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 10 Examinations at 16+
[pages 158 - 168]

518 Throughout our discussion of the teaching of mathematics in schools we have stressed the necessity of matching the level and pace of work in the classroom to the level of attainment of the pupils. We have drawn attention to the pressures which exist for as many pupils as possible to attain some form of accreditation in mathematics by the time they leave school and also to the fact that, for good or ill, the syllabus and papers of the examination which many pupils will attempt at 16+ are dominant factors in determining the content and pace of the work which is done in most secondary classrooms. It follows that it is essential that the syllabuses of examinations at 16+ and the examination papers which are set should not impose inappropriate constraints on work in secondary schools. Pupils must not be required to prepare for examinations which are not suited to their attainment nor must these examinations be of a kind which will undermine the confidence of pupils.

519 We discuss first the examinations intended for pupils whose attainment in mathematics is within the top 60 to 65 per cent of that of the school population as a whole. At present these are O Level and CSE, which are shortly to be replaced by a single system of examination at 16+. In the course of our work we have held discussions with representatives of several examination boards and have been permitted to observe the awarding procedures used by some boards which, in preparation for the introduction of a single system of examination, are offering an examination in mathematics which can be attempted by candidates for CSE as well as candidates for O Level. During these meetings we have been impressed by the skill and care with which the officers and examiners of the boards approach their task. We have also been helped by the written submissions which we have received from the examination boards.

520 We explained in Chapter 9 the problems which face those who set examinations in mathematics and the reasons why we believe that CSE and O Level examinations are not well suited to some of those who at present attempt them. We therefore welcome the government's decision to introduce a single system of examination at 16+ because we believe that it provides an opportunity to introduce examinations of a kind which will assess pupils more appropriately than is the case at present. We also welcome the statement in paragraph 16 of the government paper Secondary school examinations: a single system at 16+ (1).

The educational studies carried out by the Steering Committee led them to the conclusion that in at least some subjects it would be necessary to provide a variety of alternative examination papers and tests, at different levels of difficulty, in order to provide satisfactorily for candidates from the intended wide ability range. This is especially the case where, as in mathematics or modern languages, the range of skills involved is wide or certain concepts are within the grasp of some candidates but beyond the reach of others. The government accept this view, and consider it essential that the examination system should enable all candidates to demonstrate their capabilities. The assessment procedures must, therefore, provide for the inclusion of items suitable only for some candidates, or required only for some candidates, and in such a way that the curriculum is not distorted for others.

The proposed single system of examination at 16+

522 We do not believe it is the task of this Committee to prescribe in detail the way in which those who have the responsibility for developing the new single system of examination at 16+ should carry out their task. We believe, however, that it may be helpful to outline a possible approach which we believe would be consistent with the curricular aims for the teaching of mathematics which we have already set out. We hope that those who have responsibility for devising suitable examinations in mathematics will give this careful consideration.

523 We outline the possible approach which we are suggesting in terms of the grades which will be awarded in the new single system. Seven grades of success will be certificated. Of these, grades 1, 2 and 3 will correspond to the existing grades A, B and C at O Level, and will subsume the existing grade 1 in CSE; grades 4, 5, 6 and 7 will correspond to the existing grades 2 to 5 in CSE (2). We repeat that the new system is intended to cover the range of pupils for whom the O Level and CSE examinations are designed at present, ie about the top 60 to 65 per cent of the ability range in any subject. It is only for this group of pupils that we make the suggestion which follows.

524 The scheme of examination which we wish to outline would provide a range of papers which would enable each candidate to attempt a combination of these papers 'focused' at one of three grades on the scale. For instance, the combination of papers focused at grade 6 would be one on which a candidate who was awarded grade 6 would be able to obtain about two thirds of the marks which were available. A rather higher mark would achieve grade 5 and a lower mark (but not, we would hope, below about 50 per cent) would merit grade 7. It would be possible for a candidate who did exceptionally well and who achieved a very high mark on these papers to be awarded grade 4.

525 We believe that it would be appropriate to provide combinations of papers which would focus at grade 2 (appropriate for candidates expected to gain grades 1,2 or 3), at grade 4 (for candidates expected to gain grades 3, 4 or 5, though with the possibility of achieving grade 2) and at grade 6. We would envisage that the syllabus for the papers focused at grade 4 would be more extensive than that for the papers focused at grade 6 and that some of the questions asked would be of a more demanding type than those included in the papers focused at grade 6. There would similarly be a further increase in syllabus content and difficulty of question in the papers focused at grade 2.

526 It would also be necessary to make provision for the award of grades below those suggested as appropriate for the combinations of papers focused at grade 2 and grade 4. We would, however, expect teachers to advise pupils and their parents as to the combination of papers to be attempted in such a way that only in the most exceptional circumstances would it be necessary to award a grade lower than 4 on the papers focused at grade 2, and lower than 6 on the papers focused at grade 4.

527 We have been given to understand that there are some teachers who are expecting that the introduction of a single system of examination at 16+ will remove the necessity of advising pupils and parents as to the papers within the examination which pupils should attempt. However, the whole of our argument for a differentiated curriculum implies that the same set of examination papers in mathematics cannot be suitable for all pupils. If follows that those who teach mathematics must accept responsibility for giving such advice. We believe that the scheme which we are proposing will help teachers in this respect by giving them ample scope for formulating appropriate advice and will not require decisions to be taken at too early a stage.

528 We believe that it would be possible to implement the suggestion which we are making in a number of different ways. One way would be to set a series of four papers graded in difficulty and content so that, for example, papers 1 and 2 could provide an examination focused at grade 2, papers 2 and 3 an examination focused at grade 4 and papers 3 and 4 an examination focused at grade 6. An alternative arrangement would be to provide three pairs of papers, one pair focused at grade 2, one at grade 4 and one at grade 6, with some questions included in more than one pair of papers if this was felt to be necessary in order to establish comparability between the same grade awarded on different pairs of papers. If our suggestion for the introduction of 'Extra Mathematics' were to be adopted, the relevant paper would be available only to candidates attempting the papers focused at grade 2.

529 We wish to point out that the provision of a paper which would be taken by all candidates would not accord with the fundamental principles which we have set out in paragraph 521 unless it was suitable for inclusion in a combination of papers which was focused at grade 6.

530 The two 'reference levels' to which we referred in paragraph 472 [in chapter 9] would correspond to the syllabuses for papers focused at grade 6 and grade 2. We have noted with approval the recent moves of some boards to cease to offer both 'traditional' and 'modern' mathematics syllabuses for O Level and CSE Mode 1 examinations and instead to adopt a single syllabus in mathematics. We hope that this policy will be followed by all boards when devising the syllabuses for the new 16+ examination and that alternative 'traditional' and 'modern' syllabuses will not be offered. When drawing up these syllabuses we believe that it will also be necessary to consider whether certain topics which are at present included in many O Level and CSE syllabuses should continue to be included in the new syllabuses. We cite as an example multi-base arithmetic. Although this is a topic which offers opportunity for interesting and often challenging work at a variety of levels in the hands of a skilled teacher, and which can therefore appropriately find a place in some classrooms, we do not believe that a question of the kind 'evaluate 27 x 3 in base 8' is suitable as an examination question at any level.

Teacher assessment

531 We wish now to discuss a further matter relating to methods of assessment in public examinations. Throughout our discussion of mathematics teaching in both the primary and secondary years we have stressed the importance of presenting mathematics in the classroom in such a way that pupils of all levels of attainment are made aware of the applications of the mathematics which they are studying. We have pointed out that, in order to do this, it is necessary for pupils to undertake relevant practical work, problem solving and investigations.

532 Examinations in mathematics which consist only of timed written papers cannot, by their nature, assess ability to undertake practical and investigational work or ability to carry out work of an extended nature. They cannot assess skills of mental computation or ability to discuss mathematics nor, other than in very limited ways, qualities of perseverance and inventiveness. Work and qualities of this kind can only be assessed in the classroom and such assessment needs to be made over an extended period.

533 It is possible to go further. Not only do written examinations fail to assess work of the kind we have described in the previous paragraph but, in cases in which they comprise the only method of assessment, they lead teachers to emphasise in the classroom work of a kind which is directly related to the type of question which is set in the examination. This means that, especially as the examination approaches but often also from a much earlier stage, practical and investigational work finds no place in day by day work in mathematics.

534 We have noted with interest that the first APU report on secondary testing at 15, when comparing the results of its written and practical tests, draws attention to the fact that different methods of testing 'can highlight different aspects of pupils' performance, and can assess complementary features of their mathematical knowledge' (3). It therefore seems clear that, if assessment at 16+ is to reflect as many aspects of mathematical attainment as possible, it needs to take account not only of those aspects which it is possible to examine by means of written papers but also of those aspects which need to be assessed in some other way.

535 We believe it is now widely acknowledged that it is appropriate to combine teacher assessment and written papers in the examination of pupils who are likely to attain the lower grades in CSE; some CSE boards include an element of teacher assessment for all candidates who attempt an examination in mathematics. Teacher assessment is, however, less common in O Level examinations even though, in some cases, provision exists for this to be included or for pupils to submit project work which can be taken into account during the awarding procedure. Because, in our view, assessment procedures in public examinations should be such as to encourage good classroom practice, we believe that provision should be made for an element of teacher assessment to be included in the examination of pupils of all levels of attainment. The proportion of the total assessment which this should represent is a matter for discussion and might well vary, for example according to the level of attainment of the pupils concerned. We accept, however, that it is also necessary to provide a method of assessment based entirely on written work in order to meet the needs of those candidates, in practice often adults, who attempt the examination independently.

536 If teacher assessment is to be included, it will be necessary for teachers to develop the necessary expertise in carrying out such assessment and for the necessary help and guidance to be made available. The aspects of pupil performance which are to be assessed will also need to be made clear; it will not, for instance, be satisfactory for teacher assessment to be based on testing of the same kind as is used in the written papers set by the examination board. Suitable procedures for moderation will also need to be established. Although some mathematics teachers are already accustomed to assessing their pupils for the purposes of public examinations, many are not and time will be needed to develop the necessary skills. For this reason, we believe that it will be necessary in the early stages for teacher assessment to play a smaller part in overall assessment procedures than will be possible and desirable as teachers develop experience in this work. It has been pointed out to us that an increased amount of teacher assessment within examinations at 16+ will require increased expenditure for training and travel. This, too, may delay the implementation of our suggestion while financial resources are limited.

Evidence of achievement in mathematics for lower-attaining pupils

537 We turn now to what we believe to be the more difficult problem of providing evidence of achievement in mathematics for pupils whose attainment is below that for which O Level and CSE, and the new single system of examination at 16+, are intended. The choice of 60 to 65 per cent as the proportion of pupils for whom O Level and CSE examinations are designed at present, and for whom the single system of examination at 16+ is intended, is an arbitrary one: it is based on the recommendation which was made when the introduction of CSE was proposed nearly twenty years ago. Figure F (paragraph 195 [in chapter 5]) shows that rather more than this proportion of pupils at present obtain a graded result in mathematics in O Level or CSE (and that the proportion increased from 1977 to 1979), and that a very much higher proportion obtain a graded result in English. If, as a result of following the more suitable courses which we have suggested, more pupils than formerly were to attain the standard at present represented by CSE grade 5 (and we would hope that this might be the case) the proportion of the school population who attained a graded result would increase. It would therefore be necessary either to accept that the examination was appropriate for this higher proportion of the school population or to raise the standard required for the attainment of particular grades in order to keep the proportion of the school population who achieved a graded result to some predetermined level.

538 It has been suggested to us that, because of the demand on the part of pupils and their parents for some evidence of mathematical achievement, the new single system of examination should cater, in mathematics if not in other subjects, for a larger proportion of the school population than is at present intended by making possible the award of a grade or grades lower than 7. We wish at this stage to make two comments about this suggestion.

539 The first is that we believe it to be most important that the papers focused at grade 6 should not be distorted and stretched to accommodate pupils whose attainment is below the level at present represented by the award of CSE grade 5. Any such distortion could result either in papers being set which were less well suited to their purpose than they should be or to a lowering of the level at which these papers were focused. We would not wish either of these to happen.

540 The second is that very careful consideration needs to be given to the means by which the mathematical achievement of lower-attaining pupils should be assessed. It should not be thought that the existing pattern of timed written papers towards the end of the fifth year, even if accompanied by a substantial element of teacher assessment, is necessarily appropriate for these pupils. We discuss this further in the following paragraphs.

Schemes at present in use

541 Schemes which provide evidence of achievement on the part of lower-attaining pupils are already in existence in different parts of the country. Several schools have sent us details of 'certificates' which they have devised to provide for the needs of such pupils. In some cases a group of schools in the same area has adopted a common scheme, sometimes after consultation with local employers as we described in paragraph 97 [in chapter 3]; in at least one case the tests which provide evidence of achievement are not only taken by lower-attaining pupils but are also taken by pupils who are expected to obtain a graded result in O Level or CSE. There are also schemes on a somewhat larger scale which have been introduced by local authorities, for example Hertfordshire, which any school in the authority may use. A national scheme is provided by SLAPONS (School Leaver's Profile of Numerical Skills) to which we referred in paragraph 98 [in chapter 3]. One of the purposes of this scheme is to provide tests which can be taken earlier in the year than CSE and O Level examinations. The results of the tests are then available to the many employers who recruit from the fifth forms of secondary schools in the months before CSE and O Level results are known. It is hoped that in this way the necessity for employers to set their own tests can be reduced. Certificates awarded as a result of other forms of school-based tests very often serve a similar purpose, in addition to providing pupils with tangible evidence of their achievement.

542 There is considerable variation in the ways in which these schemes operate. One pattern is of testing at one level only, sometimes with a requirement that pupils must demonstrate success on more than one occasion. Another pattern provides a selection of tests based on a 'core' together with 'options' from which pupils may choose. A third pattern is of a range of tests at different levels which pupils may attempt in succession. Schools which operate schemes of this third kind have told us that they consider such schemes to increase the motivation of pupils because they provide both incentive and evidence of progress over a period of several terms.

543 There is also considerable variation in the content of the tests which are used. A small number of those which have been sent to us cover a reasonably wide range of mathematical topics but most, including the SLAPONS tests, concentrate wholly or in great part on the testing of computational skills, very often divorced from any context. We find this a cause for concern because we believe that the use of such tests encourages teaching of the kind which attracted adverse comment in the report of the National Secondary Survey (4). 'Lessons seen were often narrowly conceived and in 60 per cent of the schools visited HMI considered that new courses should be developed for the less able pupils ... There is a tendency to restrict the courses provided for the less able to routine calculation divorced from context and to fail to provide a sufficient range of applications of the mathematical ideas within the understanding of the pupils The improvement of these courses is a matter requiring urgent attention.'

544 We therefore hope that schools which are at present using tests of some kind for their lower-attaining pupils will review the content of their tests so as to make sure that they do not lead to, or result from, a narrow curriculum but instead contribute to its widening in the ways we have suggested. Any schools which may be considering the introduction of tests at this level should consider carefully both the form and content of such tests and also the influence which they are likely to have on the mathematics curriculum which is provided.

Our own view

545 We have considered the principles which should, in our view, govern any tests which are used for lower-attaining pupils in schools. We believe that the principles which we have set out in paragraph 521 in respect of the single system of examination at 16+ should apply to any other form of testing which is used in secondary schools. Any tests should be within the capacity of the pupils who attempt them and should not undermine their confidence. Furthermore, those who succeed in the tests should do so as the result of a good performance on the questions which have been asked. The method of testing should reflect a curriculum which is suited to the needs of the pupils and, if possible, encourage them to persevere. For this reason we believe that it is likely to be more helpful for lower-attaining pupils to be offered a series of short-term targets, success at each of which provides evidence of achievement, rather than to have to wait for a 'one-off' test when they are about to leave school.

546 In paragraph 455 [in chapter 9] we stated our view that the foundation list which we set out in paragraph 458 should constitute the greater part of the syllabus for lower-attaining pupils. It follows that the mathematical content of tests should not be limited to computation only but should be based on the foundation list. Thus any tests should include such things as the reading of graphs, charts and tables, mensuration, geometrical representation in two and three dimensions (plans, elevations, nets, etc), the interpretation of flow-charts and of other types of information given in mathematical form, and the use of a calculator. Computation should be tested at appropriate levels but, in our view, this should be done within a series of applications to problems in defined areas such as shopping, travel etc and not by examples testing the four operations in isolation. Practical and oral testing should also be included and a principal aim of any scheme of testing which is used should be to help pupils to acquire the feeling for number and measurement to which we have referred earlier in the report.

547 It would be possible for a school to satisfy the requirements which we have set out in the two preceding paragraphs by using not a single test but a series of tests which pupils could attempt in succession, perhaps from the age of about 14. An arrangement of this kind would make it possible for the tests to be criterion referenced, with each level of test related to the understanding of defined concepts and the ability to apply them to appropriate problems. It would also make possible a requirement that, in order to succeed at any level, it was necessary to achieve a high mark of the order of, for example, 70 per cent. The mathematical content of the lowest level, the number of levels and the size of the steps between them would require discussion. In our view from four to six levels would probably be the greatest number which would be practicable; and we feel that it should be possible within this number of levels to devise a realistic progression in terms of content and depth of understanding. Content could increase from level to level both in breadth and depth but the increase could be gradual; the lowest level might well consist only of oral questions and practical tests such as the measurement and estimation of length, weight and capacity, the reading of dials and very simple tables, and the giving of change.

548 In paragraph 542 we said that schools which operate tests at different levels had told us that such testing provides a valuable means of motivation for many pupils. We feel that this can be the case, especially during that part of the secondary school course when this is often most difficult to achieve, provided that the tests are of an appropriate kind and that the goal of success in a test is perceived by pupils to be possible of attainment in the not too distant future. For many pupils, too, a series of tests is better than one final test of mathematical attainment which allows no opportunity to make another attempt if the result has not been satisfactory. Furthermore, there is no need for a pupil to start at the lowest level of graduated tests nor to attempt every level; pupils can attempt the appropriate level of test when they are ready to do so and can have more than one attempt if this proves to be necessary.

549 The amount of work involved in developing graduated tests which would fulfil the requirements which we have set out in previous paragraphs would be very considerable. It would not be an economic use of resources for a large number of schools to attempt such a task individually or even by working in groups. We therefore feel that it would be preferable for any development work to be undertaken on a larger scale so as to provide a resource, perhaps in the form of a central bank, which any school could use if it wished to do so.

550 Development work of this kind could result in the availability to schools of tests at various levels, together with instructions as to the way in which performance in the tests should be assessed. We believe that, in order to ensure that the amount of testing in any one school should not be excessive, some limits would need to be imposed on the availability of tests; these would have to be discussed. In any case, we do not consider that pupils should be allowed to attempt such tests until they were in the third or perhaps fourth year of the secondary course, assuming the age of transfer to secondary school to be 11, or at the corresponding stage within other forms of secondary organisation. In this way pupils of high attainment, for whom the tests would not be designed, would not be able to enter for them at a very early age.

551 It seems likely that, if graduated tests were to become available, a number of schools might include a record of achievement in the tests as part of profiles which they provide for the benefit of employers and those concerned in further education. It could also be the case that a record of achievement of this kind, which could be made available to prospective employers, would mean that pupils did not have to attempt a succession of mathematics tests set by different employers if they applied for several jobs. Furthermore, if the results of tests from a central source were used in this way, there would be no difficulty in making specimen tests available to employers who wished to inform themselves of the type and level of the questions which were being asked. In this way employers would be enabled to have a clear idea of the mathematical achievement of pupils who succeeded at the different levels. Moreover, schools could arrange the timing of the tests in such a way that their results were available by the time at which pupils were starting to apply for jobs.

552 We believe that the availability from a central source of graduated tests based on the principles we have outlined could offer many advantages to schools. It could encourage the provision of better curricula and the motivation of pupils, and assist in the construction of pupil profiles. In particular, it could help to prevent an increase in the use of unsuitable tests; indeed, we would hope for a diminution in the number and use of such tests. Moreover, the availability of a more appropriate alternative for lower-attaining pupils should help schools to persuade these pupils, their parents, and employers that entry for CSE examinations in their present form, or for the examinations which will replace them, might not be desirable. On the other hand, we accept that there could also be disadvantages. The availability of tests, however well constructed, would not of itself lead to the increase in mathematical understanding and to the 'at-homeness' with mathematics which we seek. We are well aware of the dangers of subjecting pupils to more, and more frequent, testing and we accept that the combination of certain kinds of testing and teaching could produce results which are the opposite of those we desire.

553 Nevertheless, we believe that consideration should be given to undertaking development work which could investigate the possibility of providing evidence of the achievement of lower-attaining pupils in a way which would assist and encourage the provision of suitable mathematics courses for these pupils, which could operate under the control of an individual school and which would enable use to be made of material produced outside the school. We therefore recommend that a study should be commissioned to consider whether it is possible to devise a means of providing evidence of achievement in mathematics for lower-attaining pupils in ways which will support, and not conflict with, the provision of suitable mathematics courses in schools. It should take account of the views we have expressed in the preceding paragraphs and also of other schemes which already exist. It would be necessary to make some assessment of the costs of any scheme which was suggested and to consider whether any procedure would be required to monitor and regulate the way in which the tests were used. If it were to be thought that some suitable scheme might be devised in time to make it available to schools no later than the date at which the single system of examination at 16+ was introduced, and that the necessary finance was likely to be available to do so, then the study should be undertaken as a matter of urgency. Meanwhile, we hope that individual schools which are using tests of their own will examine those tests in the light of the views we have expressed. We believe, however, that any more general development of tests of the kind we have in mind, at either national or regional level, should await the outcome of the study we are recommending.

554 If it were decided to make use of a central resource of some kind, we believe that those whose task it would be to set this up should take steps to draw on the experience of teachers throughout the country. The establishment of local working groups which could assist in developing appropriate questions would provide very valuable opportunities for the in-service training of teachers, especially those who were concerned with the teaching of lower-attaining pupils. In this way, the money spent on setting up such a central resource would also contribute to the in-service training of teachers.

555 Our suggestion that any tests which might be devised should reflect the content of the foundation list which we set out in paragraph 458 [in chapter 9] means that they would cover a large part of the syllabus which we have recommended for 16+ examinations focused at grade 6. However, we do not consider that in devising the tests any attempt should be made to relate their syllabus content to that of the 16+ examinations in any specific way. Nor do we believe that initially there should be an attempt to establish any direct correspondence between one of more levels in any graduated tests and specific grades in the 16+ examination. However, we believe it would be appropriate that the levels in any graduated tests should be such that, in general, candidates awarded grade 6 in the new single system at 16+ would also be able to achieve the highest level in the tests; we would expect that some of those who were awarded grade 7 would also be able to achieve this level.

556 If at some stage it were to be decided that it would be desirable, within the single system of examination at 16+, to introduce the award of a grade or grades for mathematics lower than grade 7, we believe that experience gained in the study we have proposed would assist in the development of suitable assessment procedures.

Footnotes

(1) Secondary school examinations: a single system at 16+ Cmnd 7368. HMSO 1978.

(2) In particular this means that the existing CSE grade 4 - the grade awarded to the 16 year old of average ability - will become grade 6 in the new single system.

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

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