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Cockcroft (1982) Notes on the text
Part 1
Part 2 Chapter 5 Mathematics in schools
Part 3 Chapter 12 Facilities for teaching mathematics
Appendices Appendix 1 Statistical information
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The Cockcroft Report (1982)
Mathematics counts 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
ISBN 0 11 270522 7
Chapter 11 Mathematics in the sixth form
557 Although A Level courses account for the major part of mathematics teaching in the sixth form, several other kinds of mathematics course are also provided in most sixth forms. These include 'service' courses for students whose A Level studies require a knowledge of mathematics beyond O Level standard but who are not studying mathematics at A Level; O Level and CSE courses for students who wish to improve the result which they have obtained in the fifth form; and courses designed primarily for 'new' sixth formers, that is those who are not studying any subject at A Level.
A Level mathematics 558 We start our discussion of A Level mathematics courses by drawing attention to an important change which has taken place in recent years. Thirty years ago, and for many years before that, it was usual to combine the study of single-subject mathematics with the study of two other science subjects, usually physics and chemistry; double-subject mathematics was usually combined either with physics or, less often, with both physics and chemistry. It was rare to combine mathematics with non-science subjects. Although mathematics is still very often combined with physics and chemistry, it has become increasingly common in recent years for students to combine the study of mathematics at A Level with subjects other than these. It is therefore not surprising - indeed, it is to be expected - that additional syllabuses and changes to existing syllabuses have been introduced which reflect the differing needs of those who combine mathematics with subjects other than physics or chemistry. In addition, pressure to avoid undue specialisation at the age of 16 has caused some people to ask whether it is educationally desirable to devote two thirds of A Level study time to mathematics, as can be the case if double-subject mathematics is chosen; we return to this point in paragraph 586. 559 A Level courses in mathematics, as in other subjects, are designed as two-year courses for students aged about 16 to 18. We have tried to discover how many of those who study mathematics at A Level go on to further study of some kind. A study (1) carried out recently for the Standing Conference on University Entrance (SCUE), which has been made available to us, suggests that in 1977 and 1978 just under 60 per cent of those who passed mathematics at A Level were accepted by a university in the United Kingdom; this number represents some 45 per cent of all who attempted A Level mathematics. Some who failed mathematics at A Level also gained a university place, but the information provided by the study suggests that their number was small. It seems, therefore, that between 45 and 50 per cent of those who studied A Level mathematics in these two years gained university places. Tables 28 and 29 of Appendix 1 show that in 1977 and 1978, as also in 1979, almost 60 per cent of all entrants to universities in England and Wales who had an A Level qualification in mathematics chose degree courses in engineering and technology, the physical sciences or mathematical studies; about half of these chose engineering and technology, which is the largest single 'user' of university entrants with A Level mathematics. 560 Some of the others who studied mathematics at A Level will have been accepted for degree courses in the non-university sector and for other forms of further study, but it has not been possible to determine their number. However, though many of those who study mathematics at A Level go on to degree courses or other forms of higher education, by no means all do so; nor do all of those who proceed to higher education continue their study of mathematics. It is therefore essential that an A Level course in mathematics should not only provide a basis for further study but also provide a course which is balanced and coherent in its own right and which reaches suitable 'stopping points' for those who will, at least for the time being, go no further. This is not always easy. Sixth form teachers have told us that, in their view, some A Level syllabuses are over-extensive so that, although the work involved is within the capability of more able students, others find the work over-demanding. As a result, pressure to cover the whole syllabus can lead these students to become confused and to perceive mathematics only as a series of apparently disconnected techniques related to particular examination questions. The teaching of A Level courses 561 We consider first the aims of mathematics teaching at A Level and the teaching methods which we believe to be appropriate. It should be one of the aims of sixth form teaching in mathematics, as in all other subjects, to enable students to develop the study skills which will prepare them for adult styles of learning; and the need to develop these skills exists equally for those who will go on to higher education and for those who will not. It is very easy for A Level teaching in mathematics to depend too much on exposition by the teacher and for students to adopt passive styles of learning. However, it is as important for students in the sixth form as for pupils of all other ages to develop problem solving techniques, to pursue independent investigations and to discuss and communicate their ideas. It is by working in these ways that students develop the confidence which they will require in order to be able to make use of mathematics in their future studies and careers. A Level mathematics is almost always taught to groups which are a good deal smaller than those in which mathematics is taught to younger pupils and sixth form teachers should exploit the opportunities which this provides to work with their students in a variety of ways. 562 It is also possible for mathematics at A Level to be presented as a very technical and somewhat arid subject with little relation to other school subjects except, perhaps, physics or to the activity of the world at large. It is therefore important that teachers should seek to counteract this impression by making use of opportunities which arise to emphasise the broader role of mathematics. The very varied applications of mathematics should be stressed and illustrations of these applications drawn from as wide a range as possible. The increasing use of 'mathematical modelling' in, for example, the social sciences provides many possibilities for an enterprising teacher and many more traditional applications are to be found in the physical sciences. Reference to the historical background of some of the topics which are being studied can both help to explain their importance and also add interest and depth to the A Level course. A microcomputer can provide a stimulus to adventurous thinking, very often initiated by the students themselves; the investigative work which can arise in this way should be encouraged. Occasional discussion of some of the assumptions underlying mathematics and of the nature of the knowledge it provides is necessary if students are to be enabled to talk about mathematics in ways which others will understand. There is at present a lack of teaching materials which assist sixth form teachers to work in these ways and more are required. 563 Most mathematics courses in sixth forms require students to undertake far less reading than is the case with courses in other subjects, including science. Much less use is made of libraries and individual reading and study is very often confined to a single textbook. It is not easy to learn mathematics from a book and the necessary ability can take a long time to develop. In order to read a mathematics text it is usually necessary to sit at a table with pencil and paper in order, for example, to fill in the intermediate working which is often omitted; students have to realise this. A helpful 'way in' can be to ask students to make use of the library to look at the way in which a different textbook deals with a topic which has just been studied; the knowledge of the topic which the students already possess will help them to follow the text. A next stage can be to ask students to approach a new topic by studying for themselves the relevant chapter in the textbook which they are using. The chapter can then be re-read and discussed by student and teacher working together so that the students can begin to appreciate the way in which it is necessary to set about reading mathematics. Students should also be encouraged to 'read around' the subject and to acquire knowledge on their own initiative. 564 Throughout this report we have emphasised the necessity of enabling pupils of all ages to achieve as much as they are able. It is especially important that students whose ability is high should be helped and encouraged to extend their work beyond the confines of the single-subject syllabus. Proper use of the library can contribute significantly to this end; it is by no means unknown for mathematically able students, with help from their teacher, to be able to cover the double-subject syllabus in the time which their fellows require to study the single subject. The applications of mathematics 565 We have referred many times in earlier chapters to the importance of applying mathematics to the solution of problems. We therefore believe that all A Level mathematics courses should contain some substantial element of 'applied mathematics' so that all who are studying the subject, whether for its own sake or because of its usefulness as a 'service subject' for their other studies, are able to gain a balanced view of mathematics. It follows that we do not favour single-subject A Level courses which consist of pure mathematics only. 566 It is at this point that we encounter difficult and intractable problems. Thirty years ago no problem arose. The applied element of both single- and double-subject mathematics courses was almost always Newtonian mechanics; some syllabuses also included a certain amount of applied calculus, usually related to problems in mechanics. However, the study of Newtonian mechanics, although related very directly to the study of physics and engineering, is of little relevance in fields such as economics, geography and social studies; in these areas topics such as probability and statistics are of much greater application. The problem, therefore, is to decide how best to provide, often within the same teaching group, for the differing needs of students who may be studying mathematics at A Level for very different reasons. Some may have chosen mathematics because of its value as a service subject in the study of physics and engineering; some may wish to make use of mathematics in their study of, for example, biological or social science; some may wish to continue with mathematics as a principal study in higher education; in many schools there will be some for whom the end of the A Level course marks the end of their full-time education. It is necessary to remember also that, at the age of 16, many students will not yet have decided on their future area of study in higher education. 567 There are strong arguments that, in an ideal situation, all who study mathematics at A Level should have the opportunity to gain some knowledge of Newtonian mechanics as well as of probability and statistics. 'Applications of mathematics, such as Newtonian mechanics, are part of our cultural heritage and of the human activity of mathematics. To learn calculus without understanding what led to its development and how it was used by Newton and others, is like learning to play scales at the piano without being shown any compositions.' (2) Equally, the increasing use which is being made of statistical techniques in so many fields makes it highly desirable that those who study mathematics at A Level should have some understanding of probability and statistics. 568. It is the experience of many teachers that the early stages of both mechanics and probability and statistics need to be taught slowly and with great care, allowing ample time for discussion and for the underlying ideas to sink in and develop. Attempts to go ahead too fast can too easily result in failure to develop understanding of the fundamental ideas and so lead to difficulty and failure at later stages of the course. This means that, in practice, the 'ideal situation' which we described in the previous paragraph is very difficult to realise. Within the time available for the study of a single A Level subject it is extremely difficult to cover essential pure mathematics and also to go a sufficient distance in the study of probability and statistics as well as mechanics to reach suitable 'staging points' in each; that is, to go far enough to enable the work which has been covered to form a reasonably coherent whole. Some of the A Level syllabuses which have been introduced in recent years include probability and statistics as well as mechanics, but these attempts to include a wider range of applications have resulted in syllabuses which are very full and hence in courses which can be over-demanding for many students. In consequence, many fail to obtain a secure grasp of either of the applied elements of their course. The same arguments do not apply to those who study mathematics as a double subject. We believe that all double-subject A Level courses in mathematics should include the study of mechanics and also of probability and statistics. 569 Although, for the reasons outlined above, we would have liked to recommend that single-subject courses should include both mechanics and also probability and statistics, we have concluded with regret that, because such courses make great demands on both teacher and students, it is not at the present time possible for us to recommend that all single-subject courses should be of this kind. On the other hand, we in no way wish to discourage those who are following such courses successfully from continuing to do so. It has been suggested to us that it would be possible to include rather less of both mechanics and probability and statistics within the single-subject A Level syllabus and still provide a coherent course. It has also been suggested that the use of computer simulation might enable essential ideas in mechanics and in probability and statistics to be assimilated more rapidly and in this way enable sufficient of each to be included in a single-subject syllabus. We believe there is a need for curriculum studies into each of these possibilities. 570 We have considered whether we should recommend the study of one area of application rather than another but have decided that it is not possible to do this because of the diversity of students' future needs and interests. The increasing use of statistical techniques, not only in the study of other subjects but also in business, commerce and manufacturing industry, provides many students with a strong reason for wishing to study probability and statistics. In addition, some who wish to study A Level mathematics may not have studied physics even to O Level standard and may be discouraged from choosing mathematics if the course includes only mechanics. It has been suggested to us that girls are more likely to study mathematics at A Level if they do not have to do mechanics, but we have no direct evidence about this. 571 Some of the greatest pressure for the inclusion of mechanics in mathematics syllabuses comes from those in higher education who are responsible for courses in engineering. However, almost all those who enter engineering courses at this level have also studied physics. Figures supplied to us by the Universities Statistical Record show that, of those who entered degree courses in engineering and technology at universities in England and Wales on the basis of A Level qualifications, almost 98 per cent had an A Level in physics or physical science in both 1978 and 1979. A great deal of the mechanics which is commonly included in A Level mathematics syllabuses is also included within most A Level physics syllabuses. This means that many students who study physics as well as mathematics at A Level cover many topics in mechanics twice, albeit from somewhat different standpoints. It could be argued that such duplication should not be necessary. However, for very many years most of those entering engineering degree courses have studied mechanics as part both of mathematics and of physics. In consequence, teaching for degree courses in engineering is based on an expectation of the level of competence in mechanics which this double study provides. Although some engineering departments accept individual students whose A Level mathematics course has not included mechanics, we have gained the impression that they do so with reluctance, believing that such students are likely to experience difficulty in covering the content of their courses satisfactorily. A recent research study (3) into the mathematical education of engineers at school and university classifies A Level courses as 'traditional', 'compromise' and 'modern'. It points out that in some syllabuses there is only a restricted coverage of mechanics and that in others it is possible to avoid mechanics entirely. However, no evidence was found that students who had followed one type of A Level mathematics course performed consistently better or worse at university than students who had followed another type of course. 572 In passing, we wish to draw attention to the fact that, so far as we are aware, there is no European country outside the British Isles in which mechanics forms part of school courses in mathematics; it is considered to be part of physics. However, there are also considerable differences in the time required to complete first degree courses in these countries and in the structure of the examinations which are taken by students in preparation for university entrance. 573 The study of mechanics is not only of relevance to those who proceed to higher education courses in engineering; it is also of value to those who study mathematics and physics in higher education. We believe, too, that there are A Level students who, even though they will not continue their study of mathematics after the age of 18, nevertheless enjoy the study of mechanics and derive benefit from it. 574 One further factor needs to be considered. Partly as a result of changes in the structure of some mathematics degree courses in recent years, there are some teachers who find difficulty in teaching mechanics. On the other hand, some teachers, especially those who completed their degree courses some years ago, do not have sufficient knowledge of probability and statistics to teach in these areas effectively. A concentration on the teaching of one aspect of applied mathematics only could, therefore, add to the problems of deployment of mathematics teaching staff in some schools. 575 We realise that our support for either mechanics or probability and statistics or both to remain available in single-subject syllabuses does not resolve problems of choice of A Level syllabus which now exist in schools. In schools and colleges in which there are sufficient students studying mathematics at A Level for it to be necessary to form more than one teaching group, it may be possible to offer alternative courses in order to provide for the differing needs of students. However, it has been pointed out to us by some who teach in such larger establishments that a solution of this kind may fail to take account of the advantages which can accrue from teaching the same course to all students and gaining from the mix of A Level specialisms within the same teaching group which this makes possible. If there are only sufficient students to form a single teaching group, it must be for each school to decide on the course it will offer, taking account of the needs of the students, the capabilities of the mathematics teaching staff, and the mathematical balance of the course which is chosen. We believe that the greatest pressure for a particular type of course to be provided may come from those who wish to study engineering at a later stage. Since these students will almost certainly be studying physics as well, we do not believe that their needs should necessarily be considered to be paramount when choosing the course to be followed. A Level core 576 Although we have come to the conclusion that it is necessary for there to be variety in the applied mathematics element of A Level syllabuses we believe that different considerations apply in respect of the range of pure mathematics which is included. In the submissions which we have received there has been a great deal of support for the proposal that a 'core' of pure mathematics should be agreed which would form part of the syllabuses of all GCE boards. The few submissions which have argued against such a core have in general done so because of concern that A Level syllabuses are already becoming overloaded and that syllabus change and development would be inhibited; we believe there is substance in this concern. 577 Although some submissions have given specific support to the 'core' proposed by the Standing Conference on University Entrance (SCUE) and the Council for National Academic Awards (CNAA), many have expressed the view that the content of the SCUE/CNAA core is too large if, as is expected by those who formulated the core, 'students should not only have been taught the whole of the core content but should be very well versed in it' (4). We are aware that, during the time for which we have been working, the GCE boards have been engaged in consultation among themselves as a result of their study of the SCUE/CNAA proposal. We understand that they are moving towards agreement on a core of pure mathematics which is somewhat smaller than that originally proposed by SCUE/CNAA and which will constitute some 40 per cent of a single-subject syllabus in pure and applied mathematics. We welcome this move and accept that the 40 per cent suggested forms a reasonable proportion of the total syllabus content. We would not wish to see the core become much larger because of the need to include a substantial proportion of applied mathematics and of the danger that any further increase would prevent the freedom of presentation and development which exists in the best sixth form teaching. 578 We believe it to be essential that, once a core has been agreed, it should be subject to regular review so that it can be changed as necessary to accommodate current developments. A suitable body will need to be set up for this purpose. We suggest that such a body should be responsible to the A Level subcommittee of the Mathematics Committee of Schools Council or to such comparable committee as may be formed as a result of the current review of the works of Schools Council. Variety of A Level syllabuses 579 In the last three or four years, there have been many adverse comments on the apparently very large number of A Level mathematics syllabuses which exist. However, those who have quoted a variety of figures between fifty and sixty have very often failed to point out at the same time that there are eight GCE boards. If each board were to offer only three syllabuses - 'pure mathematics' and 'applied mathematics' to provide the double subject and 'pure and applied mathematics' for the single subject - there would be a total of twenty-four syllabuses. Since it is now necessary to offer the single subject in two alternative forms (so that the 'applied' component is either mechanics or probability and statistics), this figure rises to thirty-two. It is therefore always likely to be the case that the number of A Level syllabuses in mathematics will appear high. 580 In fact, arithmetic calculations of this kind are not very helpful, because there is considerable variation in the way in which the GCE boards draw up and name their syllabuses. For example, some boards include probability and statistics and mechanics as alternative options within the same single-subject syllabus while others provide two syllabuses (with different titles), one of which contains probability and statistics and the other mechanics. We therefore wish to discuss the variety of A Level syllabuses in more general terms and to draw attention to some of the factors which have contributed to this variety. 581 One factor, to which we have already referred, has been the increase in the number of students combining the study of mathematics at A Level with subjects other than physics and chemistry, and the consequent need to provide syllabuses which include probability and statistics rather than mechanics. Another factor has been the world-wide movement towards modern mathematics which started in the 1960s. This has led to the introduction of new courses in many schools and, in consequence, of new A Level examinations which reflect the different content and approach of these courses. In the first place modern A Level syllabuses were associated with the School Mathematics Project (SMP) and Mathematics in Education and Industry (MEI). The SMP and MEI examinations were (and still are) available 'across the boards'; that is, they can be taken by candidates entering through any of the GCE boards. They did not, therefore, in the first instance lead to a great increase in the number of A Level syllabuses since the two sets of examinations served eight boards. However, in due course seven of the GCE boards introduced their own modern syllabuses for single and double subject and the total number of A Level syllabuses increased sharply. We believe that the number of syllabuses reached its highest point in the mid-1970s. 582 The number of different syllabuses is now decreasing. Syllabus changes during the last ten years have lessened the differences between the content of 'modern' and 'traditional' syllabuses and many feel that it is no longer appropriate to attempt to distinguish between them. We have noted with interest that none of the groups which were commissioned some six years ago to propose specimen syllabuses and examination papers in mathematics as part of the feasibility studies for the N & F proposals for examination at 18+ differentiated between traditional and modern mathematics. We support the view that the distinction should no longer be maintained. It should then be possible for the number of A Level syllabuses to be reduced further. 583 However, differences in examinations are not only a question of syllabus content, but are also concerned with the approach to the teaching of mathematics which the syllabus implies and the kind of papers and questions by means of which it is examined. These factors can be very relevant to a school's choice of syllabus and we believe it is desirable that such choice should exist. Indeed, the difference between some 'modern' and 'traditional' syllabuses is as much in the approach which is used as in the topics which are covered. If, as we hope will be the case, agreement is reached on a core of mathematics to be included in all A Level syllabuses, existing syllabuses will presumably be revised as necessary to accommodate it, and any new syllabuses which may be introduced in the future will include the core. This should reduce the differences between the content of syllabuses and so help to resolve the problems which, we have been told, are caused for both students and their teachers at the beginning of higher education courses by the variations in the content of syllabuses which exist at present. However, it should not necessarily be expected that the introduction of a core syllabus will resolve all problems. Differences in the ways in which A Level courses have been taught may result in greater differences in the performance of students who have followed the same course than result from differences of content between syllabuses. 584 We do not therefore consider that there are strong arguments for an arbitrary reduction in the number of syllabuses, except on the grounds of duplication. Indeed, the possibility of introducing new syllabuses is essential if curriculum development is not to be inhibited. There is of necessity a limit to the change which can be made to existing syllabuses in mathematics, as in other subjects, at anyone time and it is often easier to reflect curriculum development by the introduction of a new syllabus. 585 We believe the monitoring of A Level syllabuses which is at present carried out by the A Level sub-committee of the Mathematics Committee of Schools Council is both valuable and necessary. This includes the review and approval of new syllabuses and schemes of examination proposed by GCE boards or individual projects, and would provide a means of ensuring that any core which had been agreed was included, and assessed appropriately, in any new syllabus. The monitoring process also includes an annual scrutiny of the examinations set by at least two GCE boards; this provides a valuable means of exchange between boards and also of sharing expertise in examining. This work should continue at not less than its present level. Double-subject mathematics 586 The fact that mathematics at A Level can be offered as either one or two subjects causes a number of problems. Some have argued to us that double-subject mathematics leads to too great a degree of specialisation and that a more broadly based course, with mathematics as one rather than two out of three A Level subjects, provides a more balanced diet. We recognise the force of this argument, the more so since for many students the choice of double-subject mathematics has to be made at the age of 16 when future plans may be uncertain. Again, except in very large sixth forms, teaching groups for double-subject mathematics are almost always small and so, in comparison with mathematics teaching groups in other parts of the school, not economic in their use of staff. This can be a justifiable cause of concern at a time when, as we point out in Chapter 13, there is a shortage of suitably qualified teachers of mathematics. A further problem can arise from the fact that some entrants to degree courses which require a knowledge of mathematics have taken double-subject mathematics while others have taken the single subject, so that it is difficult to establish a suitable starting point for all at the beginning of a degree course. However, there are very able students who profit from the double-subject course and we believe that it should continue to be offered to these students in schools and colleges in which the necessary staffing can be made available to teach the course without disadvantage to those who are studying mathematics at lower levels. 587 In our view, too much teaching time is often given to double-subject courses. We do not consider that it is either necessary or desirable to allocate to able students who take double-subject mathematics twice as much teaching time as is allocated to the single-subject course. We accept that those who are studying double-subject mathematics may need the time of two A Level subjects for their own work, but it should not be necessary for their teacher to be present throughout the whole of this time. Indeed, it is arguable that students who require this amount of attention from their teacher have not been well advised to choose the double-subject course. 588 We have already drawn attention in paragraph 180 [in chapter 4] to the fact that there has been a marked decrease in recent years in the proportion of university entrants to degree courses in mathematical studies who have a double-subject qualification at A Level; in 1979 this figure was 55 per cent. It is most important that schools and colleges should be aware of this and should point out to their students that it is possible in many universities and other institutions of higher education to follow mathematics degree courses successfully on the basis of a good performance in the single subject. It is equally important that those responsible for the early stages of these courses should take account of the fact that many of their students may not have studied the double subject. We hope, too, that those who select students for admission to higher education will recognise that there are sound educational as well as economic reasons for offering only single-subject mathematics at A Level and will not put either direct or indirect pressure on schools which have only limited teaching resources in mathematics to provide the double-subject course, especially for students to whom it is not well suited. 589 In recent years the pattern of some double-subject examinations has changed from that of 'pure mathematics' and 'applied mathematics' as separate subjects to that of 'mathematics' and 'further mathematics'. In the latter pattern, 'mathematics' is the single-subject A Level, containing both pure and applied mathematics, and 'further mathematics' consists of more advanced work which presupposes a knowledge of the single-subject syllabus. In our view the pattern of mathematics and further mathematics is preferable because it does not imply a distinction between pure and applied mathematics, which, as we have already pointed out, we consider to be undesirable at school level. The use of formula sheets in A Level examinations 590 We drew attention in paragraph 562 to the need to avoid a style of teaching which concentrates on the acquisition of techniques at the expense of the development of a broader approach to mathematics. However, this does not mean that it is unnecessary for students to acquire fluency in the routine processes which form part of A Level work. A number of the submissions which we have received from those who teach in higher education have drawn attention to a lack of confidence and accuracy in the routines of algebra, calculus and trigonometry on the part of many who start degree courses. This has been stressed especially by professors of engineering, who have also drawn attention to an over-reliance on the use of formula sheets on the part of many students and a consequent inability to recall simple formulae which are used frequently. 591 In recent years it has become the practice of almost all GCE boards to supply formula sheets which can be used during the A Level examination. There seem to be two main arguments for such provision. Firstly, formula sheets serve as a 'safety net' at a time of possible examination stress which can lead to sudden and damaging lapses of memory, sometimes in respect of quite elementary formulae. Secondly, they are a means of providing candidates with a list of formulae which are either difficult to remember or of relatively infrequent use; these are formulae which, when not working under examination conditions, students would probably obtain from a reference book. 592 We consider both of these arguments to be valid. However, there seems no doubt that the provision of formula sheets has led some teachers to assume, and to allow their pupils to assume, that there is no need to commit to memory any of the formulae which appear on the sheet. This is a view with which we cannot agree. While it is important that students should not be expected to memorise results which have not been adequately derived and discussed, we consider that many students handicap themselves because they do not have rapid recall of certain results which are of fundamental and recurring use in the development of the subject. At all levels understanding must carry with it a degree of remembering and it is our view that, unless students have confident recall (5) of such results as the trigonometrical addition formulae and the derivatives and integrals of simple algebraic and trigonometrical functions, they will lack the 'building blocks' which they need to develop their study of mathematics satisfactorily. They will also be handicapped in their study of other curricular areas which make extensive use of mathematics both in the sixth form and also in higher education. 593 It has been suggested to us that because, so long as formula sheets continue to be provided, they are liable to be misused we should recommend either that they should no longer be provided by examination boards or that the more elementary formulae should not appear on them. We do not accept this solution. Quite apart from the difficulty of securing agreement on which formulae should be included and which should not - and a formula which one student may be able to remember with ease may for some reason present difficulties to another - the formula sheet would not serve its purpose as a 'safety net' unless it was complete. 594 We believe that formula sheets should continue to be provided at A Level and that formulae which relate to any agreed 'common core' should appear on the formula sheets of all boards. However, we reiterate that the reasons for this are those which we have already given in paragraph 591 above. The fact that formula sheets are provided should not be regarded by either teacher or student as replacing the necessity for memorising and developing confident recall of fundamental and frequently used results.
I Level mathematics 595 Since our Committee was set up, the government have published proposals for the development of free-standing intermediate examinations (I Levels) (6) as a means of broadening the studies undertaken by some of those who currently take full A Level courses. An I Level course would last two years and occupy about half the time normally given to a full A Level course. 596 We support the suggestion that a course of this kind should be available for students who are not studying mathematics as a full A Level subject. We do not, however, believe it will necessarily be easy to design a suitable course and we expect that considerable development work will be required. In our view an I Level course should not be envisaged merely as a replacement for the 'service' courses which are at present provided in some sixth forms, though it would serve some of the purposes of these courses. Nor should it be a course which is comparable to existing AO Level courses in mathematics which would serve the needs of those who find a full A Level too demanding; AO Level courses should continue to exist for this purpose. We believe that the aim of an I Level course should be to develop mathematical ideas and extend previous knowledge without setting ambitious targets in terms of manipulative competence. For example, although calculus would be included, students should not be expected to spend time acquiring facility in the differentiation and integration of complicated functions. The course should illustrate the many ways in which mathematics can be applied and also include some study of the ways in which the subject has developed. We are not aware of any existing course which would be suitable, though we believe that use could be made of some of the ideas which are contained in the Mathematics Applicable (7) course and in the N Level study entitled Mathematical Awareness. An I Level course of the type we would wish to see would require skilled teaching and this would have staffing implications for schools and for in-service education. 597 We believe that there would also be a place for an I Level course in statistics. Such a course could serve the needs of many students, especially those who are studying A Level courses such as biology, geography, sociology or economics, in which there is an increasing emphasis on the critical examination and analysis of numerical data. In evidence to us the Royal Statistical Society and the Institute of Statisticians have stressed that statistics is not merely a collection of techniques but is a practical subject devoted to obtaining and processing data; and that the study of statistics should not become separated from the origins of that data. They have also pointed out that statistics in schools frequently ignores the practical situation and concentrates on formal manipulation. Within such an I Level course as we propose there should be time and opportunity to adopt a practical approach and to place emphasis on the application of statistical techniques to data which the students themselves have collected in the course of their own laboratory and field work. In this way it would be possible to demonstrate clearly the application of statistics to the analysis of data arising from study in several different areas of the curriculum and to develop a course which did not concentrate mainly on techniques. We believe that in many sixth forms it might be preferable to provide an I Level course rather than a full A Level course in statistics, since such a course would serve the needs of a much greater number of students.
Sub I Level courses 598 The consultative paper (8) which contains proposals for the introduction of I Level examinations also proposes the development of a pre-vocational course and an associated examination to be taken by students in either schools or colleges at 17+. This is intended to replace the proposed Certificate of Extended Education (CEE) examination for which some pilot examinations in mathematics have already been developed. When planning the mathematical component of the proposed new course we believe that account should be taken not only of the pilot CEE courses but also of the findings of the research studies into the mathematical needs of various types of employment which we have discussed in Chapter 3 and of our own proposals for a differentiated curriculum up to the age of 16+. O Level and CSE courses in the sixth form 599 Some have argued to us that the decision not to implement the proposal for the introduction of CEE means that a sixth form examination which, in its pilot form, has been well suited to some students will no longer be available. Some sixth form students who have not achieved O Level grade A, B or C or CSE grade 1 in mathematics in the fifth form have taken CEE pilot examinations in mathematics, on which CSE grades have been awarded, instead of repeating the O Level or CSE examination. Because the proposed pre-vocational course is not a single subject examination but a 'package', it will not provide separate certification in mathematics. Students who wish to improve their O Level or CSE grades will therefore need to take these examinations again. 600 However, pilot CEE courses have not been available in all parts of the country and in practice many students repeat O Level or CSE examinations in mathematics in the sixth form. These 'retake' courses are often among the least satisfactory in a school or college. The time allowance is sometimes meagre, the level of attainment of the students very varied and their examination targets diverse. It is not unusual for students in the same teaching group in a sixth form or tertiary college to be preparing for three or four different O Level and CSE examinations. Furthermore, problems of timetabling sometimes mean that students are not able to be present for all the mathematics periods in the week. In these circumstances the task of the teacher is very difficult indeed and there can be a temptation merely to practise past examination papers in the hope that improved performance will result. 601 We consider that one of the reasons why some pilot CEE courses have been successful is that they have required students to approach the mathematics course in a different, and often more adult, way. In our view a course which includes the introduction of some new work approached in an adult way is more likely to succeed than one which is based on the practising of past examination papers. Furthermore, if 'retake' courses for CSE or O Level are to be effective, it is essential to provide sufficient time for them, to ensure that students are able to attend all the periods and to consider carefully the approach which is used.
Footnotes (1) The study was carried out by analysing a 10 per cent sample of applications for university places made through the Universities Central Council for Admissions (UCCA) in 1977 and 1978 by home-based candidates who had attempted mathematics at A Level. (2) Griffiths HB and Howson AG Mathematics: society and curricula Cambridge University Press 1974. (3) Heard TJ The mathematical education of engineers at school and university Department of Engineering Science. University of Durham 1978. (4) Standing Conference on University Entrance and Council for National Academic Awards A minimal core syllabus for A Level mathematics 1978. (5) We draw attention to our discussion of memory in paragraph 234 [in chapter 5]. The learning of formulae and standard results should be associated with the use of suitable checking procedures; for example, 'sine is an odd function', or 'put x = 0'. (6) Examinations 16-18 A consultative paper. DES and Welsh Office 1980. (7) Schools Council Project MA 1601. (8) Examinations 16-18 A consultative paper. DES and Welsh Office 1980. 
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