Primary mathematics : extending knowledge in practice PDF
Preview Primary mathematics : extending knowledge in practice
Extending Knowledge in Practice Primary Mathematics Extending Knowledge in Practice Primary Mathematics Alice Hansen Firstpublishedin2008byLearningMattersLtd. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrieval system,ortransmittedinanyformorbyanymeans,electronic,mechanical, photocopying,recording,orotherwise,withoutpriorpermissioninwritingfromLearning Matters. (cid:1) 2008AliceHansen BritishLibraryCataloguinginPublicationData ACIPrecordforthisbookisavailablefromtheBritishLibrary. ISBN:9781844450541 TherightsofAliceHansentobeidentifiedastheAuthorofthisWorkhasbeenassertedby herinaccordancewiththeCopyright,DesignsandPatentsAct1988. CoverdesignbyTopics.TextdesignbyCode5DesignAssociatesLtd ProjectmanagementbyDeerParkProductions,Tavistock TypesetbyPDQTypesettingLtd,NewcastleunderLyme PrintedandboundinGreatBritainbyCromwellPressLtd,Trowbridge,Wiltshire LearningMatters 33SouthernhayEast ExeterEX11NX Tel:01392215560 [email protected] www.learningmatters.co.uk Contents The author vi Introduction 1 1 Using and applying mathematics 5 2 Counting and understanding number 18 3 Knowing and using number facts 32 4 Calculating 48 5 Understanding shape 67 6 Measuring 83 7 Handling data 99 Objectives Index 115 References 123 Index 131 v The Author Alice Hansen is a Principal Lecturer in Primary Mathematics Education at the University of Cumbria. She has taught extensively at primary level in England and abroad. Alice has particular interests in how children construct geometrical definitions,thedesignofmathematicaltasksandtheuseofICTtoenhancemathe- matics teaching andlearning. vii Introduction This book hasbeen designedto help youidentifyhowyourmathematicssubject knowledge and pedagogical knowledge support each other in order for you to become a better teacher of mathematics. Through case studies, a number of mathematical aspects will be considered. These cover some of the Early Learning Goals (DfES, 2007c) and a range of Programmes of Study in Key Stages 1 and 2 of the National Curriculum for Mathematics (DfEE, 1999). However, they are arranged in chapters according to the strands of the Primary FrameworkforTeachingMathematics(DfES,2006).Thesecasestudiesaredrawn from a wide range of year groups, from working with young children in nursery settings to working with more able children in Year 6. Other case studies focus specifically on relevant and interesting research that covers an important part of children’smathematicaldevelopment.Accompanyingeachcasestudyisadiscus- sion aimed at developing your pedagogical and subject knowledge while it highlights the subject knowledge that the teacher requires in order to teach the objectives. Where appropriate, consideration is also given to how the teacher’s subject knowledgedevelopedwhile working withthechildren. Althoughthebookissetoutinthisway,itisessentialtoconsidercasestudiesfrom yeargroupsotherthantheone(s)youaredirectlyplanningfor.Inmostcasesthe discussionthatensuesrelatestoawideraudiencethantheteachersofthespecific yeargroupinthecasestudyandthereforetheideasthatyoufindthroughoutthe chapterscanbeadaptedformostattainmentlevels. Throughout the book you will see a number of cross references, shown by this symbolinthemargin.Thesewillhelpyoutofindotherplacesinthebookwhere similaraspectsarebeingdiscussed.Thehighnumberofcrossreferencesdemon- strateshowtheareasoftheEarlyYearsFoundationStage,NationalCurriculumand the Primary Framework for Teaching Mathematics are interrelated, and how you cannot effectively teach one objective in isolation. One way to be able to make meaningful links between thes e areas of mathematics is to have good subject knowledge. Inadditiontothecasestudiesandthediscussion,youwillfindstickynoteremin- ders scattered throughout the chapter. These are useful prompts for your own subject knowledge, links to curriculum content or related reading/research for you to follow up if you require more information. There are also links to further readingtosupportyourpedagogicalknowledgeandyoursubjectknowledgeofthe aspectof mathematics being discussed. This book aims to help you develop your mathematical subject knowledge and pedagogicalknowledgesothatyourconfidenceinteachingmathematicsdevelops, thusimprovingthelearningandmathematicaldevelopmentofthechildreninyour care. 1 Introduction What is the difference between mathematical subject knowledge and mathematical pedagogical knowledge? Shulman (1986, 1987) defined three categories of teacher knowledge on which manyhavebasedtheirresearchintomathematicalknowledge.Theseare:subject matter content knowledge (the amount and organisation of knowledge per se in themindoftheteacher),pedagogicalcontentknowledge(thatgoesbeyondknowl- edge of subject matter per se to the dimension of subject matter knowledge for teaching) and curricular knowledge (the full range of programs and materials designed for the teaching of particular subjects and topics at a given level) (Shulman, 1986,p9). Many others have also considered the knowledge that teachers require to teach mathematics (Morris, 2001). For example, Ball and Bass (2000) identified the connections between mathematics content knowledgeand teaching in what they termed ‘mathematics knowledge for teaching’. They explained how this involved two elements, both ‘common’ knowledge (which any educated adult should possess)and‘specialised’knowledge(theknowledgeofteachers)(Balletal.,2005). Whatis clearfrom theresearchisthatthereare a numberof elements thatinter- relate to enable teachers to effectively develop children’s mathematical understanding. Whatever the labels different researchers assign to these, within this book the edges of the elements have been intentionally blurred in order to discussthemostappropriateaspectswithinthechaptersmoreeffectively,without being constrainedby discrete(and sometimesartificial) groupings. Do I really need good subject knowledge to teach mathematics well? Forthelast25yearsthesubjectknowledgeofteachersandstudentteachershas been a focus of government and initial teacher educators and it has recently becomeembedded within the Professional Standards for Teachers (TDA, 2007a). Indeed,ifteachinginvolveshelpingotherstolearn,thenunderstandingthesubject contenttobetaughtisafundamentalrequirementofteaching(Aubrey,1997,p3). Within initial teacher education the current focus on developing students’ subject knowledgeisevident.Forexample,itisarequirementforQualifiedTeacherStatus that student teachers pass their numeracy skills test (TDA, 2007a) and the TDA currently funds subject knowledge booster courses for students entering PGCE teacher training (TDA, 2008). ResearchbyRowlandetal.(2001)demonstratedthatsecureknowledgeofmathe- matics (including and extending beyond the areas of the primary curriculum) is stronglylinkedwithmorecompetentmathematicsteachinginstudentteachers.In addition to this, they found that poor subject knowledge was connected to less competentteaching. Ball et al. (2005, p16) remind us that although many studies demonstrate that teachers’mathematicalknowledgehelpssupportincreasedstudentachievement, theactualnatureandextentofthatknowledge–whetheritissimplybasicskillsat 2
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