Mathematics counts Report of theCommitteeof InquiryintotheTeaching of Mathematics inSchools under the Chairmanshipof DrW H Cockcroft ". "j-:i, 'II '1' 1', ! ~ .-:0 ".. '/JI r"' I.. -:'( ' JI ,:.1 I "~ ,-. .. II ,' r:-= I ~i " . - " I : London:Her Majesty's Stationery Office iii Foreword BY THE SECRETARY OF STATE FOR EDUCATION AND SCIENCE AND THE SECRETARY OF STATE FOR WALES Fewsubjects intheschoolcurriculumareasimponanttothefuture of the nation asmathematics;andfewhavebeenthe subject of more comment and criticisminrecentyears.Thisreporttacklesthatcriticismhead00.It offers constructiveandoriginalproposalsforchange.Itshouldbereadbythose responsibleforschool mathematics at all levels. Themainmessageisfortheeducationservice.Thereportidentifiessi.:< agencieswhoseactiveresponseisrequired.Thecontributionof allwillbe necessaryif weareto make headway,To the extentthat thereport callsfor extra resources, progress isbound to be conditioned by the continuing needto restrain public expenditure;but many recommendations involve no such ca.ll. Wehopethat therewillbe widespreaddiscussionof thereport's conclusions and that actionwillfollow. TheCommittee'stermsof referenceinvitedittoconsidertheteachingof mathematicswithparticularregardtothemathematicsrequiredinfurther andhighereducation,employmentandadultlifegenerally.Theearly chaptersof thereportare concernedwiththeseaspects.They willbe of in-teresttomany both withinandou tside the education service.They revealthe hesitantgraspmanyadultshaveofevenquitesimplemathematicalskills. They are particularly valuableinexamining closelythemathematicswhichis infactneededindifferentkindsof employmentandineverydaylife,and itto what istaught inthe schools.The Committee's fIndingspoint to the need for teachers to devote more time to the use oJ mathematics in applica-tions taken from real life. Thisisafirst-classreport.Wearegreatly indebtedtoDr Cockcroftandthe members of hisCommittee,and commend their work to allthoseconcerned about the quality of mathematics taught inour schools. .-January 1982 v The CommiUee of Inquiry in10 the teaching of Mathematics inprimary and secondary schools in England ondWales MEMBERSHIP OF THE COMMITTEE Dr \VHCockcrofl (Chairman) Mr AGAhmed Professor MFAtiyahFRS MissKCross Mr CDavid Mr GDavies MrK TDennis Mr T Easingwood Mr HR Galleymore Mr R PHarding CBE Mr JWHersee Mrs MHughes Mr AJ McIntosh Mr HNeill Mr PReynolds Mr 0G Saunders MBE Mr HPScanlon MissHBShuard Dr PGWakely Councillor DWebster Mr LDWigham MI P HHalsey (Assessor) Mr WJAMannHMI (Secretary) Mr EL Basire (Assistant Secretary) Vice Chancellor,New University of Ulster,Coleraine. Head of Mathematics Department,FairchildesHighSchool, Croydon. RoyalSocietyResearchProfessor. AssistantPrincipalandHeadof theFacultyofMathematicsandScience, Accrington andRossendaleCollege,Accrington. Headteacher 1Dyffryn Comprehensive School,Port TaJbot. PoUcy Unit,Prime Minister'sOffice(resigned April 1980). Teacher,Dunmore County Junior School.Abingdon. Reader in Mathematical Education,Derby Lonsdale College of Higher Edu-caHon,Derby. FormerDirector,Procter and GambleLimited (appointed August1979). Ch.ief Education Officer,Buckinghamshire County Council. ExecutiveDirector,SchoolMathematicsProject;Chairman,Schools CouncilMathematics Committee. Headteacher,Yardley Junior School,Birmingham. Principal Adviser inMathematics,Leicestershire County Council(resigned March19'80). Lecturer inMathematics,University of Durham. Mathematics Adviser, Suffolk County Council (appointedMarch1980). WelshArea Secretary,Associationof Professional,Executive, Clericaland Com.puter Staff (appoi.ntedMarch1979). President,AmalgamatedUnion of Engineering Workers (resignedJanuary 1979). Deputy Principal,Homerton College,Cambridge. ChairmanandManagingDirector,AssociatedEngineeringDevelopments Limited. Chairman, Education Committee,Newcastleupon Tyne Borough Council. PostgraduateCertificateinEducationStudent,UniversityofLeeds (appointedNovember1978). Departm.ent of Education and Science. Her Majesty's Inspectorate of Schoo]s. Department of Education and Sciencc. The styles, decorations and appointments shown are those held by members at the time of their appointment totheCommittee. 10November1981 Dear Secretaries of Slate On behalf of the Commitlee of Inquiry into the teaching of mathematics in primary and secondary schools in Englandand Wales,I havethe honour to s.ubmitour repon 10you. TheRtHan Sir KeithJosephBtMP Secretary of StateforEducation andScience TheRtHonNicholasE d w a r d ~MP Secretaryof State forWales Yours sincerely WHCOCKCROFT vii Contents IntroductioDpageix Explaoatory notexv Part ODeIWhy teach mathematics?I 2The mathematical needs of adult life5 3Themathematic.a1 needs of employment12 4The mathematical needs of further and higher education42 Parttwo5Mathematics in schools56 6Mathematics in the primary years83 7CalcuJators and computersJ 09 8Assessmentand continuity121 9Mathematics in the secondary years.128 10Examinations at16+158 11Mathematics in the sixth form169 Part three12Facilities for teaching mathematics183 13The supply of mathematics teachers188 14Initial training courses203 15In-servicesupport for teachers of mathematics2] 7 16Someorher mailers232 17The way ahead242 Appendices1Statistical information246 2Differences inmathematicalperformance between guls andboys by Miss HB Shuard273 3List of organisations and individuals whohave made subm.issions to the Commi Uee288 4Visits to schools, industry and commerce; meetings with teachers301 5Abbreviations used within the text of the report303 Index305 Terms of reference Mee[ingsand v ~ s i t s ix Introduction InitsreportpubjjshedinJuly1977,theEducation,ArtsandHome Office Sub-Committee of the Parliamentary ExpenditLlre Cpmmittee stated that' 'it is clear from the points which were made over and over again by witnesses that thereisalargenumberof questions aboutthe mathematical attainments of childrenwhichneedmuchmore carefulanalysisthanwehave beenableto giveduringour enquiry.Theseconcerntheapparentlackof basiccompu-tation skills in many children, the increasing mathematical demands made on adults,the lackof qualified matbs teachers,the multiplicity of syllabuses for old,newand mixedmaLhs,the lackof communicationbetween further and highereducation,employersandschoolsabouteachgroup'sneedsand viewpoints,the inadequacy of information on job content or test results over aperiod of time,and the responsibility of teachers of mathematics and other subjects to equip children withthe skillsof numeracyn. The Committee re-commended as"possibly the most important of our recommendations" that the Secretary of State forEducation and Science should set up an enquiry into the teaching of mathematics.In their reply presented to Parliament in March 1978,theGovernmentagreed"thatissuesofthekindlistedinthe Committee'sreportneedthoroughexamination"andannouncedtheir decisiontoj(establish anInquiry to consider the teaching of mathematics in primary and secondary schools inEngland and Wales, with particular regard to its effectiveness and imelligibility and to the match betweenthe mathema-tical curriculum and the skills required in further education, employment and adult life generally' 1.They further undertook that the Inquiry would examine the suggestion that there should be afullanaJysis of themathema(icai skills required in employment and [he problem of the proliferation a f mathematics syllabuses atA-level and at 16 + . Our Committee met for[he firsttime on 25September1978withthe follow-ing terms of reference: Toconsider [he[eaching of mathematicsinprimary andsecondary schoolsin England andWales,with part;cuJar regardto (he mathematics required infurther andhigher education, employment andadultlife generally,andto make recom mendations. The fun Committee has met on 64 days,whichhave included three residential meetings.ItsWorkingGroupshavemet on 143daysinall,andtherehave been less formal discussions on many occasions. 54 schools and 26 companies of variouskinds in England and Waleshavebeenvisitedbymembers of the Committee andthere have beensix meetingswithgroups of teachersindif-x Submissions of evidence Research studies Imroduction ferent parts of the country. SmaJi groups of members have visited the Scottish EducationDepartmentinEdinburgh,[heInstitutefortheDevelopmenlof MathematicsEducation(IOWO)atUtrecht,Holland,the[ostituteforthe Teaching of Mathematics atthe University of Bielefeld,West Germany and theRoyalDanishSchoolofEducationalStudiesinCopenhagen;two membershavevisitedindustrialcompaniesin Nuremberg,WestGennany. Severalmembers of theCommittee werepresent at the Founh International Congress on Mathematical Education held at the Universiry of California at Berkeley in August1980.Individualmembers of theCommitteehavebeen invitedtoattendtheconferences andmeetings of anumber of professional bodies. Throughout our work we ha vebeen greatly encouraged bythe welcome which manypeoplehavegiventothe setting up of theInquiry andbythehelpful response which wehave receivedto our requestsfor information and written evidence.Wehave receivedwritten submissions, m(iny of them of able length.from930individuals andbodies of manykinds.73individuals and groups have met members of the Committee for discussion. A list of those who have submitled evidence and who have met members of the Committee for discussion isgiveninAppendix 3. Whenwestartedtoconsiderhowbestwemjghtrespondtoourtermsof reference, we became aware that we needed more detailedinformation about themathematicalneedsof employmentandof adultlife generallythanwe were likely either to receivein written evidence or to be ableto obtain by our ownefforts.Wetherefore requested theDepartment of Education and Sci-ence (DES) to commission two complemenrary studies into the mathematical needsof employment and also asmallstudy into themathematical needs of adult Ii fe.One of the studies into tbe mathematical needs of employment was based attheUniversity of Bath under the direction of Professor DE Bailey, assisted byMr A Fitzgerald of the University of Birmingbam, and the other at [heShellCentreforMathematicalEducation,Universityof Nottingham, under the direction of Mr R L Lindsay. The sludy into the mathematical needs of adult life was carried out by Mrs B Sewell on behalf both of the Committee and of the Advisory Council for Adult and Continuing Education. The DES alsoagreed to commission a review of existi.ngresearchonthe teaching and learning of mathematics which was carried out by Dr A Bell of tbe University of Nottingham and Dr ABishop of the University of Cambridge. The Steer-ing Groups for all these studies have included members of the Committee, and relevant evidence which hasbeen received has beenmade available on a con-fidentialbasistothose engagedinthestudies.The reportswhichhave been produced have proved to be of very considerable help to us; we refer tothem and draw on their conclusions ina number of the chapters which follow.At a tater stage the DES commissioned a small survey of mathematics teachers in secondaryschoolswhowereintheirfirstthree yearsofteaching;t.hiswas carriedoutforus by [he NationalFoundationforEducational Research. Publications and ann.ouncements Imroduclionx.i Since we staned our work a considerable number 0f official reports and other publjcations have been issued wh.ichrelate wholly or in pan: LOthe teaching of mathematics inschools. These include fromtheDES, A1athematical developmem'.Primary survey reports l'ios / and 2 (The APU Primary Surveys.) Mathematicaldevelopment.SecondarysurveyreportNo/(TheA PU Secondary Survey) Local authority arrangemenlsjor the school curriculum Abasis for choice Proposals for a Certificateof Extended Education(TheKeohaneReport) Secondary school examinations: a single sysl,emat16 plus A framework for the school curriculum The school curriculum Examinations /6- J8: aconsultative paper Education for J6-J9 year olds; fromHMInspectorate, Primary educorionin England(Repon of theNationalPrlmary Survey) Aspects of secondary education inEng/alld(Report of [heNational Secondary Survey) Aspects of secondary educlJtion inEngland: supplementary informalion on mathematics Marhemacics 5-J I:ahandbookof suggestions Developments inthe BEd degreecourse PGCE inthe public sector Teacher trainingand the sxondary school Aview of the curriculum; fromthe Schools Council, ,Wathematics and theIO-yeor old (SchooJs Council Working Paper 61) lvfathema(ics ill school and employment: a study of liaison activities (Schools Counc.ilWorking Paper 68) Statistics inschools/1-16:a review(Schools CouncilWorkjng Paper 69) The practical curr;culum(SchoolsCouncilWorkingPaper 70); fromother sources, Engineeringour future (Report of the Finniston Committee) Thefunding and organisationof courses inhigher education(Report 0f (he: Education} Science andArt.sCommiuee of the House or Commons) ThePGCEcourseandthetrainingof specialistteachersforsecondary schools(Universities Councilfor theEducarion of Teachers) Aminimal core syllabus for A-level malhemalics (Standing Conference on University Entra.nceandCouncilforNationalAcademic Awards) Children's underslanding Q1f mathematics:/1-16 (Repon of the Concepisin Secondary Mathematicsand ScienceProject). \Vehavestudied all these documents and make reference to several oflhem in {he course of {hisreport. xii Slalisticalinformation Viewsfrom{hepast (nt roduci ion SincewestartedworK,theGovernmenthaveannouncedthat(bepresent O-level and CSE examinations are [0bereplacedbya single system of exam-ining. at16 +,thatGCEA-levelsare toberetained,thattheCertificateof ExtendedEducationwillnotbeintroducedbutthattherewillbeapre-vocational examination at17+, andthat considerationisbeing giventothe introductionof Intermediate levels(l-levels).Wehave consideredthe impli-cations of these announcements sofa!asmathematicsisconcerned. Asaresultof theworkwhichhasbeencarriedoutforusbytheSlatistics Branchof theDESandby theUniversitiesStatisticalRecordwe havebeen able to obtainaconsiderable amount of information whichhasnothitherto been available.We refer [0 this, and to other existing information, from tlme 'totimethroughoutthereport.Ingeneralwequotethisinformalionin roundedterms or presentitindiagrammatic form.The detailedtables [rom which the information is taken are set outinAppendixI, whichalso givesin eachcase[hesourcefromwhichtheinformationhasbeenobtained.This Appendix also contains some ta bles to which no direct reference is made in the textbutwhichwebelievetobeof interest.TheAppendixdiscusses,where appropriate, any assumptions which i[ has been necessary to make in order to prepare {betablesandincludesbelef commems on some of them. In the light of present day criticism of standards, it is interesting to assemble a collectionof quotationsfromdocumentsof variouskinds.some of which datebacktothelastcentury,whichdrawattentiontotheallegedlypoor mathematical of the day.\Ve content with examples from approximately acentury,ahalf-century and aquarter-century ago. [narithmetic,1regret[0sayworseresullsthane.verbeforehavebeen obtained -this ispardy auributable, no doube, [0 my having so framed my sums as torequirerathermore intelligence thanbefore:the failuresarealmost invariably traceable toradically imperfectteaehing. The failuresinarithmetic aremainlyduelOthe scarcity of goodceachersof it. Thoseare laken fromreportsby HM Inspecrorswritten in1876. Manywhoare inapositiontocritidse the capaci1yof youngpeople whohave passedthroughthepublic elementaryschoolshave experiencedsome uneasiness about the condition of arithmeticalknowledge and teaching atLhepresent lime.It hasbeen said,for instance,lhataccuracy inthemanipulationof figuresdoe&not reachthesamestandardwhichwasreachedtwentyyearsago.Someemployers expresssurprise andconcernat I he:inability of youngDersons[0perform simple numerical operations involved in business. Some evening school teachers complain thallhe knowledge of arithmetic shown by their pupils does nor reach their expec-rations.Itissometimesallegedinconsequence,[houghnotasarulewi rhthe suppon of definite evidence, rha[ the teacher no longer prosecutes his atrack on this subjecrwithI heenergyorpurpose fulnessforwhichhispredecessorsaregiven credit. That extract comesfromaBoard of Education Reportof 1925. The standardof mathematicalabilityof entranrs t.otrade coursesisofren very low..... Experience shows that a large proportion of entrants have forgotten how LOdeal with simple vulgar and decimal fractions, have very hazy ideas on some easy General approach Acknowledgemen ts IntroductionXlll arithmericalprocesses,andretainnotraceofknowledgeofalgebra,graphsor geomeuy, if, infact,they ever did possess any. Some improvements in this position maybe expected as a result of the raising of the schoo! leaving age, but there isas yet noevidenceof anymarkedchange. Our final quotation comes from a Mathematical Association Report of 1954; the 5ch 001lea vi ng agewasraisedto15in1947. Itisthereforeclearthatcriticismofmathematicaleducationisnotnew. Indeed,throughout the timeforwhichwe havebeenworkingwehavebeen conscious thatformany years agreat deal of adviceto teachersabout good practice in mathematics teaching has been available in published form from a varietyofsources.TheseincludethepublicalionsoftheDES,ofHM .Inspectorate,of theScho01sCouncilandof theprofessionalmathemati.cal associations;therehave alsobeenreferencestomathematics teaching inthe reports of Committees of Inquiry,for example that of the Newsom Commit-tee.Muchof thisadvice isstillrelevant today andservesasabackground to our ownwork. Inwriting ourreport wehavetriedsofarasispossibletoavoidtheuse of technical language and to put forward our views in away whjch we hope will beintelligibletomathematicianandnon-mathematicianalike.Forthis reasonwehaveattimesomiueddetail which,hadwebeenwriting onlyfor those engaged inmathematical education, wewould have included.We hope thatthosewhowouldhavewishedustodiscusscertainmaltersingreater detailwlllunderstand(hereasonwhywehaveincertalnpJacesuseda somewhat 'broad brush'. We hope that our attempt to draw attention to those aspects of the teaching of mathematics which we believe to be of fundamental importance willbeof usebothinside andoutside the classroom. We wish to s[ress that many of [he chapters inour report, and especially those in Part 2,are inter-related.For example,inChapters 5 and 6wediscuss the elementsofmathematicsleachingatsomelength;thefactthatwedonot repeal [his discussioninChapter 9 (!vlathematics in the secondary years),but deal mainly with matters ofsyUabus content and organisation, does not mean that the teaching approaches we have recommended in earlier chapters are not equally applicable at the secondary stage.We therefore hope that those who read our reportwillviewit asawhole. In our report wehave not considered the needs of pupils with severe learning difficulties.We hope,however,thatthose who teachpupils of this kindwill find that our djscussion of mathematics teaching in general, and of the needs of low-attaining pupilsinparticular,asweUasourdiscussionof themath-ematical needs of adultlife,willbe of assistance. We would liketoexpress our thanksto allthose who have writtento us and with whom we have talked both formally and informally, We are grateful for (hehelp we have receivedfromtheheads, staff andpupils of the schools we havevisited,fromtheteacherswhom wehavemetatmeetingsindifferent pans of the country and fromthose at allleveJs whom we have met during our XlVIn! roduction visits to commerce and industry. We are grarefull too, to those who were kind enoughto arrange thesevisitsandmeetings.Wewishtothankthe Scottish Education Department for arranging our visitto Edinburgh and those whom we visited in Denmark, Holland and West Germany forrhe help they gave us andthe arrangements theymade on our bebalf. Our thanks are due to allthose who have carried out the various research stu-dies and also to those inDES Statistics Branch and the Universities Statistical Recordwho have undertaken agreat deal of workrorus andfoundways of answering Ourquestions.We are grateful forthe help we have received from many officersin theDES,inpanicular aUf Assessor,MrPHHalsey;and frommembersof J-IMInspectorate,especially Mr TJ Fletcher whohas,at Our invitation,attendedmany of our meetings. We wishalsoto express our thanks forthehelpand support whichwehave receivedfromthemembers of tbe Committee's secretariat.MrWMWhite hastakenmajorresponsibilityforobtaining,puttinginorderandpreting avery great dealof statisricalinformation. Mr EL Basire, our tant Secretary.MissEKirszberg andMrR W Le Cheminant,inaddition to theirotherduties,havegivenmuchpersonalhelptomembersofthe Committee. Among those to whom we are indebted, our Secretary) Mr W J AMann HM r. stands out. We are conscious of {he burden we have placed upon his shoulders andof the conscientious wayinwhichhehas accepredtheload.Most of all, we are gra[efuito hjrofor the able and efficient way in wh ich he has taken the outcome of our many diffuse and varied deliberations andmoulded itinto a coherentwhole. xv Explanatory note Throughout the report there are certain passages whicb are printed in heavier type.Inselecting these passages, we have chosen [hose which either relate to matters we consider to be of significance for allour readers or which call for actionby those who are outside the classroom. This means that, especially in Chapters 5 (0IIwhichareconcerned in particular withthe leaching of math-emallcs, we have nOIpicked outmany of I he passages inwhichwe make sug-gestions relating toclassroom praclice.Aswehave pointed outinrhe[mro-ductjon,Cbaplers5 toIIareinrer-relaledandwe havenot wishedto draw anentiononly !Ocertainpassages in[hem, exceptin sofar asthese passages fulftlthe purposes we have already selOuL Wedonotregardthepassagesprintedinheavier type as inanywayconsti-tuting asummary of thereport. Part 11Why teach mathematics? 1Therecanbe110doubtthatthereisgeneralagreementthateverychild shouldstudymathematicsatschool;indeed,thestudyofmathematics, together \II it h that of Engl ish.isregarded by most people as being essential.[t mightthereforebe argued[hal thereisno needto answer the question which wehaveusedasourchapterheading.Itwouldbeverydifficult-perhaps impossible-toliveanormallifeinverymanypartsof theworldinthe twentieth century without mak iog use of mathematics of some kind. This fact initself couldbethoughttoprovide asufficientreasonforteachingmath-ematics, and in one sense this isundoubtedly true.However, we believe that jt is of valueLOtrytoprovide amore detailedanswer. 2Mathematics isonly one of many subjects whichare included inthe school curriculum, yetthere isgreater pressurefor children to succeed atmathematics than,forexample,a[historyorgeography,eventhoughitisgenerally acceptedthatthesesubjects shouldalsoformpartof thecurriculum. This suggeststhatmathematicsisinsomewaythoughttobeofespecial importance.I f weaskwhythis shouldbe so,one of the reasonswhichisf r e ~ quently givenisthatmathematicsis'useful';Ieisclear,roo,that thisuseful-nessisinsomewayseentobeof a differentkindfromthatof many other subjects inthe curriculum. The usefulness of mathematics isperceived indif-ferentways.Formanyitisseenintermsof thearL(hmericskiJlswhichare needed for use at home or in the office or workshop; some see mathematics as thebasis of scientific development andmodern technology; some emphasise theincreasinguseofmathematicaltechniquesasamanagementtoolin commerce and industry. 3\\'e believe that allthese perceptions of theusefulness of mathematics arise from the factthatmathemalics provides ameans of communication which is powerful,concise andunambiguous.Eventhoughmany of lhosewho con-sider mathematics tobe useful would probably nOl express the reason in these terms,webelieve that itis the fact that mathematics can beused as a powerful meansof communicationwhichprovidestheprincipalreasonforteaching mathematics [0 allchildren. 4Mathematics can be usedto present information inmany ways, not only by meansoffiguresandlettersbutalsothroughtheliseof tables,chartsand diagramsaswellasof graphsandgeometricalortechnicaldrawings.F u r ~ thermore, the figures and other symbols which are used inmathematics can be manipulatedand comQinedinsystematicwayssothatitisortenpossibleto 2 IWhy(each mathematics? deducefurtherinformationaboutthesituationtowhichthemathematics relates.For example,if we are toldt.hatacar has travelledfor3 hours at an averagespeedof20milesperhour.wecandeducethatithascovereda distance of 60 miles.In order to obtain lhis result we made use of the fact that: 20x3=60. However,thismathematicalstatementalsorepresents(hecalculation requiredtofindthecostof 20articleseachcosting3p,thearea of carpet requjred to cover a corridor 20 metres long and 3 metres wide and many other thingsaswell.Thisprovides anillustrationof thefactthatthe same math-ematica1statementcanarisefromandrepresentmanydifferent situations. Thisfacthasimportantconsequences.Becausethesamemathematical statementcanrelatetomorethanonesituation,resultswhichhavebeen obtained in solving aproblem arising from one siLUation can often be seen to apply to a different situation. In this way mathematics can be used not only to explainthe outcomeof aneventwhichhasalreadyoccurredbutalso,and perhaps more importantly. to predicUhe outcome of an event which has yet to take pJace. Such a prediction may be simple, for example the amount of petrol which will be needed for ajoumey. its cost and the time which the journey will take; or it maybecomplex. such asthe pathwhichwillbe taken by arocket launched into space or t he loadwhich can be supported by abridge of given design.Indeed,itistheabilityof mathematicstopredict whichhasmade possiblemany of thetechnological advances of recent years . .5Asecondimportantreasonforteachingmathematicsmustbeits importance and usefulness in many other fields. It isfundamental [Q the study of the physical sciences and of engineering of allkinds.It is increasingly being used inmedicine and the biological sciences, in geography and economics, in business and management studies. It isessential to the operations of industry andcommerce inbothoffice and workshop. 6It is often suggested that mathematics should be studied in order to develop powersof10gicaJthinking,accuracyandspatialawareness.Thestudyof mathematics can certainly contribute tothese ends but the extent towhichit does so depends on t.heway in whichmathematics istaught.No[ isits contri-butionunique;manyotheractivitiesandthestudyof anumberof other subjects can develop these powers aswell.We therefore believe that the need todevelopthesepowersdoesnotinitself constitute asufficientreasonfor studying mathematics rather than other things.However, teachers should be aware of the contribution whichmathematics can make. 7The inherent interest of mathematics andthe appeal whichitcan have for many children .and adults provide yet another reasonforteaching mathema-ticsinschools.The factthat'puzzle corners'of variouskinds appear inso manypapers and periodicals testifiestothe factthat the appeal of relatively elementaryproblems andpuzzlesiswidespread;attemptsto solvethem can both provide enjoyment and also, in many cases, lead to increased mathema-tical understanding. For some people,laO,t he appeal of mathematics can be evengreater and more intense.Forinstance: Fynn.MislerCod,th.isisAnno. CollinsFOUOlPaperbacks1974. "'CarlSagan.Murmursoj Earth. Hodder and Stoughton1979 \J,.'hyreachmaLhematic;? 3 Anna andI had both 5e:nthaL maths wasmore than just working out problems. It wasa doorway [Qmagic,mysterious,brain-cracking worlds.worldswhere you hadtoueadcarefully,wOlrldswhere youmadeupyourownrules,worldswhere youhadto accept completefor your actions.BUt iewasexciting and vastbeyondunderstanding. '" Even thoughitmaybe givento relativelyfew[0 achieve the insight and sense of wonder of 7 year oldAnna and of the young man who inlater years wrote thebook, webelieve itto be important that opportunities to do so should not be deniedtoanyone.J ndeed,wehope thatallthose who learnmathematics willbeenabledtobecome- awareof the'viewlhrough(hedoorway'which manypiecesofmathema.ticscanprovideandbeencouragedtoventure throughthisdoorway.However,wehavetorecognisethattherearesome who, even though they may glimpse the view from dmeto time asthey become interestedinparticularactivities,seein itnolastingattractionandremain indifferent or insome cases actively hostile tomathematics. 8There are other reasonsforteachingmathematicsbesidesthosewhichwe have put forward inthis chapter.However, webelieve that the reasons which wehave givenmalce amore thansuffLcient case for teaching mathematks to allboys and girls and that foremost among them isthe fact that mathematics can beused as a powerful means of communication-to represent, to explain and topredict. 9Itisinterestingtonotetwoverydifferentuseswhkhhavebeenmadeof mathematics jnthe current Voyager .space programme. Not only has the pre-dictive power of mathematics been used [Qplan the details of the journeys of the two Voyager spacecraft but examples of mathematics have been included intheinformationaboutlifeonEarthwhichwasaffixedtoeachofthe spacecraftbeforetheywerelaunchedin1977toexploretheouterSolar System and [hentobecom,e "emissaries of Earth tothe realm of the stars". * Thereasonforincludingexamplesofmathematicsisexplainedinthese words: Sofaraswecan [ell, malthematicalrelationships should bevalidfor allplane[s, biologies,cultures,phjlosophies.Wecanimagineaplanetwithuranium fluoride inthe atmosphere or a life form thatlives mostly offincecstellar dust, even if these are extremely unmceJycontingencies. BUlwe cannOIimagine a civilization for which one and one does nOlequaltwo or ror which there is an integer interposed between eighc andnine.FQirthis reason, simple mathematical relationships may be evenbeHermeansof communicaeionbetweendiversespeciesthanreferences.to physic..; and astronomy. The earlypan of the pictorial information on the Voyager record isrichinarithmetic,whichalsoprovidesakindof dictionaryforsimple mathematical information contained inlater pictures; such as the size of ahuman being. 10Mathematics provides .ameans of communicating information concisely andunambiguouslybecauseitmakesextensiveuseof symbolicnotation. However,itisthenecessityof usingandinterpretingthisnotationandof graspjngtheabstractideasandconceptswhichunderlieitwhichprovesa stumbling blocktomanypeople.Indeed,the symbolicnotationwhichena-4 implicationsfor teachers IWhy leachmathematics? blesmathematicstobeusedasameansof communicationandsohelpsto makeit'useful'canalsomakemathematicsdifficulttounderstandandto use. 11TheprobJemsoflearning10usemathematicsasameansofcommu-nicationare notthe same asthoseof learning to use one'snativelanguage. Narive language provides ameans of communicationwhichisinuseallthe timeandwhich.forthegreatmajorityof people,4comesnaturally',even rhoughcommandoflanguageneedstobedevelopedandextendedinthe classroom.Furthermore, mistakes of grammar or of spelling do not, in gene-ral,renderunintelJigiblethemessage whichisbeing conveyed.Onthe other hand, mathematics does not 'come naturally' to moslpeople in the way which istfue of native language.Itisnot constantly being used; ithas[0 belearned andpractised;mistakesareof greater consequence.Mathematics alsocon-veys information in a much more precise and concentrated way than is usually the case with 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.Thosewhohavebeenable [0 develop suchconfidence withrelative ease should notunderestimate the d i f ~ flculties which many others ex perience, nor the extent of {he hel p wh.ich can be requiredinorder tobe abletounderstand andtouse mathematics. 12\Veconcludethischapterbydrawingthe attentionofthosewhoteach mathematics inschools towhatwebelieve to theimplications of the reasons forteaching mathematicswhichweh.avediscussed.Inour ~ i e wthemal he-maficsteacher has the task of enabling eachpupiltodevelop,withinillscapabiljties,themath-ematicalskinsandunderstandingrequiredforadultlife,for employment andforfurther study and training,while remaining aware of the difficultieswhichsomepupilswillexperienceintryingtogain such anappropriate understanding; of providing each pupil with such mathematics as may be needed for his study of other subjects; ofhelping each pupil to develop so far as is possible his appreciation and enjoyment of mathematics itseJf and hisrealisation of the role which it has prayed and will continue to play both inthe development of science andtechnology andof our civilisation; aboveall,of making eachpupilaware thatmathematicsprovideshim with apowerfulmeans of communjcation. '"MakeI'COunl.ASludyDY DavidStringer.Independent Broadcast iagAuthority1979. 5 2Themathematicalneedsof adult 'life 13TherearejndeedmanyadultsinBritainwhohavethegreatestdifficultywith even~ u c happareoLlysimpJematlersasaddingupmoney,checkingtheirchange inshops or working OUIIhe cost of five gallons of petrol.Yetthese adults arenot juStt.heunintelligent ortheuneducated.They comefrommanywalksof life and someareveryhighlyeducatedindeed,buttheyarehopelessatarithmeticand theywanttodosomethingaboutit. The abovequotation comesfromthe prefacetothe researchstudy*onthe Yorkshire TelevisjonseriesMake it count-a series of thirteen programmes foradultsbroadcastnationallyforthefirsttime.in1978.In the conclusion tothe studyweread Duringthisinvestigationthefirmimpressionhasbuiltup-inthe investigator's mind,atleast-that functionalinnumeracy isfarmore widespread thananyonehascaredtobelieve. 14A copy of this st udy wasmade available tous soon after we started work and atabout. the same time the Advisory Councll forAdult and Continuing Education(ACACE)drewourattentiontothefactthatoneofthe outcomes of the very successful adult literacycampaign of recentyearshad been anincreasing demandforadult numeracy classes.The Council passed ontoussome of theexperiencewhichhadbeen gainedinthe course of its workonadultliteracyandwearegratefulforitswillingnesstosharethis withus.In particular the Council urged that, tempting though the approach might seem, we shouldnot set our[0 try to define the mathematicaJ needs of adultlifesolelyintermsof somekindof'shoppinglist'ofnecessaryor-desirableskillsbutshouldalsoinvestigateattitudestowardsmathematics andthe strategiesusedbythose whose mathematical abilitiesare limited in their effortsto copewiththemathematicsneededin everydaylife. 15Sincethereappearedtohavebeenverylittleresearchcarriedoutto identifythemathematicalneedsofadu1ts,wedecided,asaresultofthe Make it countstudyandourdiscussionswithACACE,thatwewouldask theDEStocommissionasmallstudytobecarriedoutonbehalfbothof ACACEandof ourCommiUee.We suggested(hatthose involvedshould investigate themathemat icat needs 0f adults indajjy life,and. inparticular. trytoidentify the strategies whichwereusedbythosewhose mathematical skillsandunderstandingwerelimited.Wefeltthatsuchaninvestigation wouldbeofusetobochbodiesbecause,althoughtheproblemsand methods of teaching adults are differentfromthose of teaching children, an understanding of lhe goals to be achievedshou.ldbe of value both tothose 6 The research study *ThefL"'5ul!i>ofIhestudyare reportedindetailinUseof mafh /!molicsbyadultsindoifylife: BridgidSewell.whichmaybe purchasedfromthe CouncilforAdulrandContinuing Educa[ion.TheAdvisoryCouncil h.asaJsopublishedasummaryor (hereport,togelherwil ha summary Qr[heresuits of a Gallup Pollnationalsurvey,inAdults' m{Jfh(>lIIalical{Jbilityond peT-Ionnanef'. 2Themathematicalneedsof adulllife who teach adults and to teachers in schools. The outcome of this investigation has drawn attention to a number of matters whichwebelieve tobeworthy of note,not only byteachersbut bymany others aswell. 16Thestudywascarriedoutintwostages.Thefirststageconsisredof interviews designedtocoverfour areas: adiscussionof selected situations,relatedtoshoppingand household matters generally,inwhichmathematics mightbe involved; brief quesllons on othermatters suchasthe reading of timetables and theuse of cakulators; attitudes to mathematics; background information, These interviews were designed to give some indication of levels of mathema-tical corn petence bu t did not require speci ficmathematical calculations to be carried out,During the first stage107people were interviewed, chosen sofar as was possible to reflect thefiveoccupationaJ groups into which the Registrar Generaldividesthepopulation.However linonesense,verymanymore people were involved because there provedto be a widespread reluctance to be interviewed about mathematics_fnthe words of thereport: Bothdlrectandindirectapproachesweretried.theword'mathematics'waS replacedby'arithmetic'or"everydayuseofnumbers Ibuti[wasclear(hatthe reasonfor people's re fusal to beinterviewed was simply that the subjectwas math-ematics... Severalpersonalcontactspursuedbytheenquiryofficerwerealso adamantintheirrefusals.Evidently 1bereweresome painfulassociationswhich theyfearedmightbe uncovered.This apparently widespreadperceptionamongst adults0 fmathematicsasadaunting su bjectpervadedagreat dealof the sample selecrion;half of thepeople- approached asbeingappropriate forinclusioninthe samplerefused(0takepart. 17I n the second stage, about hal f of those who had already been in terviewed were interviewed again at greater length. They were in vitedto answer a series of mathematicalquestionsaboutarangeof everydaysituations,someof whichwere related to topics explored during the first interview;they were not pressedtorespond,unlesstheywished[0doso,toQuestionsrelatingto situationsofwhichtheydidnothavedirectexperience.Somequestions requi.red speci fie calculations tobe carried out. some required an explanation ofmethodbutnocalculation,somerequiredtheinrerprecationof informationpresented inmathematicalterms.Originaldocumentssuchas bills,pay-slips and timetables were used whenever appropriate; there were no questionswhichtestedcomputationbyirselfunrelated[Qarealsituation, Those being interviewedwerefreetowork out the answersintheir head,to use pencilandpaper or touse acalculator,asthey wished. 18Becausethesampleof adultshadbeensmall,ACACEdecidedthatit would be desirable (Qtry to validate rhefindings of the study in some way. The Findingsof theresearch study 2The marhematicalneedsof adultlife7 Advisory Council therefore made arrangements for a selection of questions. ofthekindwhichhadbeenusedinthestudy,tobeincludedaspartof a national enquiry undertaken by the Gallup Poll. This enquiry covered arep-resentative sample of the population of Great Britain aged 16 or over; almost 3000 people were interviewed. The results of this enquiry, which are included inthe ACACE booklet Adults' mathematical ability aruJperjormance,sug-gest thatthe findingsof the originalsmallstudy are innoway untypical. 19There are.of course,many people who are able to cope confidently and competentlywithanysituation whichtheymaymeetinthe course oJ thejr everyday life whichrequjres t.hemto make use of mathematics. However. the resultsofthestudysuggest.thattherearemanyothersofwhomquitethe reverse istrue. 20The extent to whichtheneed to undertake even an apparently simple and straightforward piece of malhematjcs could induce feelings of anxiety,help-lessness,fearand evenguiltinsome of those interviewedwas,perhaps,the moststrikingfeatureofthestudy.Noconnectionwasfoundbetweenthe extenttowhichthoseinterviewedusedmathematicsandthelevelof their educationalthere were science graduateswhoclaimedto use no arithmelic and others with no qualifications who displayed ahigh level of arithmetical competence.Nor did there appear to be any connection bel ween mathematical competence andoccupational group;people of widelyvaried mathematicalcompetencewerefoundineachofthefiveoccupational groups.The estimateswhichthosewhowere interviewed gaveof theirown mathematicalcompetence didnotrelate closelytotheextenttowhichthey made use of mathematics. There were some who said that they managed very wellbut who appeared to avoid numbers and others who, although apparen'-tlyhighly competent inthe conduct of their everyday affairs.wereveryhesi-tantaboulclajmingmathematicalskill.Therewerealsosomewho.while apparently ableto perform adequately in the situations whichtheynormaiJy encountered, admitted that theywere working atthe limit of their mathema-tical competence and were anxious lest anything more complicated should be required of them. 21The feelings 0rguilt to which we referred earlier appea redto be especially markedamong those whose academic qualificatjonswerehighand who. in consequence of this,fell that they 'ought' to have a confident understanding of mathematics,even though thiswas not tbecase.FurtJ1ermore,theywere awarethatothers,towhomitwasevidentthatlheywerewell-qualifiedin generalterms.[Ookitforgrantedthattheywouldbemathematically competent."Peopleassumeyou' regoodatmathsi r you'regoodatother things."Thosewhowerenot academicallywell-qualifieddidnotappearto feelguiltyinthesameway.SomeartsgraduateswhohadgainedO-Ievel passesinmathematicswereneverthelesssoawareofalackofconfident understanding of the subject that their career choiceswere seriously reduced as aresultof their determination toavoid mathematics. 22There wasanother group consisting of thosewho,althoughabletoper-2The malhemalicaJneeds of adulllire formthecakulationswhichtheynormallyrequired,fel,asenseof inadequacybecausetheywereawarethattheydidnOlusewhalI heycon-sidered 10 be [he' proper' method; in other words, they did not make use of t.he standardmethodsforsettingoutwrittencalculationswhicharenormally taughtInI he classroom.I n fact, [he study, whilst revealing a very wide variety of approaches to the questions which were asked, also found thatmany indi-vid uals appeared LOrave only one method of tackling a given problem.I r this failed, or if the calculation involvedbecame too cumbersome, they lacked the ability andconfidence to attempt a different approach.Nor,insome cases, weretheyeventhattheremightbealternativeandpossiblymore straightforwardmethodswhichcouldbeused. 23Again, just as some feltthatthere was always a(proper' method, some fel( that there should alwaysbe an exactanswer to questions involving mathema-ticsandsofoundthemselvesindifficultieswhenitbecamenecessaryto approximate or toround 0ff a result. "I get lost on long sums and never know what to do witbthe' leftovers'.' IMy mind boggles atthe aritbmel icinesti-mation. " 24Failureandconsequentdislikeof mathematicswasoftenascribedtoa specificcausewbenyoung.Suchcausesincludedchangeofteacherorof school, absence through illness,being promoted to ahigher class and ing leftbehind,havinganirascibleorunsympatheticteacherwhofailedto resolve difficulties,Oreven on thepart of parents,usually fathers.Criticismby husbands Orwivesorbyother members of thefamily, especially comment about slowness or the need to use pencil and paper instead of performing a calculation memally, also eroded confidence and contributed todecreasinguseofmathematics."I'm afraidIhavetowriteitdown.My brother can doitinhishead."('Myhusbandsays I'm stupid." 25The report alsorefers [0'''those who dreaded what they saw as the innate characteristics of learning mathema[ics such as accuracy and speed, as weJIas tbe traditional requiremem to show allworking nearly.This recalled the long buriedanxietiescausedbythepupil'sarrivingatananswerbyamental methodandbeingrequiredtoproduceawrittensolutiondemonstratinga methodwhichhadnotbeenused".Thisperceptionof mathematics,and especially arithmetic. as something whichissupposed to lead to exact answers by the use of proper methods seemed to be quite common despite the fact that thenumberswhichariseineverydaylifeveryoftenneedtoberOUIldedor approximated insomeway, 26Anotherfeaturerevealedbytheswdywasawidespreadinabilityto understandpercentages.Manyof thoseinterviewedsaidthattheydidnot understandthem or neverusedthem.(, r mhopeless atpercentages really." Others who said that they were ableto calculate 10per cent andperhaps,but withgreaterdifficulty,15percentindicated[haltheywouldnot be ableto cope with 8 per cent or12 per cent.Nor didthey seem tohave realised that the )ntroductionof adecimalisedcurrencyhadmadeiteasiertoevaluateper-centages of sums of money thanhadpreviously beenthe case. It is clear that Themathematical needs of adult life 2The malhematicai needsof adultlife9 politicians,administrators,businessmen,journalistsandadvertisersall assume that the public at large willunderstand the many statements which are made which express comparisons in percentage terms. The study suggests that this isveryfarfrombeing the case and the results of the Gall up Poll enquiry confirm this.Eventhoughthosewhosaythattheydonot understandper-centagesprobably realise that,for example,a12per cent increase isgreater than one of 10 per cent. it seems that they would certainly not be able to work out the actual size of the difference in relation totheir own salary or wage. 27Afurther question revealed that one statistic which isnormally expressed in percentage terms,that of rate of inflation, iseven more widely misunder-stood,withmanythinkingthatafallin therateof inflationoughttobe associated withan overall drop in the level of prices(even though they often did not think that this was likely to happen) rather than a lessening of the rate at whichprices were increasing. 28The reading of chartsandtimetableswasanotherareawhichpresented difficultytomany of those interviewedbutthere wasa muchhigher rate of successon aquestion which testedability to read amap andto estimate the distancebetweentwopointsonit.Understandingoftherelativesizesof imperial and metric measures in common use wasnot widespread. 29About 70 per cent of those interviewed in the first sample had accessto a calculator if they required it but one-third of them said that they neverused one.Some of the latter admitted that they did not know howtouse a calcu-lator and others expressed doubt and distrust.'( I never use it because of the risk of major mistakes." There were also those who maintained that" brains are better"or that "they make you lazy". Some who had tried to use a calcu-latorhadbeendiscouragedbythelargenumberoffigureswhichhad appearedafter the decimalpoint,fbr instance when dividingby3,andhad lacked confidence topersevereand to discoverhowto interpret the answers they had obtained. On the other hand there were some, whose computational skills were weak,for whom the use of a calculator made all the difference."I know the theory but without the calculator 1 couldn't do it." 30Manystrategieswereencounteredforcopingwiththemathematical demands of everyday life. These included always buying 10 worth of petrol, always paying by cheque. always taking far more money than was likely to be neededwhen goingshoppingsoastobecertain of beingabletopaybills without embarrassment.There wasfrequent reliance onhusbands, wivesOr children tocheckandpaybills,tomeasure or toreadtimetables;and also reliance on past experience. Sadly, it was also clear that lack of mathematical ability had prevented some people from applying for jobs or fromfollowing courses of training which they would otherwise have wished to undertake. In thissense,they had been unable tocope. 31What, then, are the mathematkal needs of adult life? In the first plaee., it is clear that there ishardly any piece of mathematics whkh everyone uses.For 10 Numeracy * 151018.Areportof (heCentral AdvisoryCouncilrorEducation (England).HMSO1959. 2The mathematjcalneeds of adultlire example,those whodonottravelbybusortrainprobablyhave noneedto consult timetables;those who do not ddve acarhave no needtobuy petrol; those who do not have meals in hotels or restaurants have no need to be able to calculate aservice charge.The study showsthatsomepeople appearto use practicallynomathematicsbecausetheyhaveorganisedtheir livessoasto avoid its use or so as to make use of the mathematical skills of others. There are, however, very fewpeople who do not at some time need to be able to read numbers,tocount,totellthetimeortoundertakeaminimalamountof shopping.This,perhaps,representsaminimum listbutitisapparentthat many of those who possess only thisminimum of mathematical skill, as well as some whose attainment isagood deal greater,frequently experience feel-ings of stress, inadequacy or helplessness, eventhoughthey may have found methods of coping with their everydayneeds. 32Therefore. whilst realising that there are some who willnotachieve allof them, we would include among tbe mathematical needs of adult life tbe ability to read numbers and to count, to telllhe time, to pay for purchases and to give change, to weigb and measure, to understand stTaightfonvard timetables and sjmple graphs and charts. and to carry out any necessary calculations a s s o c i ~ atedwithtbese. Therearemanywho,becauseof therequirements of their employment,theirhobbies or their own interest in mathematics. are able to achieve a great deal more than this.Some develop very specialised skills;for exampJe, of the kind which are frequently exhibited by those who play darts ormake useof bettingshops.Howevert webelievethattbosewhoteacb matbematics in schools sbould do alltbat ispossible to cnable their pupils to include as part of tbeir mathematical knowledge those abilities which we have listed. 33Webelievetoothat,asanecessary accompaniment tothe listwhichwe bavegiven,itisimportanttohavethefeelingfornumberwhichpermits sensibleestimationandapproximation-ofthekind.forinstance,which makes it possible to realise that the cost of3 items at 9Sp each will be a little less than3-andwhicbenablesstraightforwardmentalcalculationtobe aceo mp lisbed. 34Mostimportant of allistheneedtohave sufficient confidencetomake effective use of whatever mathematical skill and undefSta-;tding is possessed, whether tbis be little or much. 35Thewords4numeracy'and'numerate'occurinmanyofthewritten submissionswhichwehavereceived.Inthelightof ourdiscussioninthe preceding paragraphs we believe that it is appropriate to ask whether or not an ability to cope confidently withthemathematicalneedsof adultlife,aswe havedescribedthem.shouldbethoughttobesufficienttoconstitute 'numeracy' . 36Theconceptofnumeracyandtheworditselfwereintroducedinthe CrowtherReport" publishedin1959.In a section devotedtothe curriculum 2Themathematicaloeeds of adultlife11 of the sixth form,'numerate' is defined as "a word to represent the mirror im-age of literacy". Later paragraphs in the report make clear that this definition isintended toimplyaquite sophisticatedlevelof mathemat.icalunderstan-ding."Ontheonehand... anunderstandingof thescientificapproachto the study of phenomena-observation, hypothesis, experiment, verification . On the other hand...the need in the modern world to think quantitatively, to reaJise how far our problems are problems of degree even when they appear as problems of kind. Statistical ignorance and statistical fallaciesare quite as widespread and quite as dangerous as the logical fallacieswhich come under the heading of illiteracy.""However able aboy may be.. .if his numeracy has stopped short at the usual fifth form level,he is in danger of relapsing into innumeracy. " 37Innoneofthesubmissionswhichwehavereceivedarethewords 'oumeracy'orInumerate'usedinthesenseinwhichtheCrowtherReport defines them.Indeed, weare inno doubt that the words, ascommonly used, have changedtheir meaning considerably inthe last twenty years.The asso-ciationwithscienceisno10ngerpresentandthelevelofmathematical understanding towhich the words refer is muchLower.This change isreflec-ted in the various dictionary definitions of these words. Whereas the Oxford EnglishDictionarydefines'numerate'tomean"acquainted withlhebasic principlesof mathematicsaDdscience",CollinsConciseDictionarygives "able to performbasic arithmetic operations". 38Thesecondof thesedefinitionsreflectsthemeaning whichseemstobe intendedbymostof thosewhohaveusedthewordinsubmissionstous. However.if weareto equatenumeracy withanability to cope confidently with the mathematical demands of adult life, thisdefinition istoo restricted because it refers only to ability to perform basic arithmetic operations and not toabilitytomakeuseofthemwitbconfidenceinpracticaleveryday situa tions. 39Wewouldwishtheword4numerate'toimplylbepossessionoftwo attributes. The first of these is an 4at-homeness' withnumbers and an ability tomakeuseof mathematicalskillswhichenablesanindividualto cope with thepracticalmathematicaldemandsof hiseveryday life.Thesecondisan a bility tohave some appreciation and understanding of information whichis presented in mathematical terms, for instance in graphs, charts ortab1cs or by reference to percentage increase or decrease. Taken together, these imply that a numerate person should be expected to be able to appreciate and understand some of the waysinwhichmathematics canbeusedasameans of commu-nication, as we have described inthe previous chapter. We are, in fact! asking formorethanisincludedinthedefinitioninCollinsbutnot asmuchasis implied by that in the Oxford dictionary-though it will,of course, bethecase that anyone who fulfils the latter criteria will be numerate. Our concern is that those who set out to make their pupils 40umeratel shou.ld pay attention to tbe wider aspects of numeracy and Dot be content merely to d e ~ e l o pthe skiDs of computation. 12 Viewsexpressed beforetheCommittee wassetup HouseofCommons.Tentl) ReportrromtheExpend..irure Commiltee.ThearfainmenfS0/ the school leaver.HMSOJ 977. 3Themathematicalneedsof employment 40It isclear from thereport of theParliamentary Expenditure Committee*' towhichwereferredintheintroductionthatthevolumeofcomplaints wruchseemedtobecorningfromemployersaboutlackof mathematical competenceon[hepartofsomeschoolleaverswasoneoftheprincipal reasonsforitsrecommendationthatourInquiryshouldbesetup. 41We believe thatthese complaints started to come tothefore in 1973and 1974whenanumberofarticlesandletterswhichwerehighlycriticalof mathematicsteachinginschools,andof 'modern mathematics'inparticu-lar.appearedinSkill,anews-sheetpu bJishedbytheEngineeri ngInd ustry TrainingBoardforgrouptrainingschemesintheengineeringindustry. Thesecomplaintswere[oHowedin1975and1976byarticlesandlettersin Blueprint,thenewspaperoftheEngineeringlnd ustryTrainingBoard, whichalsoexpresseddissatisfactionwiththemathematicalauainmelllof someen t rantstothej nd u stry.Morewid espreadcr it icismappearedin newspaperarticlesandcorrespondence columnsduring[heseyears. 42InhisspeechmadeatRuskinCollege,OxfordinOctober1976,Mr JamesCallaghan,at thattimePrimeMinister.said: Iamconcernedonmyjourneystofindcomplainlsfromi[]dustrythatnew recruitsfromlheschools sometimes do not have thebasicLOOJS[0 dothejob that isrequired ...... Thereisconcernaboutthesrandardsofnumeracyof school leavers.[stherenora caseforaprofessionalreviewof rbemathematicsneeded byinduslry a[differentlevels?Towhat extent arethese deficienciesthe resultof insufficientco-ordinationbelweenschoolsandindustry?Indeedhowmuchof thecriticismaboutbasicskillsandattitudesisdueLOindustry'sown shorlcom-ingsralherlhantotheeducationalsystem? 43InwrittenevidencetotheParliamentaryExpenditureCommHtee,the Confederationof BritishIndustry(CB1)stated: Employersarebecomingincreasinglyconcernedthatmanysehoolleavers, particularly(hoseleavingatlhestatutoryagehavenotacquiredaminimum acceptableslandardinthefundamentalskillsinvOlvedinreading,writing, arithmeticandcommunicarion.Thisshowsupintheresultsofnearlyevery educationalenquirymadeamongstlheeBImembership,andisbackedupby continuingevidencefromtrainingofficersinindustryandfurthereducation lecturersrhat youngpeople at16+cannotpasssimple teslSinmathematics and req uireremedialtuitionbeforetrainingandfurthereduca.tioncoursescanbe started. Employers'views expressedtothe C-ommiUee )Themathematicalneedsof emptoymenc13 In oral evidence to the Expenditure Committee aCBr representative stated: Mathematics, Ithink -or arithmetic, which isreallythe primary concern rather than mathematics themselves-is the one area which is reaiJy brought up every time as aproblem. It seemsthat industry's needs are greater in this respect than almost any other. This is the way, certainly, in which shortfall in the education of children makesitself most manifestimmediately[0an employer. \VritrenevidencetotheExpenditureCommitteefromtheEngineering Industry Training Board (EITB) stated: The Engineeri ngI ndustry Training Board, over .the last two years, received from it.sindustryincreasingcriticism,withsupportingevidence,ofthelevelof attainmem, particularly in arithmetical skitls, of schoolleavers offering themselves forcraftandtechniciantraining .... ..Intheviewof the Engineering Industry Training Board theindustry needs ahigher level of attainment in basic mathematics amongrecruits thanitisnowgettingandbelievesthat, withcloserco-operation betweenschoolandindustry,childrencanwhilestillatschoolbemotivatedto aC,hieve(his . . .... Mathematicsis,however,not.simplyaquestionofbasic manipulative skills. An understanding of the concepts is also neededand these are better taught. bywith double-subject Alevelmalhemalics 19151979 26.932./ 17.518.f 35.032.d 2.33.0 1.00.9 7.05.3 3.13.3 0.60.4 6.64.6 100100 Souree:Univers.itie.s St.atisticalRecord IDegree courses includedineach subject group aregiveninAppendixI.pa.ra&raphA2L 524The mathemaricalneedsof further and higher education 179About80percentof entrantswirbadouble-subjectqualificationin mathematics a(A-levelread engineering and technology, physical sciences Or mathematicalstudies.Figure4illustratesthewayinwhichenlranlSwitha double-subjectqualificalionweredistriburedbetweenthesethreeareasof study in1973and1979.(See alsoAppendixI,Table 30). Figure4Numbers 0/ universityentrantsinEngland andWalesadmitted on thebasisof A-levels.Subjectchoicesandnumberswithdouble-subject A-level quaNftcations inmathematics:1973and 1979 8000 Numberstakingsubject Numberswithdoublema1hemotics 6,000 4000 2.000 737973797379 S o u r c e ~Universit ies5t at iSl icalRecord .. Mathema ticalstud iesincJu des degreecourses)nmathematics, statistics,compulerscience. combinationsof thcseandalsoa Yarie!yof coorseswhichcombine mathematicswithsubjectsother Ihanthese, of informalionpTOyidedbyUSR aboutfirsldegrees.awardedin 1979withinthefieldof mathema-ticalstudiesshowsthaIaboul6.5 per cent wereinma(hema(ics only, about25per(emincomputer scienee,eitherwhollyorinparI, andabout 8 per een(inslatisrics. 4The mathemalicalneedsof funher andhigher educationS3 Degree courses inmathematical studies 180We wish to draw attention to the drop inthe proportion of those reacting mathematical studies* atuniversities in England and Wales who have a dou-ble-subjectqualificationand to its implications . ln1973the proportion was almost 80 per cent; in 1979 (his had dropped to 55 per cent. Although there are someuniversitiesatwhichitisstillthecasethataveryhighproportionof thosereadingdegreesinmathematicshavetakenthedoublesubjectat A-level,ourownenquirieshaveestablishedthatthereareothersatwhich substantiallylessthanhalfofthosereadingmathematicshaveadouble-subject qualification. We believe that this information may come as a surprise to many people inbothuniversitiesand schoo1s.It is not within our terms of referenceto comment on itsimplicationsforthose who teachmathematical studies in universities; the implications for those who teach in schools are very great. 181It isverycommonlysupposedthatitisalmostessentialtohavetaken double-subject mathematics at A-level inorder to read mathematics success-fullyatuniversity.However,it is veryimportantIbat those wboleachmath-ematicsiosixtbformsandthosewhoadvisepupilsabouttheirchoiceof degreecourse sbouldreaJisethattherearenowuniversitiesinwhichmore thanhalf of thosereadingmatbematics are doing sofromabasis of single. subjectA-Jevel.11followsthattheyshouldootdissuadepupilswhobave taken only tbe single subject at A-level from applyjng to read degree courses in mathematics.We see no likelihood that the demand for mathematics gradu-ateswilldecrease-indeed,webelievethatthedemandwilJcontinueLo grow-andthosewhoseinterestsandabilitieslieinthisfieldneedevery encouragementtostudymathematicsat degree level.Aswepointedout in paragraph173,it does not followthat thosewhohave taken only thesingle subject are necessariJy less able at mathematics than those who have taken the double subject and so Jessfittedto embark on amathematics degree course. There seems no doubt that at most universities they wiJIbe increasingly likely tofindthemselvesinthe company of others who are similarly qualified. Degree courses inengineering and technology 182Aknowledge of mathematics is essential for the study of engineering and of most other technological subjects . We drew attention in paragraph 172 to the 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 has been much greater thanthe increase in total entry to all other mathema tics and science courses, which has risenbyonJy 20per cent.Furthermore,despite anoverall drop during this period inthetotaJnumber of entrants to universitiesinEngland andWales witha qualificationinmathematics,therehasbeenan increaseintheproportionof theseentrants whohave chosentoreadengi-neering and technology,and a1soasmall absolute increase in their numbers. TablesAandCshow[hatdegree courses inengineering andtechnology are attractinganincreasingproportionof universityentrantswithanA-level 54 Themathematical requirementsof professionalbodies 4The mathematicalneedsof further and highereducation qualification inmathematics and,inparticular, of (hose with double-subject mathematics.Theproportionof entrantswithdouble-subject mathematics is,however,decreasing bothwithin engineering and technology courses and overall. 183Although notdirectly within our terms of reference,we have given some attention to the mathematicalrequirements of professional bodies.Many of those engagedinprofessional activitiesseekmembership of the appropriate professional institution or association.Insome casesmembership of such a body is a necessary qualification for professional advancement; in other cases membership,althoughnotessentialforcareerpurposes,providesoppor-tunitytokeepabreastofcurrentdevelopmentsbyreadingpublications. attendingmeetingsandtakingpartintheworkof committees.Mostinsti-tutions conduct their own lexaminations, commonly in two or three parts, for admissionto membership.,whichisusuallyoffered at morethan one grade. Thepossessionofanappropriateacademicqualificationoftensecures exemptionfromsomeor;allof theseexaminationsbut admissiontohigher gradesof membershipnormallyrequiresevidenceof relevantprofessional experience. 184Anumber of the professionalbodies who have written to ushave stated themathematicalrequirementsfordirectentrytotheirvariousgradesand have also supplied details of their own examinations. When enlry is at gradu-ate level, itisusually assumed that any necessary mathematics will have been covered either at school or during the degree course and nofurther mathema-tical requirement js stipulated. However, one exception to this isthe Institute ofActuarieswhosefinal,examinationsrequireaconsiderableextensionof mathematical and statisticalknowledge and its application.When entry to a professionalbodyisatlowerlevels,anymathematicalrequirementisnor-mallystatedintermsof successatA- orO-Ievel;anyfurthermathematics whichisrequired isthen included within subsequent professional study. 185Almost alltheprofessionalbodieswhosubmitted evidence stressedthe importance of being able 1.0apply computational skills confidently inavar-ietyof ways .Theseincludeaccuracyandspeedinmentalcalculaljonand abilityto checkt hereasonablenessof answers;insome casesextendedand complex calculations are necessary_Specific calculations identified bybodies 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 manyreferences to [he needtobe able tointerpret data with understanding. 186Mostinstitutionstakeforgrantedthemathematlcalfoundationpro-videdbyanentranespTieviousstudy .Anymathematicsincludedwithin professionalexaminationsisusuallylimiledeithertotopicsof aspeciaJist nature which are unlikely to have been studied before entry to the profession Ortoappl1cationsof mathematics in unfamiliar contexts.Thisisespecially 4The mathematical needs or funher andhigher education55 true of professional bodies whose members are concerned withbusiness and commerce. The examinations of these bodies frequently include applications of statistics, and to alesser extent techruQues of operational research,which areusedwithintheparticularprofess-ion.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 factthat at thepre-senttimefewschoolleavers willhave studiedthe subjectto any depth. 56 Part 2 ;"Abrief summaryofrheReview maybepurchasedfrom(heShell C ~ n t r eforMathematicalEduca-tion,UniversityofNottingham; see alsoparagraph756. Attainment in mathema tics 5Mathematics in schools 187In the second part of our report wediscuss the teaching and learning of mathematicsinschoolsaswellasmethodswhichareusedtoassess attainment.Before turning to particular aspectssuchas mathematics in the primary and secondary years we consider some matters which are fundamen-tal to the teaching of mathematics topupils of all ages, and also certain mat-terswhicharise as a consequence of the discussion in earlier chapters, of the submissionswhichwehave recejvedand of our own experience.In order to provideabackgroundwestartbydrawingattentiontothelevelsof attainment in mathematics which are to be expected of school teavers, so that readers may bear in mindthe proportions of the school population to which thedifferentpartsofourdiscussionrelate;wealsoconsidertheattitudes tOwardsmat.hematicswhich pupilsdevelopduringtheirschooldays andthe mathematical attainmentof girls. 188In this part of our report we draw on Areview ojresearch inmathema-tical education which summarises the results of the study carried out for our Committee under the direction of Dr A Bell of the University of Nottingham and Dr ABishop of the University of Cambridge. For the sake of brevity. we shallhenceforward refer to it asthe Review oj research.'" 189We believe tbat tbere is widespread misunderstanding among the public at large as totbe levels of attainment in matbematics which are to be expected among scboolleavers. At the present time about a quarter of the pupils in each yeargroup achieve O-levelgrade A,B or C or CSE grade 1;about a further two- fifths achieve CSE grade 2,3. 4 or 5:the remain der , amou n ting to almost one-thjrdof the yeargroup, leaveschool without any mathematical qualifi-cationinO-levelor CSE.Thesefiguresare not surprising;theyreflectthe proportionsoftheschoolpopUlationforwhomO-IevelandCSEexami-nations are intended and have been designed. At a higher level, between 5 and 6 per cent of the pupils in each year group achieve an A-level qualification in mathematics; about Ipupil in 200 reads a degree course in mathematIcal stu-dies. 190Figure5illustratesinapproximatediagrammaticfonnthe'mathema-ticalattainmentprofile'of thoseinEngland andWaleswho leftschoolin 1979,andof thoseof schoolagewhocompleted A-levelcoursesinFEor tertiary collegesin that year.It isbasedon figurescollected in the annual10 per cent survey of schoolleavers (see AppendixI,paragraph A3) and on an Comparisonof exami-nationresul[sin English and mathematics SMathemaeics in schooh Figure5CMathemalical attainment profile' for leavers in1979 degreein mill hemal icelStudA- level (anvgrJde) a-levelgradeA. B.CQrCSE1 O-levelorCSE lanyI -Tot.slno ofleavers 57 26% 68% 100% estimate of the numbers completing A-level courses in mathematics inFE and tertiarycolleges,sinceinformationavailableabouttheexaminationper-formance of students inthese colleges does notidentify separately those who are of sch 001age_ 191The num ber of pupils who have been studying marhemarics at A-level in schoolsandsixthformcollegesinEnglandhasincreasedsteadilyinrecent years both in absolute terms, as apercentage of all pupils, as a percentage of all pupils taking A-levelcourses.In the schoolyear1973 -74, some 43per cent of boys and17per cent of girls taking A-level courses inthefirstyear of thesixthformwerestudyingA-levelmathematics.Intheschoolyear 1979- 80,these figures had risen to almost 5 Jper cent of boys (approximately 41000 and to23per cent of girls(approximately 17000). 192It isnot possibleto obtain comparable statistics for those of sixth-form age who are taking A-level courses in FE and tertiary colleges,but we have no reasontobelieve thatthe picture wouldbesignificantly different. 193In order to provide more detailedinformation about mathematical per-formance in eSE, O-Ievel and A-level,the DES has at our request analysedin avarietyofwaystheinformationaboutresultsobtainedintheseexami-nations which was supplied in 1977.1978 andt 979 by schools inEngland and Wales aspart of the annual10 per cent survey of school1eavers inboth main-tained and independent sectors. This has provided more detailed information aboutexaminationperformanceinmathematicsthanhashithertobeen available. Because the survey relates to schoolleavers in a given year who may be aged16,17,18 or, very occasionally, 19, the information does not relate to a complete year group. Nevertheless,because the patterns for 1977,1978 and 1979are very similar,webelievethat thepicture theyprovide isunJikelyto differ significant ly from that which would emerge if it were possible to obtain information relatingtoa complete yeargroup_ 58 % TO 20 30 40 ABC orl Dor2 1977 Engliso ABC orI 00r2 1978 Enteredexamination butwasungraded. :5Mathematics in~ c h o o 1 s 194Some of thosewhohavewrittentoushave drawnattentiontofigures published each year in DES Sialisticsojeducalion Vol2 which show that the proportion of pupUs who achieve O-level grade A, B or Cor CSE grade 1 in English at some stage during their school career ismuch higherthan that of thosewhoachieve thesegrades inmathematics.Insome submissionsithas beensuggestedthatthestandardrequiredinmatbematicsexaminationsis thereforetoohigh,Inconsequencewehavealsoobtainedinformation relating to CSE and O-levelresultsin Englishforthesethree years, 195Figure 6illustrates approximately the O-leveland CSE performances in English andmathematics or arithmetic of those leaving schoolinthesethree years.The figuresonwhichitisbasedaresetoutindetailinAppendixI, Tables] 1 and12;paragraphsA 7andA9ofthisAppendixdescribethe procedurewhichhasbeenusedtoeosurethat thosewhohavetakenboth O-level and CSE in the same subject or, forinstance, both mathematics and arithmetic are only includedonce.The letters in the columns of Figure 6 refer to O-level grades Ato E, the numbers to CSE gradesIto 5,It isreasonable to Figure6Proportionsoj pupilsawardedO-LevelgradesA10EandCSE grades 1 to5 ;nEnglish and MathematicslAn'thmetic ABC or I Oor2 1979 Didnotenter examination. M athemat ic-s/ Arithmet ic ABC orl 1977 ABC or1 1978 ABC orl 1979 % 10 20 "Aspectsof !:.ecrmdaryeduCOllon IJI/If,:/and.A byHM I orSchools.H ,\;ISO ,g; therefore, inIhefirstinstancelead to agreat increase in the number of A-level syllabuses since the two sets of examinations servedeighlboards.However,induecoursesevenof theGCEboardsin-troducedtheirownmodernsyllabusesforsingleanddoublesubject andthe toralnumberofA-levelsyUabusesincreasedsharply.Webelievethatthe number of syllabuses reachedilS highestpoim inthe mid-1970s. 582Thenumberof differentsyllabusesisnowdecreasing .SyUabuschanges during thelasttenyearshave lessened the differencesbetween the conten[ of 'modern'and'traditional'syllabusesandmanyfeelthatitisnolongerap-propriate to attempt to distinguish between them. We have noted with interest that Done of thegroups whichwerecommissionedsome sixyears agotopro-posespecimensyllabusesandexaminationpapersinmathematicsaspartof thefeasibilitystudiesfortheN&Fproposalsforexaminational18 +dif-ferentiatedbetweentraditionalandmodernmathematics.WesupportThe view[hatthedjstinctionshouldnolongerbemaintained.ItshouldIhenbe possiblefor the number of A-levelsyllabuses to be reducedfurther. 583However,differences in examinations arenotonly a question of syllabus content!butarealsoconcernedwiththeapproachtotheteachingof machematicswhichthe syUabusimplies andthekindof papers andquestions bymeansofwhichitisexamined.Thesefactorscanbeveryrelevant[0a Double-subject mathematics IIMathematics inthe sixthform 177 school'schoiceofsyllabusandwebelieveitisdesirablethatsuchchoice should exist.[ndeed,the difference between some'modern'and'traditional' syllabuses isasmuchin the approach whichisusedasinthe topicswhichare covered.If,aswehopewillbethecase,agreementisreachedonacoreof mathematicsrobeincludedinallA-levelsyUabuses,existingsyllabuses",,'ill presumablyberevisedaourCt';DES100/.SUIVL')' ,SomtCSEboardsoffer asepara.leexaminationinArithmetic;thelable shows1he combined resu liSforMalhemaLics andAri! h metic. 250AppelldixISlatislical information A 7Some pupils attemptbOlhO-level and CSE examinations in mathematics, eitherinthesameyearorindifferentyears.Anaggregationof [heO-level results given in Table 3 andthe CSE results giveninTable 4 does nottherefore giveapictureof rheoverallsimation.Table5 amalgamates[heO-leveland CSEresu1tsobtainedby schoolleaversinsuchawaythateachpupilappears only once,accordingtohis'best'grade.In the caseof Ie averswhose highest grades included both O-Ievel grade A, B, Cor O-level pass and CSE grade 1 we haveaccountof theO-Ievelresultanddiscountedthe CSEresult.We havegroupedtogetherthose whose'best'grade wa9,8 Marhemalicalstudies31 ,8 Medicruand denlal1 Bjological 22 Other sciencesS3 Businessstudies2,4 Geograph yI I Other snbjt'Cl53'9 whoweregraduates in mathemalical studies5J ,6 S 1J 127 o 17 38 8 7 33 256 49.6 74 115 420 4 47 75 42 18 71 866 48. .5 6 24 lJ4 I 39 11 IS J 8 71 342 39,2 9S 151 440 I 56 73 41 27 9\ 975 45.1 \Nomen 6

144 2 39 39 II 17 74 367 392 SO\l(ce: Un) SI31 0.;11C31Jlecord ISeeparagraphA21, Historical8ackgro u nd Appendix 2 By tvtissHBShuard 273 Differences inmathematical performance between girls an.dboys B IPupils inschools are often classifiedaccording to their sex,and discussion of their educational programme may take their sex into account, either direct-lyorbecauseof socialcustom.UntilfairlyrecentlyinBritain,jtwascom-monplaceto discussseparateJy the educationof boys and girls.,including the mathematicscoursestheyshouldfollow,andexpectationsoftheir mathematicaJattainmentwe:redifferent.Inthisappendix\differencesin mathematicalperfonnancearedescribed,andpO$s,iblecausesofthesedif-ferencesarediscussed.Historicalandstatisticalevidence,andthecon-siderablevolumeof researchoneducationaldifferencesbetween'thesexes, are drawnupon.Suggestions are madeformeasures whichmighthelpto im-prove themathematicaJperformance of girls. B2Mathematicsestablisheditselfinthecurriculumofboys'publicand secondary schools inthefirsthalf of thenineteenthcentury,cmdwhengirls' secondary schoolsbegan(Qbe foundedlaterinthe century,rhepioneers of girls'education wishedtointroduce its scudy intotheirschools.Lecturing in 1848,ProfessorF0Mauricediscussedthecurriculumofopened Queen's College;hisremarks about mathematics have often quoted out of contextasabelief('thatwomenstudentswereunlikelytolldvancefarin mathematics" f.However,thebeliefwhichheactuallyexpressedwasthat although"weareawarethatourpupilsarenot.likelytoadvancefarin Mathematics",therewereposltivebenefitswhichgirlswouldgainfromits study,and"the leastbit of knowledge.. .must be good".2.At an 01. her new girls'school,CheJtenham Ladies'College,tvtissDorothea Beale ack.nowledg-edthat although she wjshedtointroduce mathematics intothe curriculum, it wouldspelJfinancialruinforthe school,becauseparents didnotwishtheiI daughters(Qstudyit.Evenarithmeticwassuspect;around1860,afather wrote to Miss Beale, on decidilng to send his daughters to another school,"My dear lady,if mydaughters were goingtobebankers,it wouldbe very wellto teacharithmeticyoudo,but really there isno need"J.However, the open-ing of the Cambridge Local Examinations togirlsin1863 gave impetus to the teachingofarithmeticandmathematicsingirls'secondaryschools .Of tile first25candidatesfromtheNorthLondonCollegiateSchool,10failedin arithmetic;theheadmistress,MjssBuss,washorrified,andtheteachingof arithmetic at once became amatter of extreme importance jn her school J On-lythree years later,girls were doing aswellinarithmetic as in other subjects in the CambridgeLocalExaminations,andwhensubstantialnumbersof gkls' secondaryschoolswerefoundedafter1873,mathematicsbecamearegular 274Appendi.l(2DiJfercncesinmal hematicalDerformance belweengirlsandbo:-,s subjectof (hecurriculum.Atfirstthereweregreatdifficuhiesbecause of tht inadequatesupplyofteachers;thisslowlyimprovedasmorewOmen gradua[ed fromuniversities,but in1912 itwas esrimared thaI.,out.ofsome 90() womenteaching mathemaJLics insecondary schools,only about had Ihemselves studiedasfara5thecalculus.However,theboys'schoolswereno better off,foraboulthe same proporlionof menteachers of mathemalicsin secondary schools hads(Udiedcalculus. I B3InthepubJlcelementary schools,bothboysand girlslearnedarithmetic, but as[heRoyalCommissionon the Elementary Education Actsreponed in 1888,"asthetimeof thegirlsislargelytakenupwithneedlework,the time [hey can giveto arithmeticislessthan !hatwhichcanbegivenbyboys". They thereforerecommendedthalthearithmeticalrequirements of the curriculum should bemodified jnthe ,caseof girls. B4Inthe1912Reportontheteaching of mathematicsinBritain I,MissER Gwatkin,headmistressof QueenMaryHighSchool,Liverpool,notedthat mathemaclcs occupied a goodpositioninthe curriculum of mostgirlsldary schools,butthe pressure causedby the introduction of a wider rangeof su bjectsintothecurricul urnwascausingth ispositiontobequestioned. Among theobjectionstomathematics asanimportantsubjectforgirlswere thatthesubjec[wasunlnlieres{ingtomostgirls,thatirsutilitarianvaJueto them wasnegligible,whichcouldexplainthelackof interest,andthatitsdif-fiCU\lYputa strain on pupils out of aU. proportion to {hebenefit received.Miss Gwatkin presented a dear leaseagainstthese objections.using such arguments asthefollowing; Allgids oughtrogrow up reasonable beings,andmanyor themdon01;allgirls oug.hrlOacquire aknowledge of themeaning of language.anda power of using it accurately,andmany0 r themdonoloMathemaeics,propertytaught,willhelpto bothends.InaUbranchesofMal hemalics.thoughperhapsmoreespeciaUyin Geometry.itisnecessary('0be clear-headed. Womenare said1.0be iuconsist eot.aod Imoreo ...er.to be q l-l i Leuna bJeto recognise theirinconsisrency.There isnoplace where the penalties of inconsistency are more striking (haninthemathematical classroom. Ma(hematics. _.offers unique oppon.unilies 1.0the teacher forrecognising anden-couraging independenttb,ougrl.., _ Gifls,perhaps,needagrea!eTstimulusto in-dependent thoughtthanboys do. Muchharmisdonebyr.atlngtoohighlyagirl'spowertoproduce-somehow, anyhow-the correit.a(Bielefeld.1980. 1 L.H.,Theeff'eclsof sexrolesociali2a.riononr-.-1atbematicsparticipationandachieve-ment,inNf Papers inEducalion and Work No.8 (seeref.18aboye.) Fennema,E ..\l,,'omenandgirlsinma,lhemacic;-Equllyinmalhemariealeducation,Educa-lional Siudies inMarhematics,JO,1979.384- 401. i'Fox,L.H.,Brody,L.,Tobin,D.,(eds).,WomenandIhemathematicalmystique,Johns U.P.,1980. Appendix2Differences inmatht?Jl]aricalperformance bet ..... girlsandboys287 ;j.Becker,J.,Astudy oj difjerential trealmenl ofJemoles ond moles inmalhemOlics dOSSfS.UO-published doctoral Uoi\"ersity of Maryland.1979. :'Sears, 1..DeveJopmenlof GenderRole,inBeach,F.,Se:r::ondbeholliour.JohnWiley.1%5. } 'Good, T. L., Sykes.).N ' IBrophy. J. 1:..,of teaeher.sex and :;tuden[ sex on classroom in-leraclion, Jou.rnal of Educationol Psychology.65,I.1973.74-87. ). Departmentof Education and CurriClJlar difJerencesjor ooys and girlsinmi:red and smgle-Si!.X schools,Education Survey 21,HMSO,J97.5. l 'jLobban., G . inprimary schools,WomV/Speaking.4,1975, ) "Dweck,C.S.,Bush,.5.,SexdifferencesinlearnedhelplessnessI.Deye/opml'rltal Psychology.12,2.1976,147-156. Dweck, C.S.,Davidson.W ..Nelson,S.,Enna,B.,Sexdifferences inlearnedhelplessness,II andIll, Developmental Psychology.14.3.1978, 268-276. 11 Fennema,E . . Allribut!onIheMYandachievementinmathemalics,inYussen,S.R.(cd),The Developmenr oj Reflection, NewYork,AcademicPress.1981. J} Fennema,E., The sexfCIlJor notinMathematics Education, inFennema.E, (ed .), Motn(!mQlics edU{'Q/iOff re.search:Implicationsjor the 8tls.Associatioafor SupervisionandCur-riculumDeo.e\opmenl.D.C.. , lPree.ce,M .. Sturgeon , S., Malnemalit:s and girls,draftrq>ertof BP projcC[,Shef-(ield CityPotytechnic (unpublished),1981. "levine,t.,L.Identificationoj !'ease"'swll)'qualifiedy.>Omendoflotpur.;uemathematical c.areers.repon [0 theNational Scic=nceoundcHjon,USA,1976. J!BerrilJ,R. , Wallis, P .. Sexroles inmalhematics.MVlhemo{;cs in School.5,2. March1976.lS. '"Dornbusch, S., Touy or nOf10Iry.Sfa'l/ord2,2,1974,50-.54. -;BuxLOn.L., Do you panic abuul mathJ'?,Helnemann,1981. 288 Appendix3Listof organisations and individualsWilOhave madesubmissionsto the Committee *denotes those who hove met members oj the Committee jor discussion. MeJ Abramsky MrGAdlam AdvisoryUnirForComlPulerBased Educalion ProfessorM Ailken Mr AGAitken&MrsNEHughes Staffof AlbemarlePrimarySchool, London SW19 ROlary Clubof Aldridge MrsEF Allan lvtIsAAllen Mr JHAllen Amersham (OldTown) VVomen's (nst1tute MsML Andison Mr0Armer Professor AM Arthurs Asb &LacyLimired Mrs S Ashby MrERAshley * AssistaruMasrers &Mist resses Association AssociatedExamini.ngBoardforthe GeneraJCertificateorEducation AssociationforScience Education! Education (Co-ordination) Commit lee Associationof British Chambers of Commerce Association of Career Teachers Associationof Educat.lonal Psychologist s AssociaLionof Gradu31e Ca.reers AdvisoryServices Association of HeadmistJressesof Preparatory Schools t: Associa tedLancash i reSchools Examining Board Associationof PolytechniCTeachers Associationof Teachers of Domestic ScienceLimired Associationof Teachers0fGeology Associationof Teachers of Mathematics Associationof UniversityTeachers MI R Atherton MrGM Austin MissME Austin Avery Hill College MrJK Backhouse Mr CWBaker M.rD Ball Rotary Clubof Bangor Banking InformationService MrB Banks *Barbers of Fulharn MIJKBarnes BarneLLEA RotaryOu b0fBarnelandEasl Barnet Rotary Clu b of Bamstaple MissJulia Barrete Barton Conduits Limited BalhCollegeofHigherEducaLIon Rotary Club of Bath University of Bath,School of Education MrEBathgate Miss] S Batty Professor Dr HBauersfeld MrS R Beaumon t Mr TBeck BedfordCollege of Higher Education ROlary Club of Bedford Bed fordshireLEA Mr KMBed well Appendix 3Lisl of organisations and individual.s who have made submissionstothe289 Committee "'DrA\V Bell MrJ ABell M.rB DBennet L MeDHBennet t !\1r DBent BerkshireLEA Miss BJ Berry Rotary Club of Beverley 1 Bibby & Sons Limited MrR GBiddlecombe Bifurcated Engineering Limited *Miss EBiggs Rotary Club of Billingham Mr D Bird Ms MHBird Mr J G Birkett City of BirminghamLEA Rotary Club of Binningham Bishop Bunan College of Agric u It u re Bishop Grosseteste CoUege Rotary Clubof Bishop's Stortford *Dr AJBishop Ro[ary Club of Blackpoal BoltonLEA Mr CLBoltz Boots Company Limited *Mr 1 \V GBoucher Rotary Club of Bournemouth Mrs CBowler Boxfoldia Ljmiled Mrs V R Bradbury BradfordCoUege City of Bradford LEA UniversityofBrad ford.Under-graduateSchoolofStudiesin Mathematical Sciences RotaryClubof Braintree &Bocking MrF TBrawn and Mr GJohns Mr S EBray Mrs SBrennard BrentLEA Bretton HaJICollege BrightonPolytechnic Rotary Club of Brighton UniversityofBristol,Schoolof Education BristolPolytechnicFacultyof Education Rotary Clu bof Bristol BritishAirports Authority BritishAirways BritishAssociationforEarlyChild-hood Education BritishBroadcastingCorporation, BBC Education BritishBroadcastingCorporation, Open University Prodllctions BritishCouncil BritishPetroleum Company Limited British Rail BritishSocietyfortheHistoryof Mathematics British Society forthe Psychology of Learning Mathematics British SteelCorporation lvir WR Broderick *W Brooks and Sons Bromley LEA tv1r1 Brown *Mrs MLBrown lVtrR VGBrown Professor MBruckheimer BruneI University Departmen t 0f Education Rotary Club of Buckingham BuckinghamshireCollegeofHigher Education Buckinghamshire LEA Professor HBurkhardt Bury LEA BusinessEducation Council Mr LGBuxton C &A Modes Mr JCable *Mr J Cain Calderdale LEA CambridgeInstituteofEducation, Academic Board Cambridgeshire LEA University of Cambridge. Departmentof Education UniversityofCambridge,General Board of the Faculties Universityof Cambridge Local Examinations Syndicate 290Appendix)Lis!of organisations a_ndindi'Jiduals whohavemade submissionsto Ihe Commil(ee UniversityCollege,Cardiff, Departmentof Education Careers Service Advisory Council for \-Vales Mr J B Cannel Carpel Industry Training Board MrsJ E Carrick CentreforScienceEducation, ChelseaCollege Centre for StatisticalEducation Ceramics,Glass&MineralProducts Industry Training Board MrsMChadwick Mr B R Chapman CharteredInstituteofPublic Finance andAccountancy CharteredInstituteofTransport Rotary Club of Chatham ChelmerInstitUleofHigherEduca-tion,FacultyofEducation,Arr.s and Humanities Rotary Club of Chelmsford ROlary Club of Cheltenham ChemicalandAlliedProducts Industry Training Board Chemical Society Cheshire LEA *Rorary Club of Chester Staffof{heMathematicsDepart-mem,ChippingSad burySchool, Bristol Christ's CoUege.Livel1l001 City andGuildsofLondonfnslitute CiryUniversilY CivilService Commission CommercialUnionAssurance CompanyLimiled Committee forGirls and Mathematics CommineeofHeadsofUniversity GeologicalDepartments Comrninee of Professors of Statistics CommitteeofVice-Chancellorsand PrincipalsoftheUniversities0 r theUnitedKingdom *Confederation of BritishIndustry "Confederationof BritishIndustry (WaJes) Con federationfortheAdvancement of SlaleEducat ion Conference of Professors of Applied Malbematics Construc{ion Industry Training Board Professor 0E Conway MrsF CoO\\tay MrE CCooper MrsJ Cooper DrM GCooper ROLaryClub of Corby MrM L Cornelius CornwaULEA Mr G8Corston COLlonandAlliedTextilesIndustry Training Board Councilfor EduCmploymen,48 Retaill(3dt' crilici..'JTIof cmraocs to5),mathemalieal needs of])4 Revie ......of rearchinto malhemat;{:ol educorion188,205-6, 234-7,2AO-I ROIl'lIeaming238-9 RoyalSociety of An.s lRSA)98.149.161 s of work363- 4,510 Sehoolininitiallraining 687-91 Schoolkuvers,DES10%survcy of 193 SchoolMat hemal Project (SM!108,279 - lH.581 Schools CouncilMathematies Commitr,ec578.5R5 Scolland748,785 Seeondary years,mathematics inthe Chapter') Differencesinallainmenl betweenpupils436-7 D'Jferenccbel weensyllabus and curriculum438 Changes inlast (wenly years439-41 inlroducrionof CSE440-1 1n fiuencof c.xomin3tionsyllabuses442- 50 CSElimiledg.radeexaminalions 447 Cour>e;s.for11-16 yearoldpupils451-R2 309 foundalionlislof malhemauC81topic,s458.provisionfor lowe!allaining pupils459-66.provisionforverylowat-taining pupil467- 9,provisionforpupilsfor v'hom andOleclare iDiended470-3.pro\'isionforhigh all.Jining pupils474-81.'exira mathematics'480-1. ior veryhighanaining pupils482 Malhemalic!iIhe curriculum48J-5 Time alloealion486-90 Organisationof teaching groups491-501 leaching in seu.edgroups493-4,leachingi.nmixedabili-ty groups495-8,indjvidnal learning499- 50l Deploymenlor I ea,hi ngstaff 502 - r5Head of department507-17 duties508.510-16.needfor(ime toOUIduties509 ionI CSIS,employers 87- 98 Selleellgroups350.493-4 '5('\'