math for primary teachers
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Helping students: job 1TRANSCRIPT
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MathematicsforPrimaryTeachersInEngland,andinternationally,numeracystandardsinprimaryschoolsarethecauseofasmuchconcernasliteracystandards.Oftenthegreatestproblemwithmathsat primarylevelistheteachersownunderstandingofthesubject.MathematicsforPrimaryTeachersaimstocombineaccessibleexplanationsofmathematical conceptswithpracticaladviceoneffectivewaysofteachingthesubject.Itisdividedintothreemainsections: SectionAprovidesaframeworkofgoodpractice SectionBaimstosupportandenhanceteacherssubjectknowledgeinmathematicaltopicsbeyondwhatistaughttoprimarychildren.Eachchapteralsohighlights teachingissuesandgivesexamplesoftasksrelevanttotheclassroom SectionCisacollectionofpapersfromtutorsfrom4universitiescoveringissuessuchastheteachingofmentalmathematics,childrensmathematicalmisconceptions andhowtomanagedifferentiation.Theyarecentredaroundthethemeofeffectiveteachingandqualityoflearningduringthiscrucialtimeformathematicseducation. ValsaKoshyisSeniorLecturerinEducationatBrunelUniversitywithresponsibilityformathematicsinservicecourses.PaulErnestisProfessorinMathematics EducationatExeterUniversity.RonCaseyisSeniorResearchFellowatBrunelUniversity.
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MathematicsforPrimaryTeachersEditedby ValsaKoshy,PaulErnestandRonCasey
LondonandNewYork
PageivFirstpublished2000 byRoutledge 11NewFetterLane,LondonEC4P4EE SimultaneouslypublishedintheUSAandCanada byRoutledge 29West35thStreet,NewYork,NY10001 RoutledgeisonimprintoftheTaylor&FrancisGroup ThiseditionpublishedintheTaylor&FranciseLibrary,2005.
TopurchaseyourowncopyofthisoranyofTaylor&FrancisorRoutledgescollectionofthousandsofeBookspleasegotowww.eBookstore.tandf.co.uk. 2000ValsaKoshy,PaulErnestandRonCaseyselectionandeditorial matter2000individualchapterstheircontributors Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedor utilisedinanyformorbyanyelectronic,mechanical,orothermeans,now knownorhereafterinvented,includingphotocopyingandrecording,orinany informationstorageorretrievalsystem,withoutpermissioninwritingfrom thepublishers. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationData Koshy,Valsa,1945 Mathematicsforprimaryteachers/ValsaKoshy,PaulErnest,and RonCasey. p.cm. Includesbibliographicalreferencesandindex. I.MathematicsStudyandteaching(Primary)I.Ernest,Paul. II.Casey,Ron.III.Title QA135.5.K6720009933554 372.7'044dc21CIP ISBN0203984064MasterebookISBN
ISBN0415200903(PrintEdition)
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Contents Listofcontributors Acknowledgements Introduction
viii x xi
SECTIONA
1 3 21
1 Teachingandlearningmathematics PAULERNEST
SECTIONB RONCASEYANDVALSAKOSHY
2 Wholenumbers 2.1Developmentofnumberconceptsintheearlyyears 2.2Theroleofalgorithms 2.3Placevaluerepresentationofnumbers 2.4Numberoperations 2.5Factorsandprimenumbers
23 23 25 25 29 37 38 41 41 50 59 60 61 66 66 68
2.6Negativenumbers 3 Fractions,decimalsandpercentages 3.1Fractions 3.2Decimals 3.3Indices 3.4Standardindexform 3.5Percentages 4.1Sequences 4.2Series
4 Numberpatternsandsequences
Pagevi 4.3Generalisedarithmetic 4.4Functions 4.5Identitiesandequations 4.6Equations 4.7Inequalities 5.1Theconceptofmeasure 5.2Length 5.3Area 5.4Volume 5.5Weight 5.6Time 5.7Angles 69 74 77 78 82 87 87 88 90 92 93 94 94 95 100 100 104 110 114 114 117 125 125 126 127 129 129 135 135 138 143
5 Measures
5.8Theuseofscales 6 Shapeandspace 6.1Coordinates 6.2Transformations
6.3Enlargement 7 Probabilityandstatistics 7.1Probability 7.2Statistics 8 Mathematicalproof 8.1Theroleofinduction 8.2Proofbyinduction 8.3Deductionsandarguments 8.4Conjecturesandsupportingevidence 8.5Lookingforexceptions 8.6Proofbycontradiction 8.7Generalisationandproof
Selfassessmentquestions Multiplechoicemathematics
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SECTIONC
147 149 158 172 182 196
9 Effectiveteachingofnumeracy MARGARETBROWN 10 Mentalmathematics JEANMURRAY 11 Childrensmistakesandmisconceptions VALSAKOSHY 12 Usingwritingtoscaffoldchildrensexplanationsinmathematics CHRISTINEMITCHELLANDWILLIAMRAWSON 13 Differentiation LESLEYJONESANDBARBARAALLEBONE
APPENDICES
211 213 216 219 220 222 223
Answerstoselfstudyquestions Answerstoselfassessmentquestions Answerstomultiplechoicemathematics Recordofachievement Mathematicalglossary Index
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ContributorsMargaretBrownisProfessorofMathematicsEducationatKingsCollegeLondonandamemberoftheNationalNumeracyTaskForce.Shewasintheworking partywhowrotethemathematicsNationalCurriculumandhasbeeninvolvedinmanyofthemajorinitiativesinthiscountryinMathematicsEducation.Shehas directedanumberofresearchprojectsinnumeracyinthelastfewyearswhichhaveguidedshapingnationalpolicy. JeanMurrayisDirectorofPrimaryEducationatBrunelUniversity.SheisalsoresponsibleforthedevelopmentofthemathematicscomponentsofthePGCEandBA coursesandteachesonthemathematicseducationinserviceprogrammeoftheUniversity.PriortoenteringHigherEducationshetaughtinprimaryschoolsinInner London. ValsaKoshyisSeniorLecturerinEducationatBrunelUniversity.PriortojoiningtheUniversityshewasamemberoftheILEAmathematicsadvisoryteamfora numberofyears.ShecoordinatesthemathematicsinserviceprogrammesattheUniversityandteachesInitialTrainingstudents.Shehaspublishedmanypractical booksforteachers:themostrecentonesareontheteachingofmentalmathsandeffectiveteachingofNumeracyintheprimaryschool. ChristineMitchelllecturesinprimarymathematicsattheSchoolofEducation,UniversityofExeter.ShehastaughtinprimaryschoolsintheUKandhasprovided consultancyandinservicesupportinassessmentandmanagement.Sheresearchesthedevelopmentofmathematicalreasoninginyoungchildren. WilliamRawsonlecturesinprimarymathematicsattheSchoolofEducation,UniversityofExeter.Aswellashavingtaughtinprimaryandsecondaryschoolsinthe UK,hehasawideexperienceofteachinginSouthAmerica,AfricaandAsia LesleyJonesisHeadofPrimaryInitialTeacherEducationatGoldsmithsCollege,UniversityofLondon.ShejoinedGoldsmithsCollegeafterbeingateacherfora numberofyearsandiscurrentlyinvolvedinteachingmathematicseducationtoInitialtraineesandpractisingteachers.SheeditsMathematicsinSchool,the professionaljournaloftheMathematicalAssociation,andhaswrittenmanybooksonpracticalapplicationsforteachers. BarbaraAlleboneisLecturerinEducationatGoldsmithsCollege,UniversityofLondonwheresheteachesmathematicseducationtobothInitialTrainingstudents andteachers.She
Pageix hastaughtinprimaryschoolsforanumberofyearsandhasledinservicetrainingofteachersinLEAspriortojoiningGoldsmithsCollege.Oneofhermajor researchinterestsistheeducationofAbleChildrenandtheroleofquestioninginextendingchildrensthinking.
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AcknowledgementsThisbookattemptstobringtogetherthetwostrandswhichwebelievecontributetothequalityofteachingandlearningofmathematicsandraisepupilsachievementin onebookthedevelopmentofteachersSubjectKnowledgeandPedagogicalSkills.Theauthorswishtoacknowledgeandthankvariouspeoplewhohelpedtomake thisbookareality. First,wethankthelargenumbersofpractisingteachersandInitialTrainingstudentswhomwehavetaughtformanyyears,andwhoprovideduswithvaluable insightsintoaspectsofmathematicseducation.Theseinsightshelpedustoselecttheaspectsofmathematicseducationincludedinthebook.Weareparticularlygrateful tothosewholookedatthedraftsandprovidedcriticalcommentaryatvariousstagesofwritingthebook. ThankstoBarbaraAllebone,MargaretBrown,LesleyJones,ChristineMitchell,JeanMurrayandWilliamRawsonwhowrotepapersontopicsofcurrentsignificant interestforSectionCofthisbook.Thesepeople,fromfouruniversities,areactiveatbothnationalandinstitutionallevelinpolicymakingandresearchrelatingto mathematicseducation.Weacknowledgetheirwillingnesstofindtime,withintheirbusyschedules,tocontributetothebook. ThankyoualsotoProfessorMartinHughesandProfessorTonyCrockerforactingasrefereestothepapersinSectionC. TheamountofsupportandincisiveandconstructivecommentsprovidedbyHelenFairlie,formerSeniorEditoratRoutledge,hasbeeninvaluable.Wethankherfor that. ValsaKoshy,PaulErnestandRonCasey
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IntroductionAsweapproachthemillennium,primarymathematicsteachingisatacrossroads.Theprimaryteachercannotaffordtotakethewrongpath.Thisbook,wehope,will bothassistinselectingtherightpathaswellasilluminatethejourneyalongit. EvidencefromOfstedinspectionsandfindingsfrominternationalcomparisonshavecausedconcernaboutchildrensmathematicalperformances.Asaresult,raising thelevelofachievementinmathematicsisnowstronglyonthenationalagenda.OneoftherecommendationsoftheNumeracyTaskForceforimprovingstandardsand expectationsistheneedforprimaryteacherstobesupportedinordertocovermathematicssubjectknowledgerelevanttotheprimarycurriculumandpupilslater development,andeffectiveteachingmethods(DfEE,1998).TheTeacherTrainingAgency(1998)introducedaNationalCurriculumforMathematicsforInitial TeacherTrainingstudentswhichrequiresthemtodemonstrateknowledgeandunderstandingofmathematicsaswellasthepedagogicalskillsrequiredtosecurepupils progressinmathematics.FromSeptember1999,primaryschoolteachersareexpectedtointroduceastructureddailymathematicslessonoftenreferredtoasthe numeracyhouraspartoftheNationalNumeracyStrategy.Muchemphasisisplacedonfocus,pace,balanceofknowledgeandskillsanddevelopmentof processes.Themessageisclear.Teachersneedtodeveloptheirsubjectknowledgeandhaveclearideasabouthowtoteachmathematicalideas. Webelievethatopportunitiesfordevelopinggreaterunderstandingofmathematicaltopicsandconsideringthemosteffectiveteachingskillswillgreatlyenhancethe qualityofmathematicsteachinginourschools.Thisbeliefhaspromptedustowritethisbook.Inselectingthecontentandstyleofthisbook,theauthorshavedrawnon theirconsiderableexperienceofbeinginvolvedinbothinserviceandinitialtrainingofteachers.Boththesegroupshavebeenconsultedatdifferentstagesduringthe writingofthisbook. Whatcontributestotheeffectiveteachingofmathematics?Askewetal.(1997)identifiedagroupofeffectiveteacherswhotheydescribedasconnectionists. Askewsummarises(Askew,1998)thattheseteachersemphasisedtheconnectionsby: valuingchildrensmethodsandexplanations sharingtheirownstrategiesfordoingmathematics establishingconnectionswithinthemathematicscurriculum,forexamplefractionsanddecimals.
Pagexii TheresearchteamatKingsCollege,London,foundthatthechildrenintheseteachersclassesachievedhigheraveragegainsintestsinnumeracyincomparisonwith othergroups.(YoucanreadaboutthisresearchinMargaretBrownspaperinSectionCofthisbook.) InlistingaspectsofgoodpracticeinteachingandlearningmathematicstheHMI(1989)attachesmuchimportancetopupilsmotivation: Distinctive,goodworkinmathematicswasgenerallyaccompaniedbyahighlevelofmotivationandengagementinthetask:thepupilsshowedinterest, commitmentandpersistence(p.26). Howcanteacherssupporttheirchildrentobemotivated,toshowinterestandcommitment?Ateachersownenthusiasmisanimportantfactor.Theyplayacrucialrole indevelopingtherightattitudeinthechildren.Wehaveallheardpeopleattributingtheirsuccessorfailureinlearningmathematicstoaparticularteacheroragroupof teachersinaparticularschool.Theproblem,however,isthatmanyadultsexperienceanxietyandfearwhentheytalkabouttheirownlearningofthesubject. Discussionswithteachersandstudentshaveoftenhighlightedtheseanxietiesandtheirlackofconfidenceaboutteachingthesubject.Theirconcernsusuallyhaveorigins fromtheirschooldays.Thereasonsfortheirinsecuritieshaveseldombeenlackofwillingnessorabilitytolearn.Whatwehavelistedbelowaresomeofthearticulated reasonsfortheirdislikeofthesubject.Acloselookatthesereasonswillbeagoodfirststepwhenconsideringhowtodevelopandenhanceonesownmathematics teaching.Thefollowingareamongthemostoftenmentionedcomments: Ineverunderstoodmuchofthemathematicsatschool,soIdonthaveenoughconfidencetoteachittochildren. Couldnotfollowteachersexplanations. Itwasalltoofastforme,Icouldntkeepup. Ijustlearnttherulesinordertopasstheexamination. BythetimeIgottothefinalyearsofmyschooling,thegapsinmyknowledgebaseweresowidethatIgaveup. Mathslessonsweresoboringandirrelevant. Iwasafraidoffailure,especiallyofbeingshownuptobeuseless. Idonthavethebasicmathematicsknowledgetoriskgivingmychildrenverychallengingwork. Thewordmathsmakesmehaveapanicattack. Aconsiderationoftheabovelistinitselfshouldgreatlyassistyouinyourpersonaljourneytowardsimprovingyourmathematicsteaching.Besidesofferingwhatwe hopewillbetherapeuticreading,thesecommentswillraisethemostimportantquestionhowcanI,asateacher,ensurethatmypupilswillnotdevelopanyofthe anxietiesintheabovelist? Thethreesectionsofthisbookaredesignedtosupportyouinyoureffortstodevelopyourpractice.InChapter1,PaulErnestprovidesaframeworkforreflecting onmanyissueswhichshouldsupportyourunderstandingaboutmathematicsteachingandlearning.Hefocusesontheaimsofmathematicsteaching,thenatureof mathematicsteaching,teachingstylesandtherequirementsoftheNationalCurriculumandStatutoryAssessment.Otheraspectsoflearningmathematicsspecial needs,equalopportunitiesandculturalissuesarealsoconsidered.
Pagexiii Chapters2to8,inSectionB,dealwithmathematicaltopicswhichcovertherequirementsoftheNationalCurriculumbeyondKeyStage2,theNationalCurriculum forteachertrainees(TTA,1998)andtheFrameworkforTeachingMathematicsfortheNationalNumeracyStrategy.Ineachofthesechaptersweconsiderspecific areasofmathematics.Eachchapterdealswithmathematicssubjectknowledgeatyourownlevelprovidingexplanationswithexamplesaswellasinterconnections betweentopics.Keyissuesintheteachingofthesetopicstochildrenarealsoverybrieflydealtwith.Ourbeliefisthatasyoureadthissection,manymathematicalideas whichwerenotunderstoodbeforeorforgottenwillbegintomakesense.AttheendofsectionB,youareprovidedwithsomeselfassessmentquestionsandagrid forauditingyourachievementandplanningpersonallearning.Afterundertakingtheassessmentyoumayneedtorevisitpartsofthatchapterordiscusstheideaswith otherstutorsandfriends. SectionCcontainsfivechaptersdealingwithtopicswhichweconsidertobetopicalandimportantinthecontextofourpursuitofexcellenceinmathematicsteaching andlearning.Inthesechaptersmathematicseducatorsfromfourinstitutionssharetheirexpertiseandresearchfindingswiththereaderinordertofacilitatereflectionand informedchoices.Werecommendthatyoumakenotesonthekeyideasineachchapterandshareyourthoughtswithyourcolleagues.
ReferencesAskew,M.(1998)PrimaryMathematics.Aguidefornewlyqualifiedandstudentteachers,London:HodderandStoughton. Askew,M.,Brown,M.,Rhodes,V.,William,D.andJohnson,D.(1997)EffectiveTeachersofNumeracy:AreportofastudycarriedoutfortheTeacherTraining Agency,London:KingsCollege,UniversityofLondon. DfEE(1998)NumeracyMatters.ThepreliminaryReportoftheNumeracyTaskForce,DepartmentforEducationandEmployment. HMI(1989)TheTeachingandLearningofMathematics,DepartmentforEducationandScience.London:HMSO. TeacherTrainingAgency(1998)InitialTeacherTrainingNationalCurriculumforPrimaryMathematics,(DfEECircular4/98).
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SectionA
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Chapter1 TeachingandlearningmathematicsPaulErnest
Whyteachmathematics?Everyteacherofmathematicsshouldaskthemselvesthefollowingbasicquestions.Whatismathematics?Whyteachmathematics?Theimmediatereasonwhywe teachmathematicsisthatwehaveto,andourchildrenmustlearnit,becauseitisintheNationalCurriculum.Butwhyisitthere?Theremustbeareasonwhyitis thoughttobesoimportant.Answeringthisquestionexplainswhyeveryonethinksmathematicsissoimportant,andwhatweshouldemphasisetoourchildren.Also, havingaclearideawhyweteachmathematicscanserveasasourceofinspiration,avisionofwhatourchildrencangainfromlearningthesubject. Thefirstandmostobviousaimisforchildrentogainknowledgethatisuseful.Butthereareusesofmathematicsatseverallevels.Togetthefullestbenefitchildren shouldbe: learningbasicmathematicsskillsandnumeracyandtheabilitytoapplythemineverydaysituationssuchasshoppingandtheworldofwork learningtosolveawiderangeofproblems,includingpracticalproblems understandingmathematicalconceptsasabasisforfurtherstudyinmathematicsandothersubjects,includinginformationtechnology learningtousemathematicsaspartofcitizenship,aspartofacriticalunderstandingofsocietyandtheissuesofsocialjustice,theenvironment,etc.Thisinvolvesbeing abletolookcriticallyatstatisticalclaimsandgraphsinadvertising,politicalclaims,etc. learningtosuccessfullyusetheirmathematicalknowledgeandskillsintestsandexaminations,togivethemthequalificationstheyneedforemploymentandfurther studyandtraining. Thesearealreadyambitiousaimswhichgobeyondthebasicusesthatmanyhaveinmindformathematics.Buttheseskillsareneededtopreparechildrenforthe advancedpostindustrialworldofthetwentyfirstcenturyandthesocialandenvironmentalproblemsitwillbring. Second,wemustaimforchildrentogainandgrowpersonallyasindividualsfromthestudyofmathematics.Childrenshouldbe: gainingconfidenceintheirownmathematicalskillsandcapabilities learningtobecreativeandexpressthemselvesthroughmathematics,includingexploringandapplyingmathematicsintheirownhobbies,interestsandprojects.
Page4 Mathematicsshouldbecontributinginthiswaytotheeducationoffullyroundedindividualswhoareconfidentandabletousewhattheyhavelearned,sometimesin originalandcreativeways. Third,weshouldaimforchildrentogainsomeappreciationofmathematics,byunderstandingsomeofitsbigideasandappreciatingtheirimportanceinhistory, societyandtheculturesoftheworld.Weliveinaninformationsociety,andchildrenshouldappreciatethatmathematicsisthelanguageofinformationandcomputers. Weareallpartofthefamilyofhumankind,andmathematicsisoneofthemostimportantcentralthreadsthatrunsthroughourhistoryandourpresentlife. Thesearesomeofthemostimportantaimsfortheteachingandlearningofmathematics.Childrenshouldgainusefulknowledgeandskills,theyshouldgrowandbe enhancedasdevelopingpersonsbyit,andtheyshouldgainabroaderappreciationofthesubject.Together,theyprovideagoodifnotcompleteanswertothe question,whyteachandlearnmathematics?
Whatismathematics?Toofewteachersinthepasthaveaskedthemselvesthequestion,whatismathematics?Ourviewofthenatureofmathematicsaffectsthewaywelearnmathematics, thewayweteachit,andwillaffectthewaythechildrenweteachviewmathematics.Inteachingandinlearningmathematics,toooftenwemoveonfromonetopicto thenext,fromoneskilltothenext.Itisalltoorarelythatwestandbackandtakeabroaderviewofmathematics,letalonesharethisviewwiththechildrenweteach. Sothisisaveryimportantquestion,onethatisessentialtoconsider,especiallyatthebeginningofabooklikethis. Therearedifferentanswerstothequestionaccordingtowhetherweaskmathematicians,philosophers,psychologistsoreducationalresearchers.Perhapsthemost usefulanswerforteacherscomesfromthereviewofresearchontheteachingandlearningofmathematicscarriedoutbyAlanBellandcolleagues(1983).This influencedboththeCockcroftReport(1982)ontheteachingofmathematicsandtheOfstedanalysisoftheaimsandobjectivesofteachingmathematics(HMI,1985). Belletal.distinguishedthedifferentthingsthatcanbelearnedfromschoolmathematics.Theseincludethelearningoffacts,skillsandconceptsthebuildingupof conceptsandconceptualstructuresthelearningofgeneralmathematicalstrategiesandthedevelopmentofattitudesto,andanappreciationof,mathematics.These differentlearningcomponentsofmathematicsareexploredinmoredetailbelow.
FactsTheseareitemsofinformationthatjusthavetobelearnedtobeknown,suchasNotation(e.g.thedecimalpointinplacevaluenotation%)Abbreviations(e.g.cm standsforcentimetre)Conventions(e.g.5xmeans5timesxknowingtheorderofoperationsinbrackets)Conversionfactors(e.g.1km=5/8mile)Namesof concepts(e.g.oddnumbersatrianglewiththreeequalsidesiscalledequilateral)andFactualresults(e.g.multiplicationtablefacts,Pythagorasrule). Factsarethebasicatomsofmathematicalknowledge.Eachisasmallandelementarypieceofknowledge.Factsmustbelearnedasindividualpiecesof information,althoughtheymayfitintoalargermoremeaningfulsystemoffacts.Whentheyfitinthiswaytheyaremucheasierandbetterremembered,butthenthey becomepartofaconceptualstructure.For
Page5 example96=54isafact.Butwhenachildalsoknowsthat96=69,andthat97hasonemoretenandonelessunit,and95hasonelesstenandonemoreunit, and96=(101)6,andsoon,thisfactispartofthatchildsconceptualstructure.
SkillsSkillsarewelldefinedmultistepprocedures.Theyincludefamiliarandoftenpractisedskillssuchasbasicnumberoperations.Theycaninvolvedoingthingsto numbers(e.g.columnaddition),ortoalgebraicsymbols(e.g.solvinglinearequations),ortogeometricalfigures(e.g.drawingacircleofgivenradiuswithcompasses), etc. Skillsaremostoftenlearnedbyexamples:firstseeingworkedexamples,andthendoingsome.Thatis,repeatedpracticeoftheskill,usuallyonexamplesof graduateddifficulty. Seeinghowlearnersactuallyperformskillsisavaluablelesson.Foraswellaslearningskills,childrenmakeerrors,oftenonthewaytolearningtheskills.Manyof theseerrorsarepartofarepeatedpattern.Theyoftenseemtocomefromchildrenlearningsomeofthepartsoftheskillbutmissingoutapart,orputtingthemtogether incorrectly.Othererrorscomefrommisapplyingarule.Forexample,inaddingfractions,manychildrensimplyaddthetopnumberstogether,andthebottomnumbers. Researchersfoundthatabout20%ofsecondaryschoolchildrenmadethefollowingmistake:1/3+1/4=2/7(seeHart,1981).Whyshouldtheydothis?Itseemslikely thattheyaremisapplyingtheeasiermultiplicationruleforfractions,butaddinginsteadofmultiplying. Errorpatternsinskillssuggestthatchildrenabsorbsomeofthedifferentcomponentstheyhavebeentaught,andputthemtogetherintheirmindsintheirown individualways.Thisleadstotheimportantconclusionthatchildrenthemselvesconstructtheirskillsandknowledge,basedontheirteachingandlearningexperiences, andiscalledtheconstructivisttheoryoflearning.Thisalsoexplainshowsomechildreninventtheirowncorrectbutunusualskills.
ConceptsandconceptualstructuresAconcept,strictlyspeaking,isasimplesetorproperty.Thisisameansofchoosingamongalargerclassofobjectsthosewhichfitundertheconcepts.Forexample, theconceptredpicksoutthoseobjectsthatweseewhichareredincolour.Theconceptofnegativenumberpicksoutthosenumberslessthanzero.Theconcept squarepicksoutjustthoseplaneshapeswhichhavefourstraightequalsidesidesandfourequal(right)angles.Aconceptistheideabehindaname.Tolearnthename isjusttolearnafact,buttolearnwhatitmeans,andhowitisdefined,istolearntheconcept. Aconceptualstructureissetofconceptsandlinkingrelationshipsbetweenthem.Itiscomplexandcontinuestogrowasthechildaddsmoreconceptsandlinks throughlearning.Forexample,placevalueandquadrilateralareconceptualstructures.Placevalueisthesystemofnumerationweusewhichsetsthevalueofadigit, e.g.9,accordingtoitspositionorplacing.So9intheunits,tens,hundredsandtenthsplacehasthevalue9,90,900,0.9,respectively,withzerosandthedecimal pointshowingitsposition.Understandingplacevaluemeansknowingthis,andthateachcolumnisworthtentimesmorethanitsrighthandneighbour,andatenthas muchasitslefthandneighbour.Somultiplicationby10,100,1000meansmovingthewholenumbertrain(allthedigitsinanumber)one,twoorthreeplaces,
Page6 respectively,totheleft.Italsomeansknowingthatthereisnoendtothesupplyofplacestotheleftandright,andthatthatnumbersofanysizecanbeexpressedwith tendigitsandadot. Quadrilateralmakesupasimplerconceptualstructure.Butitincludesknowingtherelationshipsbetweenpolygons,quadrilaterals,trapeziums,rhombuses, parallelograms,rectangles,squaresandkites. Theconclusionthatchildrenconstructtheirownknowledgeappliesevenmoretoconceptualstructures.Ourmemoryofallthathappenstous,bothinandoutof school,isputtogetherinauniquewayinourmind.IhavecertainpicturesIassociatewiththenumbers1to100,butotherpeoplewillhavedifferentpictures,orother feelingsorassociations.Inotherwords,ourconceptualstructuresforwholenumberaredifferent.Ofcoursetheyshouldsharesomefeatures,suchasthefactthat11 comesbefore12. Someresearchersdrewaverythoroughmapofthebasicknowledgeandskillsmakinguptwodigitsubtraction,withabout50components,notcountingindividual numberfactssuchas53=2(seeDenvirandBrown,1986).Theytestedquiteafewprimaryschoolchildrenandfoundthatalthoughthemapwasausefultoolin describingpersonalknowledgepatterns,itdidnthelppredictwhatthechildrenwouldlearnnext,evengiventeachingtargetedverycarefullyatspecificskills.Many childrendidnotlearnwhattheyweretaught,butmoresurprisinglylearnedwhattheywerenottaught!Thisfitswiththeconstructivisttheorythatchildrenfollowtheir ownuniquelearningpathandconstructtheirownpersonalconceptualstructures. Mostofthemathematicalknowledgethatchildrenlearninschoolisorganisedintoconceptualstructures,andthefactsandskillstheylearncanalsobefittedinor linkedwiththem.Themoreconnectionschildrenmakebetweentheirfacts,skillsandconceptstheeasieritisforthemtorecalltheknowledgeandtouseandapplyit.
GeneralstrategiesSolvingproblemsisoneofthemostimportantactivitiesinmathematics.Generalstrategiesaremethodsorproceduresthatguidethechoiceofwhichskillsor knowledgetouseateachstageinproblemsolving. Problemsinschoolmathematicscanbefamiliarorunfamiliartoalearner.Whenaproblemisfamiliarthelearnerhasdonesomelikeitbeforeandshouldbeableto rememberhowtogoaboutsolvingit.Whenaproblemhasanewtwisttoit,thelearnercannotrecallhowtogoaboutit.Thisiswhengeneralstrategiesareuseful,for theysuggestpossibleapproachesthatmay(ormaynot)leadtoasolution.Openendedproblemsorinvestigationsmayrequirethelearnertobecreativeinexploringa newmathematicalsituationandtolookforpatterns. Thefirstareainwhichmostchildrenlearngeneralstrategiesisinsolvingnumberproblems.Ifaskedtoadd15and47mentally,childrenlearntolookforwaysto simplifytheproblem.Thustheywilloftentrytomaketenwithpartoftheunits.Theymighttake3fromthe5toaddtothe7tomake10(15+47=12+50=62)orthey mighttake5fromthe7toaddtothe5tomake10(15+47=20+42=62).Somewillsimplyaddthetensandunitsseparately(15+47=50+12=62).Thegeneralstrategy isthatofsimplifyingtheproblemthroughdecomposingandrecombiningnumbers. Thefollowingaresometypicalgeneralstrategiesthatlearnershavebeenseentouseonavarietyofmorecomplexproblemsandinvestigations:
Page7 representingtheproblembydrawingadiagram tryingtosolveasimplerproblem,inthehopethatitwillsuggestamethod generatingexamples makingatableofresults puttingtheresultsintableinahelpful(suggestive)order searchingforapatternamongthedata thinkingupadifferentapproachandtryingitout checkingortestingresults. Generalstrategiesareusuallylearnedbyexample,orareinventedorextendedbythelearner.TheyarerecognisedasimportantintheNationalCurriculumforchildren ofallages,andthefirstattainmenttargetUsingandApplyingMathematicsismainlyconcernedwithdevelopingandusinggeneralstrategies.Threetypesofgeneral strategyareincludedinNationalCurriculummathematics.Thefirstismakingandmonitoringdecisionstosolveproblemsconcerningthechoiceofmaterials, proceduresandapproachesinproblemsolving.Thesecondisdevelopingmathematicallanguageandcommunicationwhichconcernstheoralcommunicationand writtenrecordingandpresentationofproblemsolvinganditsresults.Thethirdisdevelopingskillsofmathematicalreasoningconcerningmathematicalthinking,andthe useofreasoningtoarriveat,checkandjustifymathematicalresults.
AttitudesAttitudestomathematicsarethelearnersfeelingsandresponsestoit,includinglikeordislike,enjoymentorlackofit,confidenceindoingmathematics,andsoon.The importanceofattitudestomathematicsiswidelyaccepted,andoneofthecommonaimsofteachingmathematicsisthatafterstudy,alllearnersshouldlikemathematics andenjoyusingit,andshouldhaveconfidenceintheirownmathematicalabilities.Aswellasbeingagoodthinginitself,apositiveattitudeoftenleadstogreaterefforts andbetterattainmentinmathematics.However,toomanyyoungstersandadultssadlysaythattheydislikemathematicsandlackconfidenceintheirabilities.Someeven feelanxiouswheneveritcomesup. Attitudestomathematicscannotbedirectlytaught.Theyaretheindirectoutcomeofastudentsexperienceoflearningmathematicsoveranumberofyears. However,sometimesaparticularincidentcanchangeastudentsattitude,suchasteacherencouragementandinterestinthelearnerswork(positiveeffect),orpublic criticismandhumiliationofthelearnerinmathematics(negativeeffect).However,theseeffectsareunpredictableandtheydependonthelearnersownresponseto thesituation.
AppreciationTheappreciationofmathematicsconcernsunderstandingthebigpicture.Itinvolvessomeawarenessofwhatmathematicsisasawhole(theinneraspect),aswellas someunderstandingofthevalueandroleofmathematicsinsociety(theouteraspect).Thisouterappreciationinvolvessomeawarenessofthefollowing: 1mathematicsineverydaylife 2thesocialusesofmathematicsforcommunicationandpersuasion,fromadvertisementstogovernmentstatistics
Page8 3thehistoryofmathematicsandhowmathematicalsymbols,conceptsandproblemsdeveloped 4mathematicsacrossallcultures,inart,andinallschoolsubjects. Schoolmathematicstoooftentreatsonlythefirstofthese,withalittleofthesecond.Oftentheoutcomeisthatmathematicsisseenonlyasabagoftools,asetofbasic skills,mostlyinarithmetic,tobeusedwhenneeded.Butmathematicsismuchmorethanthis.Itisacentralelementinhumanhistory,societyandculture.Mathematical symmetryhasbeenacentralelementinreligionandartsincelongbeforerecordedhistorybegan.Thedevelopmentofmathematicalperspectiveheraldeda breakthroughinRenaissancepainting.Everyculturearoundtheworldusesmathematicalpatternsanddesignsintheirart,craftsandrituals.Science,information technology,andallthesubjectsoftheschoolcurriculumdrawuponaspectsofmathematics.Sotoneglecttheouterappreciationofmathematicsistoofferthestudent animpoverishedlearningexperience.Thisneglectmaybeforthebestofreasons,tocoverthenecessaryknowledgeandskillsofschoolmathematics.Butwhenan outerappreciationisneglected,notonlydoesschoolmathematicsbecomeslessinterestingandthelearnerculturallyimpoverished,italsomeansthatmathematics becomeslessuseful,aslearnersfailtoseethefullrangeofitsconnectionswithdailyandworkinglife,andcannotmaketheunexpectedlinksthatimaginativeproblem solvingrequires.Anouterappreciationofmathematicsisnotaluxuryoranoptionalextra.Itshouldbeapartofeverylearnerseducationalentitlement. Aninnerappreciationofmathematicsinvolvessomeawarenessofsuchthingsasthefollowing: 1bigideasinmathematicssuchassymmetry,randomness,paradox,proofandinfinity 2thedifferentbranchesofmathematicsandtheirconnections 3differentphilosophicalviewsaboutthenatureofmathematics. Toooftentheteachingandlearningofmathematicsinvolveslittlemorethanthepracticeandmasteryofaseriesoffacts,skillsandconceptsthroughexamplesand problems.Thisfitswiththewellknownviewthatmathematicsisnotaspectatorsport,i.e.thatitisaboutsolvingproblems,performingalgorithmsandprocedures, computingsolutions,andsoon.Suchactivitiesareofcourseattheheartofmathematics.Butiftheybecomethewholeofschoolmathematics,studentsmaynotseethe bigideasbehindwhattheyaredoing,letalonemeetandwonderatthebigideaswhicharenotintheNationalCurriculum,suchasparadox,infinity,orchaos.Yetideas likethisarewhatfirestheimaginationofmanyyoungpeople,aswellasthegrowingreadershipofpopularmathematicsbooks.Similarly,gettinganappreciationofthe differentbranchesofmathematicsandtheirlinks,e.g.thatofalgebraandgeometrythroughCartesiangraphs,orbecomingawareofcontroversiesinaccountsofthe natureofmathematics,isnotontheagendaofschoolmathematics.Butanyunderstandingofmathematicswithoutsomeelementsofthisinnerappreciationof mathematicsissuperficial,mechanicalandutilitarian.Iamnotproposingsomethingthatisunrealisticorexcessivelyidealistic,forinmyviewthisiswithinthegraspof virtuallyalllearnersofmathematics.Forexample,manyprimaryschoolchildrenarefascinatedbytheideaofinfinity,andhaveasenseoftheneverendingnessof counting.Whyshouldwenottrytodrawandbuilduponthisinterest,andtheirwonderandawe,intheteachingofmathematics?
Page9 Viewsconcerningthenatureofmathematicsasawholeformthebasisofwhatiscalledthephilosophyofmathematics.Therearemanydifferentviewsabout mathematics,butmostfallintooneofthreegroups. First,thereisthedualistviewthatmathematicsisafixedcollectionoffactsandrules.Accordingtothisview,mathematicsisexactandcertain,cutanddried,and thereisalwaysaruletofollowinsolvingproblems.Thisviewemphasisesknowingtherightfactsandskills.Thebacktobasicsmovementwhichemphasisesbasic numeracyasknowledgeoffacts,rulesandskills,withlittleregardforunderstanding,meaningorproblemsolving,canberegardedaspromotingadualistviewof mathematics. Second,thereistheabsolutistviewthatmathematicsisawellorganisedbodyofobjectiveknowledge,butthatanyclaimsinmathematicsmustberationallyjustified byproofs.ThetraditionalmathematicsofGCEOlevelsandAlevelswheretheemphasisisonunderstandingandapplyingtheknowledge,andwritingproofs,fits withtheabsolutistview. Third,thereistherelativistviewofmathematicsasadynamic,problemdrivenandcontinuallyexpandingfieldofhumancreationandinvention,inwhichpatternsare generatedandthendistilledintoknowledge.Thisviewplacesmostemphasisonmathematicalactivity,thedoingofmathematics,anditacceptsthattherearemany waysofsolvinganyprobleminmathematics. Althoughthefirst(dualist)viewisprimitiveandnotphilosophicallydefensible,boththesecondandthirdviewscorrespondtolegitimatephilosophiesofmathematics (seeErnest,1991).However,itisimportanttodistinguishbetweenstudentsviewsofschoolmathematics,teachersviewsofschoolmathematics,andteachersviews ofmathematicsasadisciplineinitsownright,forthesemaybeverydifferent.Inaddition,teachersandlearnersviewsofthenatureofmathematicsarenotnecessarily conscious.Theymaybeimplicitviewswhichteachersorstudentshavenotstoppedtoconsiderconsciously. TheAssessmentofPerformanceUnit(1985)conductedextensiveinvestigationsintoperceptionsofmathematics,aswellastowardsattitudestoit.Theyfoundthat studentsdistinguishedmathematicaltopicsashardeasyandasusefulnotuseful,andthatthesecategoriesplayedasignificantpartintheiroverallviewofmathematics. Theyalsofoundthatstudentstendeduniformlytoregardmathematicsasawholeasbothusefulandimportant,reflectingarealisticperceptionoftheweightthatis attachedtothesubjectinthemodernworld.
MathematicsintheNationalCurriculumWhatmathematicsis,isonething,butwhatchildrenhavetolearnisanother.However,mostoftheelementsdiscussedaboveareincludedinmathematicsinthe NationalCurriculum.Thisisthepublishedcurriculumthatallchildren516yearsofageinnormalstateschoolshavetofollow.Furthermore,althoughprivateschools arenotboundbylawtofollowit,virtuallyallofthemdo,becausetheyareaimingatthesametestsandexaminationsfortheirchildren.
ThestructureoftheNationalCurriculumOveralltheNationalCurriculumisorganisedinfourkeystages(seeTable1.1). PrimaryschoolingcoversKeyStages1and2.ItincludesthefollowingNationalCurriculumsubjects:English,mathematics,science,technology(designand technology,andinformationtechnology),history,geography,art,music,andphysicaleducation.Theonly
Page10Table1.1NationalCurriculumkeystages
KeystageKeystage1 Keystage2 Keystage3 Keystage4
Pupilsages57 711 1114 1416
YeargroupsYears12 Years36 Years79 Years1011
exceptionisinWales,whichalsoincludesWelsh(andEnglishisomittedinWelshspeakingclassesforKeyStage1). Foreachsubjectandforeachkeystage,thereareprogrammesofstudywhichsetoutwhatpupilsshouldbetaught.Therearealsoattainmenttargetswhichsetout thestandardsthatpupilsareexpectedtoreachinparticulartopics.Forexample,inmathematicsthefourattainmenttargetsare:UsingandApplyingMathematics Number(includingAlgebra,forolderchildren)Shape,SpaceandMeasuresandHandlingData.Inmathematics,asinmostsubjects,eachattainmenttargetisdivided intoeightlevelsofincreasingdifficulty,plusanadditionalhigherlevelforexceptionalperformance(beyondGCSE),forgiftedstudents.
MathematicsintheNationalCurriculumAtKeystage1,forpupilsaged5to7years,theprogrammeofstudyinmathematicshas3elements,whichcanbesummarisedasfollows: 1UsingandApplyingMathematics.Pupilsshouldlearntouseandapplymathematicsinpracticaltasks,inreallifeproblemsandinmathematicsitself.Theyshould betaughttomakedecisionstosolvesimpleproblems,tobegintochecktheirwork,andtousemathematicallanguageandtoexplaintheirthinking. 2Number.Pupilsshoulddevelopflexiblemethodsofworkingwithnumber,orallyandmentallyusingvariednumbersandwaysofrecording,withpracticalresources, calculatorsandcomputers.Theyshouldbegintounderstandplacevalue,developmethodsofcalculationandsolvingnumberproblems.Theyshouldalsocollect, recordandinterpretdata(laterthisbecomespartofHandlingData). 3Shape,SpaceandMeasures.Pupilsshouldhavepracticalexperiencesusingvariousmaterials,electronicdevices,andpracticalcontextsformeasuring.They shouldbegintounderstandandusepatternsandpropertiesofshape,positionandmovement,andofmeasures. TheprogrammeofstudyinmathematicsatKeyStage2forpupilsaged7to11yearshas4elements. 1UsingandApplyingMathematics.Pupilsshouldlearntouseandapplymathematicsinpracticaltasks,inreallifeproblemsandinmathematicsitself.Theyshould begintoorganiseandextendtasksthemselves,devisetheirownwaysofrecording,askquestionsandfollowalternativesuggestionstosupportthedevelopmentof theirreasoningskills.
Page11 Thereshouldbefurtherdevelopmentoftheirabilitytomakeandcheckdecisionstosolveproblems,tousemathematicallanguagetoexplaintheirthinking,andto reasonlogically. 2Number.Pupilsshoulddevelopflexiblemethodsofworkingwithnumber,inwriting,orallyandmentally,usingvariedresources,andwaysofrecording,and calculatorsandcomputers.Theyshoulddevelopanunderstandingofplacevalueandthenumbersystem,therelationshipsbetweennumbersandmethodsof calculation,andofsolvingnumberproblems.Theyshouldbegintounderstandthepatternsandideaswhichleadtothebasicconceptsofalgebra. 3Shape,SpaceandMeasures.Pupilsshouldusegeometricalideastosolveproblems,havepracticalexperiencesusingvariousmaterials,electronicdevices,and practicalcontextsformeasuring.Theyshouldbegintounderstandandusepatternsincludingsomedrawnfromdifferentculturaltraditionsandextendtheir understandingofthepropertiesofshape,positionandmovement,andofmeasures. 4HandlingData.Pupilsshouldlearntoaskbasicstatisticalquestions.Theyshouldcollect,representandinterpretdatausingtables,graphs,diagramsandcomputers. Theyshouldbegintounderstandanduseprobability. ThissummaryoftheNationalCurriculumcontainsmanyofthedifferentelementsofschoolmathematicsdiscussedabove.Firstofall,itspecifiesindetailthefacts,skills andconceptualknowledgethatchildrenneedtolearnintheareasofnumber,geometryandmeasurement(Shape,SpaceandMeasures),andprobability,statisticsand computermathematics(HandlingData).Secondly,thegeneralstrategiesofproblemsolvingaregivenanimportantplace,bothinthespecialattainmenttargetUsingand ApplyingMathematics,butalsointheotherstoo.Threemaintypesofstrategyareincludedinthefirstattainmenttarget.First,therearestrategiesforusing mathematics,sothatitbecomesapowerfultoolforchildrentoapplyinsolvingproblemsacrossarangeofcontexts.Second,therearestrategiesforcommunicating inmathematicssothatchildrencantalk,listen,readandwritemathematicswithunderstanding.Third,therearestrategiesfordevelopingideasofargumentand proof,sothatchildrencanmakeandtestpredictions,andcanreason,generalise,testandjustifymathematicalideasandarguments. AttitudestoandappreciationofmathematicsaretheelementsdiscussedabovewhicharemissingfromtheNationalCurriculum.Butthesearethingsthatcannot easilybetaughtortested,perhapsnotatall.IntheearlydevelopmentoftheNationalCurriculum,thefirstreportoftheMathematicsWorkingGroup(Departmentof EducationandScience,1987)includedlargesectionsonattitudesandappreciation.Butintheenditwasdecidedthatbecauseitwasnotpossibletospelloutexactly howtheyshouldbetaughtandtested,theyshouldpermeatethewholecurriculum.AsupplementtotheNationalCurriculumwaspublished,calledtheNonStatutory GuidanceforMathematics(NationalCurriculumCouncil,1989a).Thisemphasisesteachingmathematicssothatlearnersdeveloppositiveattitudestoandan appreciationofmathematics.Forexampleitstatesthefollowing: Mathematicsprovidesawayofviewingandmakingsenseoftheworld.Itisusedtoanalyseandcommunicateinformationandideasandtotacklea rangeofpracticaltasksandreallifeproblems. Mathematicsalsoprovidesthematerialandmeansforcreatingnewimaginativeworldstoexplore.Throughexplorationwithinmathematicsitself, newmathematicsiscreatedandcurrentideasaremodifiedandextended(p.A2).
Page12 Afterdescribingtheusefulnessofmathematicsineverydaylife,work,andotherschoolsubjects,thedocumentcontinuesasfollows: Asacomplementtoworkwhichfocusesonthepracticalvalueofmathematicsasatoolforeverydaylife,pupilsshouldalsohaveopportunitiesto exploreandappreciatethestructureofmathematicsitself.Mathematicsisnotonlytaughtbecauseitisuseful.Itshouldalsobeasourceofdelightand wonder,offeringpupilsintellectualexcitementandanappreciationofitsessentialcreativity(p.A3). Therearealsoothersectionswhichstresstheimportanceofdevelopingmathematicalappreciation,suchassectionFontheimportanceofcrosscurricularworkfor mathematics.Thedocumentalsoincludesrecommendationsforgoodmathematicsteaching,includingthefollowing. Activitiesshouldenablepupilstodevelopapositiveattitudetomathematics. Attitudestofosterandencourageinclude: fascinationwiththesubject interestandmotivation pleasureandenjoymentfrommathematicalactivities appreciationofthepower,purposeandrelevanceofmathematics satisfactionderivedfromasenseofachievement confidenceinanabilitytodomathematicsatanappropriatelevel(p.B11). Sothisdocumentpaysparticularattentiontothedevelopmentofpositiveattitudesandappreciationinmathematics,andtheimportanceoftheseelementsforthe NationalCurriculum.Overall,itisclearthattheNationalCurriculuminmathematicsemphasisesalloftheelementsofschoolmathematicslistedabove,includingfacts, skills,concepts,generalstrategies,attitudesandappreciation,somedirectlyandsomeindirectly.
TeachingandlearningmathematicsTheprevioussectionsdiscussdifferentelementsofschoolmathematics.Eachofthemplaysanessentialpartinallmathematicalworkandthinkingincludingusingand applyingmathematics.Facts,skillsandconceptualstructuresmakeupthenecessarybasicknowledgeforapplyingmathematicsandsolvingproblems.General strategiesareconcernedwiththetacticsofapplications:whattodoandhowtousethisknowledgetosolveproblems.Appreciationandattitudesalsocontributeto usingandapplyingmathematicsbyprovidinginterestandconfidenceandthroughfosteringpersistence,imaginativelinks,andcreativethinking. ThedistinctionbetweenthesedifferentelementsofschoolmathematicsandtheirimportancewaspartofthemessageofthelandmarkCockcroftReport,which influencedthedevelopmentoftheNationalCurriculum.Thisreportarguedthateachoftheseelementsrequiresseparateattentionanddifferentteachingapproaches. Onpurelyscientificgrounds,thereportconcluded,itisnotsufficienttoconcentrateonchildrenlearningfactsandskills,ifnumeracy,understanding,andproblem solvingabilityarewhatarewanted.Sothemoreextremeclaimsofthebacktobasicsmovementineducationwererejected:basicskillsalonearenotenough.Andthis argumentstillremainsvalid.
Page13 OnthebasisofitsreviewofpsychologicalresearchtheCockcroftReportmadeitsmostfamousrecommendation. Mathematicsteachingatalllevelsshouldincludeopportunitiesfor *expositionbytheteacher *discussionbetweenteacherandpupilsandbetweenpupilsthemselves *appropriatepracticalwork *consolidationandpracticeoffundamentalskillsandroutines *problemsolving,includingtheapplicationofmathematicstoeverydaysituations *investigationalwork(Cockcroft,1982,paragraph243). Sotheteachingapproachesneededtodevelopthedifferentelementsofmathematicsatanylevelofschoolingincludeinvestigationalwork,problemsolving,discussion, practicalwork,exposition(directinstruction)bytheteacher,aswellastheconsolidationandpracticeofskillsandroutines.Figure1.1showshowtheseteaching approachescanhelptodevelopchildrensappreciationofmathematics,strategiesfortacklingnewproblems,conceptualstructuresinmathematics,aswellastheir knowledgeofmathematicalfactsandskills. TheconnectinglinesinFigure1.1showsomeofthemoreimportantinfluencesofdifferentteachingandlearningstylesonthelearnedelementsofschool mathematics,butfurtherlinescouldbeadded.ThemostimportantpointmadebytheCockcroftReportisthatifwewantalloftheoutcomeslistedontherighthand sidetobedeveloped,thenweneedtousethemixofapproacheslistedonthelefthandsideofthefigure. TheCockcroftmodelofteachingstrategiesisabalancedone,becauseitsaysthatnoonemethodshoulddominate,andthemethodwechooseshoulddependon whatwewantthechildrentolearn,andwhatissuitablefortheresourcesavailableandforthechildrenand
Figure1.1Therelationbetweenteachingstylesandlearningoutcomes
Page14 school.Nevertheless,teachingapproachescanbecontroversialaccordingtowhethertraditionalorprogressiveapproachesareinfashion. Aquicklookatthehistoryofprimarymathematicsconfirmsthis.Inthe1950schildrenmainlyworkedonarithmeticandmeasuresintheformofoldfashionedsums, notallthatdifferentfromVictorianarithmetic.Inthe1960sprimaryschoolchildrennotonlybegantostudymodernmathematics,insteadofjustarithmetic,butthere wasalsoanewemphasisonpracticalwork,problemsolvinganddiscoverylearninginmathematics.Thiswasduetotheinfluenceofanewwayofthinkingexpressed intheNuffieldprimarymathematicsprojectandHerMajestysInspectorEdithBiggswidelyreadreportonprimarymathematics.Inthe1970stherewasareaction, thebacktobasicsmovement,butthemostsignificantdevelopmentwasthewidespreadadoptionofindividualisedprimarymathematicsschemesinschool,which childrenworkedfromattheirownpace.Althoughthesepersistedinthe1980s,thisdecadealsosawtheendorsementofproblemsolvingandinvestigationalworkin mathematicsbytheCockcroftReportandHMI,andlatertheNationalCurriculumthroughtheattainmenttargetUsingandApplyingMathematics.Soprimaryschool teachersworkedhardtoincludethisinthemathematicscurriculumtoo,althoughmanywereworriedanddidnotfeeltheyfullyunderstoodwhatwasinvolved.Inthe mid1990stherehasbeenanofficialturnagainstprogressiveteachingapproachesandwholeclassinteractiveteachinghasbeenendorsedbyOFSTEDandthe governmentDepartmentforEducationandEmployment.Thusthenewnumeracyhourrequires(orratherstronglyrecommends)thatskillspracticeandwholeclass teachingshouldbeuseddailyinprimarymathematics. TheCockcroftmodelofteachingshouldsatisfyallofthesechangesinfashion,becauseitarguesthatchildrenneedtoexperienceallapproachesinabalancedway, notjustlearnercentredapproaches(problemsolving,investigationalandpracticalwork,pupiltopupildiscussion)orteachercentredones(teacherexposition, consolidationandpracticeofbasicskills,teacherleddiscussion).Furthermore,thesocalledchildcentredapproachesarenecessaryifchildrenaretobeableto makepracticaluseofthemathematicstheylearn,asteachingUsingandApplyingMathematicsmakesveryclear. Thedifferentpurposesoffourmaindifferentteachingapproachesareasfollows.Firstofall,indirectinstruction,theteacherstatesandshowstheclasstherules, skillsorconceptstobelearned,andprovidestheclasswithexercisestoapplythisnewknowledge,oftenshowingworkedexamples.Thestudentslistenandwatch, andthenapplythenewknowledgetotheexercisesset.Insodoingtheyarelearningandapplyingorpractisingandreinforcingfacts,skillsandconcepts. Inguidedinstructiontheteacherarrangespracticaltasksorasequenceofexampleswhichhaveapatternorindirectlyembodyaconceptorrule.Whatthelearner hastodoistoworkthroughthetasksandspottheruleorlearntheconceptorskillimplicitinthegivenexamples.Thelearnerhastoworktogainthenewknowledge, andaswellasdevelopingunderstandinglearnstospotpatternsandtogeneralise. Inproblemsolving,theteachersroleistopresentproblems,butleavethesolutionmethodsopentothestudents.Learnershavetoattempttosolvetheproblems usingtheirownmethods,andlearntobecomeindependentproblemsolvers,aswellasdevelopingtheirgeneralstrategies. Ininvestigatorymathematics,theteacherpresentsaninitialmathematicaltopicorareaofinvestigation,ormayapproveastudentsownproject.Thelearnersrole istoaskthemselvesrelevantquestionstoinvestigateintheprojectareaandtoexplorethetopicfreely,hopingto
Page15 developsomeinterestingmathematicalideas.Sothisapproachencouragescreativethinkingaswellastheuseofproblemsolvingstrategies. Table1.2summariseswhatisinvolvedinthesefourteachingapproaches.Foreachapproachitshowswhattheteacherdoes,whatthelearnerdoes,andthe processesinvolved. Needlesstosay,thisisaverysimplifiedpictureofwhatgoesoninthedifferentteachingapproaches.Forexample,ininvestigatorymathematicstheteacherdoes muchmorethanjustpresentinganinitialareaofinvestigationorapprovingastudentschoicesuchasmaintaininganorderlyclassroom,circulatingamongthechildren askingquestionstogetthemthinkinginnewways,orgettingthemtochecktheirwork:controllingtheuseoftimeandequipment,andsoon.Tobeworthwhilesuch activitiesmusttakeplacewithinanoverallcurriculumplanforteachingtheNationalCurriculum. Twothingsshouldbestressedaboutthisrangeofteachingapproaches.Firstofall,ineachcasethelearnersareinvolved,takinganactivepartintheirownlearning. Thisisessentialforsuccessfullearning,andisdiscussedmoreinthenextsection.Second,althoughsometimeschildcentredteachingapproachesareregardedas inefficientandwastefuloftime,theyprovidesomethingthatteachercentredapproachescannot.Thisispracticeincreativeusesandapplicationsofmathematics. Recentlytherehavebeenanumberofinternationalcomparisonsofachievementinmathematics.Britishchildrenatages9and13havecomeoutbelowaverageon numberskills.Thisissomethingthatneedstobeimprovedupon,andhasbeenmuchcriticisedinthepress.IncontrastchildreninJapan,SingaporeandotherPacific Rimcountrieshavecomeouttopinthisarea.ExpertshavebeensentouttotheFarEasttofindoutwhattheirsecretis.However,inproblemsolvingandthepractical applicationsofmathematics,Britishchildrencameoutverynearlytop.Thisissomethingweshouldbeproudof,butlittlehasbeenwrittenaboutitinthepapers. Interestingly,theJapaneseexpertsaresayingthattheirownstudentsarenotcreativeenoughintheirthinking,andfutureeconomicsuccesswilldependupondeveloping this.SotheyaresendingexpertstoBritainandtheWesttofindouthowweteachcreativeproblemsolvingsowell.Soweshouldtrytokeepupthetraditionofusinga varietyofteachingapproachesbecausethisiswhatishelpingtodevelopTable1.2Theroleofteacherandstudentindifferentteachingapproaches
Teachingapproach RoleoftheteacherDirectinstruction Guidedinstruction Problemsolving Investigatory mathematics Toexplicitlyteachrules,skillsorconcepts,andprovide exercisesforapplication Togivepracticaltasksorasequenceofexamples representingaconceptorruleimplicitly Topresentaproblem,leavingsolutionmethodsopen Topresentaninitialareaofinvestigation,orapprovea studentsproject
RoleofthelearnerToapplythegivenknowledgetoexercises
ProcessinvolvedThedirectapplicationsoffacts, skillsandconcepts
Toidentifytheruleorconceptimplicitinthegiven Generalisation,rulespotting, examples conceptformation Toattempttosolvetheproblemusingownmethods Problemsolvingstrategies Tochoosequestionstoinvestigateinprojectarea andtoexplorethetopicfreely Creativethinkingandproblem solvingstrategies
Page16 alltheskillsandcapabilitiesourchildrenwillneedfortheworldofthetwentyfirstcentury,includingcreativeproblemsolvingskills.
LearningmathematicsInthepastfewdecadeswehavecometoknowmuchabouthowchildrenlearnmathematics.TheSwisspsychologistJeanPiagetmadeextensivestudiesofchildren learning,payingspecialattentiontomathematics.Hehadthegreatinsight,basedonhisobservations,thatchildreninterprettheirsituationsandschooltasksthrough schemasthattheyhavebuiltup,whicharetheconceptualstructuresdiscussedearlier.Thesestructuresguidewhatchildrenunderstand,whattheyexpect,andhowthey actorrespond.Histheoryplacesgreatemphasisonoperations,whetherphysical,imaginedormathematical,andsuchfeaturesaswhethertheycanbeundone,once done,andwhatstaysthesameduringoperations.Theseideashavedirectapplicationstomathematicaloperationswhere,forexample,theoperationofadditioncanbe undone(subtraction),andchangingtoequivalentfractionsleavesthevalueunchanged. Piagetalsohadatheorythatchildrensdevelopmentgoesthroughfixedstages,withdifferentkindsofthinkingateachstage.Whilethereissometruthinthisonthe largescale,forexamplechildrenusuallyhavetomasterthemorebasicideasofnumberbeforetheymoveontothemoreabstractideasofalgebra,ithasbeenshown thatlanguageandthesocialcontexthavemoreinfluenceonthechildsdevelopmentthanPiagetthought. OneoftheimportanttheoriesbasedonPiagetsworkiscalledconstructivism,whichwasmentionedaboveindiscussingconceptualstructures.Thisisthetheorythat firstofall,alllearners(indeedallpersons)makesenseoftheirsituationsandanytasksintermsoftheirexistingknowledgeandschemas(conceptualstructures).So existingschemasactlikeapairoftintedspectacles,everythingseenisseenthroughthemandcolouredbythem.Secondly,allnewknowledgeisbuiltupfromexisting ideasandknowledgeextendedorputtogetherinanewway.Thismeansthatweonlyunderstandnewthingsintermsofwhatwealreadyknow.Thirdly,alllearningis active,althoughthisactivityisprimarilymental,sobeingtoldorshownthingsmaysuggestnewwaysforthelearnertointerpretorconnectherexistingknowledge,but cannotdirectlygivetheknowledgetothelearner,Childrendonotjustreceiveknowledge,theyhavetoreconstructit.Inotherwords,forproperlearningweneed tofullyunderstandnewideasbeforewecanmakethemourown.Fourth,becauseoftheactivenatureofalllearning,mistakesareanaturalpartofthesamecreative processthatresultsinstandard(correct)knowledgeandskills.Learnersneedtobeguidedtotestandtoadjusttheirunderstandingstowardsthestandardknowledge. Somistakesareanecessaryandgoodthing,asstepsonthewaytoproperunderstanding.Childrenneedtofeelfreetotrythingsoutandmakemistakeswithoutany shame,fearorfeelingtheneedtohidethem,sothattheycancorrectthemandcontinuetolearnwithouttheinterferenceofanybadfeelings. Theseareimportantideas,withobviousapplicationsintheteachingandlearningofmathematics.HoweverPiagetsisnottheonlythetheoryoflearningandmany peoplealsolooktothetheoriesofthegreatRussianpsychologistLevVygotsky(1978).Hisideaisthatlanguageandsocialexperienceplayadominantroleinlearning. Hearguesthatmostnewknowledgeislearnedthroughlanguageandothersymbolicformsincludingpictures,diagramsandmathematicalsymbols,andwefirstmeet thesewhentheyarepresentedtousbyotherpersons.Sowelearnlanguagebyhearingitused,byimitating,throughbeingguidedandcorrected,andfromthiswe attainbasicmasterywhichweexpandthroughuseandpractice.Vygotskydescribeswhatalearnercandoasintermsofzones.Thefirstzone
Page17 consistsofwhatthestudentcandounaided,soitismadeupoftheabilitiesdevelopedsofar.Thesecondzoneconsistsofwhatstudentcandowithhelpfromsomeone else,theirteacher,peersorparents.Thesemakeupthetasksandabilitieswithinreach,butnotyetattained.ThiszoneiscalledtheZoneofProximalDevelopment. Vygotskystheoryisthatteachingshouldbedirectedatthiszone,becauseitextendswhattheleanercandounaided.Sothestudentcanbeshownsimpleworked examplestoimitate,andafterthisexperiencewillgraduallymastertheskillsortypesoftasks.Indeed,understandingmaynotcomeuntillater,aftertheskillhasbecome routine.
CrosscurriculardimensionsThisbookisabouttheteachingandlearningofmathematicsintheprimaryschool.However,whatyouactuallyteachischildren,andtheydonotnecessarilydoalltheir learninginseparatesubjectboxes.Mathematicsisjustoneoftheseboxes,andinteachingitwealwaysneedtobeawareofhowitlinkswithothersubjects,and withchildrensownexperiencesandtheirlives.OneimportantinnovationintheNationalCurriculumistopayspecialattentiontotheselinks,intheformofcross curriculardimensions,themesandskills(NationalCurriculumCouncil,1990).Thesearecrosscurricularlinksandideaswhicharecommontoallofthesubjectsofthe schoolcurriculum,andwhicharesupposedtoweavethemalltogetherintoaunifiedwhole.Thecrosscurricularskillsarenumeracy,literacy,oracy,information technologyskills,andpersonalandsocialskills.Clearlychildrenmustlearnnumberandtheuseofcalculatorsandcomputersinmathematics.Buttheyalsomustlearnto readandwrite,listenandspeakintheirmathematicslessons.Personalandsocialskillscomeupeverywhere,inlearningtoworktogether,inlearningtolisten,respect andvalueeachothersideasandcontributions,andsoon.Sotheseskillsarenotdifficulttoseeandincludeinmathematicsteaching. Thecrosscurricularthemesincludeeconomicandindustrialunderstanding,careers,health,citizenship,andtheenvironment.Evenintheprimaryschool,children needtobedevelopinganunderstandingoftheeconomicbasisofsocietyandanawarenessoftheworldofwork,andthecentralroleofmathematicsintheseareas. Theymustalsobegintounderstandtheirrolesasfuturecitizensandhowtheirchoicesaffecttheirhealthandtheenvironment.Somuchoftheinformationabouthealth andtheenvironment,whetherlocal,nationalorglobalisbestdisplayedmathematically,usingnumbersandgraphs.Evenfromaveryyoungagechildrencareabout whatishappeningtotheenvironmentandteacherscanbuildonthisbothtoteachthemmathematicsandtohelpthemgrowintocaringandresponsiblecitizens. ThecrosscurriculardimensionsidentifiedbytheNationalCurriculumCouncilareequalopportunities,multicultural,andspecialeducationalneeds.Equal opportunitiesareaboutthedifferentopportunitiesgiventoboysandgirls,andtheimportanceoffairnessintheirtreatment.Inthepast,mathematicswasthoughtofasa boyssubject,andoftenboyswereencouragedandgirlsdiscouragedinmathematics.Mostlythiswasdoneinunintendedways,liketeachersaskingboysmore challengingquestions,andmathsschemesshowingfewerpicturesofgirlsandwomen,andthenmostlyinpassiveortraditionalroles.Sincethe1980sthishaschanged, andnowgirlsdoaswellinmathematicsasboysthroughoutallofprimaryandsecondaryschooling,andmostteachersexpectasmuchfromgirlsasboys.However thereisstillaresidualbeliefinsocietythatmathematicsisamalesubject,andresearchshowsthatgirlsarestill,onaverage,lessconfidentabouttheirmathematical abilitythanboys
Page18 (Walkerdine,1998).Soitisjustasimportanttodaythatteachersshouldprovideequalopportunitiesintheirclassrooms,andtrytodevelopconfidenceinalloftheir children. ThemulticulturaldimensionisthesecondinthesetoflinksidentifiedbytheNationalCurriculumCouncil.Thereisamistakenviewthatmulticulturalmathematicsis aboutaccommodatingtheneedsofethnicminoritystudentsintheclassroom.Actually,multiculturalmathematicsisabouttheappreciationofmathematicsdiscussedat thebeginningofthischapter.Itincludesappreciatingthehistoricalandculturalrootsandusesofthesubject.ThroughlearningabouttheMiddleEastern(Mesopotamia) andAfrican(Egypt)originsofmathematicschildrendevelopanunderstandingoftheglobalinterdependenceofallhumankind.TheyneedtobeawareoftheHinduand Mayanoriginsofzero,withoutwhichwecouldntcalculateeffectivelyorhavecomputers,andtheroleoftheGreekandArabiccivilisationsintheinventionofgeometry andalgebra.ChildrencanlearnaboutsymmetrybymakingHinduRangolipatterns,IslamictessellationsandAfricansanddrawings,sodevelopingtheirmathematical understandingthroughenjoyablecreativework.ModernBritainisamulticulturalsociety,itispartofaunitedEuropeandpartofaglobalvillage.Amulticultural approachnotonlyenrichestheteachingandlearningofmathematicsandtheexperiencesofchildren.Italsopreparesthemtobecitizensofamulticulturalsociety,and oftheworld! Thelastcrosscurriculardimensioninthissetisthatofspecialeducationalneeds.Atanyonetime,oneinfiveschoolchildrenmayexperienceaspecialeducational need(Warnock,1978).Thismaybeevenmorecommoninmathematicsbecauseofthewidespreadofachievementlevels.Therearemanypossiblespecialneedsin mathematics.Childrenmaybelowachieversinschoolmathematics,andmayneedextraworktohelpthemunderstandandmasterconceptsandskills.Childrenmay displayexceptionalabilityinpartorallofmathematics(mathematicalgiftedness),andneedadditionalenrichmentworktokeepthemchallengedandinterested. Childrenmayhavespecificlearningdifficultiesinsomeareaofmathematics,suchasfractions,andmayneedextraattentionandworktohelpthemgetoverthis stumblingblock.Sometimespoorreadingskillsandlanguagedifficulties,includingdyslexia,makelearningmathematicsdifficult,andtheseneedspecialattention.There areyetothertypesofspecialneedsthatcansurfaceinmathematics,suchasdifficultiesduetophysicalimpairments(e.g.childrenwhoarehardofhearing),andchildren whohaveemotionalorbehaviouraldifficultieswhichinterferewiththeirmathematicallearningandperformance.Ineachcase,theteachermustfindanindividual solutionthatsuitstheneedsoftheparticularchild,callingonthehelpofothersifnecessary.Whatevertheirspecialneeds,allchildrenareentitledtoabroadand balancedcurriculumandlearningexperienceinmathematics(NationalCurriculumCouncil,1989b).Teachersmustbeespeciallycarefulnottoprejudgewhatachild cando,andtoputaceilingonit.Itistheteachersresponsibilitytobringchildrenonasfartheycango,intheirmathematicslearning.Weneverknowhowfar forwardthatisuntilweseewhattheyhaveachieved! Thischaptersummarisessomeofthemoreimportantideasabouttheteachingandlearningofmathematicsintheprimaryschool.Manyofthemaredifficultideas,but theywillcometomeanmoreasyoucontinuetousetheminteachingmathematicsandinwatchingandhelpingchildrentolearn.Beingateachermeansundertakinga lengthyandexcitingjourneyoflifelonglearning.Wewishyouluckasyoucontinueonthiscareer,andwehopetohelpyoutofurtherdevelopthemostimportantthings totakewithyou:aninformedeyeandthedesiretokeeponlearningandinquiring.
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ReferencesAssessmentofPerformanceUnit(1985)AReviewofMonitoringinMathematics1978to1982,London:DepartmentofEducationandScience. Bell,A.W.,Kchemann,D.andCostello,J.(1983)AReviewofResearchinMathematicalEducation:PartA,TeachingandLearning,NFERNelson:Windsor. Cockcroft,W.(Chair)(1982)MathematicsCounts,ReportoftheCommitteeofInquiryintotheTeachingofMathematics,London:HMSO. Denvir,B.andBrown,M.(1986)UnderstandingofNumberConceptsinLowAttaining79YearOlds:PartsIandII,EducationalStudiesinMathematics,Volume17, pp.1536and143164. DepartmentofEducationandScience(1987)TheInterimReportoftheMathematicsWorkingGroup,London:DES. Ernest,P.(1991)ThePhilosophyofMathematicsEducation,London:FalmerPress. Hart,K.(ed.)(1981)ChildrensUnderstandingofMathematics:1116,London:JohnMurray. HMI(1985)Mathematicsfrom5to16,London:HMSO. NationalCurriculumCouncil(1989a)NonStatutoryGuidanceforMathematics,York:NationalCurriculumCouncil. NationalCurriculumCouncil(1989b)ACurriculumForAll(CurriculumGuidance2:SpecialEducationalNeedsintheNationalCurriculum),York:National CurriculumCouncil. NationalCurriculumCouncil(1990)TheWholeCurriculum,York:NationalCurriculumCouncil. Vygotsky,L.S.(1978)MindinSociety.Thedevelopmentofthehigherpsychologicalprocesses.Cambridge,MA:HarvardUniversityPress. Walkerdine,V.(1998)CountingGirlsOut(2ndedn),London:FalmerPress. Warnock,M.(Chair)(1978)ReportoftheCommitteeofEnquiryintotheEducationofHandicappedChildrenandYoungPeople,London:HMSO.
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SectionBTheaimofthissectionistosupportyoutoenhanceyoursubjectknowledgeinmathematicaltopics.Thetopicsincludedinthissectionareselectedonthebasisofwhat isconsideredtobenecessaryforasoundunderstandingofthecontentsoftheNationalCurriculumatKeyStages1,2and3,therequirementsoftheInitialTraining NationalCurriculuminmathematicssetbytheTeacherTrainingAgencyandtheFrameworkforteachingmathematicstoimplementtheNationalNumeracy Strategy. Thefollowingobjectivesguidedthestyleandcontentofthechaptersinthissection.Theyaredesignedto: encourageyoutothinkaboutthemathematicsyoualreadyknow identifygapsinyourknowledgeandunderstandingofmathematicaltopics considermathematicaltopicsatyourownlevelthroughrelevantcontexts makeconnectionswithvariousstrandsofmathematicaltopics acquireorrevisethecorrectterminologyandlanguageofmathematics considersomekeyissuesintheteachingofthetopicstochildren. Eachchapterinthissectionbeginswithalistofthemathematicstopicscoveredsubheadingsareusedtoguidethereaderthroughthevarioustopicsincludedinthe chapter.Mathematicsisdevelopedthroughexamples.Emphasisisplacedonaddressingtheunderlyingprinciplesofatopicwiththeaimoffacilitatinggreater understandingofthetopic.Manyoftheprinciplesareexplainedinordertofacilitatethinking,indepth,aboutwhymanyproceduresandrulesactuallywork.Itis hopedthatyouwillreadthechaptersinthissectionslowlyandsystematicallyastheintentionistoprovideexplanationsofcomplexideasratherthanoffersuperficial discussionsaboutmathematicaltopics.Withinthetext,keyideasaboutteachingthetopicsarebrieflyreferredto,whereappropriate,butitisassumedthatyouwill refertotextbookschemesandsourcesforotherpracticalideas. Werecommendthatyoutaketimetoreadeachsectionofthechapters.Youmayreadasectionaboutatopicthatyouareteachingtoyourclass,oraboutatopic thatisbeingcoveredonyourcourse.Beforereadingasectionitisagoodideatothinkaboutorwritedowntheideasyoualreadyknowaboutthetopic,alsoaspects ofthetopicyoumayfeelanxiousaboutorhavedifficultieswith.Whileyoureadthesectionmakenotesaboutnewideasandvocabularyyoucomeacross.Asyouread throughthetext,itisalsoagoodideatogiveyourselfsomequestionstotacklebeforeyoutrytheexercisesattheendofthechapter.Teacherswhotrialledthese sectionsfounditusefultolookatsectionsofchildrenstextbooksandteachershandbooksandrelatetheideastowhatistaughttochildren.
Page22 Thechaptersdealingwithnumberarelongerthantherest.ThisisbecausethenumbersectionsinboththeNationalCurriculumandtheTTANationalCurriculum aresubstantiallylongerthantherestoftheothersections.Also,inviewoftheemphasisplacedbytheNationalNumeracyStrategyondevelopingnumericalskillsand understanding,itwasfeltthatyouwouldappreciateopportunitiestoreflectonaspectsofnumberingreaterdetailthanyouhavedoneinthepast. Finally,rememberthatlearningandunderstandingmathematicstakestime.Asyoureadthechaptersinthissection,youshouldgainmoreinsightintowhateach mathematicstopicisaboutanddevelopyourexpertiseandconfidencetoteachit.This,inturn,shouldenableyoutoteachitinsuchawaythatthechildrenyouteach willbothenjoylearningmathematicsandunderstandwhattheyarelearning.
AuditingyoursubjectknowledgeAttheendoftheChapters2to7twotypesoftasksareprovided.Thefirstisacollectionoftaskswhichenableyoutothinkabouttheimplicationsforteaching particulartopicstochildrenandtheotherisasetoftasksforyoutotry.TwosetsoftestsareincludedattheendofChapter8,whichyoumayuseforauditingyour knowledge.TheRecordofAchievementandtheauditgridintheappendicesmayalsobeofhelp.
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Chapter2 WholenumbersThischapterfocuseson: 2.1Developmentofnumberconceptsintheearlyyears 2.2Theroleofalgorithms 2.3Placevaluerepresentationofnumbers 2.4Numberoperations 2.5Factorsandprimenumbers 2.6Negativenumbers
Somethingtothinkabout: Wehavetentoesonourfeetandtenfingersonourhands.Itisnaturalforustouseacountingsystembasedonten.Thesoundsandsightsweinterpretto getinformationaboutoursurroundingsarereceivedbytwoearsandtwoeyes.Isitthereforenaturalforustouseaninformationtechnologybasedontwo?
2.1DevelopmentofnumberconceptsintheearlyyearsCardinalandordinalnumbersEarlyexperienceswithcountingmakechildrendealwithtwoaspectsofnumber:theordinalityandthecardinalityofnumber.Countingone,two,three,fourgoldfish inabowlorcountingone,two,three,fourpawsonacatinvolvesusing1,2,3and4asordinalnumbers.Recognisingthatthegoldfishandthepawsofthecathave somethingincommonthattheybothconsistoffourthingsinvolvesinsightintotheircommoncardinalitythecardinalnumber4isusedtodescribethefournessofthe goldfishandthepaws.Thesamenumbersymbolisusedforbothaspectstheordinalandcardinal.Thetwoaspectsalloweachnumbertohavetworoles.When4is beingusedincountingupto4,itisplayingitsordinalrole,butwhen4isbeingusedtoindicatethesizeofagroupof4,itisplayingitscardinalrole. Whichaspectofnumberisinvolvedwhenyouseethatapersonorteamisrankedfourth? Somehow,youneedtocounttowardsthepersonorteamtoseethatthepositionisfourthhere,theordinalaspectofthenumberfourisinvolved. Immediatelyspottingthecardinalityofagroupofthingsispossible,formostpeople,perhapsforsmallnumbers. Thecardinalityofagroupofthingsisthenumberofthingswhichareinthegroup.
Page24 Hereisanactivityforyoutotry.
Countingstrategies Thisactivityisdesignedtohelpyoutogaininsightintotheabilityofadultsandchildrentospotthecardinalityofasmallgroupofobjects.Showa fewadultsandchildrenasmallcollectionofthings,say5,6,7or8.Findouthowtheydeterminethenumberinthatsetofthings. Dotheycount1,2,3andsoon? Dotheyusetheirfingerstocount? Dotheycountintheirheads? Dotheyjustknowimmediatelybyobserving? Howlargeagroupofobjectscantheyspotimmediately,withoutconsciouslydoinganything? Youngchildrenneedtobeprovidedwithexperiencestolearnaboutthethreeaspectsofnumber.Twohavealreadybeendealtwiththecardinalandthe ordinalnumbers.Thethirdistheuseofnumbersymbols. Thecardinalaspectofanumberisusedtodescribethenumberinaset:10beadsintheset. Theordinalaspectofanumberreferstoanumberinrelationtoitspositionintheset:colourthefifthbeadred. Anumbersymbol,say9,isusedbothtoexpressthecardinalityofthenumbernineandtoshowsomethinginthe9thplace.Itisalsosometimesusedasalabel:A9 orB9(asaroad).
CountingingroupsWhenyouarecountingthenumberofobjectsinaset,doesitmakeadifferenceifthethingsarearrangedinsomesmallergroupsoftwoorthreeorfour?Doesthis allowthecardinalitytobearrivedatbyspottingmultiplesoftwo,threeorfour? Showsomechildrenthefollowingpictureandask:howmanyleavesarethere?Thenaskhowtheyworkeditout.
Youmayfindthatoneofthefollowingstrategieswereused. Recognisethattheleavesarearrangedinthreesandadd3repeatedlytogetthetotal. Usetheknowledgeofthe3timestableandworkout6lotsof3tobe18. Countinonesfrom1to18.
Page25 Hereisanotheractivityforyoutotrywithafewchildren. Giveachildalargebagofbeansabout50orso. Askthatchildtofindouthowmanybeansthereare. Aswellasobservingthechild,conductaninterviewtocarefullydeterminethestrategyusedbythechild. Inwhatwayisthechildsstrategydifferenttoyours? Repeattheexperimentusing2pcoinsinsteadofbeans. Inwhatwayshaveyourfindingschanged? Trythiswithmorechildren. Theresultsofthisexperimentshouldillustrateanimportantandveryusefulprincipleinlearningmathematics. Thebestwayofdoingsomethingdependsonthecontextandontheindividual,butchildrenneedtobeshownandtaughtarangeofstrategiesfordoing mathematicssothattheycanchoosethemostefficientstrategy. Forexample,achildwhodecidestocount50objectsinonescanbeshownthatcountingingroupsisamoreeffectivewayofcounting.
2.2TheroleofalgorithmsTheideaofanalgorithmwillbedevelopedthroughoutthetext.Sothefollowingstatementsareworthreflectingon: 1Analgorithmisaprocedurefordoingsomething.Youcanperformacalculationusingdifferentalgorithms.Forexample,toadd35+36,youmayusedouble 35+1=71oryoumaychooseastandardalgorithm,forexampleoneyouhavelearntatschoolwritingthetwonumbersverticallyasasumtoaddthem. 2Anefficientalgorithmisonewhichdoesthejobbetterthanotheralgorithms. 3Althoughlearnersofmathematicsshouldbetaughtalgorithmsforcalculations,thesecanbementalandwrittentherearetimeswhenthelearnercanjudgethecontext oftheprocedureandfindamoreefficientalgorithmfordealingwiththecalculation.Whenaskedtosubtract398from500onaworksheet,achildmaydecideto useamentalstrategywhichisbasedonanumberline:from398to400is2,400to500is100makingtheanswer102. Disciplineisrequiredforitem2above,butitem3requiresadegreeoffreedomforthelearnersothatteachersmayadoptadifferentroleoffacilitatorofacreativeand flexibleattitudeinthelearner.
2.3PlacevaluerepresentationofnumbersEfficientwaysofhandlingnumbersdependverymuchonhowthenumbersarerepresented.Understandingalgorithmsandfindingmoreefficientalgorithms,inturn, dependsonhow
Page26 wellalearnerappreciatesourpresentnumbersystem.ThisappreciationmaybeenhancedbyconsideringsomeaspectsoftheRomannumbersystemnolongerused forcomputation,butstillappearingonsomedocuments. RecallthatthefollowingsymbolsareusedintheRomansystem: Iforone Vforfive Xforten Lforfifty Cforonehundred Mforonethousand. Arethereanyadvantagesinthissystem? Inordertowriteanumberupto999inourpresentsystemaRomanneededtoknowonlyfivesymbolsratherthanten. TheRomansymbolsmakevarioussimplificationspossible: insteadofIIIIItheycouldwriteV insteadofVVtheycouldwriteX insteadofXXXXXtheycouldwriteL insteadofLLtheycouldwriteC. InsomewaysthereareplacevalueconventionsintheRomansystem.ConsiderthedifferencebetweenIX,representingnine,andXI,representingeleven.The meaningoftheIdependsonitspositionrelativetotheXtotheleftofXtheImeansonelessthanwhereastotherightofXtheImeansonemorethan.Does thisconventionalwayshold?For32theRomanswroteXXXII.TheIIindicatestwomorethan,butthe30isrepresentedbyXXX.Heretheconventionbreaks down.TheXXXmeanstenandtenandten.SpendingalittletimeconsideringthegoodaswellasthenotsogoodfeaturesoftheRomansystemwillbeuseful preparationformakingabalancedappraisalofourpresentsystemwhichusesthetensymbols0,1,2,3,4,5,6,7,8and9.Whenyounextobserveyoungchildren countingwiththeirfingersyoumayberemindedoftheRomansystem.Canyoulinkitwiththehumanhand?DoesIlooklikeafinger?WhenVwasusedforfive insteadofIIIII,wasoneopenpalmbeingsymbolisedratherthanfivefingers? IftheRomansystemcanbeimaginedaslinkedwithonehand,thenourpresentsystemcanbelinkedwithtwohands.Yetaretheresomesimilaritiesbetweenthe twosystems?Whatabout555?Thefirst5meansfivehundred,thenext5meansfiftyandtherightmost5simplymeansfiveunits.Istheresomesimilaritybetween555 andXXX?Thinkaboutit!Inthehundreds,tensandunitssystemthemeaningofa5dependsonitspositioninthenumber.Incontrast,themeaningofeachXinthe Romannumberisthesameregardlessofitspositioninthenumber.However,the555makesuseofacompactingtechniquejustlikethatusedinXXX.Itisacompact wayofwriting500+50+5,as30isthoughtofintheRomansystemasXandXandX.Learninghownumberscanbesplitintothesumofpartsisaveryusefulskill whichcanenhanceyourunderstandingofnumericalalgorithmsandwillbeconsideredinthenextsection.
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ThebasetenrepresentationofnumbersTheplacevaluenumerationsystemisbasedontwofundamentalprinciples: thegroupingaspectgroupingintens.Thesystemisreferredtoasthebasetensystem Usingtendigits:0,1,2,3,4,5,6,7,8,9,10indifferentpositionsyoucanwriteanynumber.Thepositionofdigitsfromlefttorightdeterminesthevalueofthe number.Forexample: 546means6singles,4tensand5hundreds 304means4singles,0tensand3hundreds.
KeyissuesinteachingplacevaluetochildrenAsastartingpoint,itisusefultoremindourselvesthatagoodunderstandingofplacevalueofwholenumbersanditsextensiontodecimalnumbersisvitalbecauseitis thebasisofbothourmentalandwrittencalculations.Thereisevidence(Brown,1981Koshy,1988SCAA,1997)toshowthatchildrenatallkeystageshave difficultiesinunderstandingmanyaspectsofplacevalue.Someoftheareaswhichcauseconcernincludedifficultieswithzeroasaplaceholder,readingandwriting largenumbers,problemsrememberingrulesofalgorithmsforaddingandsubtractingnumberswhichinvolvecarryingorexchanging. OnepossiblereasonsuggestedbySCAA(1997)foryoungchildrensdifficultyininternalisingtheprincipleofplacevalueisthenatureofthenamesofnumbers between10and59.ThereasonforJapanesestudentsacquiringagreaterunderstandingoftheplacevalueconceptisexplainedintermsoftheJapanesenumbersystem havingnumbernames,uptohundreds,consistentwiththenumberstheyrepresent,e.g.theJapanesefortwentytwoshowstherearetwotens.Itissuggestedthatin theEnglishsystem,thenamingofnumbersinrelationtotheirplacevaluedoesnotbegintoappearuntilnumberscontaininghundreds,e.g.threehundredandtwenty nine.Restrictingyoungpupilstonumbersupto20maybedoingthemadisservicebecauseitisnotuntilonegetstothesixtiesthattheplacevalueandnumbernames cometogether:sixty,seventy,eight(t)yandninety.Twentyandthirtyinsteadoftwotyandthreetydonotmakethestructureexplicit(SCAA,1977,p.78). Whenworkingwithnumbers,itisusefultobearinmindthetwoaspectsofnumber: thenumberlineaspectwhichdealswiththeorderinwhichnumbersappearonanumberline,and thegroupingconceptwhichfocusesonconsideringnumbersingroupsofhundreds,tensandunitsandsoon. TheFrameworkforteachingmathematics(DfEE,1999)tosupporttheimplementationoftheNationalNumeracyStrategyplacesmuchemphasisonteachingplace value.
TeachingplacevalueInthefollowingsection,somespecificteachingresourcesforteachingplacevaluearedealtwith: placevaluearrowcards basetenmaterials setsof09digitcardsfordiscussingplacevalue.
Page28 Thearrowcards,asshownbelow,havebeenfoundparticularlyusefulbyteacherstosupportchildrensunderstandingofplacevalue.Tomakeasetofarrowcards, youneedninecardsprintedwith100to900,ninecardsprintedwith10to90andninecardsprintedwith1to9. Byoverlayingthreecardsfromthedifferentsetsyoucanmakeany3digitnumber,e.g.687: Bytargetingquestionssuchas:makeanumberwith3hundredsinit,canyoumakeanumberbetween350and450,makethenumber235,youhave467 howmanymoreisneededtomake500?andsoon,youarefocusingontheimportantprinciplethatthevalueofadigitdependsonitspositioninanumber.Fora classactivityyouwillneedseveralsetsofarrowcards.Duringanintroductionofalesson,youcouldaskagroupofchildrentochoosethreecardsonefromthe hundreds,onefromthetensandonefromtheunitsandmakea3digitnumber.Askthegroupofchildrentostandinorderbasedontheirnumbers,forexample, smallestfirst.Askthechildrenwhosenumberisthenearestto400. Hereisanactivitywhichdemonstratesthegroupingandregroupingprinciplesofplacevalue. Askthechildrentomakea3digitnumberusingthearrowcards:say356wasmadebyonechild.Askthechildwhatnumberneedstobesubtracted(takenaway) inordertoshowazerointhetenscolumn.Quiteoften,childrenwillsaysubtractfiveandaresurprisedthatyouareinfacttakingaway50! Placevalueblocks,below,usuallyreferredtoasbase10material,arecommonlyusedtoshowtherelativesizesofsingles(units),tens,hundredsandsoon.Base10 blockscanbeusedformakingamodelof389as:3hundreds,8tensand9singles.TheseweredesignedbyZ.P.Dienesspecificallytomodeltheplacevaluesystem ofnumber.Mostschoolshavethesematerialstheireffectiveusewilldependonthewaychildrenareencouragedtostudyhowtheirstructurerelatestotheway numbersareconstructed.
Base10materialscanalsobeusedtodemonstratenumberoperationswhicharedealtwithlater.
Page29 Activitieswhichusedigitcards0to9andplacevalueboardsalsoprovideopportunitiesforenhancingchildrensunderstandingoftheprinciplethatthevalueofa digitisdeterminedbythepositionitoccupies.Anexampleisgivenbelow. Tryplayingthisgamewithupto4players. Youneedafewsetsofshuffled09cards,placedfacedownandaplacevalueboardforeachplayer.
Beforethegamestartsdecidewhetherthelowestorthehighestnumberwins.Takeitinturnstoplaceacardontheboardinanyposition,bearing inmindthecriterialowestorhighestselectedforwinning.Onceacardisplaced,itcannotbemoved.Changethecriteriaasoftenasyouwish andincludeanewcriterionnearesttoanumber,say450. Allthethreeteachingaidsdescribedaboveareusefulformodellingtheprinciplesofplacevalue.However,itmustbestressedthatsimplyusingmaterialsdoesnot guaranteechildrensacquisitionofconceptsappropriatequestioning,discussingandexplainingideasarealsoveryimportant.
2.4NumberoperationsSomeusefulprinciplesAsanintroductionitisusefultoconsiderthefourbasicoperationsaslinkedinpairs. Multiplicationcanbethoughtofasrepeatedaddition,whilstdivisioncanbethoughtofasrepeatedsubtraction.Letusspendalittletimetryingtounderstand whatisinvolvedinthelinksbetweenthepairsofoperations.
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Duringapublicexaminationapupilneededtofindtheresultofmultiplying,withoutacalculator,thenumbers17and35.Hewrotedownthenumber17 thirtyfivetimesandthenproceededtoaddthe17stogethertofindthetotal. Whatisyourassessmentofthispupil?Whatisyourviewofwhatisbeingattempted?Whatdoesthepupilknow?Whatisitthathedoesnotseemtoknow?Can youthinkoftwoprocedureswhichwouldbebetterthanthatadoptedbythispupilforfindingwhat1735isequalto? Thispupilcertainlyhassomedeficienciesincarryingoutnumericalworkefficiently.Whatisbeingattemptedistheadditionofthirtyfive17s,onpaper,byliningup the7sinacolumnofunitsandthe1sinacolumnoftensandproceedingtouseanalgorithmforaddition.Heseemstoappreciatethattherequiredthirtyfive17scan befoundbytheprocessofrepeatedaddition.Unfortunately,hedoesnotseemtoknowanalgorithmastepbystepprocedureforperformingtheoperationof multiplicationonwholenumbers. Thefirstthingthispupilcouldhavechosentodo,insteadofwritingdown17thirtyfivetimes,wastowritedown35seventeentimesthiswouldhavestillinvolved doingmultiplicationbyrepeatedaddition,butwiththeamountofwritingrequiredcutbyalmosthalf. Ofcourse,thesecondthingwhichcouldhavebeendonewastooptforamultiplicationalgorithmamentalmethodorawrittenone.Thiswouldhavebeenfar moreefficientinthesensethatitwouldhavebeenquickersothatmoretimecouldhavebeengiventootherpartsofthemathematicsexamination. Fromtheabovediscussionwecanderiveanotherusefulpieceofinformation.Thefactthat3517and1735arebothequalto585indicatesthatmultiplicationis commutative.Itshouldalsobepointedoutthat35+17and17+35bothequal52,showingadditionalsobeingcommutative.Thisisaveryusefulandimportant teachingpoint. Theoperation+performedonintegers(wholenumbers)iscommutativemeansthatadditioncanbeperformedinanyorder,whetheritiscarriedouthorizontallyor vertically.Inusingverticaladdition,whethersomebodywritesthe17abovethe35orthe35abovethe17doesnotmatter!
AdditiveidentityYoumaynothaveheardofitbefore,butitisworthspendingalittletimeonthissectionbecauseitcontainsanimportantprinciplewhichhelpsunderstandingsome complexmathematicalideas. Thinkofanumber.Whatnumbercanyouaddtoitsothattheresultisthesamenumberyouoriginallythoughtof?Regardlessofthenumberyouthoughtofthesame numberdoesthetrick.Thatnumber,ofcourse,isthenumber0(zero). Add0toanynumberandtheresultisthesamenumber. 0iscalledtheadditiveidentityortheidentityforaddition.
Page31 Askingchildrentoadd0isuseful,infactitsroleasanadditiveidentityturnsouttobeveryusefulwhentryingtounderstand,forexample,operationswithnegative numbers.
MultiplicativeidentityAgain,thinkofanumber.Whatnumbercanyoumultiplyitbysothattheresultisthesamenumberyouoriginallythoughtof?Regardlessofthenumberyouthoughtof thesamenumberdoesthetrick.Thatnumber,ofcourse,isthenumber1(one). Multiplyanynumberby1andtheresultisthesamenumber. 1iscalledthemultiplicativeidentityortheidentityformultiplication. Again,askingchildrentomultiplyanumberby1tooisuseful,asthenotionofamultiplicativeidentityturnsouttobeveryusefulinunderstanding,forexample,some operationsonfractions. Letusnowtakealookattheconnectionbetweensubtractionanddivision,restrictingtheexamplesconsideredforthetimebeingtothoseinvolvingthesubtractionof positiveintegersfrompositiveintegersgreaterthanthem. Considerthecaseofsubtracting3from18.Theresultis15.Take3awayagain,butthistimefromthe15andtheresultis12.Repeattheprocesstoget,in succession,9,then6,then3,then0.Whathashappened?Startingwiththe18,thenumber3hasbeensubtractedsixtimesuntil0remains. Whatis18dividedby3?Itis6. Sodivisioncanbethoughtofasrepeatedsubtraction.Justhowusefulisthislink?Cansomeonewhodoesnotknowanalgorithmfordivision,neverthelessdo divisionbychangingitintoarepeatedsubtractioneventhoughthedivisionwilltakelongertodothatway? Inwhatcaseswilltherebeasnag? Asimpleexampleofsuchacasewouldbe2344.Theresultof5leavesaremainderof3.Itmaybeofinteresttocheckwhatchildrenunderstandaboutthis remainderconceptwhentheyhavemasteredadivisionalgorithm. Soistheoddoneout.Whereastheotherthreeoperationsonwholenumbersaddition,subtractionandmultiplicationalwaysyieldanexactintegerresult,the operationofdivisioncansometimesproducearesultwhichisnotexactlyanintegerbecauseoftheremainder. Haveyounoticedthatalltheexamplesconsideredsofarhaveinvolvedtwonumbersbeingoperatedonbyoneofthefouroperations:+,,and.Isitpossibleto performoneoftheseoperationsonthreepositiveintegers?Thisbringsustoanotherlinkbetween+and. Adding8+6+29ispossibleandcanbedoneintwoways.Theresultof8+6couldbeobtainedfirstandthentheresultof14addedtothe29toproducethefinal resultof43.Alternatively,the6and29couldfirstbeaddedtoget35whichcanthenbeaddedtothe8toagainproducetheresult,43,ofaddingthethreenumbers8, 6and29.Themathematicallanguagefordescribingthisfeatureofadditionisgivenbythefollowing: +isabinaryoperation +isassociative. Abinaryoperationisanoperationwhichisperformedontwothingsatatime.Checkthatxisalsoabinaryoperation. Sayingthat+isassociativesimplymeansthatifmorethantwonumbershavetobeadded,thenanytwomaybeadded(associated)togetherfirstbeforethenext numberisaddedtothetotal.Theprocedurecanbeclarifiedwiththeaidofbracketsasfollows:
Page32
or
Youshouldnowbeabletocheckthatxisalsoanassociative,binaryoperation. Theothertwooperationsofsubtractionanddivisionare,ofcourse,binaryoperationsbuttheyarenotassociativeasiseasilyillustrated. Considertheexampleof26175. Sincesubtractionisabinaryoperationtwonumbershavetobechosenforthefirstsubtraction.Theselectiondoesaffecttheresultasthefollowingdemonstrates.
whereas
Selectonesimpleexampletoconvinceyourselfthattheoperation,division,isalsonotassociative. Thefactthatdivisioncanproducearemainderleadsquitenaturallytotheconsiderationoffractions,sinceafractioncanbethoughtofintermsofdivision.
FlexiblecalculationsWhenaskedtoadd563+99mentally,childrenandadultsarelikelytousestrategieswhicharenotthesameastheywoulduseinwrittenalgorithms. Tocarryouttheabovecalculationmentally,forexample,onemayuse:563+1001asapossiblestrategy.Itisquitecommontoseechildrenconditionedinsucha waythatifaskedtodotheabovecalculation,theywoulduseawrittenalgorithmasaverticaladditionsum.This,ofcourse,involvesamorecomplexmethodologyfor thejobinhand. Thesameappliestootheroperations.Sometimesyoumayfindchildrenwritingoutaverticalsumforcarryingoutthesubtraction:50001=andspendalotoftime workingitout,evenwhentheyareperfectlycapableofcarryingouttheoperationintheirheadwithinseconds.Itmakessensetohighlighttoachildwhoisspending considerabletimeworkingout2520asalongmultiplicationor200025asalongdivisionsumthattheymaybeabletodotheseoperationsmuchfasterandwith accuracyiftheyusedfactstheyalreadyknowsuchas25times2is50orthatfour25smakeonehundred.TheMathematicsFrameworkprovidedfortheNational NumeracyStrategyprovidesagreatdealofsupporttoteachersindevelopingflexibleandefficientmethodsofcalculations.Focusingontheuseofthemostefficient methodforthejobinhandshoulddiscouragechildrenfromselectingataughtmethodautomaticallywhentherearemoreeffectivealternatives.Thechapteronmental
Page33 mathematicsinSectionCofthisbookprovidesaverydetailedexpositionoftheissuesrelatingtoteachingcalculationstochildren. Althoughwerecommendtheuseofmentalcalculationsandtheneedforchildrentobeflexiblewhenengagedincalculations,webelievethatchildrenshouldalsobe familiarwithwrittenalgorithms.Whencalculatinglargernumbersbothwholenumbersanddecimalswrittenalgorithmsprovidechildrenwithanotheroption,which alwaysworks.
StandardwrittenalgorithmsAddition Thestandardwrittenalgorithmtaughtinschoolsisbasedonthegroupingprincipleofplacevalue.Inthewrittenalgorithm,additionisconventionallycarriedoutfrom righttoleft.Theideathattensinglesorunitscanbeexchangedforonetenandtentenscanbeexchangedforahundredandsoonisthebasisforthe carryingaspectofaddition. Whenadding
youaddthetwosetsofunits,tensandhundredsthereisnocarryingbecausenoneofthecolumnsproducesananswerofover9,whichnecessitatescarrying.But intheexample
however,youadd7and8toget15,whichisonetenand5units.Theteniscarriedtothetenscolumn.Adding6tensand5tensgivesyou11tens,thenofcourse youneedtoaddthecarriedonewhichgivesyou12tens.Tentensmakesonehundredsoonehundrediscarriedtothehundredsplaceandaddedtothetotalof3 hundredsand5hundreds.Thisprinciplecanbeusedforaddinganyplacevalueswithincreasingnumberofdigitsorforanynumberofrowsofnumbers. Whenteachingchildrenhowtoaddvertically,itisusefultostressandreinforcetheprinciplesofplacevalueusedintheoperation,sothatchildrenrelatetheword carryingtowhatisactuallyhappeningratherthanlearnitasarulethathelpstoproducecorrectanswers.Periodically,whenengagedinawrittensum,itisagoodidea toaskchildrentowritenexttoit(inacircle)whattheestimatedanswerwouldbe.Thisprocessofcheckingforreasonablenesscanbeusedforalloperationsandhas manybenefits.First,itremindschildrenwhattheoperationisallaboutsothattheydonotadoptamechanicalmodeandperformaskillwithoutthinkingaboutwhatis actuallyhappening.Secondly,itprovidesacheckingmechanismfor
Page34 childrenwhichreducesthenumberofmistakes.Manymistakesaremadebecauseofforgottenorpartlyrememberedrules.Childrensmistakesarediscussedindetail inSectionCofthisbook.
SubtractionThewrittenmethodcommonlyusedinschoolsforsubtractionisbasedondecomposition.Textbookstoousethismethod.Thismethod,asinthecaseofcarrying,is basedonthegroupingandexchangeconceptsofourplacevaluesystem.Thisalgorithmcanalsobeexplainedusingbase10material.
Inthisexampleofverticalsubtractionyoucantake6unitsfrom8unitsand4tensfrom5tenswithoutanyrearrangementorexchange.But,whenyouhavetocarryout thesubtraction:
thedecompositionprocedureisused:takeeightunitsfrom6,youcannotdothiswithoutsomerearrangement,soyoubreakoneofthetenstakenfromthe6tens intotenunitsandshowtherearrangement.8from16unitsis8,4tensfrom5tensis1ten. Whencarryingoutaverticalsumwithhundreds
noexchangeisnecessary,buttoperformthewrittenverticalalgorithm
youhavetodecomposethehundredsandtens,(orbreakitdowninto)andrearrangethenumbertoenableyoutocarryoutthecalculation.
Page35 Theequaladditionmethodwhichwasusedcommonlyinthepast(someteachersstillusethismethodfortheirowncalculations)isnotbasedontheexchange principle,butonrememberingarule. Forexampleinthefollowingexample
take8from6cantdoit,soyouborrowonefromthenextcolumnwhichmakesthe6into16.16takeaway8givesyou8.Asyouhaveborrowedaoneyoupay backaonewhichisaddedto5whichis6,6takeaway6is0.Thisisapaperexercisewhichwaspopularintheoldendays,butislosingitspopularitybecauseitis difficulttoexplaintochildrenwhyitworksintermsoftheplacevaluesystem.Nevertheless,itisusefulforateachertobefamiliarwiththisforcommunicatingwith parentsandforhistoryssake!Bewareofthewordborrowwhenyouusethedecompositionmethodbecausethereisnoborrowing,onlyexchangingandsome rearrangement.
MultiplicationWhenteachingchildrentocarryoutmultiplicationusingaverticalmethod,itisusefultoremindthemthatthisalgorithmusesthecarryingaspectalreadydealtwithin addition.
Hereyoumultiply6unitsby8=48,carrythe4tensandplacethe8intheunitcolumn.Thenmultiplythe4tensby8whichis32,addthecarried4tensto36,givingthe answer368. Someteachersteachchildrenthetensfirstmethodwhichisthenusedasabasisforcarryingoutlongmultiplication.Here
Multiply4tensby8=320thenmultiplytheunits68=48.Add320+48=368.
Page36 Thetraditionalwayofmultiplyingby2digitnumberscanbebasedonthis:
Someteachersteachthisprocedurestartingwithunitsandaddingazerowhenyoustartmultiplyingwiththetens:
Ifyouareusingthismethod,itisimportanttomakechildrenthinkaboutwhythezeroisadded.Itisalsogoodpracticetoaskthemmakeareasonableestimateof whattheanswerwillbelikebecausemanymistakesaremadeasaresultofforgettingtheruleofaddingazero.
DivisionYouhavealreadybeenintroducedtotheideaofdivisionbeingrepeatedsubtractionearlierinthischapter.Writtenproceduresfordivisionareusuallytermedshort orlongdivision.Traditionally,shortdivisionisusedfordividingbya1digitnumber.Whencarryingoutdivision,thedecompositionaspectofsubtractionisalsoin usethiscanbepointedouttochildren. Divide455by8bytheshortmethod:
Page37 Itcanalsobecarriedoutbythelongmethodwhichshowsdivisionasrepeatedsubtraction.
Todivide2457by56usinglongdivisionwewillneedtorelyonchildrenlearningamethodassistedbytheprinciplesofplacevalueandtheunderstandingthatthe divisionoperationisbasedonrepeatedsubtraction.
2.5Factorsandprimen