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Teaching and learning geometry 11-19Report of a Royal Society / Joint Mathematical Council working group

JMC

page

Preparation of this report v

Chairman’s preface vii

Summary xi

1 Introduction 1

2 Geometry and its teaching and learning 3

3 The place of geometry in the curriculum 5

4 The 11-16 curriculum 7

5 The development of the curriculum 9

6 Status and allocation of time for geometry 13

7 Geometry 16-19 15

8 The role of assessment 17

9 Teaching of geometry 19

10 Improving the take up of mathematics 21

11 Conclusion 23

12 References and glossary 25

Teaching and learning geometry 11-19

Contents

Contents (continued)

Appendix 1 The working group 27

Appendix 2 National and international contexts for mathematics 31

Appendix 3 Some recent government initiatives in education 35

Appendix 4 Expectations of geometry in education 37

Appendix 5 Geometry in history and society 41

Appendix 6 Geometry in the current 11-16 curriculum 45

Appendix 7 Geometry in the Key Stage 3 mathematics strategy 49

Appendix 8 Spatial thinking and visualisation 55

Appendix 9 Proof – ‘why and what?’ 57

Appendix 10 Examples of applications of geometry 65

Appendix 11 3-dimensional geometry 69

Appendix 12 Frameworks for developing schemes of work for the curriculum 75

Appendix 13 Integrated approaches to geometry teaching 81

Appendix 14 Bibliography and guide to resources 87

© The Royal Society 2001Requests to reproduce all or part of this document should be submitted to:Education ManagerThe Royal Society6 Carlton House TerraceLondon SW1Y 5AG

The members of the working group were:

Dr Tony Barnard Department of Mathematics, King’s College LondonDr Richard Bridges King Edward’s School, BirminghamProfessor Margaret Brown School of Education, King’s College LondonSandy Cowling Mathematics Inspector, Merton LEACaroline Dawes Assistant Head Teacher, Wembley High SchoolMargaret Dawes (Secretary) Formerly tax partner, KPMG Professor Robin Forrest School of Information Systems, University of East AngliaJane Imrie NANAMIC, formerly Head of Mathematics, Exeter CollegeDr Keith Jones School of Education, University of Southampton Professor Anthony Kelly CBE DL FREng FRS Department of Material Science, University of CambridgeMary Ledwick Our Lady & St. John High School, BlackburnProfessor Adrian Oldknow (Chairman) Emeritus Professor, University College ChichesterDr Sue Pope Froebel College, University of Surrey RoehamptonDr John Rigby School of Mathematics, Cardiff University Professor Chris Robson School of Mathematics, University of LeedsSir Christopher Zeeman FRS Formerly Principal, Hertford College, Oxford

Observers

Richard Browne Qualifications and Curriculum AuthorityNigel Thomas The Royal SocietyAlan Wigley National Numeracy Strategy

The working group wishes to record its thanks to Professor Geoffrey Howson who made valuable contributions tothe group’s work and also to the following individuals who acted as ‘readers’: Dr Bob Burn; Tandi Clausen-May;David Fielker; Dr Tony Gardiner; Professor Celia Hoyles; Mike Ollerton; and Dr Leo Rogers (who also provided advicerelating to Appendix 5).

The Royal Society Teaching and learning geometry 11-19 | July 2001 | v

Preparation of this report

This report has been endorsed by the Council of the Royal Society and the JMC. It has been prepared by the Royal Society / Joint Mathematical Council working group on the teaching and learning of geometry.

“About binomial theorem I’m teeming with a lot of news,With many cheerful facts about the square of the

hypotenuse”(WS Gilbert, The Pirates of Penzance)

The mathematical content for pupils following theNational Curriculum in secondary schools in England isdescribed under the headings of: Number and algebra;Shape, space and measures and Handling data.However the term ‘numeracy’ has become increasinglyused in place of mathematics in relation to schooleducation. This is an unfortunate practice since itdownplays two areas, algebra and geometry, which areof major importance in school mathematics. Theteaching of each of these aspects of mathematics hasnow been the subject of commissioned reports from theRoyal Society and the Joint Mathematical Council of theUnited Kingdom, and the Qualifications and CurriculumAuthority is currently engaged in a three year project ondeveloping the teaching of both algebra and geometry.

A past President of the Royal Society, Sir Michael Atiyah,provided some succinct background to the developmentof algebra and geometry in a lecture given in Toronto inJune 2000:

I want to talk now about a dichotomy inmathematics, which has been with us all the time,oscillating backwards and forwards... I refer to thedichotomy between geometry and algebra.Geometry and algebra are the two formal pillars ofmathematics; they both are very ancient. Geometrygoes back to the Greeks and before; algebra goesback to the Arabs and the Indians, so they haveboth been fundamental to mathematics, but theyhave had an uneasy relationship.Let me start with the history of the subject.Euclidean geometry is the prime example of amathematical theory and it was firmly geometrical,until the introduction by Descartes of algebraiccoordinates, in what we now call the Cartesianplane. That was an attempt to reduce geometricalthinking to algebraic manipulation.(Reprinted in Mathematics Today, 37(2), April 200146-53.)

At school level, algebra can seem quite abstract andcerebral. In Fitzgerald’s studies for the Cockcroftcommittee it was algebra which was most frequentlycited as the part of mathematics where adultsremembered losing touch with mathematics. On theother hand, there are clear links in geometry to theworld of our senses and experience. For example, wecan easily perceive when objects are parallel, or

perpendicular, or symmetrical - such as recognisingwhen a minute adjustment is needed to the way apicture hangs. Sir Michael offers the followingcomments on our capacity to perceive, and itsrelationship with geometry:

Our brains have been constructed in such a waythat they are extremely concerned with vision.Vision, I understand from friends who work inneurophysiology, uses up something like 80 or 90percent of the cortex of the brain...Understanding, and making sense of, the worldthat we see is a very important part of ourevolution. Therefore spatial intuition or spatialperception is an enormously powerful tool andthat is why geometry is actually such a powerfulpart of mathematics - not only for things that areobviously geometrical, but even for things that arenot. We try to put them into geometrical formbecause that enables us to use our intuition. Ourintuition is our most powerful tool... I think it isvery fundamental that the human mind hasevolved with this enormous capacity to absorb avast amount of information, by instantaneousvisual action, and mathematics takes that andperfects it.

Geometry is of far reaching importance beyond theworlds of professional mathematicians and ofmathematics teaching. Geometry is frequently used tomodel what we call the ‘real world’ and has manyapplications in solving practical problems. (It isinteresting to note that the French term for a surveyor is‘un expert géomètre’.) Geometry is makingcontributions to many important scientificdevelopments such as the Human Genome Project,Buckminster-Fullerene research, and whole-bodytomography. Through media such as film, television andcomputer games we encounter computer generatedgeometric images of great complexity, and children andadults alike derive pleasure from creating designs andpatterns exhibiting geometric forms.

So geometry is an important subject, with wideapplications and a long history. It deals with matters wefind attractive and for which we have a strong visualcapacity. On the surface, then, it would appear thatgeometry should be one of the easiest branches ofmathematics to teach. But this is not the case - neither inEngland nor in much of the developed world. This RoyalSociety / JMC study set out to identify why this is so.

Geometry is one of the oldest branches of mathematics- itself one of the oldest of mankind’s intellectual

The Royal Society Teaching and learning geometry 11-19 | July 2001 | vii

Chairman’s preface

Adrian Oldknow

studies. No wonder, then, that it suffers from anembarrassment of riches in terms of theories, results,techniques and applications. Many of these are wellwithin the grasp of most, if not all, students in 11-19education. We might refer to this, not unwelcome,problem as one of abundance. Clearly, then, choiceshave to be made on what material to include in thecurriculum. At one extreme there is a danger ofchoosing eclectically from this abundance in a way thatleads to the teaching of a lot of apparently unconnected‘bits’. At the other extreme there is a danger ofdeveloping a tightly organised body of knowledgewhich addresses only a very small part of geometry. Ourchallenge has been to combine breadth with botheducational and mathematical coherence - a problemwe refer to as coherence.

One of the less obvious difficulties in teaching geometrylies in the abstractions we make – we illustrate pointsand line segments through drawings and diagrams andyet neither object can be visible, except in our ‘mind’seye’. We do not often choose to discuss such a difficultissue! Frequently however, teachers will draw rapidsketches purporting to represent objects in their ownimagination which may actually not be recognised assuch by their pupils.

The geometry of the ancient Greeks, as recorded byEuclid, was far more than a summary of known facts - itwas an organised body of knowledge starting with anumber of definitions and assumptions (axioms) whichused logical deduction to establish a series of results inthe form of theorems together with proofs. It is throughthe teaching of geometry that most pupils stillencounter at least one theorem, that of Pythagoras,together with one or more proofs, and maybe someapplications. My non-scientific guess is that most adultswill remember the name Pythagoras, and probably thathis theorem has to do with right angled triangles andwords like ‘hypotenuse’, but that it would beextraordinary if they could remember a proof of thetheorem. The role of proof, and the range of pupils forwhom it is relevant, remains a major issue in geometryteaching. Despite the long tradition for the inclusion ofgeometrical proof in school curricula there is littleevidence that we have developed effective methods forits teaching. Nevertheless, the working group supportsthe inclusion of proof in school geometry both becauseof its central role in mathematics, and as a contributionto developing more general skills of argument andcriticism.

In order to address the issue of coherence the workinggroup has followed on from a previous review of thegeometry curriculum [Wynne Willson, 1977] andformulated a set of objectives for the teaching ofgeometry in the 21st Century. Against these objectiveswe have concluded that the geometrical content of theNational Curriculum does provide a reasonable basis for

the 11-16 curriculum, but needs strengthening in twomain areas. These concern work in 3-dimensions, and inthe educational application of Information andCommunications Technology (ICT). Leaving thegeometrical content relatively unchanged for now willallow scope for dealing with the issue which theworking group has identified as by far the mostimportant one for 11-16 geometry. That is to ensurethat teachers have the knowledge, understanding, skillsand resources to teach geometry in a way whichgenuinely captures pupils’ interest and imagination,while developing their thinking and reasoning skills,their powers of visualisation, their ability to apply andmodel, and their understanding.

The 11-16 geometry curriculum in England continues toconcentrate on techniques for working in 2 dimensions,such as the plane geometry derived from Euclid,together with elements of transformation, vector andcoordinate geometry. Yet little of this finds its way intocurrent AS/A-level specifications in mathematics, whosegeometrical content has been drastically reduced overtime. Similarly, the kind of geometry studied bymathematics undergraduates bears little resemblanceto that studied either pre- or post-16. We refer to thisissue as one of progression.

While the working group is optimistic about thepossibility for significant improvement in teachinggeometry 11-16, (which is not to underestimate thechallenges to be addressed), it is far less sanguine aboutthe state of geometry in 16-19 education. Thegeometrical content of the current AS/A-levelspecifications in pure mathematics is very small andoffers little by way of progression from what has comebefore. But there is little point in advising contentchanges at this level when the whole basis of 16-19qualifications in mathematics and all other subjects hasjust undergone a series of changes, the consequencesof which have yet to be fully felt. Our view is that thegeneral position of mathematics in 16-19 educationneeds a fundamental review before geometry can beaccorded an acceptable place.

It is widely recognised that secondary schools haveproblems recruiting and retaining mathematicsteachers. Many of those currently teachingmathematics in secondary schools are not mathematicsgraduates. Due to the problem of progression, it cannotbe assumed that even trained mathematics graduatesare adequately equipped to teach geometry in the waythe working group envisages. To remedy this will requirea substantial programme of well planned continuingprofessional development for teachers which improvesboth their subject knowledge in geometry and theirapproaches to teaching it. The current Key Stage 3mathematics strategy provides substantial opportunitiesfor the professional development of mathematicsteachers in secondary schools and has the potential to

The Royal Societyviii | July 2001 | Teaching and learning geometry 11-19

make a valuable contribution to improvements in theteaching of geometry. But this alone will not be sufficientto improve teaching throughout the 11-19 sector.

Computer software, particularly that known as DynamicGeometry Software, has the potential to makesignificant improvements in how geometry is learnt andtaught. But such software is not widely available inschool mathematics classrooms, as is the case withcomputing resources in general. In order for suchresources to have maximum effect on improving theteaching and learning of geometry we need to findways which allow talented teachers the time to developa range of effective Information and CommunicationTechnology based approaches. In addition to ICT, thereis a need for a range of good materials to support the

teaching of geometry in school. These, too, need to becarefully prepared and tried out, and that will alsorequire time and effort.

The working group has been challenged to articulate itsvision for geometry teaching. I believe that what weseek is a coherent, stimulating, rewarding andchallenging geometry curriculum which is taught in away which captures students’ interest and imaginationand which attracts them towards mathematics as asubject for further study. The achievement of our visionrequires a significant improvement in the quality ofteaching, and this has major consequences - both forthe continuing professional development of teachersand for the provision of high quality supportingresources.

The Royal Society Teaching and learning geometry 11-19 | July 2001 | ix

The Royal Society Teaching and learning geometry 11-19 | July 2001 | xi

This report presents the findings of a broadly basedworking group established by the Royal Society and theJoint Mathematical Council to consider the teachingand learning of geometry in schools and colleges. Thestudy was initiated following the publication of resultsof international educational comparisons, the 1999revision of the National Curriculum for English schools11-16, and at a time of several major policy initiatives ineducation.

The working group considered the rationale for ageometry curriculum, its possible content and issuesconcerned with its effective teaching. This reportreflects its agreed views on the state of geometryteaching 11-19 and the major issues needing to beaddressed to bring about improvements. It is supportedby additional materials, some of which are printed hereas appendices, and others of which are accessible fromthe Royal Society’s website at www.royalsoc.ac.ukThese additional materials are intended to help illustratesome of the points in the report, and to offer examplesof approaches which might be taken by schools andcolleges. They are sometimes attributed to an individualor groups of members and are then not claimed torepresent the views of the whole group.

In order to help identify major issues raised, the report isstructured around a number of agreed Key Principles. Inthe main body of the report these are presented togetherwith explanations, supporting arguments and, whereavailable, evidence. Additional information andexemplification is provided in the appendices. One or morerecommendations are associated with each Key Principle.

Overall, for mathematics 11-16, we conclude that thegeometrical content of the new National Curriculum,with a few adjustments, forms an appropriate basis fora good geometry education. In order for this to beachieved, however, considerable changes are needed inthe way geometry is taught. It is vital that thoseworking to improve mathematics education ensure thattheir work contributes significantly to improvements ingeometry (as well as mathematics) teaching. Bringingabout improvements in geometry teaching will require asignificant commitment to a substantial programme ofcontinuing professional development alongside thedevelopment of appropriate supporting materials.

For mathematics post-16 we conclude that there areinsufficient opportunities for students to build on their11-16 studies in geometry. Those concerned withcurriculum design need to review the structure of post-16 qualifications in mathematics to ensure they provideimproved opportunities for students to continue tostudy geometry. The provision of challenging andinteresting geometry should help make mathematics a

more attractive subject of study for more students. Thisin turn would contribute to overcoming the currentshortage of those with good mathematical skills.

Key Principles

Key Principle 1: Geometry should form a significantcomponent of the mathematics curriculum for allstudents from 11 to 19.

Key Principle 2: Any choice of curriculum should beunderpinned by a rationale.

Key Principle 3: The geometry curriculum shouldmaintain breadth, depth and balance, and be consistentwith Key Principle 2 and the objectives inRecommendation 3.

Key Principle 4: Geometry should be given a higherstatus, together with a fair share of the teaching timeavailable for mathematics.

Key Principle 5: Students in 16-19 education shouldhave the opportunity to continue further their studies ingeometry.

Key Principle 6: The assessment framework for thecurriculum should be designed to ensure that the fullrange of students’ geometrical knowledge, skills andunderstanding are given credit.

Key Principle 7: The most significant contribution toimprovements in geometry teaching will be made by thedevelopment of good models of pedagogy, supported bycarefully designed activities and resources, which aredisseminated effectively and coherently to and by teachers.

Key Principle 8: It is a matter of national importance thatas many of our students as possible fully develop theirmathematical potential. Geometry, with its distinctiveappeal, should make mathematics attractive to a widerrange of students.

Recommendations

Recommendation 1: We recommend that curriculumand assessment specifications be reviewed to ensurethat geometry forms a significant component of themathematics curriculum for all students from 11 to 19.

Recommendation 2: We recommend that the title ofthe attainment target Ma3 of the National Curriculumbe changed from ‘Shape, space and measures’ to‘Geometry’.

Summary

The Royal Societyxii | July 2001 | Teaching and learning geometry 11-19

Recommendation 3: We recommend that the geometrycurriculum be chosen and taught in such a way as toachieve the following objectives:a) to develop spatial awareness, geometrical intuition

and the ability to visualise; b) to provide a breadth of geometrical experiences in 2-

and 3-dimensions;c) to develop knowledge and understanding of and the

ability to use geometrical properties and theorems;d) to encourage the development and use of

conjecture, deductive reasoning and proof;e) to develop skills of applying geometry through

problem solving and modelling in real worldcontexts;

f) to develop useful Information & CommunicationTechnology (ICT) skills in specifically geometricalcontexts;

g) to engender a positive attitude to mathematics; andh) to develop an awareness of the historical and cultural

heritage of geometry in society, and of thecontemporary applications of geometry.

Recommendation 4: We recommend that the currentgeometrical content of the English secondary schoolmathematics National Curriculum be regarded as areasonable basis for an appropriate and rewardinggeometry education for all pupils.

Recommendation 5: We recommend that themathematics curriculum be developed to encouragestudents to work investigatively, demonstrate creativityand make discoveries in geometrical contexts so thatstudents develop their powers of spatial thinking,visualisation and geometrical reasoning.

Recommendation 6: We recommend that themathematics curriculum be developed in ways whichrecognise the important position of theorems andproofs within mathematics and use the study ofgeometry to encourage the development of logicalargument appropriate to the age and attainment of thestudent.

Recommendation 7: We recommend that themathematics curriculum be developed to provide ampleopportunities for students to use geometry for practicalproblem solving through mathematical modelling inboth 2- and 3-dimensions.

Recommendation 8: We recommend that the geometrycurriculum be developed to give greater emphasis towork in 3-dimensions and to make better use ofInformation and Communication Technology (ICT).

Recommendation 9: We recommend that the use of theword ‘numeracy’ in government publications andannouncements be replaced by ‘mathematics’ to ensurethat geometry is accorded its rightful position.

Recommendation 10: We recommend that geometryshould occupy 25% - 30% of the teaching time, andhence a similar proportion of the assessment weighting,in the 11-16 mathematics National Curriculum.

Recommendation 11: We recommend that the total timeallocated to mathematics 11-16 be monitored to ensurestudents spend at least 3 hours a week on mathematics,so that sufficient time is given to the teaching of geometry,and to other aspects of mathematics.

Recommendation 12: We recommend that afundamental review be made of all 16-19 mathematicsprovision. This should include considering how:a) the structure and content of the current AS/A-level

Mathematics and Further Mathematics specificationscan better meet the needs of students and include agreater emphasis on geometry; and

b) other post-16 mathematics qualifications, such as FreeStanding Mathematics Units (FSMUs) and AS-level Useof Mathematics, can enable students to have theopportunity to continue their study of geometry.

Recommendation 13: We recommend that in the 16-19curriculum the key skill, ‘Application of Number’, be re-titled ‘Application of Mathematics’ and that the rangeof qualifying mathematical studies be broadened sothat students continue their study of geometry.

Recommendation 14: We recommend that a review bemade of the methods of assessment and examinationused in mathematics at Key Stage 3, at GCSE and in post-16 qualifications to ensure that appropriate credit is givenfor the attainment of specific geometrical objectives.

Recommendation 15: We recommend that the relevantgovernment agencies work together, with bodies suchas the mathematics professional associationsrepresented on JMC, to provide a coherent frameworkfor supporting the development of teaching andlearning in geometry. This will involve:a) the recognition and development of good practice in

geometry teaching through pilot studies andresearch;

b) the design of programmes of continuing professionaldevelopment and initial teacher education;

c) the production of supporting materials; andd) the establishment of mechanisms to provide

supporting resources, including ICT.

Recommendation 16: We recommend, in terms ofmathematics in general, that:a) better publicity and information be provided to

schools, students and parents about the careeropportunities afforded by studying mathematics; and

b) ways be sought to encourage schools and colleges toattract more students to study mathematics post-16,particularly at A-level.

This report presents the findings of a working groupestablished by the Royal Society and the JointMathematical Council (JMC) to consider the teachingand learning of geometry in schools and colleges. Theworking group, chaired by Professor Adrian Oldknow,met fourteen times between February 2000 and May2001. The membership of the working group is given atthe front of this report and its terms of reference can befound in Appendix 1.

The study was initiated following publication of theresults of international educational comparisons, the1999 revision of the National Curriculum for Englishschools 11-16 and at a time of several major policyinitiatives in education. Some of this background is setout in Appendices 2 and 3.

Membership of the group was carefully chosen to includethose with experience in: (a) teaching mathematics instate and independent schools and colleges, in initialteacher education and in higher education; (b) conductingresearch in mathematics (including geometry) and inmathematics education; (c) applying mathematics(including geometry) in disciplines such as science,engineering, IT and finance, and; (d) planning andimplementing mathematics curricula in Local EducationAuthorities (LEAs) and government agencies. A variety ofgroups have expectations of the mathematics curriculumand its geometrical content; some of these are consideredin Appendix 4.

The Royal Society Teaching and learning geometry 11-19 | July 2001 | 1

1 Introduction

Geometry is one of the longest established branches ofmathematics. It has an extensive range of applicationsand we give some selective historical and culturalbackground in Appendix 5. Geometry has beenaccorded a central place in mathematical education inWestern culture for a considerable period of time. Oneof the major achievements of classical geometry was thesystematic collection by Euclid of the geometricalknowledge of the ancient Greeks. This has, untilcomparatively recently, formed the basis for much of thegeometry taught in schools.

During a period of educational reforms in mathematicsin the 1950s and 1960s some new syllabuses(sometimes called ‘the new maths’) were developedwhere the emphasis was on formal structures whichwere predominantly algebraic. At the same time, therange of approaches to geometry was broadened fromits traditional Euclidean base (which was reduced indepth) to include the use of transformations, vectors,matrices and some topology.

In recent years many countries have been reviewing theaims, content and approach of their geometry curricula.The 1995 study by the International Commission on

Mathematics Instruction (ICMI) [Mammana and Villani,1998] revealed that no clear consensus was emergingabout the outcome of these reviews. The small scaleresearch study into the geometry curricula of a number ofcountries commissioned in 2000 by the Qualifications andCurriculum Authority (QCA) for England confirmed this.

Against this background the working group consideredthe rationale for a geometry curriculum, its possiblecontent and issues concerned with its effectiveteaching. Our report sets out a number ofrecommendations on issues where the working groupreached a consensus view. There are some matters onwhich the working group did not reach a conclusion,and which others may wish to pursue further. There arealso some matters which the working group did notaddress. In order to help identify major issues raised, thereport is structured around a number of agreed KeyPrinciples. These are presented together withexplanations, supporting arguments and, whereavailable, evidence. Additional information andexemplification are provided in appendices and on theRoyal Society website at www.royalsoc.ac.uk One ormore recommendations are associated with each KeyPrinciple.


2 Geometry and its teaching and learning

Key Principle 1: Geometry should form a significantcomponent of the mathematics curriculum for allstudents from 11 to 19.

This is a simple proposition to express yet it has manyfacets. First we consider some issues about the role ofgeometry in education. Then we consider the relation ofgeometry to other aspects of the mathematicscurriculum. We review some of the problems associatedwith teaching aspects of geometry and pave the way forother key principles which stem from this.

A valid case for the study of geometry may be made onseveral grounds. Geometry is a central part ofmathematics, and geometrical thinking is afundamental way to engage with mathematics.Geometry can be used to develop students’ spatialawareness, intuition and visualisation. It can also beused to solve practical problems. There are manyapplications of geometry relevant to employment andeveryday life. Other subjects in the curriculum, such asscience and technology, make use of geometrical ideasand techniques. Geometry is well established in ourculture and has an interesting history of its own. It hasan important place in the development of aestheticsand design. It can be used to encourage thedevelopment and use of conjecture, deductivereasoning and proof. Geometry can also be used to layfoundations for further studies in mathematics.

It is our view that all of these grounds, which have oftenbeen cited in the past, remain valid reasons for theinclusion of geometry as a significant part of the currentcurriculum. There are additional grounds that reflectrecent changes in our society.

The rapid development in a range of technologies meansthat citizens now and in the future will interact with avariety of forms of displayed images. These may berequired by their work, be needed in order to exchangeinformation or just be associated with leisure. A case canthus be made that geometry has a role to play within thedevelopment of citizenship in enabling students tointerpret, manipulate, control and create such images.

In recent years there has been a major shift in the UK

economy from manufacture to services. Associated withthis has been a marked increase in demand for thosewith good skills in flexible thinking and the use ofInformation and Communication Technology (ICT),together with the ability to apply mathematics(inadequately referred to as ‘numeracy’ skills). A directconsequence has been the much publicised problem inrecruiting and retaining mathematics teachers. In orderto fulfil the skills needs of industry, commerce and theprofessions - including teaching - we need to encouragemore students to engage positively with mathematicsand to choose to continue their studies in it, or relateddisciplines. We believe that geometry is a subject ofmathematical study which has its own appeal andsatisfaction and which, well taught, could encouragemore students to continue with the study ofmathematics beyond 16.

Breadth of study in geometry needs to be provided tomeet the demands outlined above. To ensure studentsalso receive appropriate intellectual challenges andstimuli it is important to provide depth in a number oftopics. The challenge, of course, is to do both within afair share of the time which should be allocated tomathematics teaching.

We conclude this first Key Principle with tworecommendations. We believe that geometry hasdeclined in status within the English mathematicscurriculum and that this needs to be redressed. It shouldnot be the “subject which dare not speak its name”.

Recommendation 1:

We recommend that curriculum and assessmentspecifications be reviewed to ensure that geometryforms a significant component of the mathematicscurriculum for all students from 11 to 19.

Recommendation 2:

We recommend that the title of the attainmenttarget Ma3 of the National Curriculum be changedfrom ‘Shape, space and measures’ to ‘Geometry’.


3 The place of geometry in the curriculum

Key Principle 2: Any choice of curriculum should beunderpinned by a rationale.

Here we work towards defining a set of objectivesagainst which to evaluate the geometrical content of acurriculum. First we summarise the current positionregarding the 11-16 curriculum for the maintainedsector in England.

The English educational system, centrally administeredby the Department for Education and Skills1 (DfES), isorganised around a number of relatively autonomousagencies and units. These include the Qualifications andCurriculum Authority (QCA), the Office for Standards inEducation (Ofsted), the Teacher Training Agency (TTA)and the British Educational and CommunicationsTechnology Agency (BECTa).

Schools and colleges have already had to adapt toconsiderable changes in very short time scales. So,rather than attempting to develop a geometrycurriculum from first principles, we have chosen toreview the current curriculum.

First we consider issues concerned with the teaching ofmathematics in England in secondary schools to pupilsaged 11-16. The QCA published a revised version of theNational Curriculum for England in 1999 forimplementation in schools and colleges fromSeptember 2000. The mathematics curriculum at KeyStages 3 (ages 11-14) and 4 (ages 14-16) differs fromthe earlier version in a number of respects. In particularthe new version details the curriculum separately foreach Key Stage, whereas the earlier version combinedKey Stages 3 and 4. It also divides the Key Stage 4curriculum into two programmes of study called‘mathematics foundation’ and ‘mathematics higher’.The geometrical content of the new curriculum isdescribed within the section Ma3 Shape, space andmeasures. It is described in much greater detail than inthe previous version. It has been suggested that theearlier version gave scope for teachers to address theitems of geometrical content found in the 1999 version.However it is our experience that some significantaspects of geometry in the new version, particularly inthe higher programme at Key Stage 4, are not currentlytaught extensively in secondary schools.

The working group considered the way in which the 11-16mathematics curriculum is presented, and currentlyexamined. At Key Stage 3 there is a single curriculum forall pupils. At Key Stage 4 it is divided into two. It isanticipated that roughly half of pupils will not study

many of the additional aspects of mathematicscontained only in the higher programme of study.Currently the examinations for mathematics in theGeneral Certificate of Secondary Education (GCSE) areset in three tiers: foundation, intermediate and higher.The introduction of separate programmes of studyalongside the use of three examination tiers raises anumber of issues. The working group chose not toconsider alternative ways of packaging the curriculumas these structures have implications for the wholemathematics curriculum, not just geometry.

The original National Curriculum has gone through twosets of revisions; neither of these has provided arationale for the content of the mathematicscurriculum. In our discussions, we identified a clear needto provide a set of objectives against which curriculumcontent should be evaluated and which we now providein the form of a recommendation to improve the focus,coherence and relevance of geometry teaching.

Recommendation 3:

We recommend that the geometry curriculum bechosen and taught in such a way as to achieve thefollowing objectives:a) to develop spatial awareness, geometrical

intuition and the ability to visualise; b) to provide a breadth of geometrical experiences

in 2- and 3-dimensions;c) to develop knowledge and understanding of,

and the ability to use, geometrical propertiesand theorems;

d) to encourage the development and use ofconjecture, deductive reasoning and proof;

e) to develop skills of applying geometry throughproblem solving and modelling in real worldcontexts;

f) to develop useful Information &Communication Technology (ICT) skills inspecifically geometrical contexts;

g) to engender a positive attitude to mathematics;and

h) to develop an awareness of the historical andcultural heritage of geometry in society, and ofthe contemporary applications of geometry.

From the analysis of the current geometry curriculum anumber of questions emerged to which the workinggroup has worked to find answers:• How should the geometrical content be determined?• Does the content of the revised Ma3 curriculum form


4 The 11-16 curriculum

1 The DfES was created in June 2001. Before this, the English education system was administered by the Department for Education andEmployment (DfEE)

an appropriate basis for the teaching of geometry insecondary schools at Key Stages 3 and 4?

• If not, how should it be modified?• What suppositions are made about the time available

for teaching mathematics, and its geometrycomponent, and are these acceptable?

• What issues need to be addressed if it is to be taughteffectively?

• How do assessment procedures impact on teaching?• What are the implications for teaching geometry pre-

11 and post-16?

There is no requirement for the development of theNational Curriculum to be based on evaluated field trialsand experiments to test feasibility. Nor are suchdevelopments necessarily linked with any associatedprofessional development for teachers, or with anydevelopment of appropriate teaching materials orassessment. Thus, in the absence of evidence, it has tobe a matter of judgement whether the geometryselected for inclusion in the content of the NationalCurriculum defines an attainable curriculum.

The Ma3: Shape, space and measures component ofthe 1999 National Curriculum certainly exhibits abreadth of study in geometry (see Appendix 6). It is theview of the working group, led by the experiencedschool teachers amongst us, that given the rightcircumstances it can provide an appropriate, interestingand attainable curriculum. We shall discuss what wemean by the right circumstances later in this report.

Before considering the curriculum content in greaterdetail we consider some recent changes in the way themathematics curriculum is implemented anddeveloped. The first of these is the model followed bythe National Numeracy Strategy (NNS) in primaryschools, which is now being extended to mathematicsat Key Stage 3 in secondary and middle schools. Thesecond is the 3 year project concerned with theteaching of algebra and geometry now beingconducted by the QCA.

The National Numeracy Strategy is managed by theDfES’s Standards and Effectiveness Unit (SEU). It hasdeveloped a year by year framework for teaching KeyStages 1 and 2 of the mathematics National Curriculum.This is based on work carried out in over 200 pilot

schools. Associated with the detailed teaching schemeshas been a large scale professional development exerciseinvolving LEAs, headteachers, mathematics coordinators,classroom teachers, governors etc. It has thus served as anational medium for the interpretation andimplementation of the established curriculum. Thegovernment has extended the work of the Strategy firstinto Year 7 (the year of entry to most secondary schools),and more recently into the whole of Key Stage 3. The KeyStage 3 mathematics strategy comes into national effectin September 2001 after a short pilot stage. Apart fromthe way it is being introduced, there are other differencesbetween the primary and secondary strategies, amongthe most pressing of which is the current shortage ofqualified mathematics teachers in secondary schools. Wewelcome the opportunities for the improvement inmathematics teaching in secondary schools which thislarge scale development has the potential to stimulate. Inthe latter stages of our work an observer from the KeyStage 3 mathematics strategy joined the working groupin order to ensure better linkage between our conclusionsand recommendations and the way the Key Stage 3strategy will implement the teaching of Ma3. Briefexamples from the current framework for mathematics inYears 7, 8 and 9 appear in Appendix 7, and there are linksto the full document on the Royal Society website atwww.royalsoc.ac.uk

An earlier working group of the Royal Society and JMCproduced a report on the teaching of algebra pre-19[Royal Society / JMC 1996]. The DfES is now supportinga 3 year study by the QCA into the teaching of Algebraand Geometry. This has already commissionedinternational studies into the teaching of those aspectsof mathematics. See also Appendix 2 for a briefdiscussion of international trends. We welcome theopportunity which this new project offers to implementour recommendations for the teaching of geometry. Wealso welcome the extended time scale for this project.

Recommendation 4:

We recommend that the current geometricalcontent of the English secondary schoolmathematics National Curriculum be regarded as areasonable basis for an appropriate and rewardinggeometry education for all pupils.

The Royal Society8 | July 2001 | Teaching and learning geometry 11-19

Key Principle 3: The geometry curriculum shouldmaintain breadth, depth and balance, and be consistentwith Key Principle 2 and the objectives inRecommendation 3 and Appendix 11.

In reviewing the curriculum we paid particular attentionto a number of important aspects of geometry relatedto our recommended set of objectives.

Through tackling, and solving, problems in geometry(both closed and open ended) pupils can develop‘thinking skills’ of reasoning, enquiry (which includesproblem posing and conjecturing) and creativity. Theycan also develop their geometrical intuition and extendtheir powers of visualisation and spatial thinking. Theseaspects are considered further in Appendix 8.

An important aspect of geometry is concerned with thedevelopment of deductive reasoning and proof. Ofcourse proof is not confined to geometry alone, andthere can be interactions, such as algebraic resultsproved geometrically and vice versa. However the use ofgeometry as a vehicle for the development of theunderstanding and use of deductive reasoning hasreceived relatively little emphasis in the English schoolcurriculum over the last 30 years.

For a variety of reasons, the whole issue of proof withinschool geometry has become emotive. In some minds itis associated with a particular style of teaching andexamining sometimes pejoratively, and erroneously,associated with the name Euclid. In others, it is regardedas the essential difference between mathematics andthe experimental sciences, and as an essential tool forthe further study of mathematics. The working grouphas had many interesting discussions about the place ofgeometrical proofs within mathematics, particularly atKey Stages 3 and 4. We have also received advice fromindividuals and bodies representing many shades ofopinion - with a strong representation in favour ofgeometrical proof from correspondents in HigherEducation and from some school teachers. We haveconcluded that it is important for all students toencounter proof during their study of geometry, whilealso recognising that some aspects of proof may bemore accessible in other mathematical contexts. For adiscussion of what we mean by proof see Appendix 9.

There is no suggestion here to attempt an axiomaticapproach to school geometry. Indeed we note that suchattempts have been made, unsuccessfully, in the past.Rather we are arguing for the use of logical argument,which builds upon what is already known by the pupil inorder to demonstrate the truth of some geometricalresult, possibly one conjectured by the pupil afterconducting a well chosen experiment. The results

concerned (i.e. the theorems) should be chosen as far aspossible to be useful, interesting and/or surprising. Thelevel of sophistication expected in the logical argumentwill depend upon the age and ability of the pupilconcerned, and the proof produced might equally becalled an ‘explanation’ or ‘justification’ or ‘reason’ forthe result. Many pupils may never reach the level ofproviding formal proofs of results (although the moreable should), but all should understand deductivereasoning and that it is more than simply stating a beliefor checking that the result is valid in many specific cases.Encouraging pupils to be critical of their own, and theirpeers’, explanations will help them develop thesophistication and rigour of their arguments. Theemphasis at all times should be on understanding, andanalysing a proof of a standard theorem has a positiverole in understanding too. Without doubt, the endresult of a proof at this level should be an understandingof why the result is true, not simply that a formalargument proves it. However, we accept that it is not aneasy matter to determine how to achieve this with eachpupil and each result and that a careful choice ofapproach will be needed. Some examples of proof in avariety of areas and styles appear in Appendices 9 and11.

We are aware that there are considerable difficulties tobe overcome in achieving our objective “to encouragethe development and use of conjecture, deductivereasoning and proof” (Recommendation 3d). We donot have a successful experience base to fall back on,nor have we found that other countries have positivelessons to offer. We have found that many teacherscurrently in post or in training do not have experience inusing geometrical reasoning themselves. We considerthe implications of these issues in Key Principle 7 below.

Mathematical reasoning is one strand of a fundamental,and unusual, area of the mathematics NationalCurriculum called Using and Applying Mathematics. Wenow consider its other strands which relate tocommunication and problem solving. In previousversions of the National Curriculum this area wasdescribed as a separate component, Ma1. In the currentversion it has been integrated within each of the otherthree components, including Ma3 Shape, space andmeasures. The working group accepts that it isimportant for all students to appreciate the power ofmathematics in the way it is applied in modellingimportant phenomena and solving practical problems -and that this applies equally in geometry as in otherareas. We have concluded that it is important for allstudents to experience the applicability of geometrythrough engaging in mathematical modelling in 2- and3-dimensions. Geometrical ideas are used in manymodels which pupils will encounter and use in the


5 The development of the curriculum

future. We are aware of the possibly confusing nature ofthe word ‘model’ in this context. By a ‘mathematicalmodel’ we mean a representation through the languageof mathematics of a real world problem. ‘Modelling’ isthe process of translation into mathematics, usuallyinvolving simplification and idealisation. This modernuse of the word recognises that this process oftranslation is itself a skill to be learned. It also recognisesthat the resulting mathematics will never be a perfectdescription of the original problem. We give someexamples of the ways in which geometry can be used inschool to model familiar situations in Appendix 10.

Another important feature of the geometry curriculumis that it provides opportunities for pupils to drawsketches, diagrams and accurate representations. Wegive just a few examples. Pupils can learn to use theproperties of figures, such as isosceles triangles,rhombuses and kites, to develop mathematically exactconstructions on paper with straight-edge andcompass. They can explore whether sets of the samefigures can be arranged to tile the plane. They canexplore and apply properties of standard figures, as wellas constructions, on computers using suitable geometrysoftware. They can produce plane sections of 3-Dobjects to apply their knowledge of 2-D figures insolving problems in 3-D. They can sketch perspectivedrawings of 3-D objects from different viewpoints. Theycan make nets from which to construct 3-D solids.

The section on Transforming Secondary Education in theGreen Paper, ‘Schools, Building on Success’ of February2001 includes the following:

4.29 The goals of our Key Stage 3 strategy are toensure that by age 14, the vast majority of pupils have:....... Learnt how to reason, to think logically andcreatively and to take increasing responsibility fortheir own learning.

Responsibility for the development of pupils’ thinkingskills now comes within the ‘Teaching and Learning inthe Foundation Subjects’ component of thegovernment’s Key Stage 3 strategy. The working groupwelcomes this recognition of the importance ofthinking skills and recommends that the national KeyStage 3 strategy makes use of the geometry componentof the mathematics curriculum for the development ofsuch skills.

So, consistent with our stated objectives, the workinggroup advocates striking a balance between thecreative, deductive and applicable aspects of geometry.

Recommendation 5:

We recommend that the mathematics curriculumbe developed to encourage students to work

investigatively, demonstrate creativity and makediscoveries in geometrical contexts so thatstudents develop their powers of spatial thinking,visualisation and geometrical reasoning.

Recommendation 6:

We recommend that the mathematics curriculumbe developed in ways which recognise theimportant position of theorems and proofs withinmathematics and use the study of geometry toencourage the development of logical argumentappropriate to the age and attainment of thestudent.

Recommendation 7:

We recommend that the mathematics curriculumbe developed to provide ample opportunities forstudents to use geometry for practical problemsolving through mathematical modelling in both2- and 3-dimensions.

We now consider what might be missing from thecurrent curriculum. The first matter we identified is theneed for much greater attention to 3-D geometry ateach stage of the curriculum for all pupils whatever theirability. It is simplistic just to note that we live in a 3-Dworld and need to be able to develop the geometricalskills to represent 3-D objects and to solve problemsinvolving them. Clearly 3-D modelling is of greatimportance in a wide range of disciplines, such asscience, engineering and design. We now come intocontact with a much wider range of 2-D representationsof 3-D objects than was previously the case. Spatialawareness, powers of visualisation and realistic meansof applying geometry cannot be developed successfullywithout paying greater attention to work in 3-D. So wepropose that in the 11-16 curriculum students shouldextend their understanding, skills and knowledge ofgeometry in the plane to solve problems in 3-D. Ofcourse, some 3-D work relies on 2-D results which willneed to be established first.

The revision of the National Curriculum by QCA in 1999gave the opportunity for greater exemplification of theways in which Information and CommunicationTechnology impacts on many subjects and theirteaching. Yet there is very little specific reference to theuse of ICT in the mathematics National Curriculum ingeneral, and in geometry in particular. Geometricalsoftware is now widely used in, for example,engineering and design. Through government andcommercial initiatives many secondary schools andcolleges have acquired powerful Computer AidedDesign and Computer Aided Manufacture (CADCAM)packages for use in teaching Design and Technology. By


contrast relatively few schools have access to softwarefor teaching geometry in mathematics. Yet by usingsuch software in appropriate ways, pupils can applytheir ICT skills to increase their knowledge andunderstanding of geometry. The software also providesthem with the opportunity to acquire and practisegeometrical skills. Opportunities occur when pupilscreate, analyse and interpret dynamic spatial images;make and test conjectures about geometricalrelationships that they can manipulate; and record andpresent the results of their investigations.

As with any approach to teaching, the educational use ofICT needs to be well thought through and carefullyplanned. The TTA has produced documentation toaccompany the current programme of lottery funded ICTtraining for all teachers in which it emphasises theimportance of a critical approach to the use of ICT. Thisexpects teachers to know where, when and how to applyICT to enhance the teaching and learning of theirsubjects. This advice is particularly important in geometrywhere a variety of approaches is needed includingmental, practical, and ICT enhanced work. Increasinglypowerful software is becoming available in education,such as that designed for simulations in science andgeography, much of which relies on sophisticatedmathematical algorithms. Pupils and teachers in allsubjects need to be cautious about accepting computerproduced results without question, and mathematics isprobably the subject best placed in the curriculum inwhich to engender a critical approach. In teachinggeometry, caution is particularly needed to avoid making

assertions based solely on computational illustrations.

Thus the working group would like to see furtherdevelopment of the curriculum with respect to work in3-D and the use of ICT. Appendix 11 on 3-D geometrygives examples of five topics that are suitable forschools. We recognise that this will have implications forresources, materials, assessment and teachers’professional development, as will the effectiveteaching of proof, modelling, problem solving and otheraspects of geometry. In many respects we need todevelop a completely new pedagogy in geometry. Weconsider such issues further below. We also recognisethat we are advocating an extension of the currentcurriculum, even before it has been fully implemented.Conscious of the potential criticism for proposing toextend an already crowded curriculum we address theissue of time allocation for mathematics below.Experienced teachers have developed their ownmechanisms for setting out the curriculum in such a waythat links can be made and time used most effectively. InAppendix 12 we include an extract from a possibleframework for the extended geometry curriculumdevised by some members of our working group.

Recommendation 8:

We recommend that the geometry curriculum bedeveloped to give greater emphasis to work in 3-dimensions and to make better use ofInformation and Communication Technology (ICT).


Key Principle 4: Geometry should be given a higherstatus, together with a fair share of the teaching timeavailable for mathematics.

Recently there has been a tendency to replace the word‘mathematics’ with ‘numeracy’, as if the two wereequivalent. This has sent out mixed messages about therelative importance of different aspects of themathematics curriculum. For example, within the pagesof the DfES The Standards Site on the Internet(www.standards.dfee.gov.uk/numeracy/) the followingdescription of the National Numeracy Strategy may befound:

Framework for teaching mathematicsThe Numeracy Framework helps teachers raisenumeracy standards nationwide by providingthem with a set of yearly teaching programmes,key objectives and a planning grid.

Similarly the introduction to the government GreenPaper, Schools: Building on Success, published inFebruary 2001, contains the following:

Every secondary age pupil must be competent inthe basics of literacy, numeracy and ICT andexperience a broad curriculum beyond.

We are concerned that this concentration on numeracyshould not result in the sidelining of geometry.

Recommendation 9:

We recommend that the use of the word‘numeracy’ in government publications andannouncements be replaced by ‘mathematics’ toensure that geometry is accorded its rightfulposition.

While accepting that the area called Ma2 Number andalgebra in the secondary school curriculum should havethe greatest amount of teaching time, we regard 25%of the available mathematics time as the minimumnecessary for the teaching of geometry in Ma3. We areconcerned about reports from some secondary schoolsthat there has been an erosion in the total time availablefor teaching mathematics, particularly in Key Stage 3 -perhaps exacerbated by teacher shortages. Primaryschools following the NNS have a daily mathematicslesson which, by the end of Key Stage 2, lasts one hour.Secondary schools following the new Key Stage 3strategy have been given guidelines for the time to beallocated to mathematics - at least 3 hours per week.Members of the working group have also expressed theview that if mathematics is to have parity of esteem with

the other core subjects of science and English then itshould be available as a double award at GCSE.

We do not have specific proposals to make about theteaching of geometry in primary schools. The NationalCurriculum at both Key Stage1 (pupils aged 5-7) andKey Stage 2 (pupils aged 7-11) has the Ma3 Shape,space and measures component. The NNS’s Frameworkfor teaching mathematics from Reception to Year 6provides an interpretation of this within the frameworkof the daily mathematics lesson. Provided that thiscurriculum is effectively implemented, then pupilstransferring from primary schools to Year 7 in secondaryschools should have a suitable basis on which todevelop their study of geometry.

The working group is aware that the effective teachingof the secondary school geometry curriculum which itadvocates is likely to require rather more time forgeometry than is currently normally the case. Therenaming of Ma3 to ‘Geometry’ should imply that thework on non-geometrical measures, such as time andspeed, is relocated in Ma2. Some aspects of Ma2Number and Algebra could be developed withingeometrical contexts, such as Pythagoras’s Theorem.

Questions have been raised about the time, andassessment, allocation to Ma4 Handling Data, and evenas to whether it should be part of the mathematicscurriculum at all. However we do not wish to make anyrecommendations in respect of the content of this, orother, parts of the mathematics curriculum. That is notto duck the issue but to record that it is for others toassess the strength of our claims for geometry againstthose of other parts of the curriculum. We have alreadypointed to the lack of a sound experience base for anappropriate pedagogy for significant aspects of thegeometry curriculum, for which there is an urgent need.It could well be that with the right approach, supportedby appropriate materials and resources including ICT,the teaching of geometry could also be made moreefficient. In summary we believe that a broad, coherentand demanding geometry curriculum can be effectivelytaught within a fair and reasonable time allocation. Thismay require some review of the balance between thecomponents of the mathematics curriculum. It willcertainly require the development of more effective andefficient teaching approaches.

Recommendation 10:

We recommend that geometry should occupy 25%- 30% of the teaching time, and hence a similarproportion of the assessment weighting, in the 11-16 mathematics National Curriculum.


6 Status and allocation of time to geometry

Recommendation 11:

We recommend that the total time allocated tomathematics 11-16 be monitored to ensure

students spend at least 3 hours a week onmathematics, so that sufficient time is given to theteaching of geometry, and to other aspects ofmathematics.


Key Principle 5: Students in 16-19 education shouldhave the opportunity to continue further their studies ingeometry.

The government has implemented reforms in the post-16 sector called ‘Curriculum 2000’. Students are nowencouraged to follow a programme of study whichincludes key skills, among which is the ‘Application ofNumber’. For most students this will be the only courseof mathematics they study post-16. Consistent with ourfirst Key Principle, we propose that its title should bechanged and its content extended so that studentsstudy material from a wider range of topics inmathematics, including geometry. There should bemore compulsory elements of geometry which areassessed through tests, and which make explicit theopportunities to develop geometrical ideas in greaterdepth for inclusion in the portfolio.

The QCA have also recently revised their criteria for themathematics General Certificate of EducationAdvanced Subsidiary and Advanced Level (AS- and A-level). Awarding bodies have now producedspecifications that are being taught for the first time inthe current academic year. The geometry in thecompulsory part (core) of pure mathematics for A-levelconsists of a very small amount of coordinate geometry(lines and circles), some trigonometry and someelementary work with vectors. Within the currentframework there is little scope for more geometry in thecore, but more use could be made of geometricalcontexts, say in the application of calculus. The workinggroup doubts that the current geometrical content of A-level mathematics forms a suitable foundation for thosestudents who go on to study science or engineering. Inparticular there should be greater emphasis on work in3-D.

The working group has discussed the possibility ofintroducing one or more optional modules at AS- and A-level, outside the core of pure mathematics, whichcould include extensions in geometry. Potentialdrawbacks of such a solution would be the increase inthe variety of routes to an award - leading to problemsof comparability of standards, and also a greater varietyof mathematical backgrounds of students taking thesame course in higher education. We do not have aninstant solution to propose with regard to improving thegeometrical content of AS- and A-level mathematics intheir current format. When a fundamental review ofthese qualifications takes place the working grouprecommends that careful consideration be given toextending the amount of geometry in the A-level core.Candidates for an extension to such content includeplane curves, such as conics, further vector geometryand a greater emphasis on parametric representations.

We would expect a greater emphasis to be given to theimportant role of coordinate geometry as a linkbetween algebra, graphs and functions, and calculus.

More generally there is reason to believe that theexisting choice of optional modules (mainly inmechanics or statistics) does not meet the needs orinterests of all potential candidates for A-levelmathematics - such as those with an interest inaesthetics, or an intention to pursue careers in the ITindustry. Currently about 60 000 of the c.230 000 A-level candidates enter for A-level mathematics. Theuptake in mathematics at this level might be increased ifthere was a wider choice of modules, one or more ofwhich included interesting geometry. We understandthat evidence is beginning to emerge that the numberof students taking AS-level mathematics as a fourthsubject in the lower sixth-form (Year 12) and then notchoosing to progress to A-level mathematics in thefollowing year is higher than anticipated. However theworking group is aware that much of this is speculation,and so we recommend that research be undertaken intothe mathematical needs and interests of post-16students, and the implication for the curriculum. Ourconclusion is that both the structure and the content ofAS- and A-level mathematics are in need of afundamental review.

Candidates who wish to extend their post-16 study ofmathematics can study A- or AS-level FurtherMathematics, although numbers doing so have beendwindling. In the 1980s around 12 000 students weretaking ‘double mathematics’ each year at A-level,whereas in the late 1990s this had levelled out at justover 5000 taking A-level Further Mathematics, andaround 2000 taking AS-level Further Mathematics –despite the growth in the numbers within the A-levelcohort. A major factor has been the problem ofmaintaining financially viable group sizes in schools andcolleges. The Gatsby Foundation is supporting a projectto make these courses more widely available to studentsthrough distance learning. We welcome such initiativesto encourage greater take up of these courses, asstudents taking such qualifications are much betterplaced for success in undergraduate studies inmathematics, physics and engineering. Similararguments apply about making these courses moreinteresting and challenging by including both moregeometry and the greater use of geometrical contexts.

There is now a range of post-16 qualifications called‘Free Standing Mathematics Units’ (FSMUs) which aredesigned to support students in their other studies andwhich should enable more students to pursuemathematics post-GCSE. The FSMUs are available atthree levels. There are some FSMUs which include


7 Geometry 16-19

geometry, but none at level 3, the advanced level. Werecommend that a geometry unit be developed at level3. In Autumn 2001 a new AS-level qualification called‘Use of Mathematics’ is to be introduced, based onadvanced level FSMU modules. The working group hadunderstood that this new qualification was intended forstudents who would not otherwise take a post GCSEmathematics course, such as those specialising in thearts, humanities and social sciences. At the time ofwriting, the qualification is still in development but itappears that it will now be predominantly a course inmathematical modelling with no geometrical content atall. Thus there would still appear to be a gap in themarket for an AS-level qualification in mathematicswhich will appeal to students specialising, say, in the artsand humanities and for whom geometry might be anattractive element of study.

The working group believes that there is still more to bedone to ensure that there is a sufficient range of level 2and 3 mathematics qualifications to attract greaternumbers of students to continue their studies inmathematics post-16. We recommend that the range oflevel 3 mathematics qualifications be reviewed toensure that students have the opportunity to studygeometry further. In particular the cultural, aestheticand historical aspects of geometry, such as thedevelopment of perspective, should be of considerableappeal to many of those students from the arts andhumanities who currently drop mathematics.

The Royal Society website provides links to currentspecifications of post-16 mathematics qualifications.

We consider that Curriculum 2000 may have an adverseeffect on mathematics, stemming partly from itscomplexity and rigidity. However as major changes to16-19 education are currently being implemented, it isnot the time to make any detailed recommendationswith respect to specific mathematics qualifications.Thus we make the following general recommendations.

Recommendation 12:

We recommend that a fundamental review bemade of all 16-19 mathematics provision. Thisshould include considering how:a) the structure and content of the current AS/A-

level Mathematics and Further Mathematicsspecifications can better meet the needs ofstudents and include a greater emphasis ongeometry; and

b) other post-16 mathematics qualifications, suchas Free Standing Mathematics Units and AS-level Use of Mathematics, can enable studentsto have the opportunity to continue their studyof geometry.

Recommendation 13:

We recommend that in the 16-19 curriculum thekey skill, ‘Application of Number’, be re-titled‘Application of Mathematics’ and that the range ofqualifying mathematical studies be broadened sothat students continue their study of geometry.


Key Principle 6: The assessment framework for thecurriculum should be designed to ensure that the fullrange of students’ geometrical knowledge, skills andunderstanding are given credit.

We do not believe that many of the geometricalobjectives in Recommendation 3 can be adequatelyassessed within the current framework of timed testsand examinations. Indeed, the current assessmentframework is one of the major reasons why importantaspects of geometry, such as work in 3-D, geometricalreasoning and the use of ICT have not been givensufficient attention in classrooms. It is only natural thatteachers concentrate on aspects of a curriculum whichcarry the greatest assessment weighting - especiallywithin the current climate of target setting in schools.

A number of questions set in national tests andexaminations which are apparently about geometry arein practice mainly exercises in algebra. We would like tosee national tests and examinations incorporatequestions which test geometrical reasoning andapplications of geometry. If, as we suspect, there is littleexperience in doing so, then we recommend that theQCA commissions work to develop more appropriateforms of examination questions in geometry.

GCSE examinations in 2003 will contain a compulsorycourse work element. This will consist of two extendedtasks each contributing 10% of the total marks. One ofthese has to be from Ma4 Handling data. The other is todemonstrate skills from Using and applying mathematicsin the context of either Ma2 Number and algebra or Ma3Shape, space and measures. The working groupwelcomes to some extent this extension of assessmenttechniques but queries the rationale used to make Ma4compulsory and not Ma3. We consider that theopportunity afforded for extended work in Ma3 would bean effective way to ensure that some of our objectives forgeometry teaching are more effectively fulfilled. We areconcerned that teachers faced with a choice between

Ma2 and Ma3 may reject geometry in favour, say, ofalgebra - perhaps because of their own subject confidenceor because they judge the more algorithmic nature ofsome forms of algebraic enquiry to be a ‘safer bet’. Thuswe recommend that GCSE mathematics should includesome compulsory course work in geometry.

The examinable course work element in the new AS/A-level mathematics course is almost entirely restricted tothe applications, such as statistics and mechanics. If, aswe recommend, greater opportunity is afforded tostudents to extend their study of geometry on thesecourses then a review of the appropriate means ofassessment is also needed. By contrast the FSMUs havetheir own forms of assessment - usually 50%examination and 50% coursework. Many units specifyand assess the use of appropriate ICT.

If the assessment framework for the curriculum ingeometry can be developed to include both betterexamination questions and a reasonable contributionfrom extended course work, then we believe thatteachers will also be encouraged to develop formativeassessments in geometry for their students. It is a widerquestion than this working group’s remit to considerwhether teachers’ assessments of students’ progressshould contribute to National Curriculum assessmentand to public examinations. Such a change wouldrequire the kind of reinstatement of teachers’professional judgements which is discussed in therecent government Green Paper [DFEE, 2001].

Recommendation 14:

We recommend that a review be made of themethods of assessment and examination used inmathematics at Key Stage 3, at GCSE and in post-16 qualifications to ensure that appropriate creditis given for the attainment of specific geometricalobjectives.


8 The role of assessment

Key Principle 7: The most significant contribution toimprovements in geometry teaching will be made bythe development of good models of pedagogy,supported by carefully designed activities and resources,which are disseminated effectively and coherently toand by teachers.

We did not enquire further into the impact of differentforms of school organisation, such as ‘setting’ andmixed-ability teaching, nor did we reach a consensusabout the issues arising from providing a more inclusivecurriculum or from providing some more differentiatedcurricula. We anticipate that pilot studies in goodpractice may provide some helpful guidance in respectof these issues.

We now turn to the description of the 11-16 NationalCurriculum. The phraseology used throughout is “pupilsshould be taught to...”. The National Curriculumhandbook sets out in some detail what should betaught, but not why, or how. A good deal of scope forinterpretation still rests with the teacher. We are awarethat a tendency has recently developed in teaching themathematics National Curriculum which breaks it downinto a large number of very limited objectives -sometimes known as ‘bite-sized chunks’. Such anapproach can, and often does, result in fragmentationand in the failure to develop important links betweencurriculum areas. We believe that the successfulimplementation of the Ma3 component in theclassroom will only be achieved if teaching programmesare focused and coherent, and if they develop linkswithin geometry and mathematics generally whereappropriate. In Appendix 13 we give some examples ofways in which aspects of the geometry curriculum couldbe integrated within a particular theme, and also whereaspects of geometry could be linked with other areas ofmathematics such as algebra and handling data.

Individual teachers implement the curriculum by planningschemes of work and lessons for their classes. So it is amatter of the greatest importance to ensure that teachershave the necessary information, skills and resources tointerpret the aims and objectives of the curriculum.Recent moves to ameliorate problems of recruitment andretention of teachers, together with the government’sintention to modernise working practices in health andeducation, mean that there is now a much morefavourable climate for improving the system of teachers’continuing professional development (CPD). The workinggroup welcomes the declared intention to provide CPDsupport to improve and update teachers’ subjectknowledge and related pedagogy.

Research, such as that reported by Hoyles and Healey,has confirmed the views of experienced teachers in

schools that there are many teachers of mathematicswho have large gaps in their knowledge of geometry.Similarly, we believe that there are also many teacherswho have been taught geometry through styles ofteaching which we would not advocate as appropriate.Thus our view is that in respect of geometry teachingthere is a need for a significant CPD initiative.Government is giving greater attention to spreadinggood practice between teachers and schools throughinitiatives, such as the beacon schools. The DfES (whilststill DfEE) recently launched its CPD strategy. This has itsown website at: www.dfee.gov.uk/teachers/cpd whereCPD initiatives are presented under the strap line“Learning from each other... Learning from what works”.The working group welcomes this approach. We regard itas vital that pilot studies should be carried out withoutdelay to identify and enhance good practice in theteaching of geometry. At the same time planning shouldtake place for a national system of provision for CPD ingeometry and its teaching. This could be within theframework of the Key Stage 3 strategy. One idea whichhas received some support is the provision of two weekgeometry summer schools for serving teachers, teacherscurrently in training and those about to embark on acourse of initial teacher education (ITE). Financialinducements may be needed to encourage attendance invacation time. By concentrating on subject knowledge ingeometry, as well as its teaching, such courses shouldgenerate enjoyment of mathematics and thus help as onepart of the long term process of sustaining or renewingteachers’ enthusiasm for it.

New graduates entering courses of initial teachereducation have very varied backgrounds in geometry.Many will have experienced little, if any, geometry atsixth form or university level. Within the currentstatutory curriculum for the initial training of secondarymathematics teachers there is little scope to provide therich overview of geometry that we believe is essentialfor effective teaching. Further professionaldevelopment for teachers early in their career isessential, but is most likely to concentrate on thedevelopment of their teaching skills. So it is importantthat, in parallel with developments in CPD to supportthe teaching of geometry, there is a recognition of theneed to improve the geometrical background of thoseintending to enter mathematics teaching during, orbefore, their initial training.

In order to support the developments in the effectiveteaching of geometry which we seek there is a need fora variety of materials in both printed and digital form, aswell as resources such as models, posters, activity kits,videos, libraries of digital images, computer softwareand the like. Some of this already exists and we provide afar from exhaustive list of these in Appendix 14. An


9 Teaching of geometry

important activity will be to review the current provisionand to develop new materials and resources asappropriate.

Geometry teaching outside primary schools can be, andhas been, conducted with a minimal amount ofequipment - such as a stick of chalk and a piece of string(echoing images of ancient Greeks drawing in the sand).Our view is that teachers should now have at theirdisposal an appropriate variety of equipment fromwhich to select, depending on fitness for purpose. Inparticular we wish to see the potential of ICT realised insupporting the teaching and learning of geometry.There is already software available, such as for dynamicgeometry (DGS), but its use is not widespread. Manyschools do not have licences for the software. There isalso a need for the development of additional software,such as to support work in 3-dimensions. Increasingnumbers of schools and colleges are now beingequipped with interactive whiteboards - where acomputer image is projected onto a touch sensitivescreen. This medium has considerable potential forinteractive whole-class teaching of geometry. We wouldlike to see the funding to schools for ICT being usedmore effectively to support the geometry curriculum.

The mathematics professional associations have a keyrole to play in each of these developments inpartnership with the newly formed General TeachingCouncil (GTC) and other bodies such as the RoyalSociety, Higher Education institutions, the NationalNumeracy Strategy, QCA, Ofsted, TTA and BECTa.

Recommendation 15:

We recommend that the relevant governmentagencies work together with bodies, such as themathematics professional associationsrepresented on JMC, to provide a coherentframework for supporting the development ofteaching and learning in geometry. This willinvolve:a) the recognition and development of good

practice in geometry teaching through pilotstudies and research;

b) the design of programmes of continuingprofessional development and initial teachereducation;

c) the production of supporting materials; andd) the establishment of mechanisms to provide

supporting resources, including ICT.


Key Principle 8: It is a matter of national importance thatas many of our students as possible fully develop theirmathematical potential. Geometry, with its distinctiveappeal, should make mathematics attractive to a widerrange of students.

In launching UK Maths Year 2000, the Prime Minister madeclear the importance of mathematics in the education ofthose creative and flexible thinkers on whom our nationaleconomic prosperity will depend. The demands ofcommerce and industry for articulate graduates withmathematical skills far outstrips the supply of graduates inmathematics and related fields. One consequence of this isthe current severe shortage of new teachers, especially formathematics in secondary schools. For a variety of reasons,insufficient numbers of our ablest students are choosing topursue mathematics as a specialism following both GCSEand AS/A-level. There is no universal panacea. So it is vitalthat we take any opportunity to review where and how thesubject could be made more interesting, attractive,relevant, challenging, rewarding and engaging to allstudents. We are convinced that geometry has a lot to offerin this respect. For some students it may be the logicalaspects which are the most appealing, for others it may bethe visualising, or the modelling, or the historical andcultural, or the visual and aesthetic aspects.

In general we believe that students are not givenenough information about the importance ofmathematics in the world of work, and the significantadvantages a mathematical education can bestow interms of employability. The ways in which the

performance of secondary schools and colleges arepublished through examination results takes no accountof the relative national economic importance of somesubjects over others. So, for example, there is a positiveincentive for institutions to persuade students to choosesubjects in which it is easier to achieve high grades at A-level than those subjects judged harder, which includemathematics and physics.

Overall the profile of mathematics needs to be higher inschools, colleges and universities if we are to attractmore students at all levels of attainment to realise theirpotential in the subject. This means that still more needsto be done to improve the status of mathematicsteaching, and to attract (and retain) good recruits. It alsomeans that students should be made more aware of therelationships between mathematics and the othersubjects they study. We believe that geometry is a goodvehicle for achieving this aim.

Recommendation 16:

We recommend, in terms of mathematics ingeneral, that:a) better publicity and information be provided to

schools, students and parents about the careeropportunities afforded by studyingmathematics; and

b) ways be sought to encourage schools andcolleges to attract more students to studymathematics post-16, particularly at A-level.


10 Improving the take up of mathematics

For mathematics 11-16, we have concluded that thegeometrical content of the new National Curriculum,with a few adjustments, forms an appropriate basis fora good geometry education. In order for this to beachieved, considerable changes are needed in the waygeometry is taught. It is vital that those working toimprove mathematics education ensure that their workcontributes significantly to improvements in geometry(as well as mathematics) teaching. Bringing aboutimprovements in geometry teaching will require asignificant commitment to a substantial programme ofcontinuing professional development, together withthe development of appropriate supporting materials.

For mathematics post-16 we have concluded that thereare insufficient opportunities for students to build ontheir 11-16 studies in geometry. Those concerned withcurriculum design need to review the structure of post-16 qualifications in mathematics to ensure they providebetter opportunities for students to continue to studygeometry. More generally there is at present a severeshortage of those with good mathematical skills - andthe provision of challenging and interesting geometricalcontent and contexts should be a valuable means tomake mathematics a more attractive subject of study formore students.


11 Conclusion

References

Cockcroft, W.H. (1982), Mathematics counts. Report of the Committee of Inquiry in Teaching of Mathematics.London: HMSO.

DfEE & QCA (1999), Mathematics: The National Curriculum for England. London: HMSO.

DfEE (2001), Schools: Building on Success. Green Paper. London: The Stationery Office.

Hoyles, C. & Healey, L. (In press) Curriculum change and geometrical reasoning, in Boero, P. (ed) Theorems in School,Dordect: Kluwer

Mammana, C. & Villani, V. (Eds.) (1998), Perspectives on the Teaching of Geometry for the 21st Century. Dordrecht:Kluwer.

National Numeracy Strategy (2000), Framework for teaching mathematics: Years 7 to 9. London: DfEE (availablefrom http://www.standards.dfee.gov.uk/ keystage3/strands/mathematics/ )

Oldknow, A.J. & Taylor, R.N. (1999), Engaging mathematics: gaining, retaining and developing students’ interest.London: Technology Colleges Trust.

Royal Society / JMC (1996), Teaching and Learning Algebra pre-19. Report of a Royal Society / JMC working groupchaired by Professor R Sutherland. London: The Royal Society.

Willson, W.W. (1977), The Mathematics Curriculum: geometry. London: Blackie / Schools’ Council.

Glossary

AS level Advanced Subsidiary Level – a qualification between GCSE and A-levelBECTa British Educational and Communications Technology AgencyCADCAM Computer Aided Design and Computer Aided ManufactureCPD Continuing Professional DevelopmentDfEE Department for Education and Employment (now replaced by the DfES) DfES Department for Education and SkillsDGS Dynamic Geometry SoftwareFSMU Free Standing Mathematics UnitGCSE General Certificate of Secondary EducationGTC General Teaching CouncilHE Higher EducationICMI International Commission on Mathematics InstructionICT Information and Communications TechnologyIT Information TechnologyITE Initial Teacher EducationITT Initial Teacher TrainingJMC Joint Mathematical Council of the United KingdomKS Key Stage (of the National Curriculum)LEA Local Education AuthorityMa3 “Shape, space and measures” component of the mathematics National Curriculum NC National CurriculumNNS National Numeracy StrategyNOF New Opportunities Fund (a Government funding initiative)Ofsted Office for Standards in EducationQCA Qualifications and Curriculum Authority


12 References and glossary

QTS Qualified Teacher StatusSEU Standards and Effectiveness Unit (of the DfES / DfEE)TIMSS Third International Mathematics and Science StudyTTA Teacher Training Agency


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