Cockcroft (1982)

Notes on the text
Preliminary pages Foreword, Membership, Contents, Introduction

Part 1
Chapter 1 Why teach mathematics?
Chapter 2 The mathematical needs of adult life
Chapter 3 The mathematical needs of employment
Chapter 4 The mathematical needs of further and higher education

Part 2

Chapter 5 Mathematics in schools
Chapter 6 Mathematics in the primary years
Chapter 7 Calculators and computers
Chapter 8 Assessment and continuity
Chapter 9 Mathematics in the secondary years
Chapter 10 Examinations at 16+
Chapter 11 Mathematics in the sixth form

Part 3

Chapter 12 Facilities for teaching mathematics
Chapter 13 The supply of mathematics teachers
Chapter 14 Initial training courses
Chapter 15 In-service support for teachers of mathematics
Chapter 16 Some other matters
Chapter 17 The way ahead

Appendices

Appendix 1 Statistical information
Appendix 2 Gender differences in mathematical performance
Appendix 3 List of those who made submissions
Appendix 4 Visits and meetings
Appendix 5 List of abbreviations

Index

Report of the Committee of Inquiry into the teaching of mathematics in schools under the chairmanship of Dr WH Cockcroft

London: Her Majesty's Stationery Office 1982
© Crown copyright material is reproduced with the permission of the Controller of HMSO and the Queen's Printer for Scotland.

ISBN 0 11 270522 7

Chapter 9 Mathematics in the secondary years
[pages 128 - 157]

436 As we have continued our work we have become increasingly aware of the implications for mathematics teaching of the differences in attainment in mathematics which exist among pupils of any given age and of the extent to which these differences increase as pupils become older. We drew attention in paragraph 342 to the 'seven year difference' which exists among 11 year olds. If we relate this to work in the secondary years, it means that the mathematical understanding of some pupils who transfer to secondary school at 11 is likely already to be greater than that of some pupils who have just left school at 16. On the other hand, some of those who arrive at the same time may not, while at school, attain the understanding which some of their fellow 11 year olds already possess. When considering work in the secondary years it is also necessary to remember that pupils learn at very different speeds and that, in the sense we have explained in paragraph 228, mathematics is a hierarchical subject.

437 Thus there are three factors - level of attainment at the beginning of the course, speed of learning and the need to obtain a sufficient understanding of certain topics before being able to proceed to others which depend on them - which must be fundamental to our consideration of the content of the mathematics curriculum at the secondary stage. Unless the great differences which exist between pupils are recognised in the provision which is made, those whose potential is high will be denied the opportunity to make the progress of which they are capable and those whose attainment is limited are liable to experience continuing and dispiriting failure. It is from this starting point that we discuss the teaching of mathematics in the secondary years.

438 We start by explaining the differing senses in which we use the words 'syllabus' and 'curriculum'. We use 'syllabus' to denote a list of mathematical topics to be studied but 'curriculum' to include the whole mathematical experience of the pupil; in other words, both what is taught and how it is taught. The curriculum therefore includes the syllabus; it is concerned with the way in which the syllabus is presented in the classroom as well as with other matters which are important. For example, problem solving, logical deduction, abstraction, generalisation, conjecture and testing should play a part in the work of all pupils. However, the kind of work which is undertaken and the methods which are used will of necessity vary with different groups of pupils; nor will the level of sophistication and the depth of understanding which are achieved be the same for all.

439 At the present time the mathematics syllabuses which are followed by most pupils during the secondary years are very strongly influenced by the content of the O Level syllabuses intended for only the top 20 to 25 per cent of pupils in terms of their attainment in mathematics. We believe it is necessary to explain how this has come about. Twenty years ago there were very great differences in the mathematics courses which were followed by pupils in different types of school. There were very few comprehensive schools and most pupils attended either grammar, technical or secondary modern schools. Pupils in grammar and the few technical schools and a small number of pupils in secondary modern schools followed a mathematics course leading to O Level which included arithmetic, algebra, geometry and trigonometry; some of these pupils also studied 'additional mathematics' as a second mathematical subject at O Level before entering the sixth form. Other pupils were usually taught arithmetic only, which sometimes included elements of geometry arising from calculations of area and volume, geometrical constructions, ratio and scale drawing and also simple graphs of a non-algebraic kind; most of these pupils left school at 15 and attempted no nationally recognised examination. Some of this latter group attempted regional or local examinations which varied in scope and status.

440 CSE, which pupils were able to attempt for the first time in 1965, was intended to replace these regional and local examinations and, together with O Level, to provide a public examination system for pupils aged 16 whose attainment in the subject concerned was within that of about the top 60 per cent of the school population. In consequence, many pupils in secondary modern schools, who had hitherto been taught only arithmetic, started to follow the broader CSE mathematics course, even though not all of these pupils stayed at school long enough to attempt the examination (1). Since the raising of the school leaving age in 1973, the number of pupils who attempt public examinations, especially CSE, in mathematics has increased greatly; the report of the National Secondary Survey (2) shows that, in some of the comprehensive schools which were visited, over 80 per cent of the pupils in the fourth and fifth years were following courses leading to O Level or CSE examinations in mathematics.

441 Success at grade 1 in CSE has always been accepted in lieu of a pass (now grade A, B or C) at O Level for purposes of qualification for entry to further and higher education as well as to many kinds of employment. From the outset, therefore, this has encouraged the use of similar lists of content for both CSE and O Level syllabuses. When CSE was first introduced, relatively few schools offered courses for both O Level and CSE but, as the number of comprehensive schools has increased, so has the number of schools in which it has become necessary at some stage to decide whether a pupil should attempt the O Level or CSE examination. Although in the case of some pupils the choice presents no difficulty, many schools prefer to postpone the choice for as long as possible for some of their pupils and for it to be possible for a pupil to change courses if the first choice proves to be wrong. This has led to understandable pressure that O Level and CSE syllabuses in mathematics should be 'compatible', that is that they should contain substantially the same mathematical topics, and has lessened still further the differences between the syllabus content of O Level and CSE examinations. Our own study of a selection of CSE syllabuses in mathematics from the earliest years of the examination up to the present day shows that the content of most of these syllabuses has gradually increased. In particular, there has been an increase in the number of more difficult topics of an algebraic kind which have been included and a consequent increase in the conceptual difficulty of the mathematics included in the examination and hence in the courses which pupils are required to follow.

442 We believe it is clear from the preceding paragraphs that the changes in the examination system and in the organisation of secondary schools which have taken place in recent years have influenced the teaching of mathematics in ways which have been neither intended nor sufficiently realised. At the present time up to 80 per cent of pupils in secondary schools are following courses leading to examinations whose syllabuses are comparable in extent and conceptual difficulty with those which twenty years ago were followed by only about 25 per cent of pupils. We have therefore moved from a situation in which, twenty years ago, there was in our view too great a difference between the mathematics syllabuses followed by those who attempted O Level and those who did not to one in which, at the present time, there is far too little difference in the mathematics syllabuses which are followed by pupils of different levels of attainment. Because, for the reasons which we have explained, it is the content of O Level syllabuses which exerts the greatest influence, it is the pupils whose attainment is average or below who have been most greatly disadvantaged.

443 The difference in the level of mathematical attainment between a pupil who achieves CSE grade 1 and one who achieves grade 5 is very considerable yet, in CSE Mode 1 examinations (3), both have to attempt the same examination papers. In some subjects other than mathematics it is possible to set questions which candidates of a wide range of attainment are able to attempt because the marks or grades which are awarded to the candidates can be determined according to the quality of their answers. In mathematics examinations of the kind which are most commonly used, an approach of this kind is not usually possible. This is because increase of attainment in mathematics is demonstrated not only by knowledge of additional topics but also by ability to tackle more demanding questions on topics which have been covered earlier in the course. An examination question on an 'elementary' topic which enables candidates to exhibit the level of understanding required for the award of a high grade is likely to be much too demanding for candidates who are able to attain only a low grade; such candidates may not even be able to start the question. Conversely, a question on the same topic which is suitable for candidates who will attain a low grade provides the examiner with little evidence of the capability of those candidates who are able to attain a high grade. It follows that an examination paper in mathematics which is attempted by candidates of a wide range of attainment is bound to contain questions which are too difficult for those who attain low grades. This means that low grades are awarded as a result of a very poor performance in an examination which draws attention not to the extent of a candidate's knowledge and understanding, but rather to those things which he does not know.

444 As we explained in paragraph 196, CSE regulations state that, in any subject, 'a 16 year old pupil of average ability who has applied himself to a course of study regarded by teachers of the subject as appropriate to his age, ability and aptitude may reasonably expect to secure grade 4'. It may not, however, be generally realised that the mark required to achieve grade 4 in mathematics is very often little more than 30 per cent on the papers which the candidate is required to attempt and that grade 5 - 'the performance just below that to be expected from a pupil of average ability' - is likely to be awarded to a candidate who scores little more than 20 per cent of the marks. We cannot believe that it can in any way be educationally desirable that a pupil of average ability should, for the purpose of obtaining a school-leaving certificate, be required to attempt an examination paper on which he is able to obtain only about one third of the possible marks. Such a requirement, far from developing confidence, can only lead to feelings of inadequacy and failure. Although most pupils attempt the CSE examination itself on only one occasion, it must be remembered that during the year leading up to the examination they will almost certainly practise past papers and probably also take some form of 'mock' examination. Those pupils who will achieve the lowest grades or will be ungraded cannot therefore fail to become aware of how little they are likely to be able to achieve on the examination questions which they will be required to attempt. Reference to Figure F (paragraph 195 [in chapter 5]) will show how large is the proportion of pupils to whom this applies.

445 We wish now to draw attention to a further difficulty which arises for both examiners and teachers as a result of the fact that CSE examinations are not suitable for many of the pupils who now attempt them. Examiners have a duty to set papers which cover as much of the syllabus as possible. Because they are aware that many low-attaining candidates will attempt the papers, they feel obliged to include within them a number of trivial questions on those topics in the syllabus which are conceptually difficult so that low-attaining candidates may find some questions which they are able to attempt. Teachers in their turn feel obliged to cover as much of the syllabus as possible so that their low-attaining pupils may be able to answer such questions, even though some of the topics which are included are conceptually too difficult for these pupils. This leads to teaching of a kind which, instead of developing understanding, concentrates on the drilling of routines in order to answer examination questions. We therefore have a 'vicious circle' which is difficult to break.

446 Many teachers are aware of this problem but feel unable to do anything about it. Their dilemma was expressed vividly at one of the meetings which we held with groups of teachers. 'I know that I should not be teaching in this way and I would much prefer not to do so, but I know also that I have a responsibility to do all that I can to make it possible for my pupils to obtain a CSE grade.' However, there can be little doubt that teaching of this kind can lead to disenchantment with mathematics on the part of many pupils and also provide the reason for the comments of young employees, noted during the Bath and Nottingham studies (4), that some of the mathematics which they had done at school had made no sense to them.

447 Some teachers have endeavoured to resolve their problem by devising limited grade CSE examinations under the Mode 3 procedure. Such examinations usually have a reduced syllabus content and also a maximum grade (commonly grade 3) which can be awarded. This enables the content of the examination to be matched more suitably to the needs of those who are likely to achieve the lower grades. More recently some CSE boards have themselves introduced Mode 1 examinations in mathematics or arithmetic with a maximum of grade 3. We support provision of this kind and regret that there are some CSE boards which do not permit limited grade examinations in mathematics. We hope that, in the light of the discussion in this chapter, they will reconsider their position and permit examinations of this kind until such time as a single system of examination at 16+ is introduced. However, even the use of limited grade CSE examinations cannot resolve the problem of providing a suitable curriculum for those pupils for whom CSE is not intended; we return to this point in paragraph 455.

448 We have discussed CSE examinations first of all because it is in the teaching of those pupils who enter for this examination and obtain low grades or are unclassified that we believe the greatest problems to exist. We must, however, emphasise that pupils who achieve grade E at O Level are also unlikely to have been provided with a course which has been suited to their needs. We have been told that schools sometimes find themselves under pressure from parents to prepare pupils whose attainment is not very high for O Level rather than CSE and that the change in the method of recording examination results on O Level certificates, which was introduced in 1975, has increased this pressure (5). In our view such pressure is misguided and can be harmful to the mathematical development of the pupils concerned. However, we recognise that it is very difficult for teachers and schools to withstand it in view of the higher status which O Level enjoys and the greater acceptability which is often attributed to it by parents and employers. As an example of this attitude we instance one employer who told us that he preferred to take applicants who had followed an O Level course but had not even achieved grade E rather than applicants with a good grade in CSE. This is an attitude which we believe to be mistaken and which we regret.

449 We have considered the question of mathematics examinations and their effect on the mathematics curriculum in some detail because we believe that it is essential to appreciate the way in which existing examination syllabuses have come to influence work in the classroom, especially in the later secondary years, and that this influence can have harmful effects on the mathematical development of pupils. In our view, very many pupils in secondary schools are at present being required to follow mathematics syllabuses whose content is too great and which are not suited to their level of attainment. Efforts to introduce pupils to as much of the examination syllabus as possible result in attempts to cover the ground too fast for understanding to develop. The result is that very many pupils neither develop a confident approach to their use of mathematics nor achieve mastery of those parts of the syllabus which should be within their capability.

450 This situation has arisen because the syllabuses now being followed by a majority of pupils in secondary schools have been constructed by using as starting points syllabuses designed for pupils in the top quarter of the range of attainment in mathematics. Syllabuses for pupils of lower attainment have been developed from these by deleting a few topics and reducing the depth of treatment of others; in other words, they have been constructed 'from the top downwards'. We believe that this is a wrong approach and that development should be 'from the bottom upwards' by considering the range of work which is appropriate for lower-attaining pupils and extending this range as the level of attainment of pupils increases. In this way it should be possible to ensure both that pupils are not required to tackle work which is inappropriate to their level of attainment and, equally importantly, that those who are capable of going a long way are enabled to do so.

Courses for 11-16 year old pupils

451 We cannot set out in detail the content of courses for 11-16 year old pupils; we can only indicate essentials and point to certain principles which should govern decisions on the curriculum. We believe that there should be a core of content which should be included in the mathematics course for all pupils; we discuss this further in paragraphs 455 to 458. Beyond that, both the topics to be included and the teaching approach to them must be judged in the light of the needs of particular groups of pupils. We believe it should be a fundamental principle that no topic should be included unless it can be developed sufficiently for it to be applied in ways which the pupils can understand. For example, we see no value in teaching and examining, in isolation and as a skill, the addition and multiplication of matrices to pupils whose knowledge of algebra and geometry is not sufficient for them to be able to appreciate contexts within which matrices are of use. On the other hand, some pieces of work may claim a place in syllabuses for higher-attaining pupils because of needs beyond the age of 16. It is, for example, hard to achieve the degree of facility in algebraic manipulation which is required for A Level work in mathematics if pupils have not started to develop it before they enter the sixth form and so this aspect of algebra must receive due attention for such pupils.

452. We therefore consider that there should be both similarities and differences in the mathematics courses which are followed by pupils of different levels of attainment. The similarities will be in the common core of mathematics of which all will have experience and in the approach which underlies the mathematics teaching. The differences will be in additional content, in depth of treatment, in methods of assessment and in the emphases given to different aims. For example, while some pupils will achieve a high degree of competence in simple arithmetic almost incidentally, others will continue to experience difficulty throughout the secondary years and so will need a course which takes account of this. In the paragraphs which follow we discuss the way in which we believe it is necessary to approach the formulation of such a 'differentiated curriculum'; in other words, the provision of different courses to meet the differing needs of pupils in the secondary years.

453 Since we started our work, the report of the first Secondary Survey carried out by the Assessment of Performance Unit (APU) (6) and the report of the Concepts in Secondary Mathematics and Science (CSMS) Project (7) have been published. These provide valuable information about levels of performance in a wide range of mathematical topics. The CSMS work concentrates in particular on the performance of lower-attaining pupils. Both of these reports illustrate clearly that large numbers of pupils have difficulties in understanding and applying many mathematical processes which are commonly thought to be quite elementary and that these difficulties are much greater than is generally realised either by the public at large or by some of those who teach mathematics. The very low marks which are attained by many pupils who attempt CSE examinations reinforce the evidence which is provided by the APU and CSMS reports.

454 Failure to understand a mathematical topic can result from the fact that the mathematical concepts involved are too difficult for the stage of mathematical development which a pupil has reached. It can also result from teaching which pays too little attention to practical experience or which moves ahead too rapidly so that understanding does not have time to develop. If pupils are following a syllabus which is too large and too demanding, both reasons can contribute to their poor performance. We therefore believe that, when designing a curriculum which is suitable for lower-attaining pupils, the syllabus should not be too large so that there is time to cover the topics which it contains in a variety of ways and in a range of applications. We believe that it is by approaching the teaching of mathematics in this way that understanding will best be developed and pupils enabled to achieve a reasonable degree of mastery of the work which they have covered, so that they will develop sufficient confidence to make use of mathematics in adult life. We realise that there may be some who fear that a reduction in the content of the mathematics syllabus will lead to an undesirable narrowing of the curriculum but this is a view which we do not accept. If the approaches to teaching are as varied as we have recommended in Chapter 5, we believe that such a reduction can enable a wider mathematics curriculum to be provided. Teachers will have more time to develop methods which evoke the best response from their pupils. Pupils will be freed from pressure to cover too much ground and will be able to undertake revision and consolidation on a regular basis to the extent which is necessary.

Foundation list

455 We have therefore decided to set out a 'foundation list' of mathematical topics which, while it should form part of the mathematics syllabus for all pupils, should in our view constitute by far the greater part of the syllabus of those pupils for whom CSE is not intended, that is those pupils in about the lowest 40 per cent of the range of attainment in mathematics. This group includes those who at present attempt CSE but achieve minimal or no success. We recognise that the content of the list which we propose is considerably less than the content of the syllabuses which many low- attaining pupils now follow. We believe, however, that the smaller syllabus content will make it possible for the teacher to develop the mathematics curriculum for these pupils in the way which we have set out earlier.

456 Some people have urged us not to attempt to put forward such a list in view of two possible dangers. One is that members of the public will expect that all pupils will be able to master everything which is in the list. The other is that some teachers will regard the list as comprising the total mathematics syllabus for lower-attaining pupils and will teach only those things which are specifically mentioned in it. We are aware of both these dangers but feel that we should nevertheless suggest a list of topics which we expect to be within the capability of most pupils and which we believe that all pupils should attempt so far as they can. We would not wish that any section of the list should be omitted completely, even for those whose attainment is very low, though it must be accepted that such pupils may not be able to tackle all the items in each section.

457 We have not found it easy to draw up the list nor do we suggest that it should be regarded as definitive in every detail. However, we believe it to be a sound basis on which to build; as with all lists of content it will, in any case, need to be reviewed regularly. In preparing it we have started from the premise that mathematics should be presented and taught in a context in which it will be applied to solving a variety of problems. Once a skill has been identified as worthwhile, it is appropriate to practise it. However, such practice should be carried out in order to enable the skill to be used in the solution of problems and not as an end in itself. It is for this reason that in drawing up our list we have added comments (in italics) which both amplify the list and also draw attention to ways in which we believe the various topics should be presented. These comments concentrate especially on the needs of those pupils for whom the foundation list will constitute the greater part of the mathematics syllabus. We do not intend that they should in any way be seen as limiting the range of work which should be attempted by pupils whose attainment is higher.

Foundation list of mathematical topics

458 Throughout their mathematics course, pupils should

• read, write and talk about mathematics in a wide variety of ways;
• carry out calculations in a variety of modes - mentally, on paper and with a calculator;
• associate calculation with measurement in appropriate units and become familiar with the relative size of these units.

NUMBER

Count, order, read and write positive and negative whole numbers and use them in context; eg what is the rise in temperature from -3°C to 10°C?

Understand place value in numbers of up to 4 digits.

Possess confident recall of addition and multiplication facts up to 10 + 10 and 10 x 10 and the related subtraction and division facts.

Understand interrelationships of the kind 13 x 8 = (10 x 8) + (3 x 8).

Be able to select the appropriate operation (addition, subtraction, etc) for use in the solution of a variety of practical problems.

Pupils should possess some reliable method (however unconventional) of carrying out calculations without the use of a calculator when the numbers are small, and with a calculator when larger numbers are involved. There is a need to develop and encourage intuitive methods of both written and mental calculation.

Pupils should appreciate the implications of movement of figures relative to the decimal point as a result of multiplication or division by a power of 10.

Pupils should be able to perform calculations involving the word 'of' such as 1/3 of £4.50.

Know the decimal equivalents of 1/4, 1/2, 3/4, 1/10, 1/100 and also that 1/3 is about 0.33. Be able to convert fractions to decimals with the help of a calculator.

The emphasis should always be on the use of number skills in everyday situations. Fluency in computation using methods whose basis is understood should be developed by short and frequent spells of practice; prolonged spells of routine practice are liable to be counter-productive.

Recognise coins and notes and know that 100p = £1. Handle money with confidence.

Carry out simple transactions, performing necessary calculations either mentally or on paper.

Add and subtract small sums of money without a calculator.

Multiply or divide a sum of money by a single digit without a calculator. Perform more complex calculations involving money using any appropriate method, including ready reckoner or calculator.

The emphasis in teaching should be on the use of money in everyday activities such as shopping, leisure activities, 'do-it-yourself', and on budgeting and household accounts.

It is clear that very many pupils find great difficulties with the concept of percentage. We recommend that teaching of percentage should be based on the idea that 1 per cent means '1p in every pound' or 'one in every hundred', and not on the use of particular formulae.

Appreciate the use made of percentages in everyday life.

The use of percentages should be linked to activities of the kind listed under 'money'. The examples used should be realistic and relevant to the pupils concerned. Emphasis should also be given to the way in which percentages are used both for comparative purposes in many everyday situations and also as a numerical measure based on a 100-point scale of reference.

Use a calculator efficiently to add, subtract, multiply and divide, and to convert a fraction to a decimal.

Appreciate the need for careful ordering of operations when using a calculator.

Be able to select from the calculator display the number of figures which is appropriate to the context of the calculation.

Simple calculations should in general be carried out mentally or on paper and not with a calculator. Emphasis should be placed on using appropriate procedures for checking the reasonableness of the answer which has been obtained. It is not intended that the calculator should be used for calculations such as 3.52 x 7.04 which are unrelated to a practical context.

gross wage = wage per hour x number of hours worked;
total cost = number of units used x cost per unit + standing charge.

When the numbers involved are straightforward, pupils should also be able to carry out calculations of this kind without using a calculator.

Be able to tell the time and understand times expressed in terms of 12 and 24 hour clocks.

Be able to calculate the interval between two given times, and the finishing time given the starting time and duration.

Use a bus or train timetable.

Solve simple problems involving time, distance and speed.

Emphasis should be placed on mental calculation where appropriate and on the use of both digital and traditional 'clock' display.

Measure length, weight and capacity using appropriate metric units.

Understand the relationship between millimetres, centimetres, metres, kilometres; grams, kilograms, tonnes; millilitres, centilitres, litres; know that 1 litre is equivalent to 1000 cubic centimetres.

Have a 'feel' for the size of these units in relation to common objects within the pupils' experience.

Use the following imperial units: inch, foot, yard, mile; ounce, pound; pint, gallon; and know their approximate metric equivalents, ie that 3 feet is about equal to 1 metre, 5 miles is about 8 kilometres, 2 lb is about 1 kilogram, 1 gallon is about 4½ litres.

Understand and use simple rates; eg £ per hour, miles per gallon. Read meters and dials of various types.

Emphasis should be given to practical activities involving measurement and estimation. Opportunity should be taken to relate teaching to measurement of the kind encountered in other curricular areas such as science, geography, home economics, craft and sport.

Organise systematically the collection and tabulation of simple data.

Read and interpret simple graphs and charts and extract specific information from them; construct them in simple cases.

Extract information presented in tabular form; eg cost of making a telephone call.

Pupils should be introduced to a wide variety of forms of graphical representation and use should be made of published information. Pupils should be encouraged to discuss critically information presented in diagrammatic form, especially in advertising. The drawing of graphs should wherever possible be based on information collected as a result of practical activities.

Work with flow-charts should include discussion aimed at developing the logic used in mathematical arguments, such as 'if ... then ...'.

Recognise and name simple plane figures; understand and use terms such as side, diagonal, perimeter, area, angle.

Understand and use terms relating to the circle; centre, radius, diameter, circumference, chord.

Appreciate the properties of parallelism and perpendicularity; be able to measure angles in degrees.

Draw a simple plane figure to given specifications.

All pupils should learn to use drawing instruments such as ruler, compasses, set square, protractor. For some pupils this may require considerable practice. Plane and solid shapes should be available which pupils can handle and measure, and use for building.

Attention should be paid to the representation of three-dimensional figures in two dimensions. Pupils should be able to recognise the two-dimensional drawing corresponding to a given solid, and be able to recognise and build solids from such drawings. Use should be made of such things as plans of models, dress patterns, scale drawings, photographs, maps.

Find the volume of a rectangular solid.

Understand and use the fact that the circumference of a circle = pi x diameter; know that pi is a little more than 3.

Calculations relating to plane and solid shapes provide opportunity to practise mental calculation, written calculation and the use of a calculator in a variety of ways and at different levels. Emphasis should be placed on the use of correct units for area and volume and on the use of estimation to ensure that an answer is reasonable.

There is likely to be considerable overlap between this topic and work on graphical and pictorial representation, especially in respect of the interpretation and construction of simple plans and elevations.

Be able to visualise and understand simple mechanical movement, including the working of simple linkages.

Bicycle gears, car jacks of all kinds, up-and-over garage doors are a few examples.

Understand the use of ratio as applied to such things as mixtures, eg 2 parts sand to 1 part cement; and recipes, eg work out the quantities required for 6 people from a recipe which serves 4.

This work provides opportunity to discuss ideas such as 'best buy'. Applications to craft work and model making provide overlap with work on scale drawing.

For example, as speed increases, distance travelled in a given time increases; as speed increases, time to travel a given distance decreases. For some pupils it will not be appropriate to attempt detailed calculation.

One aim should be to encourage a critical attitude to statistics presented by the media.

Emphasis should be placed on the relevance of probability to occurrences in everyday life as well as to simple games of chance. For many pupils it will not be appropriate to undertake work involving combined probability.

Attention should be paid to the different uses of the word 'average' in newspaper reports, but it is not intended that pupils should necessarily be expected to use the words mean, median and mode.

459 Having set out our foundation list we wish to consider the way in which, in our view, the mathematics curriculum of the secondary years should be developed 'from the bottom upwards' and so we consider first the provision of courses for pupils for whom O Level and CSE are not intended, that is those pupils in about the lowest 40 per cent of the range of attainment in mathematics. It is clear from the report of the National Secondary Survey (8) and also from many submissions which we have received that such provision presents considerable problems in many schools. Many have urged us to stress that mathematics courses for these pupils should be 'relevant to the requirements of everyday life'. However, this aim is easier to state than to achieve. For example, we read in the report of the National Secondary Survey that 'it was very common for the schemes of work in the schools visited to refer to the need to relate the mathematics taught to the problems of everyday life, but the convincing realisation of this aim was much more rare'.

460 We stated in paragraph 455 that in our view the foundation list should constitute by far the greater part of the syllabus of lower-attaining pupils. In order to complete the syllabus and curriculum for these pupils it will be necessary to add a small number of additional topics and to set out in detail the teaching approaches which should be used; our comments in italics may be regarded as a first step towards carrying out this latter task. The topics which are added may be chosen for a variety of reasons. There are, for example, schools in which the pupils take part in sailing and orienteering, both as a school activity and in their leisure time. Although trigonometry as such is not included in the foundation list, it would be possible in such schools to include work relating to bearings and distances, and to discuss methods by which it is possible to fix one's position. Problems arising from navigation might lead some pupils to the beginnings of trigonometry.

461 The necessity of fulfilling the tasks which we listed at the end of Chapter 1 may also lead to the inclusion of some topics, for example topics which are required by pupils for their studies in other subjects. It may be necessary to introduce certain pieces of work as a vehicle for achieving the curricular aims which we set out in paragraph 438. For example, we believe that efforts should be made to discuss some algebraic ideas with all pupils. For some this will entail little more than the substitution of numbers in a simple formula given in words, discussion of problems of the kind 'think of a number ...' and the identification of the patterns in a variety of number sequences; other pupils will be able to go further. Some people would maintain that by considering alternative methods of solving the same arithmetic problem one has embarked on the beginnings of algebra and we would also encourage work of this kind. However, as has been pointed out in official publications of various kinds over many years, formal algebra is not appropriate for lower-attaining pupils.

462 We also wish to draw attention to an extract from one of the submissions which has been made to us.

Mathematics lessons in secondary schools are very often not about anything. You collect like terms, or learn the laws of indices, with no perception of why anyone needs to do such things. There is excessive preoccupation with a sequence of skills and quite inadequate opportunity to see the skills emerging from the solution of problems. As a consequence of this approach, school mathematics contains very little incidental information. A French lesson might well contain incidental information about France - so on across the curriculum; but in mathematics the incidental information which one might expect (current exchange and interest rates; general knowledge on climate, communications and geography; the rules and scoring systems of games; social statistics) is rarely there, because most teachers in no way see this as part of their responsibility when teaching mathematics.

463 As pupils grow older, the practical applications which are used in mathematics lessons should take account of this fact. It is here that the word 'relevant' becomes important, because illustrations and applications of mathematics which are well suited to 12 and 13 years olds are not necessarily appropriate for 15 year olds; the reverse is, of course, also true. Many of the topics which occur in our foundation list relate directly to topics which are commonly included in courses with titles such as Design for living which very often form part of curricula for less able pupils in the later secondary years. Where courses of this kind exist, every effort should be made to relate work in mathematics to the content of these courses. It may be possible to arrange that part of the mathematics teaching takes place within such courses.

464 Especially as pupils become older, a great deal more time is often given to written work than to discussion and oral work. This situation very often arises from the fact that pupils who have become disenchanted with mathematics as a result of lack of success over the years can present problems of control in the classroom which make it difficult to continue oral work for any length of time. However, lack of discussion almost certainly leads to further failure and so the problem is compounded. If the approach which we are recommending is followed throughout the secondary course it is to be hoped that, as pupils grow older, they will not become disenchanted with mathematics because they will have been able to experience success and develop confidence. Problems of control should therefore be lessened.

465 In order to present mathematics to pupils in the ways we have described it will be necessary for many teachers to make very great changes in the ways in which they work at present. This may not be easy; and unless help and support is available both within schools and by means of in-service training, little may be achieved. We therefore welcome the decision of the Mathematical Association to establish a Diploma for teachers of low-attaining pupils in secondary schools. We believe that there will also be an urgent need for development work leading to the availability of additional classroom materials related to the content of the foundation list and making use of the approach which we are advocating.

466 At the present time considerable pressure exists for lower-attaining pupils to be provided with some evidence of their achievement in mathematics before they leave school. Since existing public examinations are not suited to the needs of these pupils, schools are increasingly devising schemes of their own. If we may judge from examples of such schemes which have been made available to us, there appears to have been a considerable increase in the practice during the time for which we have been working. We discuss this matter further in the following chapter.

Provision for pupils whose attainment is very low

467 Before we conclude our discussion of provision for lower-attaining pupils we wish to consider the teaching of those whose attainment is very low indeed. Such pupils are very often withdrawn from normal classes for much of the week and taught in a separate 'remedial' department. Much time is given to developing skills of reading and writing, and the teachers who work in remedial departments are usually experienced in work of this kind. These teachers frequently also teach mathematics to the pupils in their charge but are often less confident in this area; nor is the mathematics syllabus always planned in consultation with the mathematics department. For these reasons syllabuses can sometimes be excessively narrow. It is essential that there should be consultation between the head of mathematics and the head of the remedial department with regard to the mathematics which is taught and the teaching approach which is used. We stress yet again that a great deal of time needs to be given to oral work; mathematics should not be regarded as something which pupils can get on with by themselves while the teacher is hearing other pupils read. We also repeat our remark in paragraph 456 that the mathematics course should be broadly based and related to all the sections of the foundation list.

468 We consider that those who teach mathematics in other parts of the school should from time to time join in with the work of the remedial department, perhaps in a 'non-teaching' period or, preferably, on a timetabled basis, in order both to assist with the teaching and also to become aware of the problems encountered by pupils whose mathematical attainment is very low. Such an arrangement can also act as a safeguard to ensure that due attention is paid to the mathematical needs of pupils who, although weak in language skills, are not correspondingly weak in mathematics. The needs of such pupils, for some of whom English may not be the mother tongue, are sometimes overlooked.

469 It is relatively uncommon for pupils to remain in a separate remedial department until they leave school. Pupils who have been withdrawn from normal mathematics teaching, sometimes for three years, often find themselves in the lowest mathematics sets in years 4 and 5. Unless there has been effective liaison between the mathematics department and the remedial department this can lead to serious lack of continuity for these pupils. Because of the problems of teaching pupils in bottom sets, such sets are often small and contain some twelve to fourteen pupils. It has been suggested to us that, instead of having two small sets of very low-attaining pupils, it is sometimes better to combine the pupils into one set which is taken by two teachers. We believe that an arrangement of this kind can offer considerable advantages, especially if it is possible to arrange for one of the teachers to be a mathematics specialist while the other is experienced in dealing with slow learners. We suggest that some schools may wish to consider the possibilities of working in this way.

Provision for pupils for whom CSE and O Level are intended

470 We turn now to the provision of courses for those pupils for whom CSE and O Level examinations are intended. We have already stated that the foundation list, which we have proposed, should be included in syllabuses for these pupils as well as in syllabuses for those whose attainment is lower. However, many will be able to achieve mastery or near mastery of the content of the foundation list well before they are 16 and the syllabuses which they follow should be enlarged accordingly. Such enlargement will both add further topics and also increase the depth of treatment of many of the topics in the foundation list.

471 The syllabuses of the examinations which pupils will eventually attempt should not be allowed to exert too much influence on the curriculum of the early secondary years. We believe that, at the beginning of the secondary course, differentiation of content should not proceed too rapidly; instead, the differing needs of pupils should be met by differences in approach and in depth of treatment. In this way it is possible to provide maximum opportunity for any pupils for whom change of school or change of approach to mathematics results in markedly improved performance. It is necessary also to realise that the content of the examination syllabus and and the content of the teaching syllabus should not be the same; the teaching syllabus should be wider than the examination syllabus. The extra topics which are included should be chosen in the light of considerations of the kind which we have already discussed in paragraphs 460 to 462 and also in order to make the examination syllabus cohere. It is by 'embedding' the examination syllabus in a wider context in this way that mastery of the syllabus is enabled to develop as pupils increase their network of interconnections and so the level of their understanding.

472 We do not propose to discuss the content of examination syllabuses at 16+ in any detail but instead to indicate two 'reference levels'. We consider that the content of the examination syllabus for those pupils who at present achieve around CSE grade 4 should not be very much greater than that of the foundation list; that is, of a size markedly smaller than that of many existing CSE syllabuses. There should certainly be some increase in the geometrical content beyond that of the foundation list to include, for example, ideas which underlie trigonometry, such as scale drawing and comparison of right-angled triangles, and the use of Pythagoras' theorem. Experience may show that the availability of calculators will enable some elementary trigonometry to be included. Some simple algebraic work on formulae and equations which involves symbolisation is also desirable. At a higher level, we consider that a syllabus whose extent is comparable to that of existing O Level syllabuses represents a suitable examination target for pupils in the top 20 per cent of the range of attainment in mathematics. We believe that there is also need for an examination syllabus whose content lies between that of the two reference levels we have indicated; we discuss these matters further in the following chapter.

473 At this point, we wish to stress that, in recommending a differentiated curriculum and an associated difference in examination syllabuses, we are in no way advocating any lowering of standards. A considerable number of people have written to us to say that they did not understand much of the mathematics which they met at school. This has been supported by the evidence from the research studies into the mathematical needs of adult life and of employment. The researchers were told many times that too many topics were covered in mathematics, that the pace of teaching was too fast, that too little time was given to important aspects of the subject. We believe that if pupils follow a course whose content is better matched to their level of attainment and rate of learning, they will achieve not only greater confidence in their approach to mathematics but also greater mastery of the mathematics which they study. This should contribute to improvements in attainment, attitudes and confidence and so to a raising of standards overall.

Provision for high-attaining pupils

474 At all stages of the secondary course those whose attainment is high should have opportunity to undertake work which will enable them to extend and deepen their mathematical knowledge and understanding. Although it is to be expected that such pupils will be able to move through the syllabus at a faster rate than many of their fellows, this is not of itself sufficient. Steps must also be taken to develop powers of generalisation and abstraction, of logic and proof, of problem solving and investigation, as well as the ability to undertake extended pieces of work. Pupils should also be encouraged to increase their fluency in routine manipulation and to work fast. It should not be supposed that pupils automatically fulfil their potential or even realise themselves how much they are capable of achieving. Many pupils need help, and sometimes judicious pressure, to discover the 'over-drive' which they possess but of which they are not fully aware.

475 It is especially important that attention should be paid to the needs of high-attaining pupils in the early stages of the secondary course. There is a danger that the fact that they are likely to be coping successfully with the work which they are asked to do may mask the fact that the work is not sufficiently demanding and that these pupils are not being extended. In these circumstances, interest and enthusiasm can be lost and, once lost, may not be easy to regain later.

476 As with the teaching of all other pupils, variety of approach is essential. Although high-attaining pupils may well be able to work profitably on their own for quite long periods of time, they also need the stimulus of regular discussion with each other and with their teacher. They should be encouraged, too, to read books about mathematics and to learn something of the work of the great mathematicians of the past. Pupils of this kind can often appear to grasp new ideas very quickly but practice of the skills associated with these ideas is still required so that the ideas may be assimilated fully. However, the form in which the practice is undertaken should be varied and should not consist only of repetitive examples of the same kind. It is, for example, very often instructive to the pupil to be asked to compose further examples relating to the topic which is being studied. A stipulation that the examples should lead to particular types of answer can require pupils to consider more deeply the structure which underlies the work which is being done. High-attaining pupils very often relish opportunities to work with computers and work of this kind should be encouraged whenever possible.

477 The provision of suitable work requires careful planning throughout the secondary course. We estimate that between 5 and 10 per cent of all pupils, though not necessarily of the pupils in anyone school, are capable of working beyond the limits of existing O Level Mathematics syllabuses by the time they are 16. It is essential that these pupils should be enabled to continue their advance and should not 'mark time' at any stage of the course. It is therefore necessary that, throughout the course, additional provision of some kind should be made for them. There is, however, a fundamental difficulty to be overcome in most schools. Except in very large comprehensive schools or large grammar schools it is likely that the number of pupils for whom such additional provision is appropriate will not be sufficient to form a separate teaching group of normal size. This means that, because of the present shortage of mathematics teachers and the current restraints overall on staffing in schools, these pupils cannot usually be taught as a separate group but must form part of a larger group. This can pose considerable problems for the teacher of such a group, especially as pupils become older and the overall range of attainment within the group increases.

478 Some large schools provide for the needs of high attaining pupils in the fifth year by arranging for them to follow the O Level Additional Mathematics syllabus or prepare for one of the optional papers offered by at least one examination board. An arrangement of this kind presents no difficulty if there are sufficient pupils to form a separate teaching group for the purpose. However, if this is not the case, an attempt to work in this way with, for example, the whole of a top set is likely to result in a course which is too demanding for some of the pupils. This situation is clearly unsatisfactory and can lead to narrow teaching of an instrumental kind with consequent loss of confidence on the part of some pupils which may dissuade them from continuing with mathematics in the sixth form. The situation can be still worse if, in preparation for the Additional Mathematics course, pupils are pushed to take the O Level examination in Mathematics at the end of the fourth year before they are fully prepared for it. On the other hand, if no attempt is made to provide additional work for anyone in the set, those whose attainment is high will be disadvantaged.

479 There are teachers who are able to make provision for pupils to undertake more demanding work - not necessarily for examination purposes - while the remainder continue with the ordinary Mathematics course. However, we believe that there are many schools in which insufficient provision is made for high-attaining pupils. We have therefore considered ways in which it might be possible to provide for the needs of high-attaining pupils in a way which would also offer help to their teachers, so that pupils who are likely to obtain a high grade at O Level will be enabled to extend and deepen their knowledge of mathematics in appropriate ways.

Extra Mathematics

480 We believe that consideration should be given to the provision of an extra paper which could be taken by some pupils at the same time as the existing O Level Mathematics papers (or papers at the corresponding level within the single system of examination at 16+ when it is introduced). The purposes of this paper, to which we shall refer as 'Extra Mathematics', would be

• to develop a deeper understanding of the topics included in the Mathematics syllabus and to take some of them, for example geometry, trigonometry and manipulative algebra, somewhat further;
• to introduce a small number of additional topics;
• perhaps most importantly, to adopt a rather more structured approach towards mathematics, so as to help pupils towards 'an appreciation of mathematics as a self-contained logical system'.

Provision for pupils whose attainment is very high

482 A very small number of pupils whose attainment is very high will need special provision over and above that which we have already advocated for high-attaining pupils. These pupils will need to be treated individually and special arrangements made for them. Schools should be aware that they may occasionally receive pupils of this calibre. We have been pleased to note a variety of initiatives for such pupils. Some LEAs, often in association with nearby universities, organise end of term, weekend or holiday sessions for these pupils. The Royal Institution has recently completed a pilot series of Mathematics Master Classes for young people, held on ten successive Saturday mornings; further series are planned. There are now regional, national and international mathematical competitions. We believe that initiatives of this kind should be encouraged and extended.

Mathematics across the curriculum

483 When considering 'mathematics across the curriculum' it is an oversimplified view to have regard only to the 'service' aspects of mathematics, though these should not be neglected. Those who teach mathematics should be aware of the mathematical techniques which are required for the study of, for example, science, geography, craft or home economics and make provision for them. They should also try to arrange that the mathematics course and the courses in other subjects are developed in such a way that pupils will be familiar with the necessary mathematical topics by the time they are needed in other curricular areas. Furthermore, there should be liaison between teachers so that those who make use of mathematics in the teaching of their subjects do not use an approach or a language which conflicts with that which is used in mathematics lessons.

484 Conversely, those who teach mathematics should be aware of the ways in which mathematics is applied within other subject areas and should ask their colleagues who teach other subjects to provide examples of the applications of mathematics which can be used in mathematics lessons. Too little attention is often paid to this aspect. Similar considerations apply to applications of mathematics which lie outside the school curriculum. Use should be made in mathematics lessons of material gathered from newspapers and other printed sources as well as of information which pupils are themselves able to provide.

485 'Mathematics across the curriculum' is not a direct analogue of 'language across the curriculum' because mathematics is not fundamental to learning and to the development of understanding in the way which is true of language. Nevertheless, because of the ways in which mathematics can be used as a means of communication, it can play an important role in the learning process in curricular areas which may seem to be far removed from mathematics, as well as in areas with which the links are more immediately apparent. The presentation of information by means of graphs, charts and tables, the use of time scales, the use of arrows to denote relationships are only a few examples. Teachers of other subjects, as well as mathematics teachers, need to be aware of the part which mathematics can play in presenting information with clarity and economy, and to encourage pupils to make use of mathematics for this purpose.

Organisational matters

Time allocation for mathematics

486 In recent years the proportion of the teaching week given to mathematics has decreased in most schools as additional areas of study have been introduced into the curriculum. Records available within the DES show that some thirty years ago the usual allocation of time for mathematics in grammar schools was six periods per week, each of about 40 minutes. The survey of fourth year pupils in 150 modern schools, carried out in 1961 and quoted in the Newsom Report (10), showed that the average time given to mathematics in mixed schools was 215 minutes per week; the average in boys' schools was 260 minutes and in girls' schools 180 minutes. The present time allowance for mathematics in most secondary schools is the equivalent of five periods of about 35 minutes in a week of 35 to 40 periods for all pupils up to the age of 16 - some 175 minutes in all.

487 There have also been changes in the way in which the time given to mathematics has been distributed through the week. The customary pattern twenty years ago was for all pupils to do some mathematics every day - indeed, a 'daily dose' was thought by many teachers to be essential - but in recent years it has become increasingly common in comprehensive schools to timetable mathematics (as well as a number of other subjects) in double periods. This means that, in schools in which five periods are allocated to mathematics, they are very often timetabled as two double periods and a single period so that pupils are taught mathematics on only three occasions in the week; and these can sometimes occur on consecutive days.

488 We believe that the time allowance which is now usually found - between one seventh and one eighth of the teaching week - supported by appropriate homework is adequate time to devote to mathematics but that careful consideration should be given to the way in which this time is distributed within the timetable. This latter point can give rise to conflicts of interest which are not always easy to resolve. If, as we have advocated, pupils are to undertake practical and investigational work, some lessons which last for longer than about 35 minutes are likely to be required. There are some teachers who feel strongly that as many double periods as possible should be provided for mathematics in the timetable, but there are also teachers who feel that shorter and more frequent spells of mathematics teaching are more profitable. Although certain kinds of practical and investigational work require more than a single period for their successful completion, there are other kinds of practical and investigational work which can be completed satisfactorily within a shorter time. It must also be remembered that by no means all mathematics periods are used for work of this kind and that it is very easy for work to lose momentum towards the end of a double period. The number of double and single periods which are provided in the timetable needs therefore to be suited to the methods of working of the mathematics department and to the mathematics curriculum which is being used.

489 Although it is probably most common to divide the teaching week into between 35 and 40 teaching periods, other ways of arranging the teaching week exist which may or may not make it possible to timetable mathematics in either single or double periods. In practice, the way in which mathematics periods are arranged within the timetable must be the decision of each school, bearing in mind the wishes of the teaching staff and the constraints imposed both by the way in which time is allotted to other subjects and perhaps also by the geography of the school. However, whatever the pattern, we believe that it is necessary to ensure that the periods given to mathematics are suitably distributed through the week and also occur at different times during the day so that there is not, for example, an undue proportion of periods at the end of the afternoon.

490 Although we would not normally wish to see the allowance of time for mathematics reduced significantly below the level we have indicated for any pupils, we wish to draw attention to our suggestion in paragraph 463 that part of the mathematics teaching of some lower-attaining pupils could take place as part of Design for living courses. In such cases, the number of periods timetabled as 'mathematics' could appropriately be reduced for these pupils.

The organisation of teaching groups

491 When deciding on the way in which the mathematics teaching groups in a school should be organised, we believe that the overriding requirement is to achieve a form of organisation which enables pupils to work at a level and speed which is suitable for them, and also one which enables the teacher to include within his teaching all the elements which we have set out in paragraph 243 [in chapter 5]. In particular, the form of organisation should be one which enables sufficient discussion and oral work to take place. We also believe it to be essential that the timetable should not be constructed in such a way as to impose on a mathematics department a form of organisation which it regards as unsuitable.

492 The simplest way of achieving such a form of organisation is by timetabling a group of classes from the same year group simultaneously for mathematics. This provides flexibility and makes it possible to arrange the teaching groups in whatever way is best suited to the needs of the pupils; it enables the teacher or method of grouping to be changed during the course of the school year and allows individual pupils to be transferred from one group to another. It also makes possible the rearrangement of the teaching groups for special purposes or to provide cover for teachers who are temporarily absent. If the number of mathematics teachers who can be made available is more than the number of classes which are timetabled simultaneously, the degree of flexibility is greatly increased. It becomes possible, for example, to provide more teaching groups than the number of classes; this not only reduces the average size of the groups but also makes it easier to move pupils between groups other than on an 'exchange' basis. Small groups can also exist without requiring others to be unduly large. Another possibility is to use the additional teacher as a 'floating' teacher who can assist with the mathematics groups in turn or work with pupils withdrawn from their normal teaching group. It should be noted that it is not necessary for a floating teacher to be timetabled for all the periods in which the same group of classes is doing mathematics. However, if a school is short of mathematics teachers, it may not be desirable to gain additional flexibility at the expense of making use of members of staff who are not well suited to teaching mathematics.

Teaching in setted groups

493 The majority of pupils in secondary schools are taught mathematics in groups formed on the basis of attainment. These are most usually 'sets' formed on the basis of attainment in mathematics, but there are also groups formed on the basis of overall ability rather than attainment in mathematics. In some schools mathematics is taught in sets made up within two or more 'bands' formed on the basis of overall ability. It is very important to realise that within any mathematics set (and even more within groups based on general ability) there will still be marked differences in the mathematical attainment of pupils; this is especially likely to be the case if only a small number of sets spans the whole ability range in a small school or in part of a year group in a large school. It is therefore essential that the teaching takes account of these differences and is responsive to the needs of individual pupils. It should not be assumed that the same teaching approach will necessarily be suited to all in the group, that it will be appropriate for all to do exactly the same work or that pupils should always work as a single group. Teachers should also be aware of the danger that, even if unconsciously, both they and their pupils may lack expectation of what can be achieved by those in low sets and, indeed, also by those in high sets.

494 When mathematics is taught in sets it is important that there should be opportunity for pupils to be transferred between sets if their progress warrants this; and that, when it becomes apparent that a change is needed, it should not be delayed. However, the fact that transfer between sets should be possible does not imply that all sets should attempt to follow the same course. There needs to be common ground between the work of adjacent sets, but there should be differences too so that, so far as is possible, the course in each set is matched to the attainment and rate of working of the pupils in it.

Teaching in mixed ability groups

495 The mathematics supplement to the National Secondary Survey report (11) shows that half of the schools visited taught mathematics in mixed ability groups during part at least of the first year, about one quarter during the second year and one eighth during the third year. However, by no means all of these schools contained pupils covering the full range of ability; for example, some were grammar schools and some were secondary modern schools. Of the comprehensive schools which contained pupils covering the full range of ability, the proportions using mixed ability grouping for mathematics were lower; just under half in the first year, one in five in the second year and one in fourteen in the third year. The use of mixed ability grouping in the fourth and fifth years was very rare indeed - no truly mixed ability teaching across the full ability range was found in any of the schools visited, although it is known that there are a very few schools in which mixed ability grouping is used in these years. It seems, therefore, that it is only a minority of mainly younger pupils who are taught mathematics in groups which span the full ability range.

496 There are undoubtedly some teachers who are able to work in stimulating and effective ways with pupils in mixed ability groups, especially in the earlier secondary years. Where there are such suitable teachers and this method of grouping works well, it clearly provides a form of organisation which is satisfactory and we see no reason to change it. However, we believe that teachers should not be required to work in this way if they are not able to do so successfully. We believe that standards are liable to suffer if mixed ability teaching is imposed upon mathematics departments against their will.

497 We have received much less comment than we had expected about the teaching of mathematics to mixed ability groups, as about classroom organisation generally. A few of the submissions which we have received urge the advantages of this type of organisation - notably that pupils are not 'labelled' and that lower-attaining pupils are therefore likely to make better progress. The majority of the submissions which refer to mixed ability teaching draw attention to the great differences in attainment which exist between pupils and to the difficulty which some teachers find in coping with these differences within the same class.

498 A major problem to be overcome when teaching mixed ability groups is that of providing sufficient opportunity for oral work arid discussion as well as for the practice of mental computation of various kinds. It is very difficult, except perhaps when introducing a topic which is new to all the pupils in the class or an investigation which all can attempt, to work with the whole of a mixed ability class for any length of time. Work with mixed ability classes in mathematics is often made more difficult for teachers because some commercially produced schemes which are commonly used with mixed ability groups are published in such a way that it is almost inevitable that work on several different topics is going on within the class at the same time (12). This is in marked contrast to mixed ability teaching in other subjects in which pupils usually work at different levels within the same theme or topic. As a result, it becomes even more difficult for the teacher of a mixed ability class in mathematics to assemble a group of pupils to discuss and consolidate the work they are doing. Pupils therefore work on a largely individual basis; we discuss this method of working in subsequent paragraphs. We recommend that teachers who work with mixed ability groups in mathematics should limit as much as is possible the range of topics on which pupils are working at anyone time in order to make class or group discussion possible. We are aware that one scheme which has recently been developed for use with secondary pupils in mixed ability classes has been designed on the basis that all pupils in the class will work at different levels within the same topic.

Individual learning schemes

499 The use of individual learning schemes for the teaching of mathematics has become more common in recent years than used to be the case. Although these schemes are often used with mixed ability classes they can, of course, be used with pupils grouped in any other way. We have received a small number of submissions from teachers who tell us that they use schemes of this kind with success and that the motivation of pupils is increased by working in this way.

500 In our view there are some major problems which need to be resolved when using such schemes. One is that of providing sufficient opportunity for oral work and discussion. Another is the difficulty of devising materials from which all pupils can learn satisfactorily and of ensuring that the necessary interconnections are established between the topic which is being studied and other pieces of mathematics. A third is the necessity for the teacher to have a detailed knowledge of all the material which is included in the scheme. For these reasons the successful operation of an individual learning scheme makes great demands on the teacher, especially in teaching groups of the size most usually found in secondary schools.

501 It is clear that considerable success can attend the use of such schemes in the hands of skilled teachers who are committed to their use, and are able to obtain a similar commitment from their pupils. We would not therefore wish to discourage teachers from working in this way if they are able to do so successfully. However, it should not be supposed that the use of individual learning schemes in mathematics is suited to all teachers or to all pupils. We believe that there are many of each who are able to work more effectively if some form of group teaching is used.

Deployment of teaching staff

502 In most secondary schools the deployment of mathematics teachers presents problems which have no easy solution. Many of these arise from the national shortage of adequately qualified mathematics teachers which exists and which we discuss in Chapter 13. The situation can be made even more difficult if there is a lack of suitably qualified mathematics teachers in a school in which some pupils are studying mathematics at A Level. In such schools there may be only one or two teachers who are able to teach the A Level work and this will deplete still further the availability of adequately qualified staff to teach the younger pupils. It must be for each individual school to deploy its staff in the best way possible in order to minimise the disadvantage to pupils. We believe that head teachers and heads of department are rightly concerned to do whatever is possible to ensure that the same pupils are not taught by inadequately qualified staff for several years running. We do, however, wish to draw attention to certain points which we believe should be borne in mind when assigning members of staff to teaching groups.

503 It seems often to be the case that teachers of other subjects who volunteer, or who are asked, to teach mathematics and who teach mathematics for only a small number of periods in each week are given groups containing pupils of relatively low attainment. In our view this is often a mistaken policy. It is not easy to teach mathematics to low-attaining pupils. We consider that, provided that the member of staff concerned is a competent teacher of his own subject, he is much more likely to be successful with a group of pupils of average or somewhat above average attainment. A group of this kind will be a great deal easier to teach than a group of low-attaining pupils, and, provided that there is suitable support from the head of department and from a well prepared scheme of work, it should be possible for both pupils and teacher to 'learn together'. This should also enable the teacher concerned to develop more quickly confidence and skill in teaching mathematics.

504 There is a further reason why it is undesirable for non-mathematicians who teach only a few periods of mathematics in the week to teach pupils whose attainment is low. When teaching such pupils, it is helpful to be able to provide examples drawn from as wide a variety of mathematics as possible to illustrate the topics which are being studied. It often happens that, within the mathematics which is being taught to other groups of pupils, a straightforward application arises which can be adapted for use with a group whose attainment is low. Those who teach mathematics to several groups of pupils are able to take advantage of opportunities of this kind when they occur. Those whose only mathematics teaching is to a group of low-attaining pupils have no comparable opportunities and so their already difficult task becomes even harder.

505 Problems of timetabling sometimes make it necessary for one class to be taught by two different teachers. When shared teaching of this kind is unavoidable we believe that it should be timetabled with a group of pupils whose attainment is high. Pupils in a group of this kind can benefit from being able to 'pick the brains' of two teachers and will quickly question any inconsistencies which may appear to exist in the teaching they receive. If, as is probable, they study different topics with each teacher they will be able without difficulty to keep work in both topics going ahead at the same time. If, on the other hand, shared teaching is timetabled for a low-attaining group, pupils will not receive the continuing reinforcement and revision which they need from period to period and, instead of questioning inconsistencies of approach between their two teachers, are likely merely to become muddled.

506 Shared teaching also provides a method by which an experienced mathematics teacher can provide help and guidance for a teacher who is weak or not mathematically qualified. It may be much better for an experienced teacher and a weak or mathematically unqualified teacher to share the teachings of two sets rather than to assign the whole of the teaching of one of these sets to a teacher who is unequal to the task and who may, in consequence, instil attitudes to mathematics which may not be easy to change at a later stage.

The head of department

507 We believe that the head of department has a crucial role to play in implementing the matters to which we have already drawn attention. Unless he or she provides positive and sustained leadership and direction for the mathematics department it will not operate as effectively as it might do and the pupils will be correspondingly disadvantaged.

508 In our view, the head of department should be responsible for

• the production and up-dating of suitable schemes of work;
• the organisation of the department and of its teaching resources;
• the monitoring of the teaching within the department and of the work and assessment of pupils;
• playing a full part in the professional development and in-service training of those who teach mathematics;
• liaison with other departments in the school and with other schools and colleges in the area.

509 Before we comment on the specific items in our list we wish to draw attention to the need for heads of department to have time to carry out their duties. This is a matter which has come through most strongly in the submissions which we have received from heads of department, as well as in our meetings with groups of teachers and in the discussions which we have held with some of the heads of department who wrote to us. Although it is possible to carry out some of the duties outside normal school hours, certain of them can only be performed while teaching is in progress and the necessary time needs to be provided within the timetable.

510 We put first responsibility for the production and up-dating of suitable schemes of work because it is by this means that the mathematics department makes clear its aims and objectives and provides guidance and help to its members as to the ways in which these may be achieved. The importance of an adequate scheme of work is too often underestimated. The report of the National Secondary Survey (13) draws attention to the fact that, although in all the schools visited there were written schemes of work of some kind, their quality and usefulness varied greatly. We believe that a suitable scheme of work, in addition to outlining the syllabus to be followed by the different year groups and by the different ability levels within these year groups, should also set out the aims and objectives of the department and give guidance on such matters as teaching method and policies for marking and assessment. It should indicate the teaching resources which are available and state the procedures to be followed for routine matters such as the issuing of text books and stationery. The preparation of such a scheme is a major task and it is one in which other members of the department can and should playa part; but the initiative and the final responsibility for both preparation and implementation must rest with the head of department.

511 We have already discussed the matters which we believe should be borne in mind when deciding the allocation of staff to teaching groups. In some schools this is almost entirely the responsibility of the head of department; in others the head of department will need to make his own views known to the member of staff who is responsible for preparing the school timetable.

512 We believe that arrangements need to be made for holding departmental meetings on a regular and frequent basis. A regular weekly or fortnightly meeting can provide opportunities for discussion of curricular matters and classroom practice but, if there are only one or two meetings in the term, these almost invariably become filled with administrative matters. Some schools already provide a timetabled period for this purpose and we hope that more will seek to do so. A record should be kept of matters discussed and decisions taken at departmental meetings. Regular meetings also assist the development of a 'team' approach within the department.

513 We suggest that copies of notes, worksheets and other material prepared by members of the department should be filed systematically so that they become available for others to use. Materials of this kind can provide considerable help to less experienced teachers and to those qualified in other disciplines who are teaching only a few periods of mathematics in the week.

514 We regard it as an essential part of the work of the head of department to be aware of the quality of teaching which is going on within the department and of the various styles and approaches which are being used. He should review the exercise books of pupils in different classes on a regular basis so that he will be able both to monitor the progress of each class and also to check that marking of written work and assessment of pupils is being carried out satisfactorily. In our view it should be normal practice for the head of department to visit lessons given by other members of the department. He should also make it possible for members of the department to see him at work in his own classroom and to see each other at work. Observation of this kind can be of special help to probationary teachers as well as to non-specialist teachers of mathematics.

515 The question of in-service training is considered in greater detail in Chapter 15, but we wish to stress here the very special responsibility of the head of department for giving guidance and support to teachers in their probationary year and also the need to encourage the professional development of the members of the department by delegating specific responsibilities to them. These may, for example, be concerned with the preparation or revision of a unit of work or an organisational task within the department. We draw attention also to the fact that the head of department must not neglect his or her own professional development; it is necessary to take steps to keep up to date with developments in mathematical education and to be aware of the journals of the professional mathematical associations and of the articles which appear in them.

516 Because mathematics is a service subject for so many other disciplines, the question of liaison with other departments assumes particular importance. The head of mathematics should not only liaise with the heads of other departments, but should also study their syllabuses and schemes of work. In this way he can become aware not only of the mathematics which will or could be used in the teaching of other subjects but also of the stage of the course at which it will be required. Liaison with the remedial department, if one exists in the school, is also important. As we have already said; it is too often the case that the two departments operate independently of each other, with no awareness of each other's syllabus or teaching methods. Liaison also needs to be established with other schools and colleges in the area, especially those from which pupils enter the school and those to which they will transfer at 16+ or 18+.

517 We realise that in many schools the head of department may already be carrying out all the duties we have outlined in the previous paragraphs. Equally, we believe that there are some heads of department who take a more limited view of their role and it is for this reason that we have attempted to delineate carefully the responsibility which should in our view be attached to this post. More generally we feel it important that responsibility should be properly defined at all levels within the educational system and that responsibility should be linked with authority, accountability and assessment. Principles of educational management are involved here which go well beyond the brief of our Committee, but we hope that continuing thought will be given to these matters and that, where appropriate, lessons may be learned from good management practice in other fields.

Footnotes

(1) By 1968 half of the pupils in maintained schools were continuing at school after the age of 15; by 1973 the proportion was approaching 60 per cent, though these figures conceal considerable regional variation.

(2) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979.

(3) The different modes of examining are described in the note to paragraph 68 [in chapter 3].

(4) See paragraph 59 [in chapter 3].

(5) Until the beginning of 1975 results at O Level were expressed in terms of simple passes or failures, and no grades of pass were recorded on the O Level certificate. Since the summer of 1975 results have been expressed in terms of five grades and an unclassified category. Candidates awarded grades A, B or C have reached the standard of the former pass; grade D indicates a lower level of attainment and grade E is the lowest level of attainment judged to be of a sufficient standard to be recorded. Performances qualifying for one of the grades A, B, C, D or E are recorded on O Level certificates but unclassified performances are not recorded.

(6) Assessment of Performance Unit Mathematical development. Secondary survey report No 1 HMSO 1980

(7) KM Hart (ed) Children's understanding of Mathematics: 11-16 John Murray 1981.

(8) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979.

(9) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979, page 112.

(10) Half our future A Report of the Central Advisory Council for Education (England). HMSO 1963.

(11) Aspects of secondary education in England Supplementary information on mathematics. HMSO 1980.

(12) This is because the set of materials does not contain sufficient copies of the material relating to anyone topic for it to be possible for more than a small number of pupils to use them at the same time.

(13) Aspects of secondary education in England A survey by HM Inspectors of schools. HMSO 1979.