Mathematics counts Report of the Committee of Inquiry into the Teaching of Mathematics inSchools under the Chairmanship of Dr W HCockcroft ~ .J....j ~,\' I ,',. " ..." :T - , I\ ~ London:Her Majesty's Stationery Office iii Foreword BY THE SECRETARY OFSTATE FOR EDUCATION AND SCIENCE AND THE SECRETARY OF STATE FOR WALES Fewsubjects intheschoolcurriculumareasimportant tothefutureof the nationasmathematics;andfewhavebeenthe subject of more comment and criticisminrecentyears.Thisreporttacklesthatcriticismhead on .It offers constructiveandoriginalproposalsforchange.It shouldbereadbythose responsibleforschool mathematics at all levels. Themainmessageisfortheeducationservice.Thereportidentifiessix agencieswhoseactiveresponseisrequired.Thecontributionof allwillbe .necessaryif wearetomakeheadway.To the extentthatthereport callsfor extra resources, progress isbound to be conditioned by the continuing needto restrainpublic expenditure;butmany recommendations involve no such call. We hope that therewiUbe widespreaddiscussionof thereport's conclusions and that actionwillfoUow. TheConunittee'stermsof referenceinvitedittoconsidertheteachingof mathematicswithparticularregardtothemathematicsrequiredinfurther andhighereducation,employmentandadultlifegenerally.Theearly chaptersof thereportareconcernedwiththeseaspects.Theywillbeof in-terest tomany both withinand au tside the ed ucationservice. They revealthe hesitantgraspmanyadultshaveofevenquitesimplemathematicalskills. They areparticularlyvaluableinexamining closely themathematicswhichis infactneededindifferentkindsof employmentandineverydaylife,and relating itto whatistaught inthe schools.The Committee's rmdings point to the need for teachers to devote more time to the use of mathematics in applica-tions takenfromreal life. Thisisafirst-classreport.Wearegreatly indebtedtoDr Cockcroftandthe members of hisCommittee,andcommendtheirworkto allthoseconcerned about the quality of mathematics taught inour schools. January 1982 v The Committee of Inquiry into the teaching of Mathematics inprimary and secondary schoots inEnglandand Wales MEMBERSHIP OF THE COMMITTEE Dr W HCockcroft (Chairman) MrAG Ahmed Professor M F Atiyah FRS MissK Cross MrC David MrGDavies Mr K TDennis Mr T Easingwood MrHR Galleymore MrR PHarding CBE MrJW Hersee MrsM Hughes Mr A JMcIntosh MrHNeill Mr PReynolds Mr0G Saunders MBE Mr HP Scanlon MissH B Shuard Dr PG Wakely Councillor D Webster MrLD Wigham Mr PHHalsey(Assessor) Mr W JA MannHMI (Secretary) Mr EL Basire (Assistant Sccretary) ViceChancellor , NewUniversity of Ul ster,Coleraine. Head of Mathematics Department,Fairchildes HighSchool, Croydon. RoyalSocietyResearchProfessor. AssistantPrincipalandHeadof theFacultyofMathematicsandScience, Accrington andRossendale College,Accrington. Headteacher,Dyffryn Comprehensive School,Port Talbot. PolicyUnit,Prime Minister' s Office(resigned April1980). Teacher , Dunmore County Junior School,Abingdon. Reader in Mathematical Education,Derby Lonsdale College of Higher Edu cation,Derby. FormerDirector,Procter and Gamble Limit ed (appointed August1979). Chief Education Offi cer , Buckinghamshire County Council. ExecutiveDirector,SchoolMathematicsProject;Chairman,Schools CouncilMathematicsCommittee. Head teacher,Yardley Junior School , Birmingham. Principal Advi ser inMathematics,leicestershire County Council(resigned March1980). LecturerinMathematics,University of Durham. Mathematics Adviser, SuffolkCounty Council (appointed March1980). WelshArea Secretary,Associationof Professional,Executive,Clerical and Computer Staff (appointed March1979). President,AmalgamatedUnion of EngineeringWorkers (resignedJanuary 1979). Deputy Principal,Homerton College,Cambridge. ChairmanandManagingDirector,AssociatedEngineeringDevelopments limited. Chairman, Education Committee,Newcastleupon Tyne Borough Council. PostgraduateCertificateinEducationStudent,UniversityofLeeds (appointedNovember1978). Department of Education and Science. Her Majesty's Inspectorate of Schools. Department of Education and Sciencc. The styles, decorations and appointments shown are those held by members at the time oftheir appointment to the Committee. vi 10November1981 Dear Secretaries of State On behalf of the Committee of Inquiry into the teaching of mathematics in primary and secondary schools in England and Wales,I have the honour to submitour reponto you. The RtHon Sir KeithJosephBtMP Secretary of State forEducation andScience The RtHon NicholasEdwards MP Secretary of State forWales Yourssincerely -W HCOCKCROFT vjj Contents Introductionpogeix Explaoatory notexv Part ooeIWhy teach mathemat ics?I 2The mathemat ical needs of adult life5 3The mathemat ical needs of employment12 4The mathemat ical needs of further and higher education42 Parttwo5Mathematics in schools56 6Mathemat ics in the primary years83 7Calculators and computers109 8Assessmenl and continuity121 9Mathemat ics in the secondary years128 10Examinations at 16 +158 IIMathemalics in the sixth form169 Part three12Facili ties for teaching mathemat ics183 13The supply of mat hematics teachers188 14Initial training courses203 15In-service support for teachers of mathematics217 16Some other matters232 17The way ahead2A2 AppendicesIStatistical information246 2Differences inmathematical performance between girls andboys by Miss HB Shuard273 3List of organisations and individuals whohavemade submissions to t he Committee288 4Visits to schools. industry and commerce; meetings with teachers301 5Abbreviations used within the text of the report303 Index305 Terms of reference Meetingsand visits ix Introduction InitsreportpublishedinJuly1977,theEducation,ArtsandHome Office SUb-Committee of the Parliamentary Expenditure Cpmmittee stated that' 'it is clearfrom the points which were made over and over again by witnesses that thereisalargenumber of questions about the mathematical attainments of childrenwhichneedmuchmorecarefulanalysisthanwehavebeenableto giveduringourenquiry .These concerntheapparentlackof basiccompu-tation skills in many children, the increasing mathematical demands made on adults, the lackof qualifi ed maths teachers, the multiplicity of syllabuses for old,newand mixedmaths, the lackof communicationbetween further and highereducation,employersandschoolsabouteachgroup'sneedsand viewpoints,the inadequacy of information on job contentor testresults over a period of time, and the responsibility of teachers of mathematics and other subjects to equip children withtheskill s of numeracy". The Committee re-commended as "possibly the most important of our recommendations"that the Secretary of State for Education and Science should setup an enquiry into the teaching of mathematics.Intheir reply presented to Parliament inMarch 1978,theGovernmentagreed" thatissuesofthekindli stedinthe Committee'sreportneedthoroughexamination"andannouncedtheir decisionto "establish anInquiryto consider the teaching of mathematics in primary and secondary schools inEngland and Wales, with particular regard to its effectiveness and intelligibility and to the match between the mathema-tical curriculum and the skills required infurther education, employment and adult life generally" . They further undertook that the Inquiry would examine the suggestion that there should be afullanalysi s of the mathematical skills required in employment and the problem of the proliferati on of mathematics syllabuses atA-level and at16 + . Our Committee met fortbefirsttime on 25September1978wi ththe foUow-ingtermsof reference: To consider theteaching of mathematics in primary andsecondary schools in England andWales,with particular regardtothe mathemalics required in further andhigher education,employmentandadultli fe generall y, and( 0make r e c o m ~ mendations. The fullCommittee has met on 64 days, whichhave included three residential meetings.ItsWorking Groupshavemet on143days inall,andtherehave been less formal discussions on many occasions. 54 schools and 26 companies of various kinds in England and Waleshave beenvisitedbymembers of the Commiuee andtherehave beensixmeetingswithgroupsof leachers indif x Submissions of evidence Research studies I mroduction ferent parts of the country. Small groups of members have visited the Scottish EducationDepartmentinEdinburgh,theInstitutefortheDevelopmentof MathematicsEducation(IOWO)atUtrecht,Holland,theInstituteforthe Teaching of Mathematics atthe University of Bielefeld,WestGermany and theRoyalDanishSchoolofEducationalStudiesinCopenhagen;two membershavevisitedindustrialcompaniesinNuremberg,WestGermany. Severalmembers of theCommiuee werepresentat theFourthInternational Congress on Mathemati cal Educat ionheld at the University of California at Berkeley inAugust1980.Individualmembers of the Committeehavebeen invitedtoauendtheconferences andmeetings of anumber of professionaJ bodies. Throughout our work we have been greatly encouraged by the welcome which manypeople havegiven tothesettingupof theJnquiryandbythehelpful response which we have receivedto our requestsforinformation andwritten evidence.Wehave recei vedwritten submissions.many of themof consider-ablelength,from930 individualsandbodiesof manykinds.73individuals and groups have met members of the Committee for discussion. A list of those who have submittedevi dence andwho have met members of the Committee for discussion isgiveninAppendix 3. Whenwestartedtoconsiderhowbestwemightrespondtoourtermsof reference, we became aware that we needed more detaHedinformation about themathematicalneedsof employmentandof adultlife generallythanwe werelikely eithertoreceiveinwritten evidence or tobe able to obtain by our ownefforls.We thereforerequestedtheDepartment of Education andSci-ence (DES) to commission two complementary studies into the mathematical needs of employment and also a smallstudy intothemathematical needs of adult life . One of the studies into the mathematical needs of employment was basedatthe University of Bath under the di rection of Professor D E Bailey, assisted by Mr A Fitzgerald of the University of Birmingham. and the other at theShellCentreforMathematicalEducation,Universityof Nottingham, under the direction of Mr R L Lindsay. Thestudy into the mathematical needs of adult life was carried out by Mrs B Sewell on behalf both of the Committee and of the Advi sory Council forAduli and Continuing Education. The DES aJsoagreedto commission a review of existingresearchon[he teaching and learning of mathematics which was carried out by Dr A BeU of the University of Nottingham and Dr A Bishop of the University of Cambridge. The Steer-ing Groups for all these studies have included members of the Committee, and relevant evidencewhich hasbeenreceived has beenmade available on a con-fidentialbasistothose engagedinthe studies.The reportswhichhavebeen produced have proved to be of very considerable help to us;we refer to them and draw on their conclusions ina number of the chapters which follow. At a later stage the DES commissioned a small survey of mathematics teachers in secondaryschoolswhowereintheirfirstthree yearsof teaching;thiswas carried out forusby the NationalFoundationfor Educational Research. Publ ications and announcements Introductionxi Since we started our work a considerable number of official reports and other publicat ions have been issued which relate wholly or in part to the teaching o f mathematics inschools. These incl ude fromt heDES, Mathemalicaldevelopmenl . Primary survey reporls Nos I and 2 (The APU Primary Surveys) Mathematicaldevelopmelil.SecondarysurveyreportNoI(TheAPU Secondary Survey) Local authority arrangemellls Jar the school curriculum Abasis for choice Proposals for a Certi/icate of Extended Education (TheKeohaneReport) Secondary school exam ina/ions: a single sysI,em al 16 plus A framework for Ihe school curriculum The school curriculum Examinations J6-18: a consullative paper Educalion for J6- 19 year aids; fromHMInspectorate, Primary educalion in England(Report of the NationalPrimary Survey) Aspecls of secondary education inEngland(Report of theNational Secondary Survey) Aspecls of secondary educatI on inEngland: supplementary information all mathematics Malhematics 5-J I: ahandbook of suggestions Developments inthe BEd degree course PGCE inthe public sector Teacher training and the secondary school Aview oj the curriculum; fromthe Schools Counci l, Mathematics and IheIO-year old (Schools Council Working Paper 61) Mathemat ics in school and employment: a study of liaison activities (Schools CouncilWorkingPaper 68) Statistics in schools 11-16:areview (Schools CouncilWorking Paper 69) The practical curriculum(SchoolsCouncil Working Paper 70); fromother sources, Engineeringour fUlure(Report of theFinniston Committee) Thefunding and organisation of courses inhigher educalion(Report of the Education, Science andArtsCommittee of theHouse of Commons) ThePGCEcourseandIheIrainingOfspecialistteachersforsecondary schools(Universities Councilfor the Education of Teachers) Aminimal care syllabus for A-level mathemalics (Standing Conference on University Entrance andCounci l forNational AcademicAwards) Children 'sunderslanding of malhemalics:1/-16 (Repon of the Conceptsin Secondary Mathematics andScienceProject). We have studied a1l l hese documents and make reference to several of them in the course of thisreport. xii Statistical information Viewsfrom(he past tnlroducri on Sincewes t.artedwork,IheGovernmenthaveannouncedthat(hepresent O-Ievel and CSE examinations are [0 bereplacedbya single system of exam-iningat16 +.thatGCEA-levelsare[0beretai ned,thattheCertificateof ExtendedEducationwillnotbeintroducedbutthattherewillbeapre-vocational examinati onat17 + , andthat consideration isbeing giventothe introductionof Intermediate levels(I-levels),We haveconsidered the impli-cati ons of these announcementssofarasmathematicsisconcerned , Asaresultof theworkwhichhasbeencarri edoutforusbytheStatistics Branchof theDESandbytheUniversitiesStatisticalRecordwehavebeen able [0 obtaina considerable amount of information whi chhasnothitherto been available. We refer to this, andto other existing information, from tjme iotimethroughout[hereport .]ngeneralweqUOlethisinformali onin roundedterms orpresentitindiagrammatic form. The detailedtablesfrom whichthe information istaken areset outinAppendix1, which al so givesin eachcasethesourcefromwhichtheinformat ionhasbeenobtained.Thi s Appendix also contains some tables to which no direct reference is made in the textbutwhichwebelievetobeof interest.The Appendi x discusses,where appropriate. any assumptions which it has been necessary to make in order to preparethe tablesandincludesbrief comments on some of them. In the light of present day cri ticism of standards, itis interesting to assemblea collectionof quotationsfromdocumentsof variouskinds.someof wh.ich datebacktothelastcentury.whichdrawattentiontotheallegedlypoor mathematical standards of the day.We cont ent oursel ves with examples from approxi mately a century. ahalf-century and a quarter-century ago. (narithmetic,Iregrettosayworticresultsthaneverbeforehavebeen obtai ned -thisispartly attributable, nodoubt, [0 my having so f ramedmy sums as torequirerathe r moreintelligencethanbefore:the failuresarealmost in\'ariably traceabletoradically imperfectteaching. Thefai luresinari thmet ic aremainlydue to the scarcityo fgoodteachersof it. Thosecomments are lakenfromreports byHMInspecrors written in1876. Manywho areinapositionto criticise thecapaci ty of youngpeople whohave passedthroughthepublic element aryschoolshaveexperiencedsomeuneasiness aboutthe conditionof arithmeticalknowledge and teaching al lhe present time.It hasbeensaid,forinstance,thataccuracy in themanipulation of figuresdoesnot reachthesamestandardwhichwasreachedtwen tyyearsago.Someemployers expresssurpriseandconcernattheinabilityof youngpersons[0performsimple numerical operations involved in business. Someevening school teachers complain thal theknowledge of arithmetic shownbytheirpupils docs not reach their expec-tations.Itissometimesaltegedinconsequence,t houghnorasarulewi ththe support of definite evidence, ('hat'the teacher no longer prosecut es his attack on this subjectwiththeenergyorpurposefulnessforwhichhisprede-cessorsaregiven credit. That exlract comes from aBoard of Education Report of 1925. The standardo fmathematicalability of ent rantstotrade coursesisof[envery [ow. . . . .Experience shows thatalargeproportion of entrants have forgolten how to deal wi thsimple vulgar and deci mal fractions, have very hazy ideas on some easy General approach Acknowledgements 100roductionxiii arithmericalprocesses,andretainnotraceof knowledgeof algebra,graphsor geomerry, if, infact,they ever did possess any. Some improvementsin this position may be expecled asa result of the raising of the school Jeavi ngage, butthereis asyer no evidence of anymarkedchange. Our final quotation comes from a Mathematical Association Report of 1954; the schoolleaving agewas raisedto15in1947. Itisthereforeclearthatcriticismofmathematicaleducationisnotnew. Indeed,throughout the timeforwhichwehave beenworkingwehavebeen consciousthatformanyyears a great deal of adviceto teachers aboutgood practice in mathematics teaching has been available inpubli shed form from a variet yofsources.TheseincludethepublicationsoftheDES,ofHM Inspectorate,of the SchoolsCouncilandoftheprofessionalmathematical associati onsitherehave alsobeen referencestomathematicsteaching inthe reports of Committees of Inquiry, for example that of the Newsom Commit-tee. Muchof this advice is stiJlrelevanttoday andservesasabackground to ourownwork. Inwritingourreportwehavetriedsofarasispossibletoavoidtheuseof technicallanguage andto putforward our views ina waywhichwe hope will beintelligibletomathemati cianandnon-mathematictanalike.Forthis reasonwehaveat limesomitteddetailwhich,hadwebeenwriting onlyfor those engaged inmathematical education, wewouldhave included. We hope thatthosewhowouldhavewishedustodiscusscertainmattersingreater detailwillunderstandthereasonwhywehaveincertainplacesuseda somewhat 'broad brush'. We hope that our attempt to draw attention to those as pects of the teaching of mathemati cswhich we believe 10beof fundamental importance willbeof use bothinside andoutsidethe classroom. We wish to stress that many of the chapters in our report, and especially those inPart2,are inter-related.For example,inChapters 5 and 6 wediscuss lhe elementsofmathematicsteachingatsomelength;thefactthatwedonot repeal this discussion inChapter 9 (Mathematics in lhe secondary years) , but deal mainly with matters of syllabus content and organisation, does not mean that the teaching approaches we have recommended in earlier chapters are not equaJJy applicable at the secondary stage. We thereforehopethat those who readourreportwillview it asawhole . Inour reportwe have not considered the needs of pupil s with severe learning difficulties.We hope,however, that those who teach pupils of thiskindwill findthat our discussion of mathemat icsleaching ingeneral, andof the needs of low-attainingpupilsinparticular,aswellasourdi scussion of themath-ematicalneeds of adult life,willbeof assistance. Wewouldlike toexpressourthanks to allthose who havewrittento us and withwhom wehave talked both formally and informall y.We are gratefulfor {hehelp wehavereceivedfromthe heads,staff andpupils of the schools we havevisiled,fromtheteacherswhom wehavemetalmeetingsindifferent parts of the country and fromthose atall levels whom wehave met during our XJ vl mroduClioJ1visi ts to commerce a nd industry. We are grateful,too, to those who were kind enoughto arrange thesevisitsandmeetings .Wewishto thankthe Scottish Education Department for arranging our visitto Edinburgh and those whom wevisitedinDenmark, Holland and West Germany forthe help they gave us andthe arrangements t hey made on our behalf. Our thanks are due to allthose who have carried out the various research stu-di es and also to t hose inDES Statistics Branch and the Universiti es Statistical Recordwho ha veundertakena great deal of workfor us andfound ways of answering our questions.We a re gratefulforthe help we havereceived from manyoffi cersintheDES,inpa rticular our Assessor ,MrPHHal sey;and frommembersof HMInspectorate,especiall y Mr TJFl etcherwho has,at our invitation,att endedmany of OUf meetings. Wewishalsoto expressour thanksfo rthehelpand support whichwehave receivedfrom themembers of the Committee'ssecretariat.MrW M White hastakenmajorresponsibilityforobtaining,puttinginorderandinter-preting avery great dealof statistical informatio n. Mr EL Basire, our Assis -tant Secretary,Mi ssEKirszberg andMrRW Le Cheminant,in addition to theirotherduties,havegivenmuchpersonalhelptomembersofthe Commi tt ee. Among those to whom we are indebted , our Secretary, Mr W J AMann HMI, stands oul. We are conscious of the burden we have placed upon hi s shoulders and of the conscientiousway inwhichhe has accepted tbeload.Most of all, we are grateful to himfor the able and efficie nt way in wh ich he has taken the outcome of our many diffuse and varied deliberations and moulded itinto a coherent whole. xv Explanatory note Throughoutt.herepon (here are certainpassageswhich are printed inheavier type.In selecting these passages, wehave chosen those which either relate to matlers weconsi der to be of significance for all our readers or which call for action by those who are outside the classroom. Thi s means that, especially in Chapters 5 toIIwhieh are concerned inparticular with the teaching of math-emalics, we have nO!picked OUImany of the passages inwhichwe make sug-gestions relatingto classroom practice.Aswe bave pointed oulinthe Intro-duction,Chapters5 toIIareinter-relatedandwebavenot wi shedto draw attention only tocertai n passagesinthem, except inso farasthesepassages fulmthe purposeswehave already SCIout. We do not regardthepass.agesprintedinheavier typeasinanyway const i-tuling asummary of Ihereport. Part 11Why teachmathematics? ITherecanbenodoubtthatthereisgeneralagreementthateverychild s houldstudymathematicsatschool;indeed,thest udyofmathematics, together with that of English , isregarded by most people as being essential.It mighttherefore be argued that there isno need to answer the question which wehaveusedasourchapterheading.Itwouldbeverydifficult-perhaps impossi ble-toliveanormalli feinverymanypartsof theworldinthe twentieth cent ur y wit houtmak ing use of mathematics of some kind. This fact initself couldbethoughttoprovideasufficientreasonforteachingmath-ematics) and inone sense [his isundoubtedly true. However, webelieve that it is of valuetotrytoprovide amore detailed a nswer. 2Mathematics isonly o ne of many subjects which are included inthe school curr iculum,yetthere isgreater pressurefor childrento succeed atmathematics than,forexampl e,athistoryorgeography,eventhoughitisgenerally acceptedIha tthesesubj ectsshouldalsoformpartof thecurriculum. This suggeststhatmathemati csisinsomewaythougbttobeofespecial importance.If we as k wby this shouldbe so, o ne of the reasons whichisfre-quently give n isthatmathematicsis 'useful';itisdear,toO,that thisuseful -nessisinsomewayseentobeof adifferentkindfromthatof many other subjects inthe curriculum. The usefulness of mathematics isperceived indif-ferentways.Formanyitissecnintermsof thearithmeticskill swhich are neededfor use athome or in the office or workshop; some seemathematics as thebasis of scienti ficdevelopmentandmodern technology; some emphasise theincreasinguseofmathematicaltechniquesasamanagementLOolin commerce and i nduslf Y. 3We believe that alilhese perceptions of the usefulness of mathematics arise fromthe fact1hatmathematics provides a means of communicaOon which is powerful ,conciseandunambiguous.Eventhoughmany of thosewho con-sider mat hematics tobe useful wouldprobably not express the reason in these terms , we believe that itis the fact that mathematics can be used as a powerful meansof communicationwhichprovidestheprincipalreasonforteaching mathemati cs(0 aJichildren. 4Ma themati cscan be used to present information in many ways, not only by meansoffiguresandlettersbUlalsothroughtheuseof tables,chartsand di agramsaswellasof graphsandgeomet ri calortech nicaldrawings.Fur-thermore, the figures and other symbols which are used in mathematics canbe manipulatedandcomqinedinsystematicwaysso thatitis oftenpossible to 2JWhy (eachdeducefurtherinformationaboutthesituationtowhichthemathematics rel ales. For example,if wearetoldthat a car hastravelledfor 3 hours atan averagespeedof20milesperhour,wecandeducethatithascovereda distance of 60 miles . In order to obtain this result we made use of the fact that: 20x3=60. However,thismathematicalstatementalsorepresents(hecalculation requiredtofindthecostof 20articleseachcosting3p,thearea of carpet required to cover a corridor 20 metres long and3 metreswide and many other thingsaswell.Thispl ovides anillustrationof thefactthat thesame math-ematicalstatementcanarisefromandrepresentmanydifferent situations. Thisfacthasimportantconsequences.Becausethesamemathemati cal statementcanrelatetomorethanonesimation,resultswhichhavebeen obtained in solving a problem arising from one si LUation can often be seen to apply to a different situation. Inthis way mathematics can be used not onl y to explainthe outcomeof aneventwhichhasalreadyoccurredbutalso,and perhaps more importanlly, to predict the out come of an event which has yet to take place. Such a predi ction may be simple, for example theamount of petrol which will be needed for ajourney, its cost and the time which the journey will take; or itmaybecomplex , such asthe pathwhichwillbetaken by arocket launched into space or the loadwhich can be supported bya bridge of given design.Indeed,itistheabilityof mathematicstopredictwhichhasmade possiblemanyof the technological advancesof recentyears . .5Asecondimportantreasonforteachingmathematicsmustbeits importance and usefulness in many other fields . It isfundamental to the study of the physical sciences and of engineering of allkinds . It is increasingly being used inmedicine andthe biological sciences, in geography and economics, in business and management studies.Itis essential to the operations of industry and commerce inbothoffice andworkshop. 6It is often suggestedthat mathematics should be studied in order to develop powersof logicalthinking,accuracyandspatialawareness.Thestudyof mathematics can certainly contribute tothese endsbut the extent to whichit does so depends onthe way in whichmathematics istaught. Nor isits contri-butionunique;manyotheractivitiesandthestudyof anumberof other subjects can develop these powers aswell . We therefore believe that theneed todevelopthesepowersdoesnotinitself constitute asufficientreasonfor studying mathematics rather than other things . However, teachers shouldbe a\vare of the contributionwhichmathematics canmake. 7The inherent interest of mathematics and the appeal whichitcan have for many children and adults provide yet another reason forteaching mathema-ticsinschools.The factthat' puzzle corners'of various kinds appear inso manypapers andperiodicals testifiestothe factthat the appeal of relatively elementaryproblemsandpuzzlesis widespread;attemptsto solvethem can both provide enjoyment and also, in many cases,lead to increasedmathema tical understanding.For some people, too, the appeal of mathemati cs can be evengreaterandmOre intense . Forinstance: -F,:nn.Mis/erGod,IhisisAlina. CollinsFount 1974. "'CarlSagan./vlurmursa/Earth. Hodder andStough Ion1979 Whyleach maLhemalics? 3 Anna andI hadbOlhset::nlhatmaths wasmore than just working outproblems. It wasa doorway tomagic.mysterious, brain-crackingworlds,worlds where you hadtoueadcarefully,worldswhere youmadeupyour ownrules,worldswhere youhadto acceptcomplece responsibi lityfor youracrjons.SUl i(wasexciting and vastbeyondundersGlnding." Even though itmay be givento relativelyfewto achieve the insight and sense of wonder of 7 year oldAnna and of the young man who inlater yearswrote the book, webelieve it tobe important that opportunities to do so should not be deniedto anyone.Indeed,wehope that allthose who learnmathematics willbeenabledtobecome awareof the'viewthroughrhe doorway'which manypiecesofmathema.ticscanprovideandbeencouragedtoVenture throughthisdoorway.However,wehavetorecognisethattherearesome who, even though they may glimpse the view from time to Lime asthey become interestedinparticularaClivities,seeinitnolastingatt ract.ionandremain indifferent orin some cases activelyhostile tomathematics. 8There are other reasonsforteachingmathematicsbesidesthosewhichwe have put forwardinthis chapter.However,we believe that the reasonswhich we have given make amore thansufficient case for teaching mathematks to allboys and girls andthat foremost among them isthe factthat mathematics can be used as a powerful means of communication-to represent, t.oexplain and to predict. 9ItisinterestingtonOtetwoverydifferentuseswhi chhavebeenmade of mathematics inthe current Voyager s pace programme. Not only hasthe pre-dict ive power of mathemati csbeen usedto plan the details of the journeys of the two Voyager spacecraft but examples of mathematics have been included intheinformationaboutlifeonEarthwhichwasaffixedtoeachofthe spacecraftbeforetheywerelaunchedin1977toexploretheouterSolar System and then to becom.e"emissaries of Earlh to the realm of the stars" .* Thereasonforincludingexamplesofmathematicsisexplainedinthese words: So faraswe can tell, rna:thematical relationships should bevalidfor allplanets, biologies.cultures,philosophies.Wecanimagineaplanetwithuraniumhexa-nuoride inthe atmosphere or a life form that lives mosll y off interstellar dust, even jf theseareexrremely contingencies.But we cannol. imagine a civili zalion forwhich one and one does not equal two or for which there is an integer interposed between eight andnine.For this reason. simple mathematical may be evenbettermeansof communicati onbetweendiversespeciesthanreferences(Q physic.o;;and astronomy. The earlypan of thepictorial information on the Voyager record isrich inarithmetic,whichalsoprovidesakindof diclionaryforsimpl e mathematical infonnationcontai nedinlater pictures, such astheof a human being. 10Mathematicsprovides .a means of communicating information con_cisely andunambiguouslybecauseitmakesextensiveuseof symbolicnotation. However,itisthenecessityof usingandinterpretingthjsnotationandof graspingtheabstractideasandconceptswhichunderlieitwhichprovesa stumbling blocktomany people.Indeed,the symbolicnotationwhichena 4 lmplications forteachers IWhy leachmalhemalics.1 biesmathemati cs[0beusedasameansof communicationandsohelpsto makeit'useful1 canalsomakemathematicsdifficult tounderstandandto use. IITheproblemsof learningtousemathemat icsasameansof commu-nicationare n01thesameasthose of learningtouse one'snativelanguage. Nativelanguageprovidesameans of communicationwhichisinuseallthe timeandwhi ch ,forthegreatmajorityof people,'comesnaturally' ,even thoughcommandoflanguageneedstobedevelopedandextendedinthe classroom. Furthermore, mistakes of grammar or of spelling do not, in gene-ral , renderunintelligible themessage whichisbeing conveyed.On the other hand, mathematics does not 'come naturally' 10 most people in the way which istrue of nat ivelanguage.It isnot constantly being used; ithasto be learned andpractised;mistakesareof greaterconsequence.Mathematicsalsocon-veys information in a much more precise and concentrated way than is usual1y the case wit h the spoken or written word. For these reasons many people take a long time not only to become familiar with mathematical skills and ideas but to develop confidence inmaking useof them. Those whohavebeenable to develop such confidence withrelative ease should notunderestimatethe dif-fic ult ies which many others experience, nor the extent of the help which can be requiredin order to be able to understand and to use mathematics. 12Weconcludethischapterbydrawingthe attentionof thosewhoteach mathematicsinschools to whatwebelieve to theimpli cat ions of the reasons forteaching mathematicswhichwehave di scussed.Inour viewthernalhe maries teacher hasthetask of enablingeachpupiltodevelop,withinhi scapabilities,themath-ematicalskiHsandunderstandingrequiredforadultlife,for employment andfor furtherstudy and training, while remaining aware of the difficult ieswhichsomepupilswillexperienceint rying10gain suchanappropriate of providing each pupil with such mathematics as may be needed for his study of other subjects; of helping each pupil to develop sofar as is possible his appreciation and enjoyment of mathematics itself andhisrealisation of the role which it has played and wi llcont inue toplay both inthe development of science andtechnology and of our civilisation; above all,of mak ingeachpupilaware thatmathematicsprovides him with apowerfulmeansof communica ti on. '"Makeitcoum.ASLlldyby Da\' ;:.Q'.I 2103 MedicalanddenlaJ139 Biological sciences61 Othersciences422 BusinessstUdiesIa5 Geography34 Othcrsubjects397 All subject groups6006 1979 1714 968 1728 161 47 281 177 19 246 5341 070of allentrants wilh doublesubject A-levclmalhematics 197)1979 16.932. / /7,5/8./ 35.032.4 2.33.0 /.00.9 7.05.3 3. /3.3 0.60.4 6.64, 6 100/00 Source:StatisticalRecord ! coursesincludedin each subject group aregiveninAppendixI.paragraphA21 . 524Tbe mathema!icalneeds of rurther and higher education 179About80percentof entrantswi !.hadouble-subjectqualificationin mathemat ics at A-levelread engineering and technology, physical sciences or mathematicalstudies.Figure4illustratesthewayjnwhichentrantswitha double-subjectqualificalionweredistributedbetweenthesethreeareasof study in1973and1979.(See also AppendixI,Table 30). Figure4Numbers oj university entrantsinEngland andWales admitted 011 thebasisoj A-levels.Subjectchoicesandnumberswithdouble-subject A-level qualijicatio1ls inmathematics:1973and1979 8000 Numberstakingsubject NumberswithdOUblemathematics 6,000 4000 2. 000 737973797379 Source:Universit iesStatisticalRecord "Mathematicalstudiesincludes degreecoursesinmathematics, statistics.comput erscience, combinationsoft heseandalsoa varietyof cOurseswhichcombine mSlhcmati cswithsubjectsother thanthese.Analysisof informati onprovidedbyUSR a boutfir stdegrees.awardedin 1979wilhinthefieldof mathema-ticalstudiesshowsthatabout65 peT centwerei nmathematicsonly, about25percentincomputer science,eitherwholl yorinpari, a ndabout 8 per eent in stalisrics. 4Themathematical needsof funher andhigher education53 Degree courses inmathematical studies 180We wishto draw attention to the drop inthe proportion of those reading mathematical studies' atuniversities in England and Wales who have a dou-ble-subject qualification and to its implications. In1973the proportion was almost 80 per Cent ; in 1979 this had dropped to 55 per cent. Although there are someuniversitiesatwhichitisstillthecasethataveryhighproportionof thosereadingdegreesinmathematicshavetakenthedoublesubjectat A-level,our ownenquirieshaveestablishedthatthereareothersatwhich substantiallylessthanhalfofthosereadingmathematicshaveadouble-subject qualification. We believe that this information may come as a surprise to many people inboth universities and schools.It is not within our terms of referencetocomment on itsimplicationsforthosewho teachmathematical studies in universities; the implications for those who teach in schools are very great. 181It isverycommonlysupposedthatitisalmostessenlialtohavetaken double-subject mathematics at A-level inorder to read mathematics success-fullyatuniversity.However,it is very i.mportantthat those wboteachmath-ematicsinsixthformsandthosewhoadvisepupilsabouttbeirchoiceof degreecourse shouldrealisethat therearenowuniversitiesinwhichmore thanhalf of thosereadingmalhematics aredoing sofromabasis of single-subjectA-level.It followsthattheysbouldnotdissuadepupilswhobave taken only Ibe single subjectal A-level from applying toread degree courses in mathematics.We seeno likelihood Ihat the demand for mathematics gradu-aleswilldecrease-indeed,webelievethaIthedemandwillcontinueto grow-andthosewhoseintereslsandabilitieslieinthisfieldneedevery encouragement 10studymathematicsat degree level.Aswepointedout in paragraph173,itdoesnot followthat those whohavetaken only thesingle subject are necessarily less able at malhematics than those who have taken the double subject and solessfitted10embark on a mathematics degree course. There seems no doubt that at most universities they willbe increasingly likely to findthemselvesinthe company of others whoare similarlyqualified . Degree COurses inengineering and technology 182A knowledge of mathematics is essential for the study of engineering and of most other technological subjects.We drew attention inparagraph 172 to Ihe fact that the number of entrants to courses in engineering and technology increased by 34per cent between 1973 and 1979 whereas the overall university entry increasedby 28per cent. This increase hasbeenmuch greater than the increase in total entry to all other mathematics and science courses, which has risenby only 20per cent.Furthermore,despi te anoverall dropduringthis periodinthe totalnumber of entrants to universitiesinEngland andWales withadouble-subjectqualificationinmathematics,therehasbeenan increase in theproportionof theseentrants who havechosentoreadengi-neering and technology , and also a smal l absolute increase in their numbers. TablesAandCshow (hat degree courses inengineering andtechnologyare attractinganincreasingproportionofuniversityentrantswithanA-level 54 Themathematical requirementsof professionalbodies 4The mal hematicalneeds of further and higher education qualification in mathematics and, inparticular, of those with double-subject mathematics . Theproportionof entrantswithdouble-subject mathematics is,however.decreasing both within engineering and tech nology courses and overall. J 83Although not directlywithin our terms of reference, we have given SOme attention to the mathematicalrequirement s of professional bodies . Many of those engaged inprofessional activities seekmembership of the appropriate professional institutionor association . In some casesmembership of sucha body is a necessary Qualification for professional advancement ; in other cases membership,althoughnotessentialfOTcareerpurposes.providesoppor-tunitytokeepabreastofcurrentdevelopmentsbyreadingpublications, attendingmeetingsandta.kingpartintheworkof committees.Mostinsti-tutions conduct their own examinations, commonly in two or th ree parts, for admission 10membership., whichisusuallyoffered at morethan one grade. Thepossessionofanappropriateacademicqualificationoftensecures exemptionfromsome orallof theseexami nat ionsbUladmi ssionto higher gradesof membershipnormallyrequiresevidenceof relevantprofessional experience. 184Anumber of the professional bodies who have written to us have stated themathematicalrequirementsfordirectentrytotheirvariousgradesand have also supplied details of their own examinations. When entry is at gradu-ate l e v e l ~itisusually assumed that any necessary mathematics willhave been covered either at school or during the degree course and no further mathema-tical requirement is stipulated. However, one exception to this is the Institute of Actuarieswhosefinal-examinationsrequireaconsiderable extensionof mathematical and statistical knowledge and itsapplication.When entry to a professionalbodyisatlowerlevel s.anymathematicalrequi rementisnor-mallystatedintermsof successatA- orO-level;anyfurthermathematics whichisrequired isthen includedwithin subsequent professional study. 185Almost alltheprofessionalbodjeswho submitted evidence st ressedthe importance of being a ble to applycomputational skillsconFidently inavar-ietyof ways .Theseincludeaccuracya ndspeedinmentalcalculationand abilityto checkthereasonablenessof answers;insome cases extendedand complex calculations are necessary. Specific calculations identified by bodies whose members are concelrnedwith commerce include interest. discount and value-addedtax,cashflow,costing andpricing,and budgetary control ; itis frequentlynecessary tobe able to dealwithbothmetric and imperialunits. There were also many references to the needto beable to interpret data with understanding. 186Mostinstitutionstakeforgrantedthemathematicalfoundationpro-videdbyanentrant'spr'eviousstudy.Anymathematicsincludedwithin professionalexaminationsisusuallylimitedeithertotopicsof aspeciaJist nature which are unlikely to have been studied before entry to the profession or to applications of mathematics in unfamiliar contexts.This isespecially 4The mathematical needs of funher andhigher education55 trueof professional bodies whose members are concerned withbusiness and commerce. The examinations of these bodies frequently include applications of statistics, andto a lesser extenttechniques of operational research,which areusedwithintheparticularprofession.Thecollection,classification, presentation and analysis of data, use of probability distributions, hypothesis testing,correlationandregressionanalysis,surveymethodsandsampling techniquesalloccurfrequentlywithinthe syllabuses of professional exami-nations . This emphasis on statisticsno doubt reflects the factthatat the pre-senttimefewschoolleavers willhave studiedthe subjectto any depth. 56 Part 2 "'Abrief summaryoftheReview maybepurchasedfrom(heShell C ~ n t r eforMathematicalEduca-[jon,Universityof Nottingham; see alsoparagraph756. Attainment jn mathematics 5Mathematics in schools 187Inthe second part of our report wediscuss the teaching and learning of mathematicsinschoolsaswellasmethodswhichareusedtoassess attainment.Before turning toparticular aspectssuchasmathematics in the primary and secondary years we consider some matlers which are fundamen-tal to the teaching of mathematics to pupils of all ages, and also certain mat-terswhicharise as a consequence of the discussion inearlier chapters, of the submissionswhichwehave receivedand of our own experience.In order to provideabackgroundwestartbydrawingattentiontothelevelsof attainment in mathematics which are to be expected of school leavers, so that readers may bear in mindthe proportions of the school population to which thedifferentpartsofourdiscussionrelate;wealsoconsidertheattitudes towardsmathematics whichpupilsdevelopduringtheirschooldays andthe mathematical attainmentof girls. 188In thispart of our report wedrawon Areview o/research inmathema-tical education which summarises the results of the study carried out for our Committee under the direction of Dr ABell of the University of Nottingham and Dr ABishop of the University of Cambridge. For the sake of brevity, we shallhenceforwardrefer to itasthe Review 0/ research . >Ilxth form shown in Table 7.The numbelTsinthjs table are of U5C essential38 I.uscinprimary years384-8, use:insecon dary years389-92, calculat ors shouldreplaeelogarithm 391.provision o f calculators in .secondary 6chools 393-4,uSeinexaminations 395 Computer stud ies396- 401 notpartof mathematics397,demands onma thematic..c; tenchers398-400.dcmands on advisory staff 401 CompOlcrs402- 13 can assislleaching of mathemalics 402-6,software required4(J7.use inprimary years408-9.usc in secon dary years410- 11,use byindividuaJ pupils412-13 Calculators, U5etnc:mployment73- 4,133.114 Canada790 Calc:ring and hotcl5141- 3 306Index Cemres rer ma(hematicaJ educ31ioll758 Certific:ue of E)tlendedEducation(CEE)598- 601 Cenificare of Secondary Education (CSE) 68,1.05,150.162, 189.194- 7. "40- 1. 443- 7.470. 472. 519-20. 515. 5)7.541. 599- 601,FigureE College758 Cilyand Guildsof Londontnsli l Ule (CeLl )li S,149,151. 159- 60.165 Classroom accOmmod