NC2ML
PRACTICE
BRIEF
THEROLEOFMATHEMATICALTASKS
WHYDOTHETASKSWECHOOSEMATTER?
Themathematicaltasksyouposeduringclassdeterminethenature
ofthemathematicsyourstudentswillengagein.Selecting“good”
tasksisnotatrivialact,itgreatlyimpactsyourstudents’learningand
theirdevelopmentofmathematicalhabitsofmind.Tasksthat
requirestudentstomemorizeaformulaorpracticeaparticularskill
leadstoadifferenttypeoflearningthantasksthatrequirestudents
toconsidermultiplesolutionstrategiesandconvinceothersthat
theirstrategiesarevalid.
QuestionsforDiscussion
1.
2.
3.
Inwhatwaysmightusinglowfloor-highceiling–
widewalltaskscreateamoreequitable
mathematicslearningenvironment?
Whataretheinstructionalchallengesyouface
whenyouusehighercognitivedemandtasks?How
mightyoumitigatethosechallenges?
Howdoyoudeterminewhattoolsyoumightmake
availabletoyourstudentstohelpcreatea“wide
wall”environmentforaparticulartask?
WHATISA“GOOD”MATHTASK?
Thedifferenttypesofthinkingrequiredofstudentstorespondtoataskisreferredtoascognitivedemand.SteinandSmith
(1998)developedaframeworkforcharacterizingthecognitivedemandofatask.Thisframeworkcanbeveryhelpfulwhen
selectingorrevisingtaskstomeetyourlearninggoals.Intheframeworktherearetwotypesofhighcognitivedemandtasks:
procedureswithconnectionsanddoingmathematics.Generally,highcognitivedemandtasksarenon-routine,requiringstudents
tothinkconceptuallyaboutmathematicalideas.Incontrast,lowcognitivedemandtasksrequirestudentstorememberfacts
andproceduresandapplythemtofamiliarsituations.Therearetwotypesoflowcognitivedemandtasksintheframework:
memorizationandprocedureswithoutconnections.Briefexamplesofeachtypeoftaskareprovidedinthefigurebelow.
Lowcognitivedemandtasks
Memorization
Example.Describethesystemas:
a)ConsistentandIndependent
b)ConsistentandDependent
c)Inconsistent
Procedureswithoutconnections
Example.Solveusingelimination.
3𝑥 + 5𝑦 = 6
3𝑥 − 5𝑦 = 18
Highcognitivedemandtasks
Procedureswithconnections
Example.Paulhas100dimesandquartersinhispiggybank.Thetotalamountof
moneyinhispiggybankis$15.25.Howmanyquartersanddimesdoeshehave?
Doingmathematics
Example.Allison’sparentshaveagreedtolethergetacellphone.However,
beforedoingso,shemustfindthebestcellphoneplanandconvinceherparents
whyitisthebest.Afterdoingsomeresearch,shewasabletonarrowherchoices
downtotwocompanies,HorizonWirelessandB-Mobile.HorizonWireless
chargesa$20feeandthen$10foreverygigabyteofdatausedeachmonth.BMobilecharges$40andthen$2foreverygigabyteofdatausedeach
month.PrepareanargumentforAllisontoconvinceherparentsthatshehasthe
bestplan.
2
North Carolina Collaborative for Mathematics Learning (NC ML) I UNCG School of Education I North Carolina Department of Public Instruction
PROVIDING“LOWFLOOR–HIGHCEILING–WIDEWALL”OPPORTUNITIES
Whilealllevelsoftasksareappropriateatdifferenttimesbasedonyourlearninggoals,ideallystudentsshouldexperiencemore
highcognitivedemandtasksthannot.Asynthesisofresearchhasshownthatcurricularmaterialsthatcontainhighcognitive
demandtasksaresuccessfulinimprovingstudents’conceptualunderstandingofimportantmathematics,inimprovingstudents’
abilitiestoreason,communicate,problem-solvingandmakemathematicalconnections,andimprovingtheirperformanceon
stateandnationalachievementtests(Boston&Wolf,2006).So,notonlyisitimportantforstudentsthatweincorporatemore
highcognitivedemandtasks,butalsothatweidentify“good”ones.Really“good”mathematicaltasksarethosethathaveentry
pointsforeverystudentinyourclassandallowspaceforextensionstokeepallstudentsmathematicallyengaged.Weoftenuse
thephrase“lowfloor,highceiling,andwidewalls”todescribesuchtasks.(Lowfloor=anyonecanaccessthetask;highceiling=
therearelotsofpossibilitiesfortakingthingsevenfurther;widewalls=spacetoexploremultiplepathways).
YOUDON’THAVETOREINVENTTHEWHEEL
Whiletheremanylowfloor–highceiling–widewalltasksavailableinexistingresources(e.g.,theNC2MLcanvassiteandthe
MRIdocuments),theyarenotalwayseasytofind.Asaresult,studentsareoftenpresentedlowcognitivedemandtaskseven
whentheydonotmatchthelearninggoalsforthelesson.However,manytaskscanbeeasilyadapted.Byprovidingless
directionandaskingbroaderquestionsataskcanbetransformedintoalowfloor–highceiling–widewallstypetask.When
consideringhowyoumightadaptataskitisimportanttoconsideritsintendedpurpose?Doyouplantousethetaskto
introduceanewmathematicalidea?Toapplyorpracticeamathematicalconcept?Orforassessmentofsomekind?Identifying
thepurposeofthetaskwillhelpyoumakedecisionsaboutadaptation.
FINALTHOUGHTS
LOWERCOGNITIVEDEMANDTASK
Jeremyhasbeenonhisparents’cellphoneplan,but
nowasheheadsoutaftergraduatingfromcollege,
heneedstogethisownplan.Hehasaparticular
smartphonethatonlyworkswithonecompany,but
thecompanyhasmultipleoptionsforplans.Jeremy
isconsideringtwodifferentmonthlycellphone
plans:
Cellphoneplan#1:noflatfee,$0.16permin
Cellphoneplan#2:flatfeeof$20+$0.10permin
Writeanequationandgenerateagraphto
representeachcellphoneplan.Useyourequation
andgraphtodeterminewhichplanisthebetterdeal
ifhetypicallytalks45minutespermonth?
HIGHCOGITIVEDEMANDADAPTATION
Jeremyhasbeenonhisparents’cellphoneplan,but
nowasheheadsoutaftergraduatingfromcollege,
heneedstogethisownplan.Hehasaparticular
smartphonethatonlyworkswithonecompany,but
thecompanyhasmultipleoptionsforplans.
Thefirsttwomonthlyplansarefairly“basic”in
Jeremy’smindandarelistedbelow.
Cellphoneplan#1:noflatfee,$0.16permin
Cellphoneplan#2:flatfeeof$20+$0.10permin
Whichisthebettercellphoneplan?Howdoyou
know?
(McCulloch,Lee,&Hollebrands,2015,CH2,p.29-30)
Itisimportanttonotethatusinghighcognitivedemandtaskstakes
carefulplanning.Beyondselectingtasksthatareappropriateforthe
mathematicalgoalsofyourlesson,youmustalsocarefullyconsider
bothhowyouwillsupportstudentsworkonthetaskandhowyou
willfacilitateadiscussionthatisfocusedonyourlearninggoals.See
Smith,Bill,&Hughes(2008)forausefulplanningprotocolasyouget
usedtoimplementingthesetasksinyourlessons.
References
Boston,M.,&Wolf,M.K.(2006).Asssessingacademicrigorinmathematicsinstruction:
ThedevelopmentoftheInstructionalqualityassessmenttoolkit.CSETechnicalReport
672.CenterfortheStudyofEvaluation,StandardsandStudentTesting,UCLA
McCulloch,A.W.,Lee,H.S.,&Hollebrands,K.F.(2015).Preparingtoteachmathematics
withtechnology:Anintegratedapproachtoalgebra(1stEdition).NCStateUiiversity,
Raleigh,NC.Availableat:https://ptmt.fi.ncsu.edu
Smith,M.S.,Bill,V.,&Hughes,E.K.(2008).Thinkingthroughalesson:Successfully
implementinghigh-leveltasks.MatheamticsTeachingintheMiddleSchool,14(3),
132-138.
Stien,M.K.&Smith,M.S.(1998).Selectingandcreatingmatheamticaltasks:From
researchtopractice.MathematicsTeachingintheMiddleSchool,3(5),344-350.
LEARNMORE
Joinusaswejourneytogethertosupportteachersandleaders
inimplementingmathematicsinstructionthatmeetsneedsof
NorthCarolinastudents.
NC2MLMATHEMATICSONLINE
FormoreinformationandresourcespleasevisittheNCDPImath
wikiforinstructionsonaccessingourCanvaspagecreatedin
partnershipwiththeNorthCarolinaDepartmentofPublic
Instructionathttp://maccss.ncdpi.wikispaces.net/
NorthCarolinaCollaborativeforMathematicsLearning
www.nc2ml.org
2
North Carolina Collaborative for Mathematics Learning (NC ML) I UNCG School of Education I North Carolina Department of Public Instruction