Developing Instructional Units

DevelopingInstructionalUnits:ApplyingWhatStudentsLearnDonnaCurry

TERC

ThenewWorkforceInnovationandOpportunityAct(WIOA)requiresadulteducatorsto helpadultlearnerspreparefortheworldofwork,notjusttopassahigh-stakes assessment.TheCommonCoreStateStandards(ofwhichtheCollegeandCareer ReadinessStandardsareasubset)aredesignedtoteachrigor:equalemphasison proceduralfluency,conceptualunderstanding,andapplication.And,researchshowsthatlearningincontextisbetterthanlearningdecontextualizedmaterial.

Accordingtocontextuallearningtheory,learningoccursonlywhenstudentsprocessnew knowledgeinsuchawaythatitmakessensetothemintheirownframesofreference. Thisapproachtolearningandteachingassumesthatthemindnaturallyseeksmeaningin contextandthatitdoessobysearchingforrelationshipsthatmakesenseandappear useful(CORD,p.1).

So,thewaytoteachmathtoouradultlearnersisthroughexperiencesthatarecontextualized.Oh,butwealreadygivethemwordproblemseveryonceinawhile. Surelythatwillsufficeforcontextualizingthemath,right?

Let’scomparethefollowingthreeexamplessoyoucandecideforyourself:

16,000÷60 =?

Tamikaplansto buyacarthatcosts$16,000. Ifshepaysforthecarovera5-yearperiod,howmuchwillhermonthlypaymentsbe?

Tamikais interestedin buyingacar.Shelearnsthatshe canhavea4-yearora5-yearpaymentplan.Shealso findsoutthat puttingmoneydownupfrontwill helpwiththemonthlypayments.Whatmightbesomepossible scenariosfor her? Whichwouldbethebestscenarioforyouifyouwereina similarsituation?Why?

Onewaytocontextualizethemaththatyouareteachingistodevelopinstructionalunits withtheendgoalbeingtheapplicationoflearning.WigginsandMcTighesuggestaprocessthatbeginswiththeendinmind;theycallthisprocessUnderstandingbyDesign, orUbD.Itisalsocalledbackwarddesign.Therearethreestepstotheprocess:

1.Identifyingthedesiredresults.Whatdowewantstudentstobeabletodowith whattheyhavelearned?Duringthisprocess,youshouldthinkabouthowstudents willusewhattheyhavelearnedinsituationsotherthanon‘thetest’since,

ultimately,weallwantourstudentstobeabletoapplywhattheyhavelearnedin ourclassroomsintheirlives,onthejob,andathome.

2.Determiningassessmentevidence.Ifwewanttoknowwhetherourstudents knowwhentousemultiplicationvs.additionorsubtractionandhowtochoosethemostefficientstrategyforthesituation,thenweneedtoprovideassessmentopportunitieswherestudentswillhavetomakethosedecisions.Youcancreatean assessmenttaskthatisdirectlyalignedtothegoalyoudevelopedinstep1.You canalsoinclude‘test-like’questionsaspartofyourassessmentplansostudents gaincomfortinusingmathinbothreal-lifeandtestsituations.

3.Planninglearningexperiencesandinstruction.Hereiswhereyouthinkaboutallthethingsyouwillneedtoteachinorderforstudentstoreachthegoalyou (andyourstudents)haveset.Thisiswheresomeofthosetraditionallessonsshow up.

WhiletheUnderstandingbyDesignprocessbeginswithabroadgoalstatementfocused onapplicationoflearning,youmighttrynarrowingthefocusabituntilyougetusedto teachingandassessinginamorecontextualizedway.Forexample,accordingtoWiggins andMcTighe,agoalstatementcouldbesomethingbroadsuchas,“applymathematicalknowledge,skill,andreasoningtosolvereal-worldproblems”(pg.3).However,itmightbebeneficialtothinkaboutamoreconcretestatementthatspecifiesthetargetknowledge/skill,themathdomain,andthetaskthatrequiresthestudenttoapplytheknowledge/skillinordertodemonstrateunderstanding.Hereisanexampleofaunitgoalstatementusingthecar-buyingscenariowesawearlier:

Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingofpercentages andoperationswithdecimalsinordertodescribeseveralcar-buyingoptions.

Bywritinggoalstatementsthisway,youalreadyhaveaclearideaaboutwhattheassessmentshouldlooklike.Thegoalandtheassessmentarethennicelyaligned.Now yourchallengeistofigureoutallthelessonsandlearningexperiencesthatyouwantto engageyourstudentsintoensurethattheygainagoodgraspofthematerial.Bytheway, contextuallearningdoesNOTmeanthateverysinglelessonhastobecontextualized.Byhavingacontextualizedgoalstatement,studentsknowhowtheyaregoingtobeapplying theirnewknowledgeandskills.But,youmayhavelessonsandlearningexperiencesthatfocusonconceptualunderstandingwithoutanycontextualizationatall.

Thereareseveralwaysthatyoucanbegintodevelopinstructionalunits.Let’slookattwo ways:startingwithaskillandstartingwithareal-lifeconcernforstudents.

Beginningwithaskill

Almosteveryadulteducationinstructorteachesfractions,solet’sbeginthere.Let’ssay youknowyouneedtoteachfractionoperationstoyourstudents.Youwantthemtobeabletoknowwhentomultiply(oruseanotheroperation)andyoualsowantthemto understandthatfractionsareusedinreal-life.Youprobablyalreadyhavemanylessons

thatyouusetoteachoperationswithfractions.But,howdoyouassesstheirabilitytousetheappropriateoperationinanygivensituation?Apageof decontextualizedfractionproblemswon’ttellyouwhetherstudentsactuallyknowthedifferencebetweenmultiplyinganddividingfractions,butreal-lifesituationswill.

So,whatmightbesomescenariosinwhichfractionoperationsmightbeused?Beforereadingon,youmightwanttojotdownyourownsampleunitgoalstatements.

Multiplication?Division?Subtraction?

Atthe endofthedinner,therewashalfapieleftintherefrigerator.Samdecidedtocuthimself1/3ofwhatwasleftlaterthatnight.Howmuchoftheoriginalpie wasleftby morning?

vs.

Atthe end ofthedinner,therewashalfapieleftintherefrigerator.Samdecidedtosharetherestofthepie withhistwofriends.How bigofaslicedid Samand histwofriends eachget?

Hereareafewexamplesofgoalstatementsillustratingreal-lifeapplicationoffractions:

Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingoffractionoperationsinordertomakeseveralbatchesofdogbiscuitsforthelocalanimalshelter.

Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingoffractionoperationsandskillinusingtapemeasuresinordertocalculatevariousattributesofbuildingcomponents.

Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingoffractionoperations,area,andperimeterinordertodesignapatiousingvarioussizesofbricks.

Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingofbenchmarkfractionoperationsindispensingliquidmedications.

Bytheendofthisunit,studentswillbeabletoadd,subtract,multiply,anddividefractionsandwriteandinterpretnumericalexpressionsinordertocreateabusiness planforasmallpizzarestaurant.

Bytheendofthisunit,studentswillbeabletousefractionsonarulerandconvertcentimeterstoinchesinordertocreateadreamhomeorhomeimprovementplaninthemostcost-effectivewayasempoweredconsumers.Theywillsharetheirplanswiththeclassbypresentingposters,photographs,orotherrepresentationsoftheirprojectuniquetotheirhousehold.

Beginningwithstudents’concerns

Anotherwaytobuildinstructionalunitsistofirstthinkaboutreal-lifeconcernsof students,thenbackintothemaththatneedstobetaught.Doaquickbrainstormyourself,jottingdownallthethingsthatareimportanttoyourstudents.

Irecentlyaskedagroupofteacherstodolikewise,andherearesomeoftheirideas:

Childcareissues

Needabetterjob

Makingendsmeetuntiltheendofthemonth

HealthcareCarissues

Caringforfamily

Lookatyourlistandtheoneabove.Arethereanytopicsthatcouldhaveamathfocus?ArethereanythatdoNOThaveamathfocus?

Onceyouhavedecidedonatopicofinteresttoyourstudents,youneedtothinkaboutsomescenarios.Let’susethechildcareexampletobrainstormsomepotentialsituations thatstudentsmightwanttoaddress:

Costperchild?

Doesageofchildmatter?

DoesitmakeadifferenceifIpayweeklyvs.monthlyvs.yearly?

Whataboutissuesrelatedtomyever-changingschedule?Doesthisimpactcost?

Howfarfromwork(orhome)isthechildcarefacility?

Aretheretaxcreditsthatsupportchildcarecosts? Howdothesecreditswork?

Foranyofthequestionsabove,thenumbersandamountofdatagiventostudents (hopefullytheywouldcollecttheirowninformationfromtheirlocalcommunity)could beadjustedsothatstudentsatvariouslevelscouldallengageinthetopic.Let’slookatthreedifferentinstructionalunitgoalstatementstoshowhowasimilarissuerelatedto childcarecostscouldplayout,dependingonthestudents’mathlevel.(I’mincludingin parenthesessomeexamplesofCCRStandardsthatmightbetaughtintheunitandthen appliedtoeachofthefinaltasks.)

ABElevel:Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingofwholenumberoperationsandsimplebargraphsinordertodeterminewhichoftwochildcareoptionsislesscostly.(PossibleCCRStandardsapplied:Representandsolveproblemsinvolvingmultiplicationanddivision.:3.OA.1,3.OA.2,3.OA.3,3.OA.4;Understandpropertiesofmultiplicationandtherelationshipbetweenmultiplicationand division.:3.OA.5,3.OA.6;Representandinterpretdata.:1.MD.4.)

Pre-ASElevel:Bytheendofthisunit,studentswillbeabletoapplytheirunderstanding offractions,percentages,andcirclegraphsinordertocreatetwodifferentbudgets basedondifferentchildcareoptions.(PossibleCCRStandardsapplied:Performoperationswithmulti-digitwholenumbersandwithdecimalstohundredths.:5.NBT.6, 5.NBT.7;Extendunderstandingoffractionequivalenceandordering.:4.NF.1,4NF.2;Buildfractionsfromunitfractionsbyapplyingandextendingpreviousunderstandingof operationsonwholenumbers:4.NF.3;Understandratioconceptsanduseratioreasoning tosolveproblems.:6.RP.1,6.RP.2.)

ASElevel:Bytheendofthisunit,studentswillbeabletoapplytheirunderstandingofalgebraicreasoningandfunctionsinordertoshowwithgraphsandequationswhich childcareoptionisthebestdeal(basedoncostonly).(PossibleCCRStandardsapplied:Generateandanalyzepatterns.:4.OA.5;Reasonaboutandsolveone-variableequations andinequalities.:6.EE.6,6.EE.7;Representandanalyzequantitativerelationships betweendependentandindependentvariables.:6.EE.9;Understandratioconceptsand useratioreasoningtosolveproblems.:6.RP.3;Analyzeproportionalrelationshipsand usethemtosolvereal-worldandmathematicalproblems.:7.RP.1,7.RP.2;Understand theconnectionsbetweenproportionalrelationships,lines,andlinearequations.:8.EE.5.)

Developinginstructionalunits,startingwithreal-lifestudentconcernsorspecificmath content,canhelpyouasaninstructorbecomemoremindfulofsituatingthemaththatyou teachinactivitiesthatrequirestudentstoapplywhattheyhavelearned.

References:

“TeachingMathematicsContextually:TheCornerstoneofTechPrep.”1999.CORD.

McTighe,Jay.andGrantWiggins.“UnderstandingbyDesignFramework.”2012. ASCD.