Math for Primary Teachers

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Helping students: job 1

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MathematicsforPrimaryTeachersInEngland,andinternationally,numeracystandardsinprimaryschoolsarethecauseofasmuchconcernasliteracystandards.Oftenthegreatestproblemwithmathsat primarylevelistheteachersownunderstandingofthesubject.MathematicsforPrimaryTeachersaimstocombineaccessibleexplanationsofmathematical conceptswithpracticaladviceoneffectivewaysofteachingthesubject.Itisdividedintothreemainsections: SectionAprovidesaframeworkofgoodpractice SectionBaimstosupportandenhanceteacherssubjectknowledgeinmathematicaltopicsbeyondwhatistaughttoprimarychildren.Eachchapteralsohighlights teachingissuesandgivesexamplesoftasksrelevanttotheclassroom SectionCisacollectionofpapersfromtutorsfrom4universitiescoveringissuessuchastheteachingofmentalmathematics,childrensmathematicalmisconceptions andhowtomanagedifferentiation.Theyarecentredaroundthethemeofeffectiveteachingandqualityoflearningduringthiscrucialtimeformathematicseducation. ValsaKoshyisSeniorLecturerinEducationatBrunelUniversitywithresponsibilityformathematicsinservicecourses.PaulErnestisProfessorinMathematics EducationatExeterUniversity.RonCaseyisSeniorResearchFellowatBrunelUniversity.

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MathematicsforPrimaryTeachersEditedby ValsaKoshy,PaulErnestandRonCasey

LondonandNewYork

PageivFirstpublished2000 byRoutledge 11NewFetterLane,LondonEC4P4EE SimultaneouslypublishedintheUSAandCanada byRoutledge 29West35thStreet,NewYork,NY10001 RoutledgeisonimprintoftheTaylor&FrancisGroup ThiseditionpublishedintheTaylor&FranciseLibrary,2005.

TopurchaseyourowncopyofthisoranyofTaylor&FrancisorRoutledgescollectionofthousandsofeBookspleasegotowww.eBookstore.tandf.co.uk. 2000ValsaKoshy,PaulErnestandRonCaseyselectionandeditorial matter2000individualchapterstheircontributors Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedor utilisedinanyformorbyanyelectronic,mechanical,orothermeans,now knownorhereafterinvented,includingphotocopyingandrecording,orinany informationstorageorretrievalsystem,withoutpermissioninwritingfrom thepublishers. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationData Koshy,Valsa,1945 Mathematicsforprimaryteachers/ValsaKoshy,PaulErnest,and RonCasey. p.cm. Includesbibliographicalreferencesandindex. I.MathematicsStudyandteaching(Primary)I.Ernest,Paul. II.Casey,Ron.III.Title QA135.5.K6720009933554 372.7'044dc21CIP ISBN0203984064MasterebookISBN

ISBN0415200903(PrintEdition)

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Contents Listofcontributors Acknowledgements Introduction

viii x xi

SECTIONA

1 3 21

1 Teachingandlearningmathematics PAULERNEST

SECTIONB RONCASEYANDVALSAKOSHY

2 Wholenumbers 2.1Developmentofnumberconceptsintheearlyyears 2.2Theroleofalgorithms 2.3Placevaluerepresentationofnumbers 2.4Numberoperations 2.5Factorsandprimenumbers

23 23 25 25 29 37 38 41 41 50 59 60 61 66 66 68

2.6Negativenumbers 3 Fractions,decimalsandpercentages 3.1Fractions 3.2Decimals 3.3Indices 3.4Standardindexform 3.5Percentages 4.1Sequences 4.2Series

4 Numberpatternsandsequences

Pagevi 4.3Generalisedarithmetic 4.4Functions 4.5Identitiesandequations 4.6Equations 4.7Inequalities 5.1Theconceptofmeasure 5.2Length 5.3Area 5.4Volume 5.5Weight 5.6Time 5.7Angles 69 74 77 78 82 87 87 88 90 92 93 94 94 95 100 100 104 110 114 114 117 125 125 126 127 129 129 135 135 138 143

5 Measures

5.8Theuseofscales 6 Shapeandspace 6.1Coordinates 6.2Transformations

6.3Enlargement 7 Probabilityandstatistics 7.1Probability 7.2Statistics 8 Mathematicalproof 8.1Theroleofinduction 8.2Proofbyinduction 8.3Deductionsandarguments 8.4Conjecturesandsupportingevidence 8.5Lookingforexceptions 8.6Proofbycontradiction 8.7Generalisationandproof

Selfassessmentquestions Multiplechoicemathematics

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SECTIONC

147 149 158 172 182 196

9 Effectiveteachingofnumeracy MARGARETBROWN 10 Mentalmathematics JEANMURRAY 11 Childrensmistakesandmisconceptions VALSAKOSHY 12 Usingwritingtoscaffoldchildrensexplanationsinmathematics CHRISTINEMITCHELLANDWILLIAMRAWSON 13 Differentiation LESLEYJONESANDBARBARAALLEBONE

APPENDICES

211 213 216 219 220 222 223

Answerstoselfstudyquestions Answerstoselfassessmentquestions Answerstomultiplechoicemathematics Recordofachievement Mathematicalglossary Index

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ContributorsMargaretBrownisProfessorofMathematicsEducationatKingsCollegeLondonandamemberoftheNationalNumeracyTaskForce.Shewasintheworking partywhowrotethemathematicsNationalCurriculumandhasbeeninvolvedinmanyofthemajorinitiativesinthiscountryinMathematicsEducation.Shehas directedanumberofresearchprojectsinnumeracyinthelastfewyearswhichhaveguidedshapingnationalpolicy. JeanMurrayisDirectorofPrimaryEducationatBrunelUniversity.SheisalsoresponsibleforthedevelopmentofthemathematicscomponentsofthePGCEandBA coursesandteachesonthemathematicseducationinserviceprogrammeoftheUniversity.PriortoenteringHigherEducationshetaughtinprimaryschoolsinInner London. ValsaKoshyisSeniorLecturerinEducationatBrunelUniversity.PriortojoiningtheUniversityshewasamemberoftheILEAmathematicsadvisoryteamfora numberofyears.ShecoordinatesthemathematicsinserviceprogrammesattheUniversityandteachesInitialTrainingstudents.Shehaspublishedmanypractical booksforteachers:themostrecentonesareontheteachingofmentalmathsandeffectiveteachingofNumeracyintheprimaryschool. ChristineMitchelllecturesinprimarymathematicsattheSchoolofEducation,UniversityofExeter.ShehastaughtinprimaryschoolsintheUKandhasprovided consultancyandinservicesupportinassessmentandmanagement.Sheresearchesthedevelopmentofmathematicalreasoninginyoungchildren. WilliamRawsonlecturesinprimarymathematicsattheSchoolofEducation,UniversityofExeter.Aswellashavingtaughtinprimaryandsecondaryschoolsinthe UK,hehasawideexperienceofteachinginSouthAmerica,AfricaandAsia LesleyJonesisHeadofPrimaryInitialTeacherEducationatGoldsmithsCollege,UniversityofLondon.ShejoinedGoldsmithsCollegeafterbeingateacherfora numberofyearsandiscurrentlyinvolvedinteachingmathematicseducationtoInitialtraineesandpractisingteachers.SheeditsMathematicsinSchool,the professionaljournaloftheMathematicalAssociation,andhaswrittenmanybooksonpracticalapplicationsforteachers. BarbaraAlleboneisLecturerinEducationatGoldsmithsCollege,UniversityofLondonwheresheteachesmathematicseducationtobothInitialTrainingstudents andteachers.She

Pageix hastaughtinprimaryschoolsforanumberofyearsandhasledinservicetrainingofteachersinLEAspriortojoiningGoldsmithsCollege.Oneofhermajor researchinterestsistheeducationofAbleChildrenandtheroleofquestioninginextendingchildrensthinking.

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AcknowledgementsThisbookattemptstobringtogetherthetwostrandswhichwebelievecontributetothequalityofteachingandlearningofmathematicsandraisepupilsachievementin onebookthedevelopmentofteachersSubjectKnowledgeandPedagogicalSkills.Theauthorswishtoacknowledgeandthankvariouspeoplewhohelpedtomake thisbookareality. First,wethankthelargenumbersofpractisingteachersandInitialTrainingstudentswhomwehavetaughtformanyyears,andwhoprovideduswithvaluable insightsintoaspectsofmathematicseducation.Theseinsightshelpedustoselecttheaspectsofmathematicseducationincludedinthebook.Weareparticularlygrateful tothosewholookedatthedraftsandprovidedcriticalcommentaryatvariousstagesofwritingthebook. ThankstoBarbaraAllebone,MargaretBrown,LesleyJones,ChristineMitchell,JeanMurrayandWilliamRawsonwhowrotepapersontopicsofcurrentsignificant interestforSectionCofthisbook.Thesepeople,fromfouruniversities,areactiveatbothnationalandinstitutionallevelinpolicymakingandresearchrelatingto mathematicseducation.Weacknowledgetheirwillingnesstofindtime,withintheirbusyschedules,tocontributetothebook. ThankyoualsotoProfessorMartinHughesandProfessorTonyCrockerforactingasrefereestothepapersinSectionC. TheamountofsupportandincisiveandconstructivecommentsprovidedbyHelenFairlie,formerSeniorEditoratRoutledge,hasbeeninvaluable.Wethankherfor that. ValsaKoshy,PaulErnestandRonCasey

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IntroductionAsweapproachthemillennium,primarymathematicsteachingisatacrossroads.Theprimaryteachercannotaffordtotakethewrongpath.Thisbook,wehope,will bothassistinselectingtherightpathaswellasilluminatethejourneyalongit. EvidencefromOfstedinspectionsandfindingsfrominternationalcomparisonshavecausedconcernaboutchildrensmathematicalperformances.Asaresult,raising thelevelofachievementinmathematicsisnowstronglyonthenationalagenda.OneoftherecommendationsoftheNumeracyTaskForceforimprovingstandardsand expectationsistheneedforprimaryteacherstobesupportedinordertocovermathematicssubjectknowledgerelevanttotheprimarycurriculumandpupilslater development,andeffectiveteachingmethods(DfEE,1998).TheTeacherTrainingAgency(1998)introducedaNationalCurriculumforMathematicsforInitial TeacherTrainingstudentswhichrequiresthemtodemonstrateknowledgeandunderstandingofmathematicsaswellasthepedagogicalskillsrequiredtosecurepupils progressinmathematics.FromSeptember1999,primaryschoolteachersareexpectedtointroduceastructureddailymathematicslessonoftenreferredtoasthe numeracyhouraspartoftheNationalNumeracyStrategy.Muchemphasisisplacedonfocus,pace,balanceofknowledgeandskillsanddevelopmentof processes.Themessageisclear.Teachersneedtodeveloptheirsubjectknowledgeandhaveclearideasabouthowtoteachmathematicalideas. Webelievethatopportunitiesfordevelopinggreaterunderstandingofmathematicaltopicsandconsideringthemosteffectiveteachingskillswillgreatlyenhancethe qualityofmathematicsteachinginourschools.Thisbeliefhaspromptedustowritethisbook.Inselectingthecontentandstyleofthisbook,theauthorshavedrawnon theirconsiderableexperienceofbeinginvolvedinbothinserviceandinitialtrainingofteachers.Boththesegroupshavebeenconsultedatdifferentstagesduringthe writingofthisbook. Whatcontributestotheeffectiveteachingofmathematics?Askewetal.(1997)identifiedagroupofeffectiveteacherswhotheydescribedasconnectionists. Askewsummarises(Askew,1998)thattheseteachersemphasisedtheconnectionsby: valuingchildrensmethodsandexplanations sharingtheirownstrategiesfordoingmathematics establishingconnectionswithinthemathematicscurriculum,forexamplefractionsanddecimals.

Pagexii TheresearchteamatKingsCollege,London,foundthatthechildrenintheseteachersclassesachievedhigheraveragegainsintestsinnumeracyincomparisonwith othergroups.(YoucanreadaboutthisresearchinMargaretBrownspaperinSectionCofthisbook.) InlistingaspectsofgoodpracticeinteachingandlearningmathematicstheHMI(1989)attachesmuchimportancetopupilsmotivation: Distinctive,goodworkinmathematicswasgenerallyaccompaniedbyahighlevelofmotivationandengagementinthetask:thepupilsshowedinterest, commitmentandpersistence(p.26). Howcanteacherssupporttheirchildrentobemotivated,toshowinterestandcommitment?Ateachersownenthusiasmisanimportantfactor.Theyplayacrucialrole indevelopingtherightattitudeinthechildren.Wehaveallheardpeopleattributingtheirsuccessorfailureinlearningmathematicstoaparticularteacheroragroupof teachersinaparticularschool.Theproblem,however,isthatmanyadultsexperienceanxietyandfearwhentheytalkabouttheirownlearningofthesubject. Discussionswithteachersandstudentshaveoftenhighlightedtheseanxietiesandtheirlackofconfidenceaboutteachingthesubject.Theirconcernsusuallyhaveorigins fromtheirschooldays.Thereasonsfortheirinsecuritieshaveseldombeenlackofwillingnessorabilitytolearn.Whatwehavelistedbelowaresomeofthearticulated reasonsfortheirdislikeofthesubject.Acloselookatthesereasonswillbeagoodfirststepwhenconsideringhowtodevelopandenhanceonesownmathematics teaching.Thefollowingareamongthemostoftenmentionedcomments: Ineverunderstoodmuchofthemathematicsatschool,soIdonthaveenoughconfidencetoteachittochildren. Couldnotfollowteachersexplanations. Itwasalltoofastforme,Icouldntkeepup. Ijustlearnttherulesinordertopasstheexamination. BythetimeIgottothefinalyearsofmyschooling,thegapsinmyknowledgebaseweresowidethatIgaveup. Mathslessonsweresoboringandirrelevant. Iwasafraidoffailure,especiallyofbeingshownuptobeuseless. Idonthavethebasicmathematicsknowledgetoriskgivingmychildrenverychallengingwork. Thewordmathsmakesmehaveapanicattack. Aconsiderationoftheabovelistinitselfshouldgreatlyassistyouinyourpersonaljourneytowardsimprovingyourmathematicsteaching.Besidesofferingwhatwe hopewillbetherapeuticreading,thesecommentswillraisethemostimportantquestionhowcanI,asateacher,ensurethatmypupilswillnotdevelopanyofthe anxietiesintheabovelist? Thethreesectionsofthisbookaredesignedtosupportyouinyoureffortstodevelopyourpractice.InChapter1,PaulErnestprovidesaframeworkforreflecting onmanyissueswhichshouldsupportyourunderstandingaboutmathematicsteachingandlearning.Hefocusesontheaimsofmathematicsteaching,thenatureof mathematicsteaching,teachingstylesandtherequirementsoftheNationalCurriculumandStatutoryAssessment.Otheraspectsoflearningmathematicsspecial needs,equalopportunitiesandculturalissuesarealsoconsidered.

Pagexiii Chapters2to8,inSectionB,dealwithmathematicaltopicswhichcovertherequirementsoftheNationalCurriculumbeyondKeyStage2,theNationalCurriculum forteachertrainees(TTA,1998)andtheFrameworkforTeachingMathematicsfortheNationalNumeracyStrategy.Ineachofthesechaptersweconsiderspecific areasofmathematics.Eachchapterdealswithmathematicssubjectknowledgeatyourownlevelprovidingexplanationswithexamplesaswellasinterconnections betweentopics.Keyissuesintheteachingofthesetopicstochildrenarealsoverybrieflydealtwith.Ourbeliefisthatasyoureadthissection,manymathematicalideas whichwerenotunderstoodbeforeorforgottenwillbegintomakesense.AttheendofsectionB,youareprovidedwithsomeselfassessmentquestionsandagrid forauditingyourachievementandplanningpersonallearning.Afterundertakingtheassessmentyoumayneedtorevisitpartsofthatchapterordiscusstheideaswith otherstutorsandfriends. SectionCcontainsfivechaptersdealingwithtopicswhichweconsidertobetopicalandimportantinthecontextofourpursuitofexcellenceinmathematicsteaching andlearning.Inthesechaptersmathematicseducatorsfromfourinstitutionssharetheirexpertiseandresearchfindingswiththereaderinordertofacilitatereflectionand informedchoices.Werecommendthatyoumakenotesonthekeyideasineachchapterandshareyourthoughtswithyourcolleagues.

ReferencesAskew,M.(1998)PrimaryMathematics.Aguidefornewlyqualifiedandstudentteachers,London:HodderandStoughton. Askew,M.,Brown,M.,Rhodes,V.,William,D.andJohnson,D.(1997)EffectiveTeachersofNumeracy:AreportofastudycarriedoutfortheTeacherTraining Agency,London:KingsCollege,UniversityofLondon. DfEE(1998)NumeracyMatters.ThepreliminaryReportoftheNumeracyTaskForce,DepartmentforEducationandEmployment. HMI(1989)TheTeachingandLearningofMathematics,DepartmentforEducationandScience.London:HMSO. TeacherTrainingAgency(1998)InitialTeacherTrainingNationalCurriculumforPrimaryMathematics,(DfEECircular4/98).

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SectionA

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Chapter1 TeachingandlearningmathematicsPaulErnest

Whyteachma...

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