcultiesandbehavioralconcerns.Thisstudyexaminedthe effectofthePALSprograminconjunctionwithcurriculumbasedmeasurement(CBM)onsecondarystudents’mathematicsperformance.Theparticipantswere92studentsfrom 10classroomsrepresentingGrades9–12.Thesestudentswere identiedasperformingsignicantlybelowgrade-levelexpectationsandreceivedinstructioninaself-containedclassroom. ClassroomswerepickedtoreceivePALS/CBMorcontinue mathematicsinstructionasusual.PALS/CBMwasimplementedtwiceweeklyandCBMwasconductedweeklyfor15 weeks.PALS/CBMstudentsimprovedincomputationmathematicsskillssignicantlymorethanstudentsinclassrooms conductinginstructionasusualwithoutPALS/CBM. Inrelationtohigherordercognitiveskillsandproblem solving,Allsop(1997)conductedastudytodeterminetheeffectsofpeer-mediatedinstructiononalgebraperformancefor at-riskandnon–at-riskstudents.Asampleof262students betweentheagesof12and15yearswasused,with38%of thesamplebeingstudentsatriskforfailureinmathematics. Resultsofthestudyindicatedthatpeer-mediatedinstruction hadasignicanteffectonstudentabilitytoperformhigher ordercognitiveskillsassociatedwithalgebraproblemsolving. Inaddition,analysisproceduresindicatedthatstudentswho wereatriskforfailureexhibitedslightlygreatergainsinproblemsolvingabilitiesthandidstudentswhowerenotatrisk. Peer-mediatedinstructionallowsforastructuredmethod oflearningbyprovidingstudentswithexplicitdetailsof theirrolesandresponsibilitiesinthetutoringrelationship (seeTable4).Inaddition,thismethodofinstructionpromotesmathematicaldiscourseandacademicengagement, whichcanbenetyouthwithinTierIIintervention(Maccini etal.,2008).Specically,engaginginmathematicaldiscourse assistsstudentswithmathematicsdifcultiesthroughuseof

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TeachingGeometrytoStudentsWithMathDifculties 21Table3. InstructionalStepsforClassroomTeacher Step numberStepExample 1Reviewpreviouslesson“Let’sreviewhowtondtheareaofasquareandtriangle.” 2Introducethepurpose“Wewillbeusingsomeoftheconceptswelearnedinworkingwithsquares andtrianglestondtheareaofatrapezoid.” 3Modelusingappropriate Concrete–Representational–Abstract stagemethods Concrete:“Let’suseourtrapezoidshapesandrulerstondthearea.” Representational:“Let’slookatthispictureofatrapezoid.” Abstract:“Findtheareausingthisinformation:b1 = 5b2 = 7h = 12.” 4PeertutoringStudentsworkindyadstopracticelearnedskillusingeachofthe Concrete–Representational–Abstractsteps. 5IndependentpracticeCurriculum-basedmeasurementmeasuresaregiventoeachstudentto measureprogressandmaintenanceofpreviousmaterial. themetacogntivestrategiesnecessarytosolvemathematical problems.Peer-mediatedinstructionrequiresthatstudentsengageinmonitoringtheirbehaviorsandengagement,aswellas thoseoftheirtutoringpartner.Thisself-monitoringpromotes students’abilitytoplanandevaluateappropriatemethodsto solvemathematicalproblemsthattheywouldfaceinageometryclass. CombiningCRAandPeer-MediatedInstruction Asdescribed,CRAandpeer-mediatedinstructionaretwo powerfulTierIIinterventionstoassistyouthinmathematics. Additionalresearchisneededtoevaluatethecombinedefcacyoftheseapproaches,particularlywithregardstogeometryinstruction.However,intheinterimitisrecommendedthat teachersemployacombinationoftheapproachestomaximize studentbenet(seeTables3and4).SimultaneoususeofCRATable4. PeerTutoringInstructionalPlan StepDescriptionExample Introducethe purpose Tuteereads € “Youwillbeusing someoftheconcepts welearnedinworking withsquaresand trianglestondthe areaofatrapezoid” Guided practice Tutorpromptstutee. Additionalprompts providedasneeded. € “Whatpartsofthe trapezoidwillweuse tocalculatethearea?” (answer:basesand height) € “Howdowendthe base?” Rolesswitchedafter eachhasonechance inarole. € “Howdowendthe height?” € “Whatistheformula thatweusetocalculate thearea?Writethis formula.Fillinthe informationwehave.” € “Whatisthearea?” andpeer-mediatedinstructionhasthepotentialtosupportthe learningneedsofstudentsviaaddressingkeyprincipleswithin theNCTMStandards(NCTM,2000).Specically,thecombinationofapproacheswill(a)provideallstudentswithahigh levelofsupportthroughincreasedchancesforfeedbackand assistance(Principle1:Equity);(b)providesacoherent,focusedmethodofinstructionthatwillallowteacherstoknow andunderstandwhatareastheirstudentsneedmoreassistancewith(Principles2&3:CurriculumandTeaching);(c) allowstudentstheopportunitytolearngeometryinsucha waythatwilldeveloptheiroverallconceptualunderstanding abilities(Principle4:Learning);and(d)provideongoingassessmentdatathatcanbeusedbyteacherstoshowstudent progressandinformdecisionsaboutfutureinterventionsand instruction(Principle5:Assessment). Byusingagraduatedinstructionalsequenceandhigh levelsofguidedpracticethroughpeer-mediatedinstruction, classroomteachersareabletoprovidestudentswithmath difcultiesincreasedexposuretocurriculumcontentand furtherdeveloptheircognitivestrategiesformathematical problemsolving.ToillustratethepracticaluseofCRA andpeer-mediatedthroughexplicitinstructionwithinan RTImodelforgeometry,weprovideadetailedlessonthat integratestheseapproachestohelpstudentssolvecontextualizedareawordproblems(seetheAppendix).Inthislesson, teachersprovideexplicitinstructionandmodelingforsolving fortheareaofspecicgeometricshapes.Thisinstruction followstheCRAgraduatedinstructionalsequencewithin eachstepoftheCRAmodel,followedbyopportunitiesfor studentstopracticeusingpeer-mediatedinstruction. Inadditiontotheinstructionaldetailsoutlinedinthe samplelesson,itisrecommendedthatteachersintegrate progressmonitoringthroughoutinstruction.AkeycomponentofRTIistheongoingassessmentofstudents’progress towardestablishedgoal(Burns,2008;Burns&Gibbons, 2008;Riccomini&Witzel,2010).Curriculum-basedmeasurescontainingitemsthatrepresenttheannualgeometry expectationsoutlinedintheNCTMstandards(2000)and district-levelstandardsshouldbeusedtomonitorstudents’ progressandgrowthtowardgrade-levelbenchmarks(seealso Foegen,2000,2008;Foegen,Jiban,&Deno,2007).Detailed skillanalysisofstudentperformanceonthesemeasurescan provideteacherswithinformationrelatedtospecicareas ofdifcultyforeachstudent(Riccomini&Witzel,2010).

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22 Dobbins,Gagnon,andUlrichMonitoringofstudentprogressshouldbecompletedduring eachphaseofCRAandpeertutoringtoguidetheteacher’s decisiontomoveontothesubsequentphaseofCRA (Hudson&Miller,2006).Specicdirectionspertainingtothe schedulingofprogressmonitoringassessmentaredetailedin thepracticalinterventionexampleintheAppendix.FinalThoughtsGeneralandspecialeducationgeometryteachersarefaced withthedifcultdailychallengeofassistingstudentswith mathematicsdifcultieswhorequireTierIIinterventions.Althoughlimitedinformationisavailableconcerningthemost efcaciousinstructionalmethodsforsupportingyouthwith mathematicsdifculties,itiscriticalthatteachersrelyonthe currentbestevidenceintheeld.Ourdiscussionandexample ofusingCRAandpeer-mediatedinstructionwithinanexplicit instructionmodelforstudentsrequiringTierIIinterventions, providesapromisingstartingpointforteachers.Thereisgreat potentialfortheseinstructionalapproachestopromoteconceptualunderstanding,provideforinteractivelearning,and providestudentswithdisabilitiesthegreatestopportunityfor successwithingeometrycoursework.AuthorNotesAngelaDobbins isapostdoctoralresearchassociateinthe ClinicalPsychologyServicesDepartmentattheUniversity ofVirginia.Herresearchinterestsaremathematicsinterventions,mathematicsprogressmonitoring,andtieredinstructionalpractices. JosephCalvinGagnon isanassociateprofessorintheSpecialEducationDepartmentattheUniversityofFlorida.Dr. Gagnon’scurrentresearchinterestsincludemathematicsinstructionforsecondaryyouthwithemotionaldisturbanceand learningdisabilities. TracyUlrich isadoctoralstudentattheUniversityofFlorida intheSpecialEducationProgram.Hercurrentresearchinterestsareearlyinterventioninmathematicsandmathematics instructionforstudentswithlearningdisabilities.ReferencesAllsopp,D.H.(1997).Usingclasswidepeertutoringtoteachbeginning algebraproblem-solvingskillsinheterogeneousclassrooms. RemedialandSpecialEducation 18 ,367–379. Aud,S.,Hussar,W.,Kena,G.,Bianco,K.,Frohlich,L.,Kemp,J., &Tahan,K.(2011). TheConditionofEducation2011(NCES 2011033) .Washington,DC:U.S.GovernmentPrintingOfce. Burns,M.,&Gibbons,K.(2008). Implementingresponsetointervention inelementaryandsecondaryschools:Procedurestoassurescienticbasedpractices .NewYork,NY:Routledge. Butler,F.M.,Miller,S.P.,Crehan,K.,Babbitt,B.,&Pierce,T.(2003). Fractioninstructionforstudentswithmathematicsdisabilities: Comparingtwoteachingsequences. LearningDisabilitiesResearch andPractice 18 ,99–111. Bryant,B.R.,&Bryant,D.P.(2008).Introductiontoaspecialseries: Mathematicsandlearningdisabilities. LearningDisabilityQuarterly 31 ,3–8. Bryant,D.P.,Bryant,B.R.,Gersten,R.,Scammacca,N.,&Chavez,M. (2008).Mathematicsinterventionforrstandsecondgradestudents withmathematicsdifculties. RemedialandSpecialEducation 29 20–32. Calhoon,M.B.,&Fuchs,L.S.(2003).Theeffectsofpeer-assistedlearningstrategiesandcurriculum-basedmeasurementonmathematics performanceofsecondarystudentswithdisabilities. Remedialand SpecialEducation 24 ,235–245. Carr,M.,Alexander,J.,&Folds-Bennett,T.(1994).Metacognitionand mathematicsstrategyuse. AppliedCognitivePsychology 8 ,583– 595. Cass,M.,Cates,D.,Smith,M.,&Jackson,C.(2003).Effectsofmanipulativeinstructiononsolvingareaandperimeterproblemsby studentswithlearningdisabilities. LearningDisabilitiesResearch andPractice 18 ,112–120. Clements,D.H.(1999).‘Concrete’manipulatives,concreteideas. ContemporaryIssuesinEarlyChildhood 1 ,45–60. Devlin,K.(2000).Findingyourinnermathematician. EducationDigest 66 ,63–66. Foegen,A.(2000).Technicaladequacyofgeneraloutcomemeasuresfor middleschoolmathematics. AssessmentforEffectiveIntervention 25 ,175–203. Foegen,A.(2008).Progressmonitoringinmiddleschoolmathematics: Optionsandissues. RemedialandSpecialEducation 29 ,195–207. Foegen,A.,Jiban,C.,&Deno,S.(2007).Progressmonitoringmeasures inmathematics:Areviewoftheliterature. TheJournalofSpecial Education 41 ,121–139. Fuchs,L.S.(2011). Mathematicsinterventionatthesecondaryprevention levelofamulti-tierpreventionsystem:Sixkeyprinciples .Washington,DC:RTIActionNetwork.Retrievedfromhttp://www. rtinetwork.org/essential/tieredinstruction/tier2/mathintervention Fuchs,D.,Compton,D.L.,Fuchs,L.S.,&Bryant,J.(2008).Making“secondaryintervention”workinathree-tierresponsivenessto-interventionmodel:Findingsfromtherst-gradelongitudinal readingstudyattheNationalResearchCenteronLearningDisabilities. ReadingandWriting:AnInterdisciplinaryJournal 21 ,413–436. Fuchs,D.,Fuchs,L.S.,Mathes,P.G.,&Martinez,E.(2002).Preliminary evidenceonthesocialstandingofstudentswithlearningdisabilitiesinPALSandno-PALSclassrooms. LearningDisabilitiesResearch andPractice 17 ,205–215. Fuchs,L.S.,&Fuchs,D.(2006).Introductiontoresponsetointervention:What,why,andhowvalidisit? ReadingResearchQuarterly 41 ,93–99. Fuchs,L.S.,Fuchs,D.,&Hollenbeck,K.N.(2007).Extendingresponsivenesstointerventiontomathematicsatrstandthirdgrades. LearningDisabilitiesResearch&Practice 22 ,13–24. Gersten,R.,&Beckmann,S.(2009). Assistingstudentsstruggling withmathematics:Responsetointervention(RtI)forelementary andmiddleschools .Retrievedfromhttp://ies.ed.gov/ncee/wwc/ publications/practiceguides Harris,C.A.,Miller,S.P.,&Mercer,C.D.(1995).Teachinginitial multiplicationskillstostudentswithdisabilitiesingeneraleducationclassrooms. LearningDisabilitiesResearchandPractice 10 180–195. Hudson,P.,&Miller,S.P.(2006). Designingandimplementingmathematicsinstructionforstudentswithdiverselearningneeds .Boston, MA:Pearson. Kashima,Y.,Schleich,B.,&Spradlin,T.(2009). Thecorecomponents ofRtI:Acloserlookatevidence-basedcorecurriculum,assessment andprogressmonitoring,anddata-baseddecisionmaking .Bloomington,IN:CenterforEvaluationandEducationPolicy. Kunsch,C.A.,Jitendra,A.,&Sood,S.(2007).Theeffectsofpeermediatedinstructioninmathematicsforstudentswithlearning problems:Aresearchsynthesis. LearningDisabilitiesResearchand Practice 22 ,1–12. 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TeachingGeometrytoStudentsWithMathDifculties 23Maccini,P.,&Gagnon,J.C.(2000).Bestpracticesforteachingmathematicstosecondarystudentswithspecialneeds. FocusonExceptional Children 32 ,1–22. Maccini,P.,Gagnon,J.C.,Mulcahy,C.,&Leone,P.(2006).Mathematicsinstructionwithinajuvenilecorrectionalschoolforcommitted youth. JournalofCorrectionalEducation 57 ,210–229. Maccini,P.,&Hughes,C.A.(2000).Effectsofaproblem-solvingstrategy ontheintroductoryalgebraperformanceofsecondarystudentswith learningdisabilities. LearningDisabilitiesResearch&Practice 15 10–21. Maccini,P.,&Ruhl,K.L.(2000).Effectsofagraduatedinstructional sequenceonthealgebraicsubtractionofintegersbysecondarystudentswithlearningdisabilities. Education&TreatmentofChildren 23 ,465–489. Maccini,P.,Strickland,T.,Gagnon,J.C.,&Malmgren,K.W.(2008). Accessingthegeneraleducationmathcurriculumforsecondarystudentswithhighincidencedisabilities. FocusonExceptionalChildren 40 ,1–32. Mazzocco,M.M.M.(2007).Deninganddifferentiatingmathematical learningdisabilitiesanddifculties.InD.B.Berch&M.M.M. Mazzocco(Eds.), Whyismathsohardforsomechildren? (pp.29–49). Baltimore,MD:Brookes. Mercer,C.D.,&Miller,S.P.(1992).Teachingstudentswithlearning problemsinmathtoacquire,understand,andapplybasicmath facts. RemedialandSpecialEducation 13 ,19–35. Miller,S.P.,Harris,C.,Strawser,S.,Jones,W.P.,&Mercer,C.(1998). Teachingmultiplicationtosecondgradersininclusivesettings. FocusonLearningProblemsinMathematics 20 ,50–70. Miller,S.P.,&Mercer,C.D.(1993).Usingdatatolearnabout concrete–semi–concrete—abstractinstructionforstudentswith mathdisabilities. LearningDisabilitiesResearchandPractice 8 89–96. Miller,S.P.,Mercer,C.D.,&Dillon,A.(1992).Acquiringandretaining mathskills. InterventioninSchoolandClinic 28 ,105–110. Montague,M.,&Jitendra,A.K.(Eds.)(2006). Teachingmathematics tomiddleschoolstudentswithlearningdifculties (pp.30–50).New York:TheGuilfordPress. Moyer,P.S.(2001).Arewehavingfunyet?Howteachersusemanipulativestoteachmathematics. EducationalStudiesinMathematics 47 175–197. Mulcahy,C.A.,&Gagnon,J.C.(2007)Teachingmathematicstosecondarystudentswithemotional/behavioraldisorders.InL.M. Bullock&R.A.Gable(Eds.), SeventhCCBDmini-libraryseries (pp.1–36).Arlington,VA:CouncilforChildrenwithBehavioral Disorders. NationalCenterforEducationStatistics.(2010). Thenation'sreportcard: Grade12readingandmathematics2009nationalandpilotstateresults (NCES2011–455).Washington,DC:InstituteofEducation Sciences,U.S.DepartmentofEducation. NationalCouncilofTeachersofMathematics.(2000). Principlesand StandardsforSchoolMathematics .Reston,VA:NationalCouncil ofTeachersofMathematics. NationalCouncilofTeachersofMathematics.(2011). Frequentlyasked questions .Reston,VA:Author.Retrievedfromhttp://www.nctm. org/standards/faq.aspx OrganisationforEconomicCo-operationandDevelopment.(2010). PISA2009results:Whatstudentsknowandcando (Vol.1).Paris, France:Author. Peterson,S.K.,Mercer,C.D.,&O’Shea,L.(1988).Teachinglearningdisabledstudentsplacevalueusingtheconcretetoabstractsequence.LearningDisabilitiesResearch 4 ,52–56. Reys,B.J.,Dingman,S.,Nevels,N.,&Teuscher,D.(2007). Highschool mathematics:State-levelcurriculumstandardsandgraduationrequirements .Columbia,MO:CenterfortheStudyofMathematics Curriculum. Riccomini,P.J.,&Witzel,B.S.(2010). Responsetointerventioninmathematics .ThousandsOaks,CA:CorwinPress. Slavin,R.E.,&Lake,C.(2008).Effectiveprogramsinelementarymathematics:Abest-evidencesynthesis. ReviewofEducationalResearch 78 ,427–515. Sowell,E.(1989).Effectsofmanipulativematerialinmathematics instruction. JournalforResearchinMathematicsEducation 20 498–505. Steedly,K.,Dragoo,K.,Arafeh,S.,&Luke,S.(2008).Effectivemathematicsinstruction. EvidenceforEducation 3 ,1–11. Swanson,H.L.,&Hoskyn,M.(1998).Experimentalinterventionresearchonstudentswithlearningdisabilities:Ameta-analysisof treatmentoutcomes. ReviewofEducationalResearch 68 ,277–321. Vaughn,S.&Fletcher,J.(2012).Responsetointerventionwithsecondary schoolstudentswithreadingdifculties. JournalofLearningDisabilities 45 ,244–256. Witzel,B.S.(2005).UsingCRAtoteachalgebratostudentswithmath difcultiesininclusivesettings. LearningDisabilities:AContemporaryJournal 3 ,49–60. Witzel,B.S.,Mercer,C.D.,&Miller,M.D.(2003).Teachingalgebra tostudentswithlearningdifculties:Aninvestigationofanexplicit instructionmodel. LearningDisabilitiesResearchandPractice 18 121–131. Witzel,B.S.,Riccomini,P.J.,&Schneider,E.(2008).ImplementingCRA withsecondarystudentswithlearningdisabilitiesinmathematics. InterventioninSchoolandClinic 43 ,270–276. Woodward,J.(2004).MathematicseducationintheUnitedStates:Past topresent. JournalofLearningDisabilities 37 ,16–31.AppendixExampleforAreaofaTrapezoid Objective: Studentswillsolvecontextualizedareaword problemswithTierIIsupportthatincludesCRAandpeermediatedinstruction. Review: Teacherengagesinreviewingpreviouslylearned skillsthatareperquisitesforcurrentlesson.Thisisusedas atimetoassessstudentunderstandingofpriorlessonsand reinforcepreviouslytaughtskills. Teachersays:“Beforewestarttoday’slessonwewillreview thefollowing:rememberthatwecanrepresentonesquareunit witheachoftheseblocks?”[Teachershowsstudentssquare unitblocks.Teacherrepresentsvariousdifferentamountsof blockstoshowstudentsthatjustbyaddingtheblockstogether, theygetthenumberofsquareunits.] AdvanceOrganizerandPurpose: Teacherstatestheobjectiveofthelessonbeforemodelingskillsforstudents. Teachersays : “Nowwewilluseourgeometrictrapezoids andsquare-unitblockstosolveforarea.” ModelingUsingConcretePhaseofCRA SampleProblem:Johnwantstoputdownnewtileinhis laundryroombecausehispuppyscratchedholesintheooring.Heneedstogureouttheareaoftheroomsothathecan ordertherightamountoftile.Theoorspaceintheroomis shownbelow.Whatistheareaoftheoorspace?

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24 Dobbins,Gagnon,andUlrichTeachersays:“Theunitofmeasurementfortheareaof theoorwillbesquaredunits.Howcanwendtheareaof thistrapezoidusingthesquareunitblocks?Let’stryusing thesquareunitblockstollinthetrapezoidshape.Now let’scountthenumberofblocksittakestocoverthewhole trapezoid.Thisisthearea.”[Teachermodelsforstudentshow tousegeometricshapesoftrapezoidsandsquareunitblocks torepresenttheproblem,makingsuretodemonstratehow tousehalfunitblockstollthecornersofthetrapezoid. Teachercontinuallyengagesstudentsduringmodelingprocess byaskingforassistance/callingonindividualstudentswhile completingstepsoftheproblem.] PeerTutoring Modeling:Teachermodelsforstudentshowtoengagein peertutoringusingthestepsinTable2and3toguidethisprocess.Followingmodeling,teacherpromptsstudentstoworkin dyadstosolveasimilarproblem(areaoftrapezoid).Students areprovidedwithavisualaidorscripttorememberprobing questionsandcorrectivefeedbacksteps. Feedback:Asthestudentsworkindyads,teachercirculates theroomprovidingfeedbackorassistanceasneeded.Student actingasthetutorpromptstheotherstudent(tutee)byasking questionssuchas“Howmanyblocksdidittaketocoverthe trapezoid.”RefertoTable4fordetailedstudentroles. ProgressMonitoring:Onceeachstudenthascompleted anareaproblemasthetutee,theteacherevaluateswhether thereisaneedforadditionalpracticebasedonstudentperformance.Studentsareprovidedwithanopportunitytoengage inindependentpracticethroughcompletingacurriculumbasedmeasurement(CBM)probethattargetsconcreteskills learned. PhaseofInstruction:Representational Teacherprovidesreviewofpreviouslessons,advanceorganizer,andobjectivebeforebeginningmodelinglesson,justas intheconcretephase. Teachersays:“Yesterdaywesawthatwecanusethesquare unitblockstorepresenttheareaofatrapezoid.Todayweare goingtousepicturesoftrapezoidsandwhatwelearnedabout squareunitblockstondthearea.” ModelingUsingRepresentationalPhase Studentsarepresentedwithapictureofatrapezoidthat hasbeenlledinwithsquareunitblocks.Teachermodelsfor studentshowtosolveareaproblemusingapictorialrepresentationofthetrapezoid(i.e.,countingupthenumberof squaresinthepicture).Teacherthenpresentseachstudent withahandoutwiththeshapeofatrapezoidonit. Teachersays:“Nowtrydrawingyourownsquareunits insidetheblanktrapezoidatthebottomofthepage.What istheareaofthetrapezoid?Isitthesameasitwasyesterday?”[Teacherprovidesstudentswithadifferenttrapezoid andpromptsstudentstoworkindyadstosolvetheproblems usingpeertutoring.] PeerTutoring Feedback:Asthestudentsworkindyads,teachercirculates theroomprovidingfeedbackorassistanceasneeded.Student actingasthetutorpromptstheotherstudent(tutee)byasking questionssuchas“Howmanysquareunitsareinthetrapezoid”or“Whatistheareaofthetrapezoid.”RefertoTable4 fordetailedstudentroles. ProgressMonitoring:Onceeachstudenthascompletedan areaproblemasthetutee,theteacherevaluateswhetherthere isaneedforadditionalpracticebasedonstudentperformance. Studentsareprovidedwithanopportunitytoengageinindependentpracticethroughcompletingacurriculum-based measurethattargetsconcreteskillslearned. Teachersays:“Sofar,wehavefoundtheareaofatrapezoid bycountingthenumberofsquareunitswithinthetrapezoid. Let’slooktoseewhetherthereisaneasierwaytondthearea ofatrapezoid.Beforewebegin,let’sreviewhowweuseda formulatondtheareaofasquare.Theareaofasquarecan befoundbymultiplyingtheheightofthesquarebythewidth ofthesquare(A = h w).” “Atrapezoidhassomesimilaritiestoarectangle.Howcan wendtheheightofthetrapezoid?” Teachermodelshowtodrawalinefromthetopofthe trapezoidtothebottomtoformarighttriangleontheside. “Justaswithndingtheareaofarectangle,wecanuse aformulatondtheareaofatrapezoid.Theformulafor ndingtheareaofatrapezoidisA =1 2(B1 + B2)h.” “B1andB2representthewidthofthetrapezoidatthetop andbottom.YoucanseethatB1andB2arenotequal.How canwendB1andB2byusingthepictureofthetrapezoid withsquareunitblocksdrawnwithinit?”[Teachermodels countingtheunitsquaresthatrepresentB1andB2].“Nowlet’s ndtheheight.”[Teachermodelscountingtheunitsquares thatrepresenttheheight.] “Nowthatwehavethemeasurementsforh,B1,andB2, tomorrowwewillusethesemeasurementstousetheformula fortheareaofatrapezoid.” PhaseofInstruction:Abstract Thealgebraicformulafortheareaofatrapezoidisvisually presentedandexplained. Teachersays : “Sofarwehavebeenusingsquareunitblocks tondtheareaofatrapezoid.Lasttimewesawthatjustlike forsquares,thereisaformulatondtheareaofatrapezoid.

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TeachingGeometrytoStudentsWithMathDifculties 25Let’sreviewthecomponentsoftheformula:Theformulafor ndingtheareaofatrapezoidisA =1 2(b1 + b2)h.” “B1andB2arethetwobaselengthsofthetrapezoid.His theheightofthetrapezoid,whichwendbydrawingaline fromB1toB2tomakearighttriangle.” “Yesterdayweusedourpictorialrepresentationofatrapezoidtondthemeasurementsofh,B1,andB2.Herearethe measurementsfromyesterday:” “Now,let’susetheformulatondtheareaofatrapezoid.” [Teachermodelsforstudentshowtollinthemeasurements intotheformulafortheareaofatrapezoid].“Whatisthe area?”[Teachergivesstudentsotherareaproblemstopractice withduringpeertutoring.] PeerTutoring Feedback:Asthestudentsworkindyads,teachercirculates theroomprovidingfeedbackorassistanceasneeded.Student actingasthetutorpromptstheotherstudent(tutee)byasking questionssuchas“Whatpartsofthetrapezoidareusedtond thearea/whatistheformulaforareaofatrapezoid.”Referto Table4fordetailedstudentroles. ProgressMonitoring:Onceeachstudenthascompletedan areaproblemasthetutee,theteacherevaluateswhetherthere isaneedforadditionalpracticebasedonstudentperformance. Studentsareprovidedwithanopportunitytoengageinindependentpracticethroughcompletingacurriculum-based measurethattargetsconcreteskillslearned.