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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 13 The supply of mathematics teachers
618 There can be no doubt that the most important resource for good mathematics teaching is an adequate supply of competent mathematics teachers. In this chapter we consider the present situation in schools and methods by which the existing stock of mathematics teachers might be increased. In the chapters which follow we consider the initial training and induction of those who teach mathematics and their subsequent in-service support. 619 The shortage of good teachers of mathematics has been a matter of concern for many years. As we hope will be clear from the earlier part of this report, mathematics is especially vulnerable to weak teaching. 'There is no area of knowledge where a teacher has more influence over the attitudes as well as the understanding of his pupils, than he does in mathematics. During his professional life, a teacher of mathematics may influence for good or ill the attitudes to mathematics of several thousand young people, and decisively affect many of their career choices. It is therefore necessary that mathematics should not only be taught to all pupils, but well taught. All pupils should have the opportunity of studying mathematics in the company of enthusiastic and well qualified mathematics teachers.' (1)
The present situation 620 The problems in primary and secondary schools are not the same. As we explained in Chapter 6, mathematics in primary schools is almost always taught by the class teacher and only a minority of primary teachers study mathematics as a main subject during their initial training. The need therefore is to increase the mathematical expertise of primary teachers overall; and also to increase the number of teachers who take mathematics as a main subject during initial training or who, at a later stage, undertake a substantial course of in-service training in mathematics, so that there will be a sufficient supply of teachers who are able to provide leadership and help for their colleagues. It is, however, necessary to recognise that the proportion of primary teachers who have taken mathematics as a main subject during their initial training is never likely to be large. 621 In secondary schools mathematics is almost always taught on a specialist basis. The shortage of well-qualified teachers of mathematics in secondary schools has increased considerably through the 1970s both as a result of the increasing number of pupils in schools and also of increasing demand for mathematicians in industry and commerce. Many of the submissions which we have received, especially those from LEAs and schools, refer to difficulties which have been experienced in appointing suitably qualified teachers of mathematics. Estimates of the extent of the shortage in secondary schools which have been made from time to time have varied considerably. In 1974 and 1976, the DES Secondary School Teacher Shortage Survey suggested that the shortage of mathematics teachers in secondary schools was of the order of 1100. A more recent estimate, given in the report of the National Secondary Survey (2) and based on visits to secondary schools during the period 1975-1978, suggested that the shortage of suitably qualified mathematics teachers in secondary schools was of the order of 3000. We give our own estimate of the shortage of mathematics teachers in paragraph 631. 622 A shortage of teachers of subjects which are at some stage optional in schools can to some extent be overcome by reducing the number of pupils who study the subject, and sometimes also the time which is given to it. However, because it is accepted that all pupils should study mathematics, shortage of mathematics teachers cannot be overcome by reducing the number of pupils to whom the subject is taught nor, other than exceptionally, the teaching time provided; it usually leads instead to the teaching of mathematics by teachers who are neither suitably qualified nor trained to do so. It seems sometimes erroneously to be thought that any qualified teacher should be able to teach mathematics adequately, especially to younger or lower-attaining pupils. Mathematics teaching is therefore especially vulnerable at a time at which rolls in secondary schools are starting to fall. For example, if one of four or five mathematics teachers in a school leaves the staff, there can regrettably be pressure not to replace him but, for the time being at least, to make use instead of teachers trained to teach other subjects, several of whom are likely to have a few teaching periods 'to spare' if the number of pupils in the school has decreased. 1977 Survey of Secondary School Staffing 623 We have therefore tried to obtain information about the levels of qualification of those who teach mathematics in secondary schools. Some information is given in the report of the National Secondary Survey but the only complete source of information on a national scale is provided by a sample survey of the staffing of some 500 maintained secondary schools which was carried out by the DES in November 1977 on the recommendation of the Advisory Committee on the Supply and Training of Teachers. The information collected included details of the qualifications of all the teachers in the schools included in the survey, the subjects which they were teaching, the amount of time given to each subject and the year group (but not level of attainment) of the pupils to whom the subject was being taught. Information was also collected about the curriculum of each school. Sixth form colleges were included in the sample as were middle schools 'deemed secondary' (that is, with pupils up to the age of at least 13) in respect of the teaching of pupils from the age of 11; tertiary colleges were not included. 624 The survey was planned and carried out before our Committee was set up, though analyses of the information which had been obtained did not start to become available until after we had started work. Although the survey was not designed to provide information about mathematics teaching in the detail which would have been possible had it been known that our Inquiry was to take place, the DES has, in addition to undertaking the analyses which were intended when the survey was planned, carried out at our request a number of further analyses of the data which is available. In this way we have been able to obtain a considerable amount of information. We realise that the figures which we quote are based on information which is now nearly four years old. However, although falling rolls in some areas and enforced economies may have produced some changes, we believe that the picture which emerges is unlikely to have changed significantly. There is, in any case, no more recent information available on a national scale. Levels of qualification of mathematics teachers 625 When considering the information about the teaching of mathematics which the survey provides, we have classified teachers in terms of their academic qualification. We are aware that the teacher who is well qualified on paper is not necessarily effective in the classroom as a teacher of mathematics; equally, a teacher who has little or no recorded qualification in mathematics may teach mathematics well. However, in the absence of any other information about the effectiveness of the teachers concerned we had no alternative but to proceed on this basis. All those teaching mathematics were assigned to one of four levels of qualification, 'good', 'acceptable', 'weak', 'nil'. The criteria used to assign teachers to these categories are set out in Appendix 1, paragraph A12. They are inevitably arbitrary, but a deliberate decision was made to err on the side of generosity in respect of qualifications which seemed to lie near the boundaries between categories. The overall picture which the analysis suggests may therefore be a little more encouraging than it should be; we do not believe that it is likely to err in the other direction. In our discussion of the outcome of the analysis, we have in some cases combined qualifications in the categories 'good' and 'acceptable' under the heading 'suitable' and qualifications in the categories 'weak' and 'nil' under the heading 'unsuitable'. 626 The results of the survey of secondary staffing show that, in November 1977, 38 per cent of all mathematics teaching in maintained secondary schools was being undertaken by teachers whose qualifications to teach mathematics were either 'weak' (17 per cent) or 'nil' (21 per cent); in other words, almost two fifths of all mathematics teaching was in 'unsuitable' hands. However, this overall figure conceals very considerable differences between different kinds of school, and also between different schools of the same kind. 627 On a national basis, the information provided by the survey suggests that there were in 1977 some 1500 secondary schools (about 35 per cent) in which at least 70 per cent of the mathematics teaching was in 'suitable' hands; and that these included some 240 schools (mainly sixth form colleges and grammar schools) in which all the mathematics teaching was by teachers with a suitable qualification. On the other hand, the results of the survey suggest that there were nearly 1300 schools (about 30 per cent) in which less than half of the mathematics teaching was in 'suitable' hands and that these included some 150 schools (mainly modern schools) in which none of the teaching was by teachers with a suitable qualification. In the remaining 1500 schools, between 50 per cent and 70 per cent of the mathematics teaching was 'suitably' staffed. 628 A more detailed analysis is shown in Appendix 1, Tables 15 -19; we commend these tables to the attention of our readers. We would repeat that, as shown in table 18, there is very considerable variation between schools of the same type so that, at the time of the survey, there were, for example, at least one modern school in which all the mathematics periods were taught by 'suitably' qualified teachers and at least two comprehensive schools in which none of the mathematics was so taught. The patterns of staffing were not directly connected with the age range of the schools concerned nor were less well qualified teachers only found in particular types of secondary school. The distribution of teaching staff may result, at least in part, from the way in which reorganisation to comprehensive education was carried out in many areas. Even though the range of ability of the pupils in a school changed markedly, there was often relatively little redistribution of teaching staff. Thus, sixth form colleges and comprehensive schools which developed from grammar schools often found themselves with a high proportion of graduate teachers, whereas comprehensive schools which developed from modern schools sometimes found themselves with many fewer graduates. Teachers in middle schools 629 It could be argued that, for teachers in middle schools, some of the qualifications classified as 'weak' on the ground that training had been for a younger age group should be regarded as 'acceptable' (see Appendix 1, paragraph A15). However, even if all teachers in middle schools whose qualifications were 'weak' were to be regarded as 'acceptable', 62 per cent of the mathematics teaching of pupils from the age of 11 in middle schools deemed secondary remained in 'unsuitable' hands. This must give particular cause for concern. In secondary schools with pupils only up to the age of 14, a high proportion of the mathematics teaching was also in the hands of 'unsuitably' qualified teachers. However, in these schools, too, it could be argued that some of the qualifications classified as 'weak' should be regarded as being 'acceptable'. Teaching of younger pupils in comprehensive schools 630 It was thought likely that less well qualified teachers might be used mainly with younger pupils. The hypothesis was examined in respect of the teaching of pupils up to the age of 16 in comprehensive schools with an entry age of 11 or 12; the full outcome is in Appendix 1, Table 19. Some confirmation of the hypothesis was found, but to a lesser extent than might have been expected. In the first year, 47 per cent of mathematics teaching was in 'unsuitable' hands; in the fifth year the figure was 37 per cent. This suggests that there are many schools in which the shortage of mathematics teachers is such that it is impossible to avoid using non-mathematicians with all age groups. 631 Appendix 1, table 16 shows that, adjusted to a 40 period week, some 178,000 periods of mathematics were taught each week in secondary schools (excluding middle schools) by teachers with a 'nil' qualification. If we assume the high teaching load of 35 periods per week, this number of periods represents the full-time contribution of more than 5000 teachers. In addition, the number of periods taught by teachers with a 'weak' qualification represents the full-time contribution of almost another 4000 teachers. These figures give a measure of the shortage of mathematics teachers in secondary schools in 1977, but take no account of the shortage in middle or primary schools. Deployment of teachers who are qualified to teach mathematics 632 In our discussion so far we have considered the proportion of mathematics periods taught by teachers with various levels of qualification. It has been suggested that the present shortage of mathematics teachers has been aggravated by the fact that some teachers who are qualified to teach mathematics are not doing so but are teaching other subjects instead. We have therefore investigated this possibility. It is important at the outset to realise that many teachers are equipped, both by qualification and training, to teach more than one subject. Teachers of mathematics, for example, are often equally well, and sometimes better, qualified to teach physics, of which there is also a shortage of teachers. If a teacher of physics were to change to teaching mathematics instead, the effect would be to increase still further the existing shortage of physics teachers and such a change might not be in the interest of the education system as a whole. When considering the question of 'misuse' of mathematics teachers, it is therefore not sufficient merely to count the total number of teachers who are qualified to teach mathematics; it is necessary at the same time to take into account the subjects which they are, in fact, teaching. 633 It proved possible to develop techniques to examine the extent of 'mismatch' in respect of the teaching of mathematics within the schools included in the sample survey. 'Mismatch' was defined as the teaching of mathematics by 'unsuitable' teachers when within the same school, teachers who were 'well' or 'acceptably' qualified to teach mathematics were not teaching either mathematics or a subject for which they were equally or better qualified. The analysis of 'mismatch' shows that in nearly a quarter of the schools in the survey there was no mismatch at all; in the great majority of the remainder the degree of mismatch was small and no greater than is to be expected as a result of the exigencies of timetabling, especially since, as we ourselves have advocated, there are significant advantages in timetabling a number of classes within the same year group to do mathematics at the same time. There were, however, a very few schools in which there seemed to be significant degree of mismatch; several of these were modern schools. It is therefore the case that attempts to redeploy teachers within individual schools are likely to contribute only minimally to improving the proportion of mathematics periods taught by 'suitably' qualified teachers. The need is undoubtedly to increase the number of teachers who are appropriately equipped to teach mathematics. Rates of entering and leaving teaching 634 We have sought information about the rates at which mathematics specialists enter and leave the teaching profession at the present time. The information about mathematics specialists in the following paragraphs relates only to graduate teachers in maintained schools, other than holders of the BEd degree, with mathematics as the only or first subject of their degree. These teachers do, however, constitute an important part of the mathematics teaching force. Figures given in the report of the National Secondary Survey (3) suggest that mathematics graduates undertake about one third of all mathematics teaching in secondary schools. 635 Figures relating to the years 1975-1979 are given in Appendix 1, tables 20 and 21. Throughout these years there were some 300 mathematics graduates working in primary schools. The number of mathematics graduates in secondary schools increased from about 7100 in 1975 to about 8100 in 1979, an increase of about 14 per cent. However, during the same period the total of all graduates working in secondary schools increased by about 30 per cent. The rate of increase of mathematics graduates was therefore very much smaller than that of graduate teachers as a whole. Indeed, whereas in 1970 mathematics graduates formed over 11 per cent of the graduate secondary teaching force, by 1979 the proportion had dropped to about 6 per cent. 636 We have been able to obtain details, by age group, of the number of graduates in mathematics only or in mathematics combined with non-science subjects who have left teaching each year since 1974-75, either on retirement or for some other reason, and the corresponding information for all graduate teachers (see Appendix 1, Table 22). Comparison of the rates of leaving shows no significant differences in respect of teachers over the age of 30 nor, until 1978-79, in respect of men aged 25 to 29. However, since 1975-76 rates of leaving have been markedly higher among men under the age of 25 who are mathematics graduates than among all male graduates of the same age. These comparative rates must be treated with caution because the numbers involved are small; nevertheless, they tend to confirm the impression that young male mathematics graduates are likely to be especially vulnerable to the attractions of alternative employment. Opportunities for such employment may well increase when the present recession ends and firms begin to recruit additional staff.
Methods of increasing the supply of mathematics teachers 637 We turn now to consideration of methods by which the supply of suitably qualified teachers of mathematics might be increased. There would seem to be three ways in which this could be accomplished. The first is to increase the number of entrants to the teaching profession who are suitably qualified to teach mathematics. The second is to take such steps as are possible to ensure that suitable teachers who are already teaching mathematics effectively will remain within the profession. The third is to improve the quality of the teaching of some of the under-qualified teachers, who are at present teaching mathematics, by means of appropriate in-service support and training. We believe that it is necessary to take action in all three of these ways. 638 Mathematics graduates form a principal source of supply of mathematics teachers in secondary schools. We have already referred to the fact that the demand for mathematics graduates is increasing from many sections of industry and commerce and will probably increase further when the present recession ends. Information provided for us by the Universities Statistical Record gives details of the occupations chosen by graduates in mathematical studies who completed their degree courses at universities in England and Wales in 1979; this is shown in Appendix 1, Table 31. These figures show that less than 10 per cent of the graduates in mathematical studies proceeded to teacher training. This figure contrasts with the fact that in 1938 over 75 per cent of newly qualified graduates in mathematics entered the teaching profession. In 1964 the proportion had decreased to 30 per cent, by 1974 to 17 per cent. There are also some mathematics graduates who enter teaching without undertaking teacher training; we return to this point in paragraph 692 [in chapter 14]. 639 In their submission to us, the Association of Graduate Careers Advisory Services (AGCAS) discussed at length the results of an inquiry which they had carried out at our request into the reasons why the number of mathematics graduates entering teaching was small. In their view, the principal factors are pay and prospects within the teaching profession and also perceived problems of discipline in schools; these factors are common to teachers of all subjects. Additional factors relating to mathematics are the fact that mathematics is thought to be a difficult subject to teach and that, because it is a compulsory subject for all pupils up to the age of 16, mathematics classes are likely to be larger and to contain a higher proportion of less well motivated pupils than is the case with many other subjects which are often optional after the age of 14. AGCAS suggest that an improvement in pay and prospects - especially the latter - needs to be an essential part of any attempt to attract more graduate mathematics teachers, together with improvements in the arrangements for induction into the profession in order to allay fears about discipline and allied problems. They also draw attention to the fact that 'low pay is seen as a reflection of the status of teaching and its importance to the community and this tends to have an off-putting effect. Students will enter low paid jobs if they are seen as attractive and confer status - eg working in the media. Teaching does not, however, have the same glamour and the low morale of the profession reinforces this view'. 640 We believe it is essential to do much more than is being done at present to improve the public image of teaching, and of mathematics teaching in particular. Too many potential teachers appear to be put off by widely reported problems of discipline, of the difficulties of teaching or of unhappy schools. Yet the results of a small survey carried out on our behalf by the National Foundation for Educational Research, to which we refer in greater detail in paragraph 671 [in chapter 14], showed that the group of mathematics teachers included in the survey, who were in their first three years of teaching in secondary schools, were generally happy in their work. A large majority believed that they were well thought of by their colleagues and that they had gained the respect of their pupils. We hope that both central and local government will respond to our report by affirming their belief in the importance of good mathematics teaching for all pupils, the need to provide good support and facilities for mathematics teachers who are already in post and, whatever the overall teacher requirement may be, the need for many more good teachers of mathematics. Good publicity is necessary in order to improve attitudes towards the teaching of mathematics on the part of the general public and, in consequence, in the minds of potential teachers. 641 In our view it is also necessary to make a much greater effort than is being made at present to recruit mathematics graduates into teaching. It has been pointed out to us that very many large employers take part in the annual 'milk round' of universities, during which representatives of companies visit universities to inform, advise and interview undergraduates, but that no comparable effort is made by the DES or by LEAs acting in concert. Some careers advisers in universities have told us that students therefore gain the impression that, despite the widely publicised shortage of mathematics teachers, LEAs are not concerned to make a sustained effort to increase recruitment. A further suggestion which has been made to us is that the DES should write to all who are starting the final year of a mathematics degree course pointing out the need for well-qualified teachers of mathematics and the attractions of teaching as a career. Such a letter should include an offer to provide further information. We believe that the possibilities of making a direct approach to undergraduates reading degree courses in mathematics and of some kind of participation in the 'milk round' should be investigated. Mathematics degree courses 642 Our terms of reference do not extend to higher education and we have not therefore made any extended study of mathematics courses in universities; nor would it be proper for us to make any recommendations relating to this area. However, because a substantial number of teachers of mathematics are, and will continue to be, mathematics graduates, we believe that it is proper for us to consider two matters. The first is the extent to which university mathematics courses in their present form provide a suitable preparation for the prospective school teacher. The second is whether significant numbers of potential mathematics teachers in schools are being lost because university courses are too demanding. We put forward our views on these two questions in the hope that universities (4) may be prepared to examine and perhaps to adjust the provision which they make. 643 In our view, the mathematical training provided at university for those who will become mathematics specialists in schools should aim
644 There are some degree courses in mathematical studies which combine mathematics and education for those intending to become teachers. While such courses can cater more specifically for intending teachers, they usually require an early commitment to teaching which many students are unwilling to make. The evidence which we have received makes it clear that most undergraduates prefer to keep their options open for as long as possible. It follows therefore that a mathematics degree course should provide within its structure a set of options which are suited to the needs of those who feel that they may wish to teach but who have not yet made a firm decision to do so. 645 We do not, of course, suggest that mathematics degree courses should be directed only to the needs of those who will enter the teaching profession. Nevertheless, the supply of an adequate number of well qualified mathematics teachers for schools must be of paramount importance to all institutions of higher education so that the level of mathematical preparation of the students who come to them may be maintained or improved. Furthermore, we believe that, for the great majority of mathematics graduates, the requirements which a mathematics degree course should fulfil are not dissimilar to those which we have suggested are needed by teachers. It is only a small minority of those on mathematics degree courses who proceed to more advanced studies in mathematics after the undergraduate stage. For this small but important minority it is essential that options should be available which prepare them for postgraduate study but we do not feel that it should be necessary for their requirements to dictate the content of mathematics degree courses for the majority of those who attempt them. 646 There can be no doubt that there is a widespread view in schools that university mathematics courses are exceptionally demanding and should only be undertaken by those who are very able. Courses at universities do, of course, differ greatly in their structure and range and we welcome this diversity. Nevertheless, there is a strong tradition, stemming from the older universities, which delineates the level of difficulty of a mathematics degree. It is possible that this is a serious obstacle to the wider recruitment of students who enrol for mathematics degree courses and hence to the provision of a larger pool from which mathematics teachers can come, as well as the mathematics graduates needed by industry and commerce. We drew attention in Chapter 4 to the large increase in the proportion of entrants to degree courses in mathematical studies who have only a single-subject A Level qualification in mathematics. Universities may wish to ask themselves whether broader and more flexible mathematics courses, designed particularly for single-subject students, might not attract a significantly larger entry. For such a policy to be effective, it would of course be necessary for schools to be made aware of the nature and purpose of such courses so that they could advise their students appropriately. 647 We draw attention to the fact that in recent years the number of girls who have studied mathematics at A Level has been only about one third that of boys; the number of women who have graduated in mathematical studies at universities in England and Wales has been some 40 per cent of the number of men. On the other hand, among these graduates the proportion of women who have entered Postgraduate Certificate in Education (PGCE) courses to train as teachers has been more than twice that of men, so that the numbers of men and women entering PGCE courses have been about the same. If the number of girls who studied mathematics at A Level were to increase, the pool of potential graduates in mathematical studies would increase and hence the pool of potential teachers of mathematics. The number of potential entrants to BEd courses who would be capable of studying mathematics as a main subject would also be increased. We believe that active steps to encourage more girls to take mathematics at A Level and then proceed to study at degree level could lead to an increase in the supply of well-qualified teachers of mathematics. Entry into teaching from other employment 648 A small number of graduates in mathematics and allied disciplines enter mathematics teaching after some years in other kinds of employment. We have received conflicting evidence as to the extent to which such transfer should be encouraged. Some argue that such entrants to the teaching profession bring with them experience which can be used in the classroom to illustrate the uses and applications of mathematics. On the other hand, we have received adverse comment on the narrow approach to mathematics and the low teaching abilities which have been displayed by some who have transferred to teaching from other occupations. It has also been pointed out that some may turn to mathematics teaching because they have not succeeded in their former jobs. It may have been too easy for graduates to enter teaching from other types of employment, without any suitable training or even any real attempt to assess potential as a teacher. However, at the present time of cutback in many areas of employment, we believe that some who have been very successful in other fields and who would make good teachers are likely to be available and that every effort should be made to recruit them. While we recognise an understandable unwillingness on the part of potential recruits to mathematics teaching from industry and commerce to commit themselves to a year's training with a relatively low income and without certainty of qualification at the end, we believe that training for such recruits to teaching is just as important as for those who have just graduated. We therefore believe that adequate financial support should be available to remove any disincentive to train. 649 There should also be adequate opportunity for those who are considering teaching, and those who might employ them, to be able to make some preliminary judgement about their suitability for teaching. We therefore welcome the courses lasting for one term or less which have been arranged by some training institutions and LEAs, in association with major companies, for staff who face the possibility of redundancy. We believe that the availability of one term of 'adjustment to teaching', followed by a further two terms of training for those whose experience of the first term convinces both them and their future employers of their potential as teachers, could lead to a desirable increase in recruitment to mathematics teaching. We also welcome the scheme operated by the Inner London Education Authority which enables mature graduates whose degrees include a substantial mathematical component and who have industrial or commercial experience to be appointed to the authority's temporary staff and paid at the Scale 1 rate for a qualified teacher whilst on secondment to a PGCE course. Those who are recruited under this scheme work in schools within the authority for those parts of the school year which are outside college terms. A further scheme at one training institution allows holders of higher national certificates or diplomas, or some equivalent qualification, to complete a BEd degree in five terms; we have been told of other institutions which would be willing to adapt their courses in order to cater for such entrants if they were forthcoming in sufficient numbers. We believe that initiatives of all these kinds should be encouraged and that they should be supported by adequate publicity. It would be helpful to establish a 'clearing house' which could offer advice on the availability of courses. 650 Some improvement in the supply of mathematics teachers has been brought about since 1977 by the government scheme for training and retraining to teach priority subjects which is operated by the Manpower Services Commission on behalf of the DES. Under this scheme special training awards are available for suitably qualified men and women over the age of 28 who wish to take a one-year course of initial training to teach shortage subjects, including mathematics. Between 1977 and 1980 more than 400 mature entrants with a suitable qualification in mathematics have been funded under the scheme while taking initial training courses to become teachers. In addition, more than 500 teachers of other subjects have taken retraining courses during this period to enable them to teach mathematics. We welcome this contribution to the mathematics teaching force and also the fact that the scheme has now been extended to include one-term or one-year courses of further training for existing teachers of mathematics. Employment of primary trained teachers in secondary schools 651 There is one source of recruitment to secondary mathematics teaching of which we have become aware which we do not believe to be desirable, nor to be in the interests of the education system as a whole. We have been told of a number of recent cases in which teachers who have taken mathematics as a main subject in BEd degree courses and who have trained to teach in primary schools have been offered, and in some cases urged to take, posts teaching mathematics in secondary schools. Not only are such teachers not suitably trained for this work but, if they enter secondary teaching, do not contribute to the increase of mathematical expertise which is so badly needed in primary schools. We believe it is important that LEAs should recognise this need and ensure that primary trained teachers with mathematical expertise are appointed to primary schools so that the teaching of mathematics in primary schools can be strengthened. As secondary rolls begin to fall we believe that every effort should also be made to enable and encourage mathematically qualified teachers who are teaching in secondary schools, but who have been trained for primary work, to transfer to primary schools. Grants during training 652 We welcome the pilot scheme which is about to be introduced to provide national scholarships for intending teachers of mathematics (5). These scholarships will provide a flat-rate payment to selected entrants to PGCE courses who are considered to be likely to become teachers of high quality. They require a commitment on the part of the recipient to enter teaching on the satisfactory completion of the PGCE course and offer a guarantee of employment as a teacher. It is, however, possible that some of these scholarships will be gained by students who would have entered teaching in any case, so the net increase in entrants to mathematics teaching may not be large. Nevertheless, this scheme demonstrates recognition at national level of the urgent need for more mathematics teachers. 653 It has been represented to us that the recruitment of mathematics graduates to PGCE courses might increase if the existing system of means-tested grants for these students were to be abolished. We accept, however, that such a change could lead to anomalies in respect, for example, of students in the final year of BEd honours courses and that the likely 'knock-on' effect may, for financial reasons, rule out such a change for several years. We believe that a system of flat-rate payments to graduates training to teach mathematics (and other shortage subjects) would prove more straightforward to operate; such payments should be conditional on the possession of suitable mathematical qualifications (see also paragraph 683 [in chapter 14]) and an undertaking to enter the teaching profession and remain as a teacher for a specified minimum time provided that training has been completed successfully. Grants under similar conditions could be given to fourth year students on BEd honours courses who were taking mathematics as a main subject. A scheme of this kind could be discontinued if the shortage of mathematics teachers ceased, though we believe that a continuing incentive for BEd mathematicians to enter primary schools would be valuable. It is, however, our view that improvement in training grants of any kind is not likely to result in significant increase in recruitment unless future prospects for pay and promotion are seen to be attractive.
Financial incentives 654 We referred in paragraph 639 to the fact that evidence which we have received from the Association of Graduate Careers Advisory Services points out that 'better pay and prospects are probably the most important potential factors for increasing the intake' of teachers. We have therefore endeavoured to compare the financial rewards of a career in teaching with those in other occupations which recruit graduates. Such enquiries as we have been able to make point not so much to the difference in initial reward as to the contrasts in perceived prospects at a later stage. This seems to be the case despite the advantages which are sometimes put forward of greater security, assured and index-linked pensions and longer holidays which teachers enjoy. The newly qualified teacher, with a good honours degree and a PGCE, starting at a salary of £5,547, can be assured of regular though not large annual increments to a total of £7,869 after 10 years service but advancement to a higher salary scale is dependent on promotion opportunities which are controlled by the Burnham salary structure. These opportunities are related to the numbers and ages of the pupils in a school. At the present time school rolls are declining nationally and we have been told that, under the existing regulations, opportunities for promotion beyond Scale 2, the top of which is currently £8,208, are likely to become less. As rolls fall, job security may also be reduced. 655 It is very difficult to quantify the salaries which can be earned by mathematics graduates who do not enter teaching or to generalise about their very varied conditions of employment. Some information about salaries is provided by a survey of its members carried out in 1980 by the Institute of Mathematics and its Applications (IMA) to which some 3600 members replied; one eighth of these were teachers in schools. The results of the survey show that, for the age range 26-35, the salaries of those members of IMA who were teachers were on average between £1000 and £2000 per annum lower that those of members of IMA employed in central government, private and nationalised industry, and commerce. It has also been pointed out to us that, in industry and commerce, the size of a salary increase is very often dependent on the competence and commitment which an employee displays, as revealed and recorded by regular assessment procedures. On the other hand, a teacher's salary advances according to a predetermined scale which, even in the case of transfer to a higher scale post, limits the total salary increase which can be gained in any year, no matter how committed and effective a teacher may be. Additional payment to teachers of mathematics 656 This raises the question of the extent to which additional payment should be made to teachers of mathematics. We have noted that the Fifth Report of the Education, Science and Arts Committee of the House of Commons (6) issued in September 1980, which deals with the funding and organisation of courses in higher education, recommends that 'a higher rate of maintenance grants for students and higher salaries for teachers in certain subjects, such as mathematics, should now be considered'. The report of the Central Policy Review Staff (7), published in May 1980, says that 'while recognising the difficulties we believe that without a pay advantage it will be very hard to overcome these chronic shortages [of teachers of mathematics, science and practical subjects] and we recommend that the nettle be grasped now'. 657 The evidence which we have received on the question of additional payment to teachers of mathematics is conflicting. The teachers' unions and the professional mathematical associations do not support additional payment. On the other hand we have received support for such payment in submissions from individual teachers as well as from outside the teaching profession. We have discussed this matter at length and are agreed that additional funding in some form is necessary if the present situation of acute shortage is to be alleviated. We suggest two approaches which we believe would be possible. 658 The first is to make greater use of the flexibility which already exists, but is little used, within the Burnham framework and also to introduce additional flexibility. LEAs are already able to offer Scale 2, 3 or 4 posts for reasons which they consider fit; merit as a teacher can be one such reason, but we believe that teachers are not promoted on these grounds as often as we would wish. We have been told that a recent suggestion to LEAs from the joint secretaries of the Burnham Committee for a limited increase in the proportion of Scale 3 posts in schools as a means of relieving the shortage of teachers of mathematics and physics appears to have received little support from LEAs; we welcome the intention behind this suggestion and regret that, perhaps because of current financial pressure, it has not been supported. 659 We wish to make a somewhat different proposal. We believe that LEAs should have discretionary power to make appointments in secondary schools at, and promotions to, incremental points above those which are defined nationally. This power exists in local government outside teaching and in many other forms of employment. The case for granting such discretion is that it could be used to encourage ambitious and competent teachers to continue as specialist teachers of mathematics rather than seek promotion by moving to other responsibilities. Such additional payment (8) would improve the pay of mathematics teachers relative to outside competition. In order to improve prospects it may be necessary for extra payment to be possible above the maxima of existing scales. 660 A major weakness of incentives which are related to the Burnham scales is the difficulty of ensuring that a beneficiary does not retain his advantage if for any reason he ceases to meet the conditions for which the award was made, for example by ceasing to teach mathematics; it should, though, be possible to devise methods of dealing with this problem. If, however, this problem cannot be overcome, we would suggest a scheme which would associate the payment of a specified additional allowance with the tenure of a particular post in a particular school. If a teacher in receipt of such an allowance changed his school or assumed other responsibilities, the allowance would automatically cease. 661 Both of these schemes could also be used as a means of attracting good mathematics teachers to schools whose mathematics departments were inadequately staffed. We suggest that the schemes should apply only to mathematics specialists who were teaching mathematics for at least a given proportion of the timetable and whose teaching was competent; we do not believe that additional payment should be made during the first two years of teaching. 662 In order to encourage good mathematics teaching in primary schools we believe that there would be merit in allowing LEAs some incremental discretion for teachers in primary schools, especially those in which numbers of pupils limit the availability of posts above Scale 1. An arrangement of this kind could assist in the appointment or retention of mathematics coordinators of good quality. 663 It might be argued that our suggestions would have little hope of success because those who were competing for the services of mathematics graduates would respond by increasing the salaries which they offer. We are not convinced that this would necessarily happen because the proportion of mathematics graduates who enter teaching is in any case small. 664 We recognise that the suggestions we make are open to objection, especially in respect of the problem of establishing competence or the lack of it. We are aware, too, that some may see a threat to relationships between teachers in proposals which they may see as divisive. We also appreciate the financial restraints which exist at present. Nevertheless, we are convinced that our suggestions would greatly help the recruitment and retention of good mathematics teachers without which the improvement in mathematics teaching which we believe to be necessary will not be attained. Any who condemn our suggestions should suggest alternative remedies to the continuing shortage of well-qualified mathematics teachers.
The need to employ newly trained mathematics teachers 665 We believe that a concerted campaign to attract more mathematics graduates into the teaching profession, backed by the financial incentives which we have proposed in the preceding paragraphs, might succeed in its objective. The prospects for success are considerably improved at the present time because the economic recession has reduced competition from other sources for mathematicians. Indeed, the immediate future may present an unrivalled opportunity to achieve a significant improvement in the size and quality of the mathematics teaching force - an improvement which we believe to be essential. Although falling rolls in schools and financial constraints on LEAs are inevitably leading to a reduction in the recruitment of teachers, we wish to emphasise the vital necessity of maintaining, and where possible increasing, the recruitment of mathematics teachers despite present financial difficulties. We believe, too, that the need for mathematics to be taught by well-qualified teachers should take priority over the need to redeploy staff. It will be of little use to mount a campaign to encourage more mathematics graduates to enter teaching if LEAs are not prepared to employ them. We therefore feel that measures should be taken which would ensure that, in the next few years, newly trained mathematics teachers will be able to obtain teaching posts. It may be that local education authorities should be subsidised, perhaps on a limited and temporary basis, by the DES so that all available suitable mathematics teachers are employed. We believe that the annual cost to central government would be small, that the benefits would be direct and immediate, and that the autonomy of local authorities to appoint their staff would be unimpaired. In addition, newly trained mathematics teachers and those who were considering training as mathematics teachers would feel convinced that their services were in demand. Measures of this kind could be reviewed annually in the light of changing circumstances and could be terminated without difficulty if and when the mathematics teaching strength in schools had reached a satisfactory level.
Footnotes (1) The Royal Society The training and professional life of teachers of mathematics November 1976. (2) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979. (3) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979. (4) We concentrate on courses in universities because it is from these that most mathematics graduates come. The degree courses in mathematics offered at other institutions are often more varied and less abstract than those offered at universities; not all of our comments may therefore apply to them. (5) The Education (Teacher Training Scholarships) Regulations 1981. Statutory Instruments 1981 No 1328. (6) House of Commons. Fifth report from the Education, Science and Arts Committee The Funding and Organisation of courses in higher education HMSO 1980. (7) Central Policy Review Staff Report Education, training and industrial performance HMSO 1980. (8) At the present time about one eighth of teaching time in secondary schools is given to mathematics. The figures which we have quoted from the 1977 survey of secondary staffing suggest that only about 60 per cent of this teaching is in 'suitable' hands. We estimate that the cost of providing an average of two additional increments on the Burnham salary scale for half of the 'suitable' teachers who were teaching a full timetable of mathematics would amount to between 0.2 per cent and 0.3 per cent of the total salary bill for secondary teachers. (This estimate is based on the payment of such increments to 1/32 assistant teachers in secondary schools.) 
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