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Math Toolkit Time!
Using Manipulatives to Help Students Develop Common Core Math Mastery
Jacqueline Burns, Global Mathematics Consultant
SessionAgenda
• Introduction&Norms• ImportanceofManipulatives• PISA• Wholenumbercomputationandmathtools(+, −, ×, ÷)• Mini-lessoncreation• Fractionsense• Mathtoolexploration• Mini-lessoncreation• Debrief• Q&A
OurNorms
• Bepresentbothphysicallyandmentally.
• Listento,andmakeroomfor,theideasofothers.
• Shareyourknowledgeandwisdom.
• Tablesidetopics.• Exercisemobilephoneetiquette.
Whyusemanipulatives?Concrete.The“doing”stageusingconcreteobjectstomodelproblems
Representational.The“seeing”stageusingrepresentationsoftheobjectstomodelproblems
Abstract.The“symbolic”stageusingabstractsymbolstomodelproblems
WhatPISA15SaysAboutMathGlobally,andinJordan
Fromhttp://www.keepeek.com/Digital-Asset-Management/oecd/education/pisa-2015-results-volume-i_9789264266490-en#page178
MathematicsPerformanceamongPISA2015participants,atnational andsubnationallevels
Country Meanperformancescore
Singapore 564Jordan 380
Highestmeanperformancescore– 564Lowestmeanperformancescore-328
WhatPISA15SaysAboutMathGlobally,andinJordan
Fromhttp://www.keepeek.com/Digital-Asset-Management/oecd/education/pisa-2015-results-volume-i_9789264266490-en#page181
MeanPerformance
2012
MeanPerformance
2015
386 380
SolvingBasicFacts:PartnerWork
TheStudent’sHatWorkontheproblems,usingmanipulativestoconcretelymodelthesolution.
TheTeacher’sHatConsiderstudentabilities:
Directmodeling,counting,derivedfacts,recall
PROBLEMS
5+7=?
12– 5=?
4+?=11
5x7=?
56÷ 8=?
PROBLEM DIRECT MODELING COUNTING DERIVED FACTS RECALL
5+7=?JoinResultUnknown
Makesasetof5countersandasetof7counters.Pushesthetwosetstogetherandcountsallthecounters.
Counts“5[pause],6,7,8,9,10,11,12,”extendingafingerwitheachcount.“Theansweris12”[Thecountingsequencemayalsobeginwiththelargernumber]
“Take1fromthe7andgiveittothe5.Thatmakes6+6,andthat’s12.”
5plus7is12.
12– 5=?SeparateResultUnknown
Makesasetof12countersandremoves5ofthem.Thencountstheremainingcounters.
Countsback“12,11,10,9,8[pause],7.It’s7.”Usesfingerstokeeptrackofthenumbersofstepsinthecountingsequence.
“12takeaway2is10,andtakeaway3moreis7.”
12takeaway5is7.
4+?=11JoinChangeUnknown
Makesasetof4counters.Makesasecondsetofcounters,counting“5,6,7,8,9,10,11,”untilthereisatotalof11counters.Countsthe7countersinthesecondset.
Counts“4[pause],5,6,7,8,9,10,11,”extendingafingerwitheachcount.Countsthe7extendedfingers.“It’s7.”
“4+6is10and1moreis11.Soit’s7.”
4and7make11.
5x7=?Makes7groupsof5countersandcountsthemall.
5,10,15,20,25,30,35 5times5is25and10moreis35.
5times7is35.
56÷ 8=?Countsout56counters.Pullsoutgroupsof8until7groupsaremade.
8,16,24,32,40,48,56 8times8is64.8lessis56.Sothat’s7.
8x7is56.
Children’sStrategiesforSolvingBasicFacts
With,orWithoutManipulatives?…ThatistheQuestion
CreateaMini-LessonCreateamini-lessonthatinvolvestheuseofatleastone mathmanipulative
• CCSS-M
• Standard(s)forMathematicalPractice
• Interdisciplinaryconnection/s
• Independent/pair/smallgroup/wholegroup?
• Howwillstudentscommunicatemathematically?
• Identifythemanipulative/sandothertools.
ManipulativesforConsideration• Counters• Connectingcubes• Basetencounters• Numberline• Randomnumbergenerator(numbercubes)
• Spinners
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
LearningthroughtheStandardsforMathematicalPractice
CreateaMini-LessonCreateamini-lessonthatinvolvestheuseofatleastone mathmanipulative
• CCSS-M
• Standard(s)forMathematicalPractice
• Interdisciplinaryconnection/s
• Independent/pair/smallgroup/wholegroup?
• Howwillstudentscommunicatemathematically?
• Identifythemanipulative/sandothertools.
ManipulativesforConsideration• Counters• Connectingcubes• Basetencounters• Numberline• Randomnumbergenerator(numbercubes)
• Spinners
Inexactlyoneminutewritedownallthenumbersyoucanthinkof…….
FractionChallenge
between0and1
“The difficulty with fractions (including decimals andpercents) is pervasive and is a major obstacle tofurther progress in mathematics. . .”
—Report of the National Math Panel, March 2008
FractionSense
Whataresomeofthebiggestchallengesstudentsfacewithfractions?
InOneMinute…
• write down everything that comes to mind when you think about or see
78
Now,inoneminute…
• writedowneverythingthatcomestomindwhenyouthinkaboutorsee
9
CompareYourResponses
• Whatsimilaritiesdoyousee?
• Whatdifferencesdoyousee?
• Anysurprisesorinsights?
TheProgressionofFractionsNFStandards,Grades3-5
Priorknowledgethroughthelensofgeometry
Developunderstandingoffractions
• 3.NF1Understandafraction1/basaquantityformedby1partwhenawholeisportionedintobequalparts:understandafractiona/basthequantityformedbyapartsofsize1/b.
TASK:Locate1onthenumberline.Labelthepoint.
Beexactaspossible.
Whileitisnotnecessarytonamealloftheintervalsonthenumberline,manystudentsmaydoso.
TASK:Locate1onthenumberline.Labelthepoint.
Beexactaspossible.Whileitisnotnecessarytonamealloftheintervalsonthenumberline,
manystudentsmaydoso.
OtherHelpfulConsiderationsforUnderstandingFractions
• Wheredoweseeandusefractionsinourdailylives?• SupportStrategy:Theuseofpicturesprovidesstudentswithrealia forunderstandingfractions
• Wholesvs.Holes• Usevisualimagesthatsupportalllearnersforunderstandingthisconcept.
• FractionBasicVocabulary(denominator)• 2ispronounced“half”• 3ispronounced“third”• 4ispronounced“fourth”(or“quarter”)• 5ispronounced“fifth”• 6ispronounced“sixth”• 7ispronounced“seventh”• 8ispronounced“eighth”• 9ispronounced“ninth”• 10ispronounced“tenth,”andsoon.
EquivalentFractions
• Themeaningoffractionequivalence• Theequivalenceofwholenumbersandfractions• Explainingfractionequivalenceingeneral
Adjectivesvs.Nouns(AdaptedfromKathyRichardson,NCTM2008)
• Youngchildreninitiallyconsidernumbersasadjectives ordescriptors• 9bears• 6cookies• 20students
• Eventually,theycometounderstandnumbersasnouns orconcepts• 9ishalfof18,• Itis1lessthan10,• Itis4.5doubled,• Itis3squared,• Itisthesquarerootof81,• ………..?
Adjectivesvs.nouns(continued)
• Studentsneedopportunitiestotransitionfromconsideringfractionsasadjectives1/2 of a pizza3/4 of an hour2/3 of a cup
• toconsideringthemasnouns5/8 is…
a little more than 1/2, but less than 1It is 3/8 less than 1It is equivalent to 10/16Itistwiceordouble 5/16Itishalfof1¼……………?
Whichisbigger...𝟏𝟑𝒐𝒓 𝟏
𝟖?
Source- https://www.youtube.com/watch?v=g0nuomCCu9A
It’sYourTurn!CreateaFRACTIONSLessonthatsupports,develops,
reinforces,applies orextends understandingoffractions• CCSS-M• Standard(s)forMathematicalPractice
• Interdisciplinaryconnection/s
• Independent/pair/smallgroup/wholegroup?
• Howwillstudentscommunicatemathematically?
• Identifythemanipulative/sandothertools.
Math Problem Solving Strategies
Act It Out Give an OpinionChoose an Operation Make a GraphChoose a Strategy Make a ListDraw a Diagram Make a ModelDraw a Picture Make a TableDraw Conclusion Solve a HOD ProblemFind a Pattern Solve a Simple ProblemFind/Create/Make a Clue Write a Math Story
GuidelinesforImplementingtheCRAApproach•Choosethemathtopictobetaught.Planwhatistobetaughtaheadoftime.Sequencethelessonssotheystartbasicandgraduallyintroducenewtopics.•Reviewabstractstepstosolvetheproblem.Ask,whatisthedesiredmathoutcomeofthegroupoflessons?Determinetheproceduralgoalofthecombinationofmathskills.Listoutthestepsorprocedures.Adjustthestepstoeliminatenotationorcalculationtricks.Changeormodifystepstocreatethemostlogicalandsequentialsetofprocedures.•Matchtheabstractstepswithanappropriateconcretemanipulative.Initialunderstandingofcontentwillbebasedoninteractionswithconcreteobjects,sobecarefulwhichonesyouchoose.Theconceptualeffectivenessofthemanipulativeobjectshouldbenotedinaccordancetothemathskillbeingtaught.Avoidconcreteobjectsthatonlycoverafewskills.•Arrangeconcreteandrepresentationallessons.Practiceconcretemanipulations.Thesamequestionsthatyouencounteryoucanbecertainyourstudentswillaswell.Practicehowtomarkpictorialrepresentationsthatappearsimilartoconcretemanipulations.Makecertainthatyourlanguagethroughoutinstructionmatchesthelanguagerequiredforthedesiredoutcome.
Guidelines,continued
•Teacheachconcrete,representational,andabstractlessontostudentmastery(accuracywithouthesitation).Modelandguidestudentsintheiruseofmanipulativeobjectsandpictorialrepresentations.Teachstudentsstepbystepgraduallyintroducingmathematicalvocabulary.Allowstudentstonameorinventtheirstepwiseprocedureswithininstruction.Movefromconcretetorepresentationaltoabstractlearninglevelsonlyafterstudentsshowaccuracywithouthesitationsinmanipulationsordrawings.•Assesseachleveloflearningaccordingtostepwiseprocedures.Helpstudentsgeneralizelearningthroughwordproblemsandproblemsolvingevents.Incorporatewordproblemsthroughoutalessontohelpshowsocialrelevanceastowhyamathskillisimportanttolearn.
Source- http://nycdoeit.airws.org/pdf/Concret-Representational-Abstraction%20Approach%20for%20Expressions.pdf
Whatisfractionsense?
“Fractionsenseimpliesadeepandflexibleunderstandingoffractionsthatisnotdependentonanyonecontextortypeofproblem. Fractionsenseistiedtocommonsense:Studentswithfractionsensecanreasonaboutfractionsanddon’tapplyrulesandproceduresblindly;nordotheygivenonsensicalanswerstoproblemsinvolvingfractions.”
JulieMcNamara
PromotingStudentLearningandMotivationCharacteristicsassociatedwitheffectivemathematicsinstruction:• Studentsareactivelyengagedindoingmathematics.• Studentsaresolvingchallengingproblems.• Interdisciplinaryconnectionsandexamplesareusedtoteachmathematics.• Studentsaresharingtheirmathematicalideaswhileworkinginpairsandgroups.• Studentsareprovidedwithavarietyofopportunitiestocommunicatemathematically.• Studentsareusingmanipulativesandothertools.
Note:Adaptedfrom“WhatDoesGoodMathInstructionLookLike?”byNancyProtheroe,September/October2007,Principal. Retrievedfromhttps://www.naesp.org/resources/2/Principal/2007/S-Op51.pdf
AnEffectiveClassroomforMathematicsIneffectiveclassrooms,teachers:
• demonstrateacceptanceofstudents’divergentideas.
• influencelearningbyposingchallengingandinterestingquestions.
• projectapositiveattitudeaboutmathematicsandaboutstudents’abilityto“do”mathematics.
Note:Adaptedfrom“WhatDoesGoodMathInstructionLookLike?”byNancyProtheroe,September/October2007,Principal. Retrievedfromhttps://www.naesp.org/resources/2/Principal/2007/S-Op51.pdf
TheCriticalLanguageofLearning
*TakenfromTeachingtheCriticalVOCABULARYoftheCommonCore:55WordsThatMakeorBreakStudentUnderstandingbyMarileeSprenger (2013)
· Kindergarten:compare,contrast,describe,distinguish,identify,retell
· 1st:demonstrate,determine,draw,explain,locate,suggest,support
· 2nd:comprehend,develop
· 3rd:organize,refer
· 4th:infer,integrate,interpret,paraphrase,summarize
· 5th:analyze
· 6th:articulate,cite,delineate,evaluate,trace
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
LearningthroughtheStandardsforMathematicalPractice
Need Interactive and eResources?• http://www.debbiewaggoner.com/math.html• http://http://www.readtennessee.org/• http://www.corestandards.org• https://ccgpsmathematicsk-5.wikispaces.com/• http://www.k-5mathteachingresources.com/• http://www.insidemathematics.org
Questions?
