Math for Primary Teachers

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CoverPageiMathematicsforPrimaryTeachersInEngland,andinternationally,numeracystandardsinprimaryschoolsarethecauseofasmuchconcernasliteracystandards.Oftenthegreatestproblemwithmathsat primarylevelistheteachersownunderstandingofthesubject.MathematicsforPrimaryTeachersaimstocombineaccessibleexplanationsofmathematical conceptswithpracticaladviceoneffectivewaysofteachingthesubject.Itisdividedintothreemainsections: SectionAprovidesaframeworkofgoodpractice SectionBaimstosupportandenhanceteacherssubjectknowledgeinmathematicaltopicsbeyondwhatistaughttoprimarychildren.Eachchapteralsohighlights teachingissuesandgivesexamplesoftasksrelevanttotheclassroom SectionCisacollectionofpapersfromtutorsfrom4universitiescoveringissuessuchastheteachingofmentalmathematics,childrensmathematicalmisconceptions andhowtomanagedifferentiation.Theyarecentredaroundthethemeofeffectiveteachingandqualityoflearningduringthiscrucialtimeformathematicseducation. ValsaKoshyisSeniorLecturerinEducationatBrunelUniversitywithresponsibilityformathematicsinservicecourses.PaulErnestisProfessorinMathematics EducationatExeterUniversity.RonCaseyisSeniorResearchFellowatBrunelUniversity.PageiiThispageintentionallyleftblank.PageiiiMathematicsforPrimaryTeachersEditedby ValsaKoshy,PaulErnestandRonCaseyLondonandNewYorkPageivFirstpublished2000 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 ISBN0203984064MasterebookISBNISBN0415200903(PrintEdition)PagevContents Listofcontributors Acknowledgements Introduction viii x xiSECTIONA1 3 211 Teachingandlearningmathematics PAULERNESTSECTIONB RONCASEYANDVALSAKOSHY2 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 66 682.6Negativenumbers 3 Fractions,decimalsandpercentages 3.1Fractions 3.2Decimals 3.3Indices 3.4Standardindexform 3.5Percentages 4.1Sequences 4.2Series4 NumberpatternsandsequencesPagevi 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 1435 Measures5.8Theuseofscales 6 Shapeandspace 6.1Coordinates 6.2Transformations6.3Enlargement 7 Probabilityandstatistics 7.1Probability 7.2Statistics 8 Mathematicalproof 8.1Theroleofinduction 8.2Proofbyinduction 8.3Deductionsandarguments 8.4Conjecturesandsupportingevidence 8.5Lookingforexceptions 8.6Proofbycontradiction 8.7Generalisationandproof Selfassessmentquestions MultiplechoicemathematicsPageviiSECTIONC147 149 158 172 182 1969 Effectiveteachingofnumeracy MARGARETBROWN 10 Mentalmathematics JEANMURRAY 11 Childrensmistakesandmisconceptions VALSAKOSHY 12 Usingwritingtoscaffoldchildrensexplanationsinmathematics CHRISTINEMITCHELLANDWILLIAMRAWSON 13 Differentiation LESLEYJONESANDBARBARAALLEBONE APPENDICES211 213 216 219 220 222 223 Answerstoselfstudyquestions Answerstoselfassessmentquestions Answerstomultiplechoicemathematics Recordofachievement Mathematicalglossary IndexPageviiiContributorsMargaretBrownisProfessorofMathematicsEducationatKingsCollegeLondonandamemberoftheNationalNumeracyTaskForce.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.ShePageix hastaughtinprimaryschoolsforanumberofyearsandhasledinservicetrainingofteachersinLEAspriortojoiningGoldsmithsCollege.Oneofhermajor researchinterestsistheeducationofAbleChildrenandtheroleofquestioninginextendingchildrensthinking.PagexAcknowledgementsThisbookattemptstobringtogetherthetwostrandswhichwebelievecontributetothequalityofteachingandlearningofmathematicsandraisepupilsachievementin 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,PaulErnestandRonCaseyPagexiIntroductionAsweapproachthemillennium,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).PagexivThispageintentionallyleftblank.Page1SectionAPage2Thispageintentionallyleftblank.Page3Chapter1 TeachingandlearningmathematicsPaulErnestWhyteachmathematics?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.ForPage5 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,fromadvertisementstogovernmentstatisticsPage8 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.TheonlyPage10Table1.1NationalCurriculumkeystagesKeystageKeystage1 Keystage2 Keystage3 Keystage4Pupilsages57 711 1114 1416YeargroupsYears12 Years36 Years79 Years1011exceptionisinWales,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,andwhatissuitablefortheresourcesavailableandforthechildrenandFigure1.1TherelationbetweenteachingstylesandlearningoutcomesPage14 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,hopingtoPage15 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 varietyofteachingapproachesbecausethisiswhatishelpingtodevelopTable1.2TheroleofteacherandstudentindifferentteachingapproachesTeachingapproach RoleoftheteacherDirectinstruction Guidedinstruction Problemsolving Investigatory mathematics Toexplicitlyteachrules,skillsorconcepts,andprovide exercisesforapplication Togivepracticaltasksorasequenceofexamples representingaconceptorruleimplicitly Topresentaproblem,leavingsolutionmethodsopen Topresentaninitialareaofinvestigation,orapprovea studentsprojectRoleofthelearnerToapplythegivenknowledgetoexercisesProcessinvolvedThedirectapplicationsoffacts, skillsandconceptsToidentifytheruleorconceptimplicitinthegiven Generalisation,rulespotting, examples conceptformation Toattempttosolvetheproblemusingownmethods Problemsolvingstrategies Tochoosequestionstoinvestigateinprojectarea andtoexplorethetopicfreely Creativethinkingandproblem solvingstrategiesPage16 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.ThefirstzonePage17 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 abilitythanboysPage18 (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.Page19ReferencesAssessmentofPerformanceUnit(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.Page20Thispageintentionallyleftblank.Page21SectionBTheaimofthissectionistosupportyoutoenhanceyoursubjectknowledgeinmathematicaltopics.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.Page23Chapter2 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, dependsonhowPage26 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.Page27ThebasetenrepresentationofnumbersTheplacevaluenumerationsystemisbasedontwofundamentalprinciples: 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.Page30 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:Page32orYoushouldnowbeabletocheckthatxisalsoanassociative,binaryoperation. Theothertwooperationsofsubtractionanddivisionare,ofcourse,binaryoperationsbuttheyarenotassociativeasiseasilyillustrated. Considertheexampleof26175. Sincesubtractionisabinaryoperationtwonumbershavetobechosenforthefirstsubtraction.Theselectiondoesaffecttheresultasthefollowingdemonstrates.whereasSelectonesimpleexampletoconvinceyourselfthattheoperation,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.ThechapteronmentalPage33 mathematicsinSectionCofthisbookprovidesaverydetailedexpositionoftheissuesrelatingtoteachingcalculationstochildren. Althoughwerecommendtheuseofmentalcalculationsandtheneedforchildrentobeflexiblewhenengagedincalculations,webelievethatchildrenshouldalsobe familiarwithwrittenalgorithms.Whencalculatinglargernumbersbothwholenumbersanddecimalswrittenalgorithmsprovidechildrenwithanotheroption,which alwaysworks.StandardwrittenalgorithmsAddition Thestandardwrittenalgorithmtaughtinschoolsisbasedonthegroupingprincipleofplacevalue.Inthewrittenalgorithm,additionisconventionallycarriedoutfrom righttoleft.Theideathattensinglesorunitscanbeexchangedforonetenandtentenscanbeexchangedforahundredandsoonisthebasisforthe carryingaspectofaddition. Whenaddingyouaddthetwosetsofunits,tensandhundredsthereisnocarryingbecausenoneofthecolumnsproducesananswerofover9,whichnecessitatescarrying.But intheexamplehowever,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,itprovidesacheckingmechanismforPage34 childrenwhichreducesthenumberofmistakes.Manymistakesaremadebecauseofforgottenorpartlyrememberedrules.Childrensmistakesarediscussedindetail inSectionCofthisbook.SubtractionThewrittenmethodcommonlyusedinschoolsforsubtractionisbasedondecomposition.Textbookstoousethismethod.Thismethod,asinthecaseofcarrying,is basedonthegroupingandexchangeconceptsofourplacevaluesystem.Thisalgorithmcanalsobeexplainedusingbase10material.Inthisexampleofverticalsubtractionyoucantake6unitsfrom8unitsand4tensfrom5tenswithoutanyrearrangementorexchange.But,whenyouhavetocarryout thesubtraction:thedecompositionprocedureisused:takeeightunitsfrom6,youcannotdothiswithoutsomerearrangement,soyoubreakoneofthetenstakenfromthe6tens intotenunitsandshowtherearrangement.8from16unitsis8,4tensfrom5tensis1ten. Whencarryingoutaverticalsumwithhundredsnoexchangeisnecessary,buttoperformthewrittenverticalalgorithmyouhavetodecomposethehundredsandtens,(orbreakitdowninto)andrearrangethenumbertoenableyoutocarryoutthecalculation.Page35 Theequaladditionmethodwhichwasusedcommonlyinthepast(someteachersstillusethismethodfortheirowncalculations)isnotbasedontheexchange principle,butonrememberingarule. Forexampleinthefollowingexampletake8from6cantdoit,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.HereMultiply4tensby8=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.2.5FactorsandprimenumbersNumber12canbewrittenastheproductof: 3and4or6and2or12and1. Inthiscase1,2,3,4,6and12arefactorsof12. Number21hasfourfactors:1,3,7,and21. Allnaturalnumberscanbewrittenasproductsoftheirfactors. Whenyoufactoriseanumberyouarewritingthatnumberasaproductofitsfactors. Aprimenumberhasonlytwofactors1anditself,whichmeansitcanonlybedividedby1anditselfwithoutaremainder. 2,3,5,7,11,13,17,19areprimenumbers. Whenyoufactorise24yougetthefollowingfactors: 1,2,3,4,6,8,12and24. Whenanumberiswrittenasaproductofprimenumbers,wecanrefertothefactorsasitsprimefactors. Forexample,considernumber12again: Startwiththefactor2 12=26Page38 Nowfactorise6 6=23 2isafactoragain. So12=223:allthefactorsof12areprime. Wehave,therefore,primefactorised12.2.6NegativenumbersAnynumbercanberepresentedonanumberline.Inthenumberlinegivenbelow,positivenumbersareshownontherightofthelineandnegativenumbersontheleft.Ifyouwanttoaddapositivenumbertoanynumberontheline,youmovetotheright:toadd6to3,youstartat3andmovetotheright3+6=9givingaresultof9.Ifyouwanttosubtractapositivenumber,youmovetotheleft.Tocarryoutthesubtraction64=2,youhavetomovetwoplacestotheleftfrom,6.Whatifyou subtract7from4?Youwillmovetotheleft,butthistimeyouwillarriveat3.Sayyouwanttoaddorsubtractanegativenumber.Toaddanegativenumber,youmovetotheleftandtosubtractanegativenumberyoumovetotheright.Ifyou add3+(4)theresultis1.Page39 Seewhathappenswhenyoucarryoutthesubtraction3(2).Youmovetwoplacestotherightandyouransweris5.SomekeyissuesinteachingnegativenumbersChildrenareoftenfascinatedwhentheyareintroducedtonegativenumbers,butthisabstractconceptneedsmuchdiscussionusingcontextssuchasmeasuring temperatureorindiving.Childrenmaycomeacrossnegativenumberswhileusingacalculatorandthismaybeanopportunityforintroducingnegativenumbers. Calculatorsprovideaveryusefulresourceforexploringnumberpatternsandsequences,positiveandnegative.Tasksrelatingtotheclassroom1Supposeyouaskedachild,sayinYear1,tocountout5objectsfromatrayfullofobjectsandmakeasetthenshowinsomewaytosomeoneelsethatthereare5 objectsinthatset.Makealistofwhatachildneedstoknowtobeabletocarrythisout. 2Studythefollowingstatement:achildwhocancountupto30correctlyshouldsurelybeabletorecognisethecardinalityofasetofobjects.Comment. 3Thefollowingaremistakesandmisconceptionscollectedfromchildrenswork.Foreachonewritedownthepossiblereasonsformakingthemistakesandwhatthe implicationsareforteaching. (a)Add1to3499:Alisonwrote4000. (b)Writefivehundredandthreeinfigures:Danielwrote5003. (c)Howmanybagsof10sweetscanImakefrom1500sweets?Jasondidasum:(d)Whatisthechangeinpriceofacarwhenasalesmanchangesthepricefrom2540to2340?Natalieanswered2.Tasksforselfstudy1Ask6peopletodo24387intheirheadsandtellyouhowtheydidit.Analysethestrategiesused. 2Estimatetheanswersbeforeworkingthemout,usinganymethod(a)1237+637 (b)350+351 (c)432179 (d)239Page40(e)7634 (f)4166 (g)376223Explaintosomeone,usingbase10material,howtodoadditionandsubtraction. 4Whichofthefouroperationsaddition,subtraction,multiplicationanddivisionarecommutative?Showanexampletoproveyourpoint. 5Usingthedigits1,2,3,4,5,and6makeupsumsfollowingtheinstructions.Usethedigitsonlyonceeachtime.Thefirstonehasbeendoneforyou:(a)Theansweris390 (b)thelargestanswerpossible (c)thesmallestanswerpossible (d)thelargestanswerpossible (e)adivisionsumwithaneven2digitanswer6Pictureanumberlineinyourmindandcarryoutthefollowingcalculations:(a)Whatintegers(wholenumbers)liebetween6and4? (b)Thetemperatureonathermometershows5.Whenthetemperaturehasgoneupby4degreeswhatreadingdoesitshow?Afteranotherriseof3degrees, whatisthenewreading? (c)Orderthefollowingnumbers:3,6,0,and8fromthelargesttothesmallest.7Writedownallthemultiplesof7between50and100 8Whichofthestatementsaretrue?(a)Allprimenumbersareoddnumbers. (b)Thereare12primenumbersbetween10and80. (c)Thereare2primenumbersinthisset.:23,59,49,91,121,Page41Chapter3 Fractions,decimalsandpercentagesThischapterfocuseson: 3.1Fractions 3.2Decimals 3.3Indices 3.4Standardindexform 3.5Percentages 3.1FractionsIntheprogressionofmathematicstopicspresentedtochildren,thestudyoffractionscanbethefirstdeparturefromtherestrictiontothepositiveintegers,whichare usuallyreferredtoaswholenumbers.Afractionisusuallyintroducedasaconceptenablingreferencetobemadetoapartofawhole. Forexample,3/8canbethoughtofasthreepartsofawholeonewhichhasbeensplitintoeightequalparts.Adiagramcanbeusedtoillustratethisparticular fraction. Whatifthetopnumberofthefractionisaninteger(wholenumber)greaterthanthebottomnumberofthefraction?Doesthisunderminethenotionofafractionbeinga partofawhole?Twoexampleswillillustrateanappropriateanswertothisquestion. Considerfirstthefraction18/3.Itstopnumberisgreaterthanitsbottomnumber.Adiagramcouldbeusedtogiveaninterpretationofthisfraction.Page42 Eachunit(whole)issplitintothreeequalparts.Oneofthoseparts,intheleftmostunit,isshaded.Thatshadedpartrepresents1/3.Intheentirediagramthereare18 suchthirds.Those18thirds,therefore,represent6units.So18/3=6. Whatabout18dividedby3?Thatisalsoequalto6. Sothefraction18/3maybeinterpretedintwoways: itmaybethoughtofaseighteenthirdsor itmaybethoughtofas183. Bothinterpretationsareequallycorrect.Theappropriatewayinwhichtoconsider18/3dependsonyourpurposeorthecontext. Forexample,ifyouhaveeighteensweetstosharebetweenthreechildrenthenyoursituationrequiresyoutothinkof18/3as18183.Asdivisionisrepeated subtraction,youareinprincipletakingawaythreefromeighteensixtimes,soeachchildgetssixsweets.Thisillustrationgivessomejustificationforsometimescalling divisionbyanothertermsharingaswellassometimescallingsubtractionbyitsalternativetakingaway. Whatifthetopnumber(calledthenumerator)issmallerthanthebottomnumber(calledthedenominator)ofthefraction?Doesthisaffectthepossibleinterpretations ofthefraction? Considerthefraction2/3forthepurposeofillustration. Thiscanalsoberepresentedbyadiagram. Theouterrectangle,aspreviously,represents1orunity.Itissplitintothreeequalpartsandtwoofthosepartsareshadedtorepresent2/3.Whatabouttheother interpretation,accordingtowhich2/3maybethoughtofas23? Ifthisisconsideredintermsofadiagram,thenthenumeratormayberepresentedbytworectangleswhichhavetobesplitintothreeequalparts.Thiscanbe achievedbysplittingeachrectangleintothree,asinpreviousdiagrams. Ofthesixsmallerrectanglesonethirdneedtobeshaded.Thiscanbedoneintwowayseitherasitisdoneinthediagramorbyshadingtwoofthesmallerrectangles withinoneonlyoftherectanglesrepresenting1.Whathasbeenachievedbythistwofoldapproach?Justthat2/3of1isthesameas1/3of1plus1/3of1.More simplytherehasbeenadiagrammaticproofofthestatementthat: 2/3=1/3+1/3.Whenthefraction18/3wasselectedabove,therewasaspecialfeaturewhichlimitedtheusefulnessofthechosenfractionthenumeratorisanexactmultipleofthe denominator.18issixtimes3.Whatifthenumeratorisnotanexactmultipleofthedenominator,butstilllargerthanit?Page43 Letsconsider,forexample,thefraction11/3.Showingthisonadiagramisawkwardbecauseelevenunits(rectangles)doesnoteasilysplitintothreeequalparts. Whyisthis?33=9and43=1211isnotanexactmultiple3or,expressedanotherway,3isnotafactorof11.Inthiscase,thealternativewayofviewingafraction canbehelpfulinpromotinganunderstandingofwhatissignifiedbythefraction. Herethefraction11/3canbeconsideredas11dividedby3,whichinturncanbethoughtofastaking3from11asmanytimesaspossible.Thesnag,ofcourse,is thatthereisaremainderof2.Lookatthediagrambelow. If11/3is11dividedby3whichcomesto3,leavingaremainderof2,whatsensecanbemadeoftheremainder?Again,lookatthesamediagram.Theremaining2 canbesplitintothreeaswasdonetoillustrate2/3.Theoutcomeofthediagrammaticrepresentationof11/3is,therefore,toshowthatitequalsthreeandtwothirds. Moreconcisely,thismaybewrittenas:Notethecontractionontherighthandsideof3+2/3.canbethoughtofasacompactwayofwriting3+2/3.TopheavyfractionsandmixednumbersAtopheavyfraction,usuallyreferredtoasanimproperfractionsuchas11/3,canimmediatelybeseentobegreaterthan1.Thedenominator,3,indicatesthe segmentsforming1,Thenumerator,11,indicatesthebatchesofthreesegmentsavailablethreebatcheswithtwosegmentsleftover.So,giveneleventhirdsasa topheavyfractionitcanbechangedintowhatiscalledamixednumber,consistingofanintegerpartandafractionpartlessthan1,bythinking: 3into11goes3times.3times3is9.Therearetwoleftoverwhicharetwothirds.Soelevenoverthreeisthreeandtwothirds. Whatofthereverseprocess?Supposeyouhaveamixednumberandwanttochangeitintoatopheavy(improper)fraction.Reasonsforneedingtodosowill beconsideredwhenoperationsonfractionsareillustrated. Considerthemixednumber The5isanindicationoffifths.Therearefivefifthsin1.Soaltogetherthereare45=20fifths,plusthreefifths,giving23fifthsintotal.Morecompactly,Page44 Sinceanyintegercanbeexpressedasafraction,integerscanbethoughtofasspecialfractions. Anintegerisafractioninwhichthenumeratorisamultipleofthedenominator. Oneimplicationofthisisthatfractionscanbeoperatedonbythesamefourbinaryoperationsaswereconsideredinrelationtointegers. Willthealgorithmsforperformingthoseoperationsbeverydifferenttothoseusedwithintegers?Itturnsoutthatthealgorithmsaresubstantiallydifferentandwill needtobeconsidereddifferently.Themultiplicationoffractionscannotbeusefullythoughtofasrepeatedadditionandthedivisionoffractionscannotbeusefully thoughtofasrepeatedsubtractionManypeoplefindoperationsonfractionsdifficulttofollow.So,readthenextsectionslowly.Itmayalsohelpyoutomakenotes whileyoureadit.OperationsonfractionsAdditionoffractions Tomakeanalgorithmfortheadditionoffractionscomprehensibleasimpleexample,illustratedwithadiagram,willbehelpful. Consideranimaginarysituationinwhichamanleaveshisfortunetohiswife,hisonlychildandaregisteredcharityaccordingtothefollowinginstruction. 7/12ofmyfortuneisbequeathedtomywife.3/8isbequeathedtomydaughterandtheremainderisbequeathedtocharity. Whatfractionofthemansfortunewaslefttocharity? Whatisneededhereisanalgorithmforfindingthesumof7/12and3/8. Toinventadiagramonwhichtheadditioncanbedisplayed,thefocusmustfirstbeonthetwodenominatorsheretwelfthsandeighthsareinvolved.Ifarectangleis usedtorepresent1(thewholefortune),thentherectanglemusteasilysplitinto12aswellas8equalparts. Whatisthesmallestnumberofsubdivisionsoftherectanglerequired? 24arerequired.Howcanyouarriveatthatnumber?Startwiththelargestofthenumbers12and8.12isnotamultipleof8soyoudoubleittoget248isafactor of24.Soyoustartwitharectanglesplitupinto24squares.Page45 Thediagramontheleft,providingthekey,enablesyoutospottwoequivalentfractions.Comparisonoftherectanglesfor1/12and1/8withthesquarerepresenting 1/24showsthat1/12=2/24and1/8=3/24. Studythediagramsforawhile.Theyillustratetheconclusionthat1/24ofthemansfortuneislefttocharity. Thediagramsaremeanttoservetwopurposes: tojustifyaprocedureforaddingtwoparticularfractionsand tosuggestageneralprocedurethatis,analgorithmwhichcanbefollowedtoaddanytwofractionswithoutadiagram. Atthisstage,recollectwhatyouknowaboutaddingfractionsyoumayremembersomerulesbutmaynothavethoughtaboutwhytherulesactuallywork! Withtheabovefourdiagramsinmind,letusseeifthestepsofanalgorithmcanbeformulated.Itturnsoutthatonlythreestepsarerequired. 1Findacommondenominatorforthefractionstobeadded. Thiscanbedonebytakingsuccessivemultiplesofthelargestofthedenominatorsuntiltheotherdenominatorsdivideintothecurrentmultiplebeingtried. 2Expresseachofthefractionstobeaddedasitsequivalentfractionwiththecommondenominatorfoundinstep1. 3Addthenewnumeratorstofindthenumeratorofthesumofthefractions.Placethatnumeratoroverthecommondenominator. Thefractionobtainedbythethreestagealgorithmisthesumoftheoriginalfractions. Step2mayrequirefurtherexplanation.Supposeyouwanttoaddthefractions1/3,5/12and7/20.Takingmultiplesof20inturnyoufindthat60istherequired commondenominator.Howdoyoufindthecorrespondingnumeratorsforeachofthefractions? Consider1/3.Remember,fromChapter2,that1isthemultiplicativeidentity.Multiplying1/3by1doesnotchangeitssize.Youneed1expressedinappropriate form.Whatisthat?Youneedtoreplacethequestionmarkin1/3=?/60.Sotheappropriateformof1is20/20.(3goesinto60twentytimes.) Youthenproceedwith:1/320/20=20/60 followedby:5/125/5=25/60(since12goesinto60fivetimes) andfinally:7/203/3=21/60 Nowyouareinapositiontosimplyaddthenewnumerators.Whatdoyoufindcuriousaboutthisresult? Ithastwocuriousfeatures:itsnumeratoranddenominatorhavecommonfactorsitisalsogreaterthan1. Thehighestcommonfactorof66and60is6.Aprocedurecalledcancellingcanbefollowedsothat6canbecancelledinto66togive11and6canbecancelled into60togive10.Thisleadstotheequality:Page46 Subtractionoffractions Thealgorithmforsubtractingfractionscanbeobtainedfromthatfortheadditionoffractionssimplybychangingstep3to: Subtractthenumeratorstoobtainthenumeratorofthedifferenceofthetwofractions. LookbacktothediagramsillustratingtheadditionoffractionsandcheckthatyoucanseethatCheckthatthethreealternativesareequal.Ofcourse,1/3isthesimplestformoftheresult. Thealgorithmsfortheothertwooperationsonfractionshavespecialfeatures.Manyofyoumayhavefeltinyourschooldaysthatthealgorithmfordivisionof fractionswasthemostdifficulttounderstand. Multiplicationoffractions Letustrytomakesomesenseofsomethinglike2/33/4. Canthisbeillustratedbyadiagram,soithasakindofphysicalmeaning?Seethediagrambelow. Whendoingmultiplicationwithintegers,suchas5times4,howcanitbethoughtof?As5lotsof4.As4repeated5times.Ifyoustartwith3/4ratherthan4,itmakes nosensetotalkof2/3lotsof3/4orof3/4repeated2/3times.Totrytogetaformoflanguagewhichhelpstomake2/33/4abitmoremeaningful,perhapsthe followingmayhelp: Thinkof54asstartwith4andthentake5ofit.Thiscouldthenleadtothinkingof2/33/4asstartwith3/4andthentake2/3ofit. Nowletusseehowdiagramscanbeusedtocalculatetheresult. Asquarecanbeusedtorepresent1itisshownsplitintoquarters.Thesequenceofdiagramsillustrateshowtheproductof2/3and3/4maybeobtained. Itislikelythatthedifficultychildrenwillhavewiththemultiplicationoffractionswillfocusonthethirddiagram.Iftheyhavebeenintroducedtotheconceptofa fractionbyaverbaldefinitionalongthelinesofafractionisapartofawholethenthatdefinitioncanbeanobstacletoappreciatingthatyoucanhaveafractionofless thanawhole.Lookatthethirddiagramandtrytothinkonlyofthegeometryofitcastoutthoughtsofthefractionsinvolved.Focusonthethreeshadedsquares.One thirdofthreesquaresisonesquare.Thatsquarerepresentsaquarter.Soathirdofthreequartersisonequarter.Howcouldthishavebeendonewithoutthediagram?Page47 Youcouldhavemultipliedthenumeratorstoget3andthedenominatorstoget12,producing3/12.Cancelling3finallygivestheresultof1/4. Amoreefficientprocedure,however,wouldhaveinvolvedcancellingthe3inthedenominatorof1/3andinthenumeratorof3/4togive:Thinkhowanalgorithmformultiplyingfractionsbeextractedfromthisillustration,sothatfuturemultiplicationscanbedonewithoutadiagram. Againathreestepalgorithmhasemerged: 1Lookforfactorscommontoanynumeratorandadenominatorcanceleachofthecommonfactorsyouhavefoundintoanumeratorandadenominator. 2Multiplytheremainingnumeratorstogetthenumeratoroftheresultmultiplytheremainingdenominatorstogetthedenominatoroftheresult. 3Checkthefractionyouhaveobtainedfortheproduct.Ifyoucanfindnofurtherfactorstocancelthenyourresultoftheproductofthefractionsis expressedinitssimplestform. Thefractionobtainedbyapplyingthisthreestepalgorithmtothegivenfractionsistheproductofthegivenfractions. Whenmultiplicationofintegerswasconsidereditwassuggestedthatitcouldbethoughtofasrepeatedaddition.Whataboutmutiplicationoffractions?Canthatbe thoughtofasrepeatedadditionoffractions?No,exceptwhenthemultiplierisgreaterthan1.Suchwouldbethecasewith,forexample, bethoughtofas .Theproductcaninvolvingthenotionofrepeatedaddition,butforthepurposeoffindingtheproductitismuchsimplertoapplythethreestepalgorithmasfollows:changingthemixednumberintoatopheavyequivalent21/5 Cancelling3into3and21givesPage48 Divisionoffractions Wewillproceedasbeforewithaparticularexampleillustratedwithdiagrams. Considerthecaseof7/82/5.Whatwouldbeanappropriatediagramforthis? Onewholeinthisillustrationisrepresentedbyarectangleenclosing80squaresasshowninthekey.Noticethatthediagramrepresenting7/8encloses70squaresand thediagramfor2/5encloses32squares.Rememberthatthedivisionofintegerscouldbethoughtofasrepeatedsubtraction.Canthisviewbeadoptedtohelptowardsanunderstandingofthedivisionof fractions?Topursuethisquestion,focusontherepresentationsof7/8and2/5intermsofsquares. Thentheappropriatequestionbecomes: Howmanylotsof32squarescanyougetfrom70squares?Sothequestionwhatis7/8dividedby2/5?hasbecometranslatedintowhatis70dividedby32? Theresultis2withaPage49 remainderof6.Whatshouldyoudowiththe6?Putitoverthe32togetthefraction6/32which,insimplestform,is3/16. Sothefinalresult,illustratedinthelastdiagramis Thinkingofthedivisionoffractionsintermsofrepeatedsubtraction,astheaboveillustrationamplydemonstrates,isverycumbersome.Letusthenexplorethe extractionofanalgorithmfromthediagrammaticillustrationoftheparticularcaseof7/82/5.Themixednumber obtainedastheresultisequalto35/16.Notice that35equals7times5and16equals8times2.Thisisverycurious,becauseifthe2/5isinvertedtobecome5/2andtheoperationofdivisionischangedinto multiplicationacorrectresultisobtainedforthatmultiplication:Thealgorithmforthedivisionoffractionsisyetagainathreestepprocedurewiththethirdsteprequiringatransformationofthedivisionintomultiplication. 1Changethedivisionsignintomultiplication. 2Invertthefractionwhichwasontherightofthedivisionsign. 3Usethemultiplicationalgorithmfortheproductoffractionsobtainedfromcarryingoutsteps1and2. Thisconcludesthepresentationofthealgorithmsforthefourbinaryoperationsonfractions.Theyhavebeenshowntoincludetheintegersasspecialcases.Their denominatorsmaybeanyoftheinfinitecollectionofintegersandareliabletobechangedfortheconvenienceofwhoeverisoperatingonthem.Inspiteofthemultitude ofusefulapplicationsoffractionsinfinancialandscientificwork,itisthosefractionswithdenominatorswhicharemultiplesoftenwhichhaveacquiredgreatpopularity. Wemayhavetenfingersandtoes,buttrynottoregardfractionswithdenominatorswhicharenotmultiplesoftenassomehowlessworthyofattention.Thefollowing linesmayhelptosustainsuitableempathywiththewholecollectionoffractionsregardlessoftheirparticulardenominators. Afractionslifeisfullofstrife Alittlelinecutsitintwo. Itstopandbottomgetcancelleddown Bylotsofpupilstoldwhattodo. Yetwithoutmorefactorsitcansurvive Insimplestformandstayalive Untildecimalsandpowersoften Begintheattackalloveragain.SomethingtothinkaboutBeforeleavingthissection,hereissomethingtoreflecton.Youoftenhearpeopledebatingwhetherweoughttoteachfractionsatall.Webelieveitisusefulformany reasonsandherearefewofthem:Page50 Inreallifecontextsweusethewordsandsymbolsforhalf,quarterandsoon.Supermarketsanddepartmentstoresusetheseallthetime.Childrenneedtobe familiarwiththeseideas. Whenwemeasureusingarbitraryunits,forexamplewhenmovingfurnitureathome,forsewingorforcooking,weoftenusetheconceptoffractions. Decimalnumbers,whichareanextensionoftheplacevaluesystem,arebasedontheideaoffractions.Percentagesandratioarealsoconceptsrelatedtofractions. Learninghowtooperateonfractionsisnecessaryfortacklingalgebraicworkandforundertakingcalculationsinprobability.3.2DecimalsDecimalnumberscanbethoughtofascombinedintegers(wholenumbers)andfractionsrestrictedtothosewithdenominatorswhicharemultiplesoften. Thedecimalnumbersformwhatiscalledthedenarysystem,basedonten,justasbinarynumbersformwhatiscalledthebinarysystem,basedontwo. Thedecimalnumbersincorporatethenotionofplacevalue,butextenditbeyondtheintegerstoitsfractionalparts.Theseparationofitsintegerandfractionalpartsis accomplishedbyasimpleandingeniousdevice,thedecimalpoint.Thisisoneexampleofasectionofmathematicsinwhichthechoiceofsigncreatesthepossibilityof enormousthinkingdevelopment. Onequestionofinterest,whichwilltobedealtwithlater,isthemathematicalconnectionbetweendecimalnumbersandtheintegersandfractions.Forthemomentlet usbeconcernedwiththepracticalimportanceofdecimalnumbersindecimalcurrencysystems. Poundssterlingisadecimalcurrency.Onepoundisequivalenttoonehundredpence.Inmoremathematicalnotationthisiswrittenas:Thisenablespricestobewrittenindecimalform,suchas4.37. Thenumberreadasfourpointthreeseven,however,iswrittenas437.Thedifferencebetweenthetwoisjustthepositionofthepoint.Thisismoreamatterof convenienceafullstoponthewordprocessorkeyboardwasusedforthefirstwhereasthedecimalpointwasimportedfromacollectionofsymbolsavailableinthe wordprocessingpackage.TheUnitedKingdomcurrencyisadecimalmoneysystemthedotseparatingthepoundsfromthepenceisessentiallyadecimalpointinits function. Theimportantaspectoftheprice4.37isthatitcanbethoughtofasadecimalnumbergiventotwodecimalplaces.Thevalueofeachdigitinthenumberdepends onitsplace.The7indicatessevenpenceor7hundredthsofapound.The3indicatesthirtypenceorthreetenthsofapound.The4totheleftofthedecimalpoint indicates4unitsor4pounds.Allthis,ofcourse,isquitefamiliartoyou.Thepurposeinmentioningitistorecallthebasicfeaturesofdecimalnumbersasapositional representationsysteminwhichthevalueormeaningofeachdigitdependsonitspositioninthenumber. Totheleftofthepointthedigitsreferto,fromrighttoleft,unitsthentensthenhundredsthenthousandsandsoonthevaluesaremultipliedbyteneachtimethe placeshiftsoneplacefurthertotheleftasshowninthediagram.Page51 Thousands1000Hundreds100Tens10Units1Tenths01Hundredths001Thousandths0001Totherightofthepointthedigitsreferto,fromlefttoright,tenthsthenhundredthsthenthousandthsthententhousandthsandsoonthevaluesaremultipliedbyone tentheachtimetheplaceshiftsoneplacefurthertotheright. Nowtomoveawayfromwhatisfamiliartowhatmaybelessfamiliar,butusefulinunderstandingtheusefulnessofusingdecimalnumbers. ConsiderthefollowingexchangeratebetweenpoundssterlingandUSAdollarswhichappearedonthe31December1998.Thisindicatesthatonepoundisworth1.6743dollars.WhyshouldtheAmericanequivalentbegiventofourdecimalplaces?TheAmericansystemisalsoadecimal systemwithonehundredcentstothedollar.Whatdothedigitsindicateintheexchangerate? The6indicates6/10ofonedollarwhichissixtycents.The7indicates7/100ofonedollarwhichissevencents.SofarthisindicatesthatoneUKpoundcanbe exchangedforonedollarandsixtysevencents.Whatabouttheotherdigits?The4indicates4/1000ofonedollarandthe3indicates3/10000ofadollar.Thereareno coinsworth1/1000ofadollaror1/10000ofadollar,sowhatpurposeisservedbytheinclusionofthe4andthe3intheexchangerate?Basically,theywillaffectvery largetransactions.TenthousandUKpounds,forexample,wouldexchangeforsixteenthousand,sevenhundredandfortythreedollarsratherthanjustsixteen thousand,sevenhundreddollarsifthe4andthe3werenotincludedintheexchangerate. Whatthisillustratesisthattheaccuracyofadecimalnumber,inapracticalapplication,doesnotbecomeabsurdjustbecausenotangiblethingscorrespondtothe degreeofaccuracy. Stayingwithinthecurrencyapplicationofdecimals,considerthelaunchoftheeuroonthe1stJanuary,1999.TheeurozonecreatedintheelevenEuropeancountries involvedbecamecontrolledbythefollowinglistofirrevocableconversionrates,sothatfromnowononeeuroequals: 13.7603Austrianschillings 40.3399Belgianfrancs 1.95583Germanmarks 5.94573Finnishmarks 6.55957Frenchfrancs 0.787564Irishpounds 1936.27Italianlire 40.3399Luxembourgfrancs 2.20371Netherlandsguilders 200.482Portugueseescudos 166.386SpanishpesetasPage52 (Sterling,notwithintheeurozoneatthetimeofthelaunchwasgivenavalueof70pencetooneeuro.Thatconversionratewillfluctuateandbeexpressedtofour decimalplaces.) Takeagoodlookattheelevenrates.Whatdoyounoticeaboutthenumbers?Doyoufeelcuriousaboutthefeaturesyouhavenoticedandcanyoumakesenseof them?Therearethreestrikingfeatures.Letusconsidertheminturn. 1Themostobviousfeatureoftheelevenconversionratesisthattheyareallexpressedindecimalnumbers.Thisgivesthemakindofuniformityandprovidesasimple basisforcalculationsabouttransactionsinvolvingtherates.Whetherornottheelevencurrenciesaredecimalsystemsofmoneycannotbeinferredfromtheir conversionratesbeingexpressedindecimal. 2Thenextfeatureyoumayhavenoticedisthattheratesvaryinthenumberofdigitstotherightofthedecimalpoint.TheItalianlirerateisexpressedtotwodecimal places,whilsttheratefortheIrishpoundisexpressedtosixdecimalplaces.Thenumbersintheconversionlist,indescendingorderofmagnitude,givingtheinteger partonlyandthenumberofdecimalplacesinbracketsafterwardsareasfollows:1936(2),200(3),166(3),40(4),40(4),13(4),6(5),5(5),2(5),1(5),0(6).Notice thatthenumberofdecimalplacesturnsouttobeinascendingorderwiththeintegerpartsindescendingorder. 3Themostcuriousfeatureoftheconversionrates,however,isthatalltheratesareexpressedtosixsignificantfigures.Thefirstsignificantfigureinadecimalnumberis thefirstnonzerodigitinthenumberasyoulookatitfromlefttoright.SothefirstsignificantfigureintheratefortheIrishpoundisthe7totheimmediaterightofthe decimalpoint. Thereasonforthesixsignificantfigureaccuracy,ratherthanauniformnumberofdecimalplaces,isconnectedwiththenotionoferrorincomputationandwillbedealt withinChapter5wherethenotionsofmeasure,accuracy,uncertaintyanderrorwillbeconsideredinsomedetail. Thefouroperationsofaddition,subtraction,multiplicationanddivisionwillnowbediscussedsoastogainsomeunderstandingofanalgorithmforperformingeach operation.AdditionofdecimalnumbersThealgorithmforperformingadditionondecimalnumbers,onpaper,isessentiallythesameasthatfortheadditionofwholenumbers.Theverticalalignmentofthe decimalpointsofthenumberstobeaddedguaranteestheverticalalignmentofallotherdigitsofcorrespondingplacevalue.Thetwinprocessesofdecompositionand carryingthenformthebasisofthealgorithmforaddition,justasfortheadditionofintegers.Theplacevaluesprovidingtheframeworkfortheadditionofdecimalsare: 10000,1000,100,10,1,1/10,1/100,1/1000,1/10000. Thethreepointsontheleftandrightindicatethattheplacevaluescontinueindefinitelytotheleftandrightaccordingtothetwoprinciples. Reminders 1Thedecompositionprocessinvolvesreducingthedigitinonepositionby1anddecomposingitintotenoftheplacevaluetoitsright.Page53 2Carryinginvolvestransferring10,oramultipleof10,oftheplacevalueinonepositiontothepositionontheleft,addingtothedigitinthatpositionthe multipleof10concerned. Theexampleoftheadditionof157.68,68.87and476.29isgivenbelowasanillustrationofthealignment,decompositionandcarryinginvolved.Noticethatthemultiplesoftenoftheplacevaluesinanycolumnarewrittenundertheanswerdigitofthenextcolumn. The2carriedintothetenthscolumnarisesfrom8,7and9havingasumof24. Whatiscarriedisthemultipleoftenwhatiswrittendownaspartofthesum(thetotaloranswer)istheremainderaftertheextractionofasmanytensaspossible. Similarly,1iscarriedintotheunitscolumn,2intothetenscolumnand2intothehundredscolumn. Haveyouspottedtheabsenceofdecompositionintheapplicationofthealgorithmforaddition?Thisisduetotherequirementofthealgorithmtoproceedfromright toleftonly. Additioninvolvesalignment,carryingbutnodecomposition,SubtractionofdecimalnumbersThealgorithmforthesubtractionofdecimalnumbersalsoincorporatesthesameprinciplesgoverningthesubtractionofintegers.Theverticalalignmentofthedecimal pointsofthenumbersinvolvedinthesubtractionguaranteesthealignmentofalltheotherdigitsofthesameplacevalue. Letusconsidertherolesofalignment,decompositionandcarryingintheprocessesofsubtractionofonedecimalnumberfromanotherbytakingtheexampleof38.9 subtractedfrom124.3. Herethereisdecompositionandnocarrying,becausethealgorithmrequiressomeprocessesfromlefttorighteventhoughtheprocedurefocusesonthedigitsofthe numbersfromrighttoleft,inincreasingorderofplacevalue. Thesubtraction,onpaper,of38.9from124.3shownbelowillustratesthealignmentanddecompositioninvolvedthelackofanyroleforcarryingshouldbecarefully noted.Noticethatthe3/10mustbecome13/10bythedecompositionofoneofthe4units,makingthe124.3into123+13/10.Whensubtractingtheunitsthe3mustbe changedinto13bythePage54 decompositionofoneofthe2tens,makingthe123into110+13.Finally,whenthetensarebeingsubtractedthe1,inthetensposition,mustbecome11bythe decompositionofthe1hundredinto10tens,makingthe110into0hundredsand11tens. Subtractioninvolvesalignmentanddecomposition,butnocarrying.MultiplicationofdecimalnumbersThealgorithmforthemultiplicationofdecimalnumbersisessentiallythesameasthatforthemultiplicationofintegers,withanextensiontotakeaccountofthedigitsto therightofthedecimalpoints.Manypeopleusealearntrule(notalwaysunderstood)formultiplyingdecimalnumbers.Hereisanexplanationastohowtheruleworks. Beforestartingtheactualmultiplication,itisadvisabletodecidetheplacevaluesoftherightmostdigitsandmultiplythemasfractionstodeterminethenumberof digitstotherightofthedecimalpointintheproduct. Considertheexampleoftheproductof251.6and43.7setoutonpaperasbelow.Therightmostdigits,6and7,represent6/10and7/10andgive42/100onmultiplication.Canyouextractaruleforpositioningthedecimalpointinthefinalproduct? Thefinalproductof10994.92isexpressedtotwodecimalplaces.Bothofthenumbersmultipliedtogivethatproduct,251.6and43.7,aregiventoonedecimal place.Thereisarulesuggestedbythis: Thefirstnumberhasonedigittotherightofthepoint. Thesecondnumberhasonedigittotherightofthepoint. 1+1=2,sotheproductofthetwonumbersmusthavetwodigitstotherightofthepoint. Letustestthissuggestedrulebeforedescribingthealgorithmformultiplication. Considertheproductof63.728and8.14.Thefirstofthesenumbersisgiventothreedecimalplacesandthesecondisgiventotwodecimalplaces.Since3+2=5, theproductmusthavefivedigitstotherightofthepoint.Isthiswhatisindicatedbyanexaminationoftherightmostdigitsofthenumberstobemultiplied? The8of63.728represents8/1000andthe4of8.14represents4/100.Theproductofthosedigitsis,therefore,32/100000.Theplacevalueofonehundred thousandthsisoccupiedbythedigitinthefifthpositiontotherightofthedecimalpoint.(The5zeroscorrespondtotheposition.)Sotheproductmaythenbefoundby settingthingsoutasfollows:Page55 Thefinalproductis,therefore,518.74592whichisanumberto5decimalplaces. Thealgorithmforthemultiplicationofdecimalnumbersmaynowbedescribedasathreestepprocedure. 1Writedownthenumberstobemultipliedoneundertheother.Itisnotreallynecessarytoalignthedecimalpointsandthedigitsofthesameplacevalue, butsuchanalignmentmayimprovethepresentation. 2Countthenumberofdigitstotherightofthepointineachofthedecimalnumbersandaddthosetwonumberstogether.Theirsumisthenumberofdigits totherightofthepointintheproduct. 3Ignorethedecimalpointsinthetwogivennumbersandjustmultiplythemasthoughtheywereintegers.Inotherwords,applythealgorithmforthe multiplicationoftwointegers.Insertthedecimalpointinthepositiondeterminedbystep2. Haveyounoticedoneimportantfeatureofmultiplicationwhichisnotsharedbyeitheradditionorsubtraction? Ifyoucannotthinkofanything,lookbackatthepreviousexamples.Comparetheaccuracyoftheanswerwiththeaccuracyofthenumbersaddedorsubtracted. Youshouldfindthattheaccuracyofthemostaccurateofthenumbersinvolvedintheadditionorsubtractionisthesameastheaccuracyoftheanswer.So,ifa numberto2decimalplacesisaddedtoanumberto3decimalplacesthesumofthetwonumbersisto3decimalplaces. Thenumberofdecimalplacesintheproductoftwonumbersisalwaysatleasttwicetheaccuracyoftheleastaccurateofthenumbersmultiplied.Theaccuracy(the numberofdecimalplaces)maybegivenbythealgorithm,butinapracticalsituationitissensibletoraisethequestion: Istheaccuracyoftheresultofacomputationgreaterthanisjustifiedbytheaccuracyofthedatausedinthecomputation? Isthereanyadvantageinconsideringthemultiplicationofdecimalnumbersasrepeatedaddition?Youcoulddo,butitwouldberathertedious.Theexampleof 251.6times43.7couldbedonebywritingdown251.6fortythreetimes,andaddingthemtoget10818.8.Thenyouwouldneedtofind0.7of251.6,whichis 176.12,andaddthatto10818.8tofinallyobtaintheproductof10994.92.Howmuchwisertoknowthealgorithmformultiplication! Letusnowconsiderwhatmaywellbethemostdifficultofthebinaryoperationsondecimalnumbers.DivisionofdecimalnumbersWhatisinvolvedinthedivisionofdecimalsmaybebetterunderstoodbyrecallingtwonotionsdealtwithinearliersections: thenotionofdivisionbeingthoughtofasrepeatedsubtractionand thenotionofafractionbeingthoughtofasthenumeratordividedbythedenominator. Letustakethefirstofthesenotionsandconsideritinrelationtothedivisionof3.4by0.2. Tounderstandhowthismaybethoughtofastherepeatedsubtractionof0.2from3.4,takealookattherepresentationof3.4inthediagrambelow.Toseehow therepeatedsubtractionmaybeperformed,useismadeoftheequivalenceof0.2,2/10and1/5.Page56 Theshadedrectanglerepresenting0.2fitsintothesquarerepresenting1fivetimes,sotherearefifteen0.2sin3.0.Addtothatthetwo0.2swhichfitintothe0.4and youobtainthetotalof17.Youhavearrivedattheresult:or17.0tothesamedegreeofaccuracyasthenumbersinvolvedinthedivision. Thisexampleofthedivisionofdecimalnumberswascarefullychosenbecauseofitssimplicity.Whatmakesitsimple?Thereisnoremainder.Thedecimalresult,or quotientasitiscalled,terminates.Thequotientis17.0exactly.Beforeconsideringadecimaldivisionwhichdoesnotterminate,letuslookatthesecondnotionreferred tothatofafractionbeingthoughtofintermsofdivision. Considerthedivisionof6.072by1.32andrepresentthisasafraction. If6.0721.32iswrittenas6.072/1.32,anunusualfeatureofthefractionisthatboththenumeratoranddenominatorarenotintegers.Recallingthatdivisionmaybe thoughtofasrepeatedsubtractionanddecidingthat1.32times4gives5.28thefractionmayberewrittenas:Nowrecallthatanynumbermultipliedbythemutiplicativeidentity,1,retainsthesamevalue.Multiplyingthefractionby1000/1000changesboththenumeratorand denominatorinto792/1320,afractionwithintegernumeratoranddenominator.Thealgorithmforthedivisionofintegersmaybeappliedtoconvertthisfractionintoa decimalnumberbysettingthedivisionoutasshownbelow:Thisisthelongdivisionprocedureextendedtodecimals,togiveaquotientof0.6.Itisaprocedurewhichrequiressomejustification.Thinkagainofthedivisionin termsofafractionandofamultiplicativeidentityintheformof10/10,sothat:Page57 Thisprovidesthejustificationofignoringthepointin792.0andthedivisionof7920by1320togetthe6totherightofthepointinthequotient. Againthedivisionterminates.Thequotientis4.6exactly.Theactual,completedivisioniscarriedoutasshownbelowwith6.072dividedby1.32changedinto 607.2dividedby132.(6072dividedby1320wouldgiveexactlythesameresult.Whychooseoneformratherthantheother?)Theproceduretoadoptwhenthedivisiondoesnotterminateisbestillustratedbymeansofanexample. Considerthedivisionof52.6by8.32.Thisworksouttobe6.322115385,waybeyondtheaccuracyofthetwonumbersinvolvedinthedivision.(Thedividendand divisorforthosewhomaybeinterestedintheterminology.)Incaseswherethedivisionproduceseitheranonterminatingquotientoraquotientwithalotofdigitsto therightofthepoint,aecisionhastobemadeattheoutsetastohowmanydecimalplacesarerequiredinthequotient.Howthisdecisionismadewillbedealtwithin Chapter5.Ifadecisionismadeinadvancethatthequotientistobeworkedoutto2decimalplaces,thenthedivisionwouldbesetoutas:Thepresentationofdivisioninvolvedsomereferencestochangingafractionintodecimalform.Thiskindoftransformationneedstobedealtwithmoregenerally.Page58TheconversionoffractionsintodecimalformAnyfractionmaybeexpressedindecimalformjustbydividingitsnumeratorbyitsdenominator.Thedecimalmaybeexactlyequaltothefractionorbeapproximately equaltoit.Letusconsidereachcaseinturn. Thefraction7/8,changedintoadecimalbydividing7by8,isfoundtoequal0.875. Thismeansthat7/8=8/10+7/100+5/1000exactly. However,whenthesameistriedwith4/7thedecimalobtainedis0.571428571 Thedecimaldoesnotterminate.Insuchacaseyoucouldoptforachosendegreeofaccuracy,ofsay2decimalplaces,andwrite:(Thesymbolmeansisapproximatelyequalto.) Alternatively,ifyourealisethatthedigits571428keeprepeatinginthedecimalexpansionwhenyoupersistwiththedivisionprocess,youcouldindicatetherecurring digitsbyplacingadotabovethefirstandlastdigitsintherepeatinggroupandwrite:Onefinalaspectofthestructureofthedecimalsystemofnumbersisworthmakingexplicitatthisstage.Recallthattheplacevalueshaveanapparentpatternof multiplyingby10aspositionsmovetotheleftawayfromthepointandmultiplyingby1/10aspositionsmovetotherightawayfromthepoint.Thereisanalternative wayofdescribingthispattern. Considerthepositionstotheleftofthepoint. Firstthereis10. Secondthereis100.Thiscanbethoughtofas1010andwrittenas102. Thirdthereis1000.Thiscanbethoughtofas101010andwrittenas103.Thesmalldigittothetoprightofthe10iscalledanindexnumber.(Thepluralofindex isindices.)Sothenextplacevaluewouldbe104anditwouldbereadastentothepoweroffour. Nowthinkofthosesamepowersfromlefttoright:4,3and2.Whatisthepatterninthissequenceofpowers?Theyaredecreasingby1eachtime.How,then, wouldyouexpectthesequencetocontinue? After2comes1andthen0.1lessthan0is1.1lessthan1is2andsoon. Thesequenceofplacevaluesmay,therefore,bewrittenas:Comparingthislistwiththepreviouswayofexpressingtheplacevaluesyieldsthefollowingequivalentexpressions:Page59KeylssuesinteachingdecimalsResearchhashighlightedthatchildrenfindtheconceptofdecimalsdifficult.Arobustunderstandingoftheplacevalueofwholenumbersandreinforcementofthe relationshipbetweencolumnseachcolumntotheleftistentimeslargerisagoodbasisforunderstandingthatthevaluesgettentimessmallertotheright. Asinthecaseofteachingofwholenumbers,itisusefultodemonstratetheconceptofdecimalsusingconcreteapparatusanddiscussion.Someofthecommon mistakesmadebychildrensuchasorderingdecimalnumbersaccordingtothenumberofdigitsregardlessofthepositionofthedecimalpointandtheuseofzeroscan beavoidedifchildrenareencouragedtodiscusstheroleofthepointanditsroleindeterminingthesizeofnumbers.Youcanreadmoreonthisinthechapteron childrensmisconceptionsinSectionCofthisbook.ThesectiononteachingandassessingofdecimalsintheSCAApublication(SCAA,1997)providesmuchsupport forconsideringissuesregardingtheteachingofdecimals.3.3IndicesTheplacevaluesystemofnumbersbothwholenumbersanddecimalsisbasedonpowersoften.Inthissectionwewillbroadenthediscussionofpowersto includepowersofanyinteger.Theindexnotationisusedtorepresent,inacompactway,theproductofanumberbyitselfmanytimes. Consider103. Thethreeabovethe10iscalledapowerandtheteniscalledthebase.Thepowerindicatesthenumberoftimesthebaseismultipliedbyitself.Thisissometimes thoughtofasthreetensmultipliedtogether,sothat103=101010.Whenverbalisingitthisway,childrenmaymistakethistomean3times10.Thinkingof10as threetensmultipliedtogethercanleadsomeonetoaskthequestionabout10: Howcanyouhavezerotensmultipliedtogetherandget10=1 Tounderstandhowthisquestionismisguided,thenotionofthemultiplicativeidentityneedstobereviewed.Letusdothisintwostages. First,letuslookintothewayinwhichtheoperationofmultiplicationworkswithnumbersexpressedaspowersofabasenumber. Considertheproductof4and8. 48=32 4=22,so4canbewrittenas22 8=222,so8canbewrittenas23 32=22222,so32canbewrittenas25 Nowlookatthesameproduct,withthethreenumberswrittenaspowersof2.Focusontheindicesandnoticethat2+3=5.Thisgivesaruleformultiplyingpowersofthesamebasenumber: Whenmultiplyingpowersofthesamebasenumber,justaddtheindicestoobtaintheproduct. Multiplyinginthisformatcanbemucheasierthanapplyingthemultiplicationalgorithmtointegers.Whichwouldyouprefertodo?2781=2187or3334=37?Page60 3+4=7ismucheasierprovided,ofcourse,youknowwhichpowersof3equalthenumberstobemultiplied(i.e.that27=33,81=34). Second,considertheruleforaddingindexnumbersinrelationtothemultiplicativeidentity,1. 1251=125becauseyouaremultiplyingbytheidentity,1.Canthisbewritteninindexform?125=555,so125canbewrittenas53.If1canbewrittenasa powerof5andtheruleforaddingindicesappliedtogivethesamepowerof5,whatpowerof5mustbeusedtorepresent1?Zero.Thatmakesitpossibletowrite 1251125intheform535=53. Thesamekindofargumentcanbeusedtojustifyanyintegertothepowerofzerobeingacceptableasarepresentationof1. Supposeyouwanttodividetwonumberswiththesamebase,perhapswithdifferentpowershereyoucansubtracttheindices. Forexample,454242=43 Checkthat45=1024,42=16,43=64and102416=64.3.4StandardindexformIftheunitofmeasurementhasbeenfixed,ameasurementintermsofthatunitcansometimesbeanextremelylargenumber.Sometimesameasurementcanbean extremelysmallnumber.Ineithercase,onlyafewdigitsofthemeasurementmaybereliable.Withsuchmeasurements,thestandardindexformofnumberscanhave considerableadvantages.Letusconsiderafewverysmallandafewverylargenumbers. Anelectron,oneofthetwelveelementaryparticlesofmodernphysics,hasamassof9.11031kg.Whyshouldsuchasmallmassbemeasuredinkilograms?(1kg isequivalenttoabouttwoandaquarterpounds).Thesmallestbacteriaareabout400nanometresinsize.(1nanometreequals109metres.)Themassofanelectron wasgiveninstandardindexform.Theinformationwasgivenintwoparts.Thefirstpartconsistedofanumberfrom1andupto10thesecondpartconsistedofa powerof10.Thesizeofthesmallestbacteriacanbechangedintostandardindexformasfollows: SomelargemeasurementsarethemassoftheSun,1.9891033gm,andthemassofourplanetEarth,5.9771027gm. HowmuchgreaterthanthemassofEarthisthemassoftheSun?ThinkingofthemassoftheSundividedbythemassofEarthintheformofafraction:SothemassoftheSunisaboutthreehundredthousandtimesgreaterthanthemassoftheEarth. So,howdoyouwritenumbersinstandardindexform?Page61 Thestandardformfor Tryrepresenting0.00000000678instandardform.Itis6.78109nowyouseehowitsimplifiesreadingthatnumber.3.5PercentagesHaveyounoticedonasemiskimmedmilkcarton,perhapsonthetop,1.7%fat? Haveyounoticed,onapacketofcornflakes,theinformationthatthecontentscontain2.2goffatper100gofthecereal?Bothitemsofinformationreferto100. Onehundredisofconsiderableculturalimportance.Thenotionofacentury,thesubdivisionsofunitsofmoney,lengthandvolumeusedinmanycountries,allrelateto thenumber100. Haveyounoticedanydifferencesinthetwobitsofinformation?Thecerealinformationisdefinitelyaboutweightitcontainstwoweights,bothingrams.Since2.2g outof100consistsoffat,itcanalsobesaidthat2.2%ofthecerealconsistsoffat.Whatabouttheinformationconcerningmilk?Isthatrelatingtoweightorvolume? Themilkissoldbyvolume.The1.7%whichisfatisofadifferentdensitytotherestofthecontentsofthemilk.So,whetherthefatcontentis1.7%byweightorby volumedoesmakeadifference.Thishighlightsoneofthegreatestsourcesofconfusionwhenconsideringpercentages.Whatisthepercentageapercentageof?Failure tobeclearabouttheanswertothisquestion,orfailuretoevenaskthequestion,canleadtoeithermisunderstandingormisrepresentationbecomingpossible. Ofcourse,notallinformationstartsoutasapercentage.Astudentwhohasscored17outof20inatestafterpreviouslyscoring41outof50maybeinterestedin judgingwhethertherehasbeenanyimprovementinperformance.Onewayofcomparingthetwoscoresistoconvertthembothintoequivalentscoresoutof100. Ifeachofthescoresisthoughtofasafractionofthetotalmarksavailabletheneachcanbemultipliedbythemultiplicativeidentity,expressedinanappropriateform. Sothefirstscorecanbeconvertedintoanequivalentscoreoutof100asfollows:5/5isthechosenformbecause20needstobemultipliedby5togivetheresultof100. Thisenablesthescoreinthefirsttesttobedescribedas85%. Thesecondscorecanbeconvertedintoanequivalentscoreoutof100bychoosing2/2astheidentity,since50needstobemultipliedby2togivetheresult100. Thisproduces:Thismaybeexpressedas82%,arelativelylowerscorethaninthefirsttest. Thesetwoexamplesofconvertingfractionsintopercentageswerequiteeasybecausetheirdenominatorswerefactorsof100.Whatif100isnotamultipleofa fractionsdenominator?Takethecaseofaclassof28having3pupilsabsent.Whatisthepercentageoftheclasswhichisabsent?Istheabsenteeismworsethanin anotherclassof25with2pupilsabsent? Thefractionofpupilsabsentinthefirstclassis3/28.Tochangethisintoapercentage,anappropriateformoftheidentityisrequired. 28timeswhatequals100?28times100/28equals100.Page62 Thisgivesthenumeratoranddenominatoroftheappropriateformoftheidentity:Thedenominatoroftheresultis,ofcourse,100. Whatisthenumerator? Itis3(100/28),whichis300/28.Cancelling4intothenumeratoranddenominatorofthisyields75/7andthisapproximatesto10.71. Thepercentageofpupilsabsentinthefirstclassis,therefore,10.71%andisgreaterthanthe8%absentinthesecondclass.Thesecondwasmucheasiertowork outas:Amajoradvantageofthenotionofpercentagesisthatitprovidesacommonstandardforcomparisons.Differentamountsoutofdifferenttotals,expressedas equivalentamountsoutoftotalsof100,canthenbecompared.Asalaryincreaseof1500perannum,forsomeoneearning15000p.a.isanincreaseof10%.The sameincrease,forsomeoneearning20000p.a.,isanincreaseof7.5%. Thisadvantageofpercentagesforcomparingquantitiesisveryusefulformanysituations.When,however,thenumbersinvolvedareverysmallorverylargethereis anotherstrategyformakingcomparisons.Thisstrategyrequiresthetwonumberstobecomparedtobeexpressedasadecimaltimesthesamepowerof10.For example,themassoftheSunis1.9891033gm,themassoftheEarthis5.9771027gm.Whatisthecomparisonofthetwomasses?Theanswercouldbeobtained usingtwostrategies. Alternatively,thestrategyusedcouldbe:1.9891033=1.989l061027,soPage63Tasksrelatingtotheclassroom1Askafewchildrentoorderthefractions:1/3,1/16,and1/20andexplainhowtheydecidedtheorder. 2ThefollowingmistakeswerecollectedfromaYear6classofchildren.Foreachofthemistakes,considerwhatmaybethereasonwhyachildmakesthis mistakeandhowyouwouldbeproactiveandaddressthisattheteachingstage?a)b) c)53=15 d)47.3727p.Usedacalculatorandgot20.37 e) 3Achildorderedthefollowingnumbers,fromthesmallesttothelargest,andbroughtittoyoutobemarked.Whatactionwouldyoutake? 111,1001,11.01,111.11Tasksforselfstudy1Dothefollowingfractionsanddecimalsums.Estimatethesolutionsbeforeyoudothem. a) b) c)2.11+41.42+0.08= d)19.314.27= 2Put>or,whichis readasisgreaterthan.Ifx 3=12,thenx>2.Inasimilarway,whenitwassaidthatthesolutionoftheequationwaslessthan3,themathematicalsymbol,

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