Partners for Mathematics Learning

1. 1 PARTNERS forMathematicsLearning GradeFive Module1 Partners forMathematicsLearning

2. 2 VideoOverview Welcometothefirstofsixmodulesof Partnersprofessionaldevelopment forteachersoffifthgrade Partners forMathematicsLearning

3. 3 NCEssentialStandards TakeabrieflookatthenewNCEssential Standards: What’snewinNumberandOperations? What’snewinAlgebra? Whatlooksfamiliar? Partners forMathematicsLearning

4. 4 Partners2009:5thGradeModules 1NumberandOperations:NumberSense andMultiplication 2NumberandOperations:Divisionand Fractions 3Measurement 4Geometry 5Data;NumberandOperations:Decimals 6Statistics(Data);ProcessStandards Partners forMathematicsLearning

5. 5 MathematicsLearning… Mathematicslearningisaboutmaking senseofmathematics Mathematicslearningisaboutacquiring skillsandinsightstosolveproblems NCTMPrinciplesandStandards2000 Philosophy Partners forMathematicsLearning

6. 6 InElementaryClassrooms… Whatshouldchildrenbelearning? Bigideasofmathematics Contentoftheessentialstandards Conceptsandprocedures Howdochildrenlearn? Throughprocessesofreasoning, communicating,representingideas,making connections,solvingroutineandnon-routine problems Partners forMathematicsLearning

7. 7 DevelopingMathematicalPower Whatdoteachersneedtodoto… Empowerstudentsto… Understandmathematics Usemathematics Enjoymathematics Haveconfidenceinthemselvesas mathematicsstudents Designinstructionaroundproblemsolving, reasoning,andsense-making Partners forMathematicsLearning

8. 8 ProblemSolving Problemsolvingmeansengaginginatask forwhichthesolutionorsolutionpathare notknowninadvance Theprocessofproblemsolvingshould permeatetheentireprogramandprovide thecontextinwhichskillsandconcepts canbelearned NCTMStandards1989,2000 Partners forMathematicsLearning

9. 9 ProblemSolving Solvingproblemsisnotonlyagoalof learningmathematicsbutalsoamajor meansoflearningmathematics Chooseproblemsthatengagestudents Createenvironmentthatencourages exploration,risk-taking,sharing,and questioning–developingconfidencein studentsengagedinproblem-solvingactivities NCTMPrinciplesandStandards2000 Partners forMathematicsLearning

10. 10 Numbersense develops… Overtime Throughmany experiences Alongside operationsense Number“glues”all strandstogether Numbersenseis foundationalto successfulproblem solving Partners forMathematicsLearning FoundationalIdeas

11. 11 ChallengesUsingNumberSense Findwaystowritethenumbers1–10 usingexactlyfour3’s Findwaystowritethenumbers11–20 usingexactlyfive2’s Writethenumbers0–50usingexactly five4’sandanysymbols: +,!×,÷,(),! Partners forMathematicsLearning

12. 12 NumberSenseIs... Aperson’sunderstandingofnumberconcepts, operations,applicationsofnumbers&operations Theabilitytousethisunderstandinginflexible waystomakedecisionsandtodevelopuseful strategiesforusingnumbersandoperations Theexpectationthatnumbersareusefulandthat mathematicsislogicalandmakessense Theabilitytousenumbersandapplicationsof numberstocommunicate,process,andinterpret information -fromMcIntosh,Reys,Reys,&Hope,NumberSENSE,Grades4-6 Partners forMathematicsLearning

13. 13 StudentswithNumberSense… Havewell-understoodnumbermeanings Understandmultipleinterpretationsand representationsofnumbers Recognizetherelativeandabsolute magnitudeofnumbers Appreciatetheeffectofoperationson numbers Havedevelopedasystemofpersonal benchmarks NCTMCurriculumandEvaluationStandards,1989 Partners forMathematicsLearning

14. 14 NumberRelationshipsCanBe… Comparedasgreaterthan, lessthan,orequal Decomposedintoa combinationofothernumbers Composedwithothernumbers tonameanewnumber Namedindifferentways Categorizedasmultiples, factors,powers,roots Partners forMathematicsLearning

15. 15 NumbersintheRealWorld Lookinnewspapers,magazines, billboards,andothersourcesfornumbers inthesecategories: Anumberinthemillions Aprimenumber Afractiongreaterthan½ Adecimalgreaterthan.75 Thefactorsof48 Amixednumber Partners forMathematicsLearning

16. 16 NumberLines 987 992 n 1017 Whatarethemissingvalues? Whatnumberisn? Howmightyoulabelthenumberline? Partners forMathematicsLearning

17. 17 PatternsinMultiples Noticeallthemultiplesofagiven numberonthe“PatternsinMultiples”Chart Comparethepatterneachnumbermakes withothernumberpatterns Whatarethesimilarities?Whatarethe differences? Whatdothesepatternstellyouaboutthe relationshipsofthesenumbers?

18. 18 DiscoveringPrimesandSquares Asatablegroup,cutoutallrectangular arraysthatcanbemadewitheachofthe numbers2through20(a3x4arrayisthe sameasa4x3array) Organizearraysandlistthedimensionsof thearraysyoumadeforeachnumber Whatdoyounoticeaboutthenumberof arraysthatcanbemadeforeachnumber? Partners forMathematicsLearning

19. 19 DiscoveringPrimesandSquares Aretherenumbersforwhichonlyone arraycanbemade? Whatarethesenumberscalled? Whatkindsofnumbershavemore thantwofactors? Partners forMathematicsLearning

20. 20 DiscoveringPrimesandSquares Forwhichofthenumbersdidyouhavean oddnumberoffactors? Howisthesetofarraysforthesenumbers differentfromothernumbers? Partners forMathematicsLearning

21. 21 Sample Classroom Questions ExploringFactors Whatisthesmallestpossible numberthathasexactly9factors? Whatisthesmallestcomposite squarenumber? Findanumberwithmorethan9factors Rollcomeinpacksof6;hotdogscomein packsof8.Howmanypacksofeach shouldwebuytouseeverything? Partners forMathematicsLearning

22. 22 WhatThenIsPrimeFactorization? Factoringanumberusingonlyprimefactors 180=15x12 3x5x3x2x2=180 Identifyingalloftheprimenumbersthat, whenmultipliedtogether,equalthevalue 30=3x2x5 18=3x3x2 or 18=32x2 Partners forMathematicsLearning

23. 23 WhatThenIsPrimeFactorization? Create2different“strings”offactorsfor eachproductbelow 24x15=360 21x12=252 48x60=2,880 Whatisthelongest“string”youcanfind? Whymaythenumber1notbeusedasa factor? Partners forMathematicsLearning

24. 24 WhatThenIsPrimeFactorization? Didyoufindthese“strings”? 24x15=360 •2x2x2x3x3x5=360 21x12=252 •3x7x2x2x3=252 48x60=2,880 •2x2x2x2x3x2x3x2x5 Whatwasyourstrategy? Partners forMathematicsLearning

25. 25 WhatThenIsPrimeFactorization? Factoringwith“trees”buildsonstudents’ knowledgeofnumberfactsanddivisibility 24 24 3 x 8 2x4 6 x 4 2x3x2x2 3 x 3 x 2x2x2 Determine:Afifthgradeexplorationor essentialstandard? Partners forMathematicsLearning

26. 26 NumericalGuessMyRule Whatarethelabels? Partners forMathematicsLearning

27. 27 NumericalGuessMyRule Whyaretheselabelsreasonable? Partners forMathematicsLearning

28. 28 NumericalGuessMyRule FindtheRules Partners forMathematicsLearning

29. 29 MysteryNumbers Iamthinkingofanumber Itisgreaterthan5x10 Itislessthan100 Itiseven Could itbe 60? Partners forMathematicsLearning

30. 30 MysteryNumbers Iamthinkingofanumber Itisgreaterthan5x10 Itislessthan100 Itiseven Itisnot70orless Itisnotamultipleof4 Itisnotamultipleof3 Nope. Itcould notbe 60! Partners forMathematicsLearning

31. 31 MysteryNumbers Iamthinkingofanumber Itisgreaterthan5x10 Itislessthan100 Itiseven Itisnot70orless Itisnotamultipleof4 Itisnotamultipleof3 Itislessthan80 Partners forMathematicsLearning Let me think

32. 32 MysteryNumbers Iamthinkingofanumber Itisgreaterthan5x10 Itislessthan100 Itiseven Itisnot70orless Itisnotamultipleof4 Itisnotamultipleof3 Itislessthan80 Partners forMathematicsLearning

33. 33 MysteryNumbers Ifyouadd5tomymysterynumberyouwill getthesameresultaswhenyousubtract mymysterynumberfrom89 Whatismynumber? = + ? - ? Partners forMathematicsLearning

34. 34 MysteryNumbers Writethemysterynumbercluesymbolically: n+5=89–n Trythese: 23+a=39–a 8xb=54–b 40–c=c+14 Partners forMathematicsLearning

35. 35 ComputationalFluency “Bytheendof[grade5]studentsshouldbe computingfluentlywithwholenumbers…. Studentsexhibitcomputationalfluencywhen theydemonstrateflexibilityinthe computationalmethodstheychoose, understandandcanexplainthesemethods, andproduceaccurateanswersefficiently. Thecomputationalmethodsthatastudentuses shouldbebasedonmathematicalideasthatthe studentunderstandswell.” FromPrinciplesandStandardsforSchoolMathematics,NCTM,2000,page152,emphasisadded Partners forMathematicsLearning

36. 36 ComputationalFluency “Flexibilitywithavarietyofcomputational strategiesisanimportanttoolforsuccessfuldaily living.Itistimetobroadenourperspectiveof whatitmeanstocompute.” “Nostudentshouldbepermittedtouseany strategywithoutunderstandingit.” JohnVandeWalle,TeachingStudent-CenteredMathematics,Grades3-5,pages101-102 Partners forMathematicsLearning

37. 37 ComputationalFluency Whatarewetryingtoachievewhen weteacharithmetictoday? RoteCalculatorsor ProblemSolverswhoexhibitarithmetic fluency* *Fluencymeanscomputing accurately,efficiently,andflexibly Partners forMathematicsLearning

38. 38 ResearchIsShowing… Teachingtraditionalalgorithmstooearly impedesthedevelopmentofnumbersense Studentsnottaughttraditionalalgorithms duringthefirst5yearsofschool(ages5-11) Acquiredanddevelopedgoodnumbersense, and Evenaftertheyweretaughtstandard algorithms,theypreferredtheirownmethods FromresearchofAlistairMcIntosh,UniversityofTasmania,andothers,reportedatICME-10,Copenagen,2004 Partners forMathematicsLearning

39. 39 ResearchIsShowing… Ratherthanalgorithms,thestudents… Performedmentalcalculations Explainedtheirownstrategiesfor computations,and Intheprocess,developednumbersense FromresearchofAlistairMcIntosh,UniversityofTasmania,Australia,andothers,reportedatICME-10,2004 Partners forMathematicsLearning

40. 40 ResearchIsShowing… Atleastshortterm,thereisevidencethata strategiesapproachtomentalcomputation Hasapositiveeffectonstudents’competence, confidenceandenjoyment Isaviablealternativeclassroomapproachto teachingmentalcomputation Isconsonantwithaconstructivistapproachto mathematicsteaching Improvesstudents’abilitytodiscuss,explain,and justifyorally FromresearchofAlistairMcIntosh,UniversityofTasmania,andothers,reportedatICME-10,2004 Partners forMathematicsLearning

41. 41 ResearchIsShowing… Anemphasisontraditionalalgorithmsandon mentalmethodsofspeedandaccuracy (ratherthanstrategies) Doesnotleadtonumbersense Providesaninefficientmethodofimproving thementalcomputationskillsespeciallyofless confident/competentstudents Inhibitsflexiblethinking FromresearchofAlistairMcIntosh,UniversityofTasmania, andothers,reportedatICME-10,2004 Partners forMathematicsLearning

42. 42 WhatAreOurGoalsandChallenges? “Theultimatepurposeofarithmetic instructionisthedevelopmentoftheability toTHINKinquantitativesituations” WilliamBrownell,1934 Partners forMathematicsLearning

43. 43 AlternativeStrategies Student-developedstrategieshave advantagesforchildrenovertraditional algorithms Theyarenumberorientedratherthandigit oriented Theyareleft-handedratherthanright-handed Theyareflexibleratherthanrigid Theyoftenemployoperationproperties adaptedfromJohnVanDeWalle,TeachingStudent-CenteredMathematics,Grades3-5,1997 Partners forMathematicsLearning

44. 44 Digitvs.NumberOrientation Traditionalapproachesemphasizeplace valueoftenmodeledonBase10materials withstandardverticalalgorithms Ratherthanbeingdigitoriented,anumber senseapproachorinventedstrategies approachisnumberoriented–not separatingadigitfromitsvaluewithin thenumber Partners forMathematicsLearning

45. 45 ResearchSays… Childrenneedinformal,reliable,butnot necessarilystandard,methodsofcomputing Childrendon'tspontaneouslydevelopthe standardalgorithms Childrenneedthechoicenottousethealgorithms -AlistairMcIntosh,UniversityofTanzania,emeritus,atICME-10,2004 Ifyouuseit,youmustunderstandwhyitworks andbeabletoexplainit -JohnVandeWalle,TeachingStudent-CenteredMathematics,Grades3-5,2006 Partners forMathematicsLearning

46. 46 HigherExpectations-NotLower Thispositiondoesnotmeanlower expectationsforcomputationalexpertise Rather,wehavehigherexpectationsfor accuratecomputingwithunderstanding Timeinvestedinunderstandingresultsin long-termlearning Partners forMathematicsLearning

47. 47 Multiplication:MentalStrategies Withoutusingthetraditionalalgorithm,how wouldyousolvethese? 14x15 325x4 333x20 Partners forMathematicsLearning

48. 48 WhatAretheMisunderstandings? Partners forMathematicsLearning

49. 49 EstimatingSolutions Whatisareasonableestimatefor 35x48? Howdidyoumakeyourestimate? Whywouldanestimatebehelpfulwhen computing? Partners forMathematicsLearning

50. 50 HowDoTheseWork?