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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 14 Initial training courses
666 At the present time most of those who teach in schools qualify for entry to teaching in one of two ways. The first is by following a course leading to the degree of Bachelor of Education (BEd). This provides both an academic qualification and also professional training as a teacher. The ordinary degree course lasts for three years, the honours course for four years. The second way is by undertaking a one-year course of professional training leading to the award of the Postgraduate Certificate in Education (PGCE) after gaining a graduate qualification other than BEd. Courses leading to entry by a third method, the non-graduate Certificate in Education, which provided academic and professional training by means of a three-year course, are now being phased out and it is no longer possible to enrol for courses of this kind. A few teachers qualify for entry to teaching in other ways but in this chapter we confine our discussion to entry to teaching by means of BEd or PGCE courses. 667 Initial teacher training in England and Wales takes place in both the university and non-university sectors. BEd courses are offered in some eighty non-university institutions, which include polytechnics and other establishments of higher education; a few are offered in universities. The structure of BEd courses varies from institution to institution. However, most BEd courses include, in addition to professional training and the study of education theory, opportunity to study one or more subjects in depth; we discuss these 'main' courses in paragraph 695. PGCE courses are offered in some thirty universities and in some sixty establishments in the non-university sector.
Entry qualifications to initial teacher training courses 668 All entrants to BEd courses are now required to satisfy the normal requirement for entry to first degree courses; that is, to have passed in at least two subjects at A Level or to have obtained an equivalent qualification. This was not the case with the Certificate in Education courses and so the minimum academic level for entry to teacher training is now higher than has been the case hitherto. A further requirement has recently been introduced in respect of entrants to teacher training who expect to become eligible to join the teaching profession in or after September 1984. These entrants are required to have obtained O Level grade A, B or C or its equivalent in both English and mathematics before starting their training course. This means that all entrants to BEd courses are now required to have obtained this qualification; it will apply to entrants to PGCE courses from September 1983. Reference to Figure F (paragraph 195 [in chapter 5]) suggests that many intending teachers are likely to find it more difficult to gain the necessary qualification in mathematics than in English. 669 The decision to require entrants to initial training to have an O Level qualification, or its equivalent, in mathematics has been welcomed in many, though not all, of the submissions we have received which make reference to it. In our view it has been right to introduce this requirement. However, as is pointed out in DES Circular 9/78, in which the decision to introduce the requirement was announced, 'formal qualifications do not necessarily guarantee possession of competence'. In addition, 'possession of a pass in mathematics at O Level of GCE or the equivalent may, of course, sometimes offer little indication of a student's potential ability to teach the subject' (1); we discuss this matter further in paragraph 679. The present reduction in the numbers being accepted for entry to teacher training means that it is not yet possible to assess the effect of the new requirements on recruitment to the teaching profession.
Recent changes in teacher training 670 The last ten years have been a time of very considerable change in the non-university sector of teacher training. The reduction in the number of training places has led to the closure of some colleges of education and the merger of others with institutions which provide higher education courses of other kinds. It has also been necessary to plan and introduce new courses to take account of the phasing out of the Certificate in Education and also of the reorganisation of colleges; in many cases reorganisation has been accompanied by the transfer of validation from a university to the Council for National Academic Awards. Reductions in staffing resulting from reorganisation have meant that there have been few appointments to the staff of training institutions in recent years; this means that the proportion of staff who have recent experience of full-time teaching in schools is decreasing. Because of the very great changes which have been taking place, much of the comment which we have received about training to teach mathematics relates to courses which either no longer exist or which have been changed very considerably, or to training which took place at a time of uncertainty or rapid change. Many comments, too, are anecdotal and, understandably, many of those who have written to us appear to have knowledge of only one training institution. We have not therefore found it easy to decide on the extent to which either criticisms or compliments which we have received relating to teacher training remain valid at the present time. We have, though, to record that we have received a good deal of comment which has been critical. Views of recently trained teachers 671 We learned that the National Foundation for Educational Research (NFER) had recently been studying the progress through sixth form and higher education of a group of students who subsequently became teachers. The final stage of the study was the administration of a questionnaire to ascertain the reaction of these teachers to their first year of teaching and to their initial training as seen in retrospect. However, the group contained very few specialist teachers of mathematics and so, in an endeavour to obtain the views of mathematics teachers, the DES, at our request, commissioned NFER to extend their study by administering a slightly amended version of their questionnaire to a sample of teachers of mathematics who were in their first three years of teaching in secondary schools. This survey was carried out in October 1980 and replies were received from 198 teachers of whom 48 were in their first year of teaching. 672 The reactions of these teachers to a number of statements about their present teaching post revealed, as we have already stated in paragraph 640 [in chapter 13], a picture of a group of teachers who were happy in their work. More than 90 per cent said that the statements 'I enjoy teaching mathematics very much', 'I am very happy teaching mathematics in my present school', 'I am very satisfied with the duties of my present job' applied at least moderately to them; about half the teachers said that these statements applied strongly. 673 In response to a group of statements about the adequacy of their initial training, these teachers felt, in general, that their training had prepared them better for the subject content of their mathematics teaching and for classroom management and organisation than for dealing with discipline problems, assessing pupils' progress or dealing with the whole range of ability, especially in mixed-ability groups. Those who had followed PGCE courses were more satisfied with the balance between the different elements of their courses than were those who had followed BEd or Certificate in Education courses. Among those teachers who commented on the balance of the course, there was a general wish for a greater emphasis on the more practical elements of the course, such as methods of teaching and classroom management, practical teaching and observation in classrooms. Those teachers who had asked themselves how time was to be found for a greater emphasis on these elements almost all suggested that the time given to education theory should be reduced. Many teachers felt that the relevance of education theory had not become apparent by the end of their initial training course. 674 Some teachers commented that their initial training courses had failed to provide adequate preparation for the teaching of mathematics to very slow learners. In our view it should not be the task of initial training courses to provide preparation for such teaching; nor do we believe it to be practicable, though students should be made aware of the variety of special needs which they may meet in their pupils. In order to be able to appreciate the special difficulties of very slow learners, as also of children with other special needs, it is necessary first of all to gain experience of teaching mathematics to pupils who do not have problems of this kind. We therefore consider that training to teach pupils with special needs should be provided by means of in-service courses undertaken after teachers have had opportunity to gain classroom experience. In our view new entrants to teaching should not be required to teach mathematics to classes of such pupils.
Professional training 675 Professional training within BEd and PGCE courses is related to the age of the pupils - primary, middle or secondary - whom the student intends to teach. Two of the elements of professional training are courses of instruction in methods of teaching particular subjects, commonly called 'curriculum courses' in BEd and 'method courses' in PGCE, and school experience. We discuss both of these. Curriculum and method courses 676 We start our discussion of curriculum and method courses by drawing attention to two matters which we believe to be of importance whatever the age range to which the course relates. We consider that the need to take account of these will require changes in a number of courses. 677 The first is the need within these courses to stress the various elements which, as we have pointed out in paragraph 243 [in chapter 5], should be included within mathematics teaching and in particular the place of oral work, discussion and practical work. 678 The second is the need to emphasise the relationship between mathematics and other areas of the curriculum. 'The contribution of a subject to the pupil's education is weakened if it is not perceived by the teacher and presented to the pupil in terms of its relation to the rest of the curriculum. There are two associated aspects of this notion. One is the extent to which the teacher sees his subject as part of the curriculum as a whole and as contributing to it. The other is the explicit making of links between subjects in such a manner that the skills and ways of thinking in a particular subject are given opportunity for development in others.' (2) The more integrated approach to the curriculum which is used in the primary years may perhaps make it easier for intending primary teachers to view mathematics in this way, provided that they have been able to develop confidence in their own knowledge and use of mathematics. It is no less important, though it is likely to be more difficult to achieve, that those who teach mathematics in secondary schools should also be led to consider the relationship of mathematics to the curriculum as a whole. Preparation for teaching mathematics in primary and middle schools 679 Because nearly all primary teachers and many of those who teach in middle schools are required to teach mathematics, all students who are training to teach at these levels take a curriculum course in mathematics. Most of them will not be studying mathematics as a main subject and many may well not have enjoyed mathematics at school. Even though the introduction of the O Level or equivalent requirement for entry to teacher training may lead to some improvement in overall competence and also in the attitude of some students to mathematics, it cannot be expected that all those who intend to teach in primary and middle schools will necessarily be looking forward to teaching mathematics; indeed, some may have given up their studies in mathematics with a sense of relief after gaining their O Level qualification. It is not therefore surprising that some students will start their training with fears about the teaching of mathematics, and that training institutions should have difficulty in giving to some of their students the positive attitude to the subject and the confidence which are necessary if these students are to be able to teach mathematics well. However, the teaching of mathematics will probably occupy about 20 per cent of the time of those who teach in primary schools. It must therefore be the major task of those who train these students to establish positive attitudes, to consolidate and deepen the students' knowledge of mathematics - a process which may involve filling some gaps - and to provide them with a firm basis from which to start teaching mathematics. Our discussion of primary mathematics in Chapter 6 indicates how much is involved in this task. 680 Only a minority of those who are now teaching in primary and middle schools have trained by means of a PGCE course (3). Indeed, there are many who feel that one year is too short to provide adequately for all the needs of PGCE students training to teach in these schools. Their mathematical needs are those which we have already described but the problems of providing within a PGCE course for students who have a sense of inadequacy in mathematics are likely to be greater, both because of the short time which is available and also because many may not have studied mathematics for at least five years. In our view the long-term solution lies in lengthening the PGCE course for primary teachers. In any event, we consider that training institutions should not offer places on PGCE courses for intending teachers in primary or middle schools to students whom they know to be mathematically weak. Preparation for the specialist teaching of mathematics 681 Although a large proportion of those who take mathematics as a main course in BEd become specialist teachers of mathematics, the majority of mathematics specialists in secondary schools train as teachers by taking a PGCE course with mathematics as their first method subject. Figures relating to the enrolment on PGCE courses since 1975 of graduates who are taking mathematics as their first method subject are given in Appendix 1, Table 24. A very sharp drop in numbers in 1978 and 1979 has been followed by a considerable upturn in 1980 but it seems that the increase is not necessarily in students who have an honours degree in mathematics. Interim findings from a survey of PGCE courses in university departments of education which is being conducted by Professor G Bernbaum at the University of Leicester show that in 1979-80 there were about 200 men and 200 women who were taking mathematics as a first method subject in PGCE courses at universities. About half of these students had an honours degree in mathematics, about 7 per cent an honours degree in a related (4) subject and about 13 per cent an honours degree in a subject which was not related. The remaining 30 per cent had joint honours, pass or ordinary degrees but there was no information about the mathematical content, if any, of these degrees. Information relating to PGCE courses in the non-university sector for the year 1980-81, which has been made available to us by the Mathematical Education Section of the National Association of Teachers in Further and Higher Education, suggests that less than 30 per cent of those who were taking first or second method courses in mathematics possessed a mathematics degree; a further 20 per cent had joint degrees including mathematics. It therefore seems clear that at the present time by no means all those who are taking mathematics as a first method subject in PGCE courses in either the university or non-university sectors are mathematically well qualified (see also Appendix 1, Table 25). 682 Two reasons have been suggested to us which may account for the fact that an increasing number of graduates in other disciplines are taking mathematics as a first subject in PGCE courses. The first is that it will be easier to obtain a post teaching mathematics than teaching some other subjects. The second is that, at a time when job opportunities are not as plentiful as used to be the case, it is a useful 'insurance policy' to obtain a teaching qualification in a shortage subject. 683 We therefore believe that the increase in recruitment to PGCE courses in mathematics should be viewed with caution. We welcome the increase in numbers of students training to teach mathematics if their degree courses have contained a substantial mathematical component. On the other hand, we view with some concern the fact that students whose degree subject is completely unrelated to mathematics should be accepted for a PGCE course in mathematics. We agree that 'these students are likely to swell the numbers of those whose qualifications are inadequate or inappropriate for the work that they are being asked to do' (5). We do not think that, within the time allotted to most method courses, they are likely to be able to learn to present mathematics effectively in the classroom. In our view, students should not be admitted to first method courses in mathematics unless their degree courses have contained a substantial mathematical component. 684 The number of teachers in secondary schools who are expected to teach more than one subject is likely to increase during the next few years because of pressures resulting from falling rolls. In some PGCE courses students preparing to teach in secondary schools take one main subject but in others they follow courses in two subjects. In some cases these courses are equally weighted but often the second subject is given less, and sometimes very much less, time. At the present time mathematics is a popular choice as a second subject. The Leicester survey showed that in 1979-80 there were about 360 students taking mathematics as a second method subject in PGCE courses at universities. Of these students, about 20 per cent had an honours degree in a subject related to mathematics and about 50 per cent an honours degree in a subject which was not related; the remaining 30 per cent had joint honours, ordinary or pass degrees but there was no information about the mathematical content, if any, of these degrees. It is highly probable that many of those who take mathematics as a second subject will be required to teach it. Indeed, it may be the case that some who choose a second method course in mathematics will do so in the hope that some training in the teaching of mathematics will help them to obtain a teaching post. It is therefore essential that these courses are substantial enough to give adequate grounding and that 'recruitment to them must be governed by exacting standards' (6). In our view students should not be admitted to a second method course in mathematics unless their degree course has contained a reasonable mathematical component or they are mathematically qualified in some other way. Time allowance for curriculum and method courses 685 From the submissions which we have received it is clear that there is very considerable variation in the amount of time which is given to curriculum and method courses in mathematics, and to their positioning within BEd and PGCE courses as a whole. A frequent complaint has been that insufficient time is given to curriculum and method work but complaints of this kind are directed at time allowances which are widely different; nor is there any apparent connection between the amount of time allowed and the intensity of the complaints. The time allotted to curriculum and method courses appears to depend on a number of factors; these include the timing of these courses in relation to teaching practices and other school visits, their relationship to other elements within professional studies which deal with classroom method and the amount of private study which is expected of students. 686 A number of submissions suggest that within primary and middle BEd courses a minimum of 90 hours spread over three or four years should be given to a curriculum course in mathematics. This suggestion was supported in oral evidence to us by the Council for National Academic Awards, which is responsible for the validation of many BEd courses. We were told that if the time allocation fell to below 90 hours, it was expected that compensation would be provided in the form, for example, of a larger than usual time allocation for curriculum science. We have also noted that 'in HMI's view a total of some 90 hours, deployed suitably in relation to school experience and block practice, would be reasonable provided that students were mathematically ready for the work' (7). This implies that some students may need to spend additional time in reaching the necessary state of preparedness. There are, however, many courses which provide considerably less time than this; we have been told of courses which give as little as 40 hours. We wish to express our concern that it is possible to spend so little time on preparation for teaching a part of the curriculum which occupies up to one fifth of a child's time during the primary years. There is also very considerable variation in the proportion of time which is given to method work within PGCE courses. We have been told of courses in which it occupies rather more than one third of the total time which is available; on the other hand, there are courses in which the proportion of time given to method work is one eighth. We refer later to the need to review the great variety of provision which exists for both curriculum and method courses. School experience 687 School experience forms part of all initial training courses. It is likely to start with short periods of observation in a school, sometimes in the company of a tutor, and will certainly include at least one extended period of teaching practice during which school staff must assume major responsibility for the student. It is therefore necessary for the staff of schools which receive students on teaching practice and the staff of the training institutions from which the students come to act together in a well defined and mutually supportive partnership; unless such a partnership exists and operates effectively students will not benefit as they should from their time in schools. This means that schools and training institutions need to explore together over a continuing period the nature of the partnership and the contribution which each can make towards it. These discussions should involve not only the senior staff of the schools concerned but also the teachers in whose classes and departments the students will be working. So far as mathematics is concerned, it is to be expected that the mathematics coordinator or the head of department will play a major part in providing support for students on teaching practice. We believe it to be desirable that they should have opportunity to discuss their role with the staff of the training institution and receive any necessary guidance. The members of staff of both school and training institution need also to be clear about their respective responsibilities in regard to the assessment of students on teaching practice. 688 It has been suggested to us that in some cases the partnership between schools and training institutions is not sufficiently well defined, with the result that schools are sometimes not sufficiently aware of the extent of their responsibilities towards the students who come to them nor of the aims, objectives and priorities of the institutions from which they come. Equally, some training institutions may not be aware of the aims, objectives and priorities of the schools to which their students go. At worst, there can be a conflict of aims between school and college in respect of mathematics teaching. If this is the case, it is essential that steps should be taken to discuss and resolve any such difficulties. 689 We have been concerned to be told that in some cases current financial restraints are limiting the choice of schools which can be used for teaching practice, so that institutions are having to send students to schools which are near at hand rather than to schools which would offer greater opportunities but which are further away. We believe it is essential that such restriction of choice should not lead to the use for teaching practice of schools in which the necessary support cannot be given to students. The same restraints are said to mean that students on teaching practice sometimes receive fewer visits from their tutors than would otherwise be the case. 690 We recognise that it is not practicable for all students who will teach mathematics in primary or middle schools to be supervised during their teaching practices by staff who are mathematically qualified. It is therefore essential that training institutions and schools working in partnership should develop reliable methods of identifying and helping those students whose work in mathematics during teaching practice is weak. For the reasons we have given in paragraph 680, students on PGCE courses may be especially at risk. 691 We draw attention finally to the fact that special attention needs to be given during practice to students preparing to teach in secondary schools who have taken a second method course in mathematics. Not only do they need opportunity to teach mathematics in addition to their main subject but their teaching of mathematics needs to be supervised adequately. Since the tutor who visits them may not be a mathematics specialist, the support given by the head of mathematics in the school will be especially important. Requirement to undertake initial training 692 Mathematics graduates are at present allowed to teach in maintained secondary schools without taking a PGCE course. A great majority of the submissions we have received which refer to this matter have urged that this exemption should cease. In our view the ability to obtain a degree in mathematics does not necessarily imply an ability to teach mathematics well without further training. As we have pointed out earlier in this report, it is not easy to teach mathematics well or to cope with the very differing requirements of pupils of different levels of attainment. The number of graduates in mathematical studies who enter teaching directly after completing their degree course has been decreasing in recent years. Information relating to the period 1974-1979 is given in Appendix 1, Table 23. 693 In our view the exemption of newly qualified mathematics graduates from the requirement to undertake initial training as a teacher should cease as soon as is practicable. A majority of us feels that, in respect of those who will in future qualify as mathematics graduates, action to remove the exemption should be taken as soon as is administratively possible; and believes that, although this might discourage some from entering mathematics teaching, the number is likely to be small so that the loss of these untrained teachers would not be sufficient to outweigh the arguments in favour of professional training for all future mathematics graduates who wish to enter teaching. We all hope that the withdrawal of the exemption would be accompanied by the incentives to undertake training to teach mathematics which we have advocated in the previous chapter. 694 Even after the withdrawal of the existing exemption, there would remain a number of men and women who would still be eligible to enter teaching in maintained schools without undertaking professional training as a teacher. We believe that, both for their own good and for the good of the pupils whom they will teach, all those who intend to teach mathematics should undertake suitable initial teacher training and, in the strongest possible terms, we urge them to do so. We believe also that LEAs should be required to make special induction arrangements for all untrained mathematics graduates which would ensure good support and thorough assessment procedures throughout their probationary period.
Main mathematics courses in BEd 695 We have been told that in 1980 some 500 students chose mathematics as a main course in the first year of BEd out of a total entry to BEd courses of rather more than 5000. The introduction of the two A Level requirement for entry to BEd courses means that main courses in mathematics should now be able to start from a base of previous A Level study in mathematics. In paragraph 643 [in chapter 13] we listed the elements which we believe should be included in the mathematics degree courses of intending teachers. These apply equally to main mathematics courses in BEd. No matter what the age of the pupils whom they may be teaching, teachers of mathematics must be able to take a wide view of the mathematics curriculum and be able to make informed judgements about the priorities within it. We believe that this aim is more likely to be fulfilled by the provision of a main mathematics course which aims at achievement over a broad front rather than by one which seeks to achieve depth by restricting its coverage to only a few areas of mathematics. In our view this should be a major consideration in the validation of main mathematics courses. 696 There is considerable variation in the size of teaching groups for main mathematics courses. We have been told that in the year 1980-81 there were nine training institutions with first year mathematics groups of less than five students, twelve with groups of between five and ten students, thirteen with groups of between ten and fifteen, and ten with groups of more than fifteen students. 697 The existence of some very small groups raises the question of whether main courses which recruit such small numbers should be allowed to continue. We have received a number of submissions in which it is argued that such courses should be maintained even although they are not economically viable. Among the reasons advanced is the fact that, because mathematics is a 'core' subject of the curriculum, it should be seen to be available as a main course in as many training institutions as possible, rather than in only a few specialist centres. It has also been pointed out that many students wish to train at an institution which is within reasonable reach of their home or, in some cases, at a particular institution and that the absence of a main course in mathematics could result in the loss of potentially good mathematics teachers. Some have argued further that, because many students take up teaching posts in schools near to the institution in which they have trained, a geographical spread of institutions which offer main mathematics courses is necessary. Many of those who are engaged in teacher training have told us that they value the opportunity to teach both main and curriculum courses in mathematics because the interaction between the two kinds of course is beneficial both to them and to their students and allows valuable links to be made between the main course, the curriculum course and the experience of students during their periods of teaching practice. 698 We accept the validity of many of these arguments for maintaining as many main courses as possible and we do not believe that the decision as to whether or not to allow a course to continue should be made only on economic grounds; we have been told that on these grounds groups of about twelve are desirable. We believe that in all cases the decision should be taken on educational grounds which take account of the need to make use of the variety of approaches to teaching which we have advocated throughout this report. Groups should not be so large as to discourage discussion and cooperative working but should be large enough to permit a sufficient interplay of ideas. We recognise that there is bound to be some fluctuation in the annual intake to most courses; however, we believe that, if a course regularly recruits less than eight students, its continued existence should be called into question. 699 We recognise that the logic of our argument means that some colleges might lose their main mathematics course and that questions of possible redundancy among mathematics lecturers could arise. We do not believe that this need happen. In our view, time previously given to the teaching of main mathematics courses could with advantage be used to assist with in-service support for teachers in the locality. In this way systematic contact between lecturers and schools could be extended, perhaps to include a regular teaching commitment in a school. We believe that such an arrangement would be beneficial to the work of the training institution concerned.
Balance within initial training courses 700 Some submissions to us have expressed concern at the apparently low status accorded to curriculum courses. Others have suggested that the introduction of the BEd degree has resulted in an increased emphasis on main subject and education theory courses at the expense of curriculum courses and other professional preparation, so that students are less well prepared than formerly for effective work in the classroom. We recognise the importance of these concerns. In our view proper professional preparation is of the greatest importance for intending teachers and should be given appropriate status. 701 We believe that requests for a greater emphasis on the more practical elements of both BEd and PGCE courses, such as those of some of the recently trained teachers who responded to the NFER questionnaire, stem from what is perceived to be a lack of relevance in some parts of the course. It is clearly important that students should feel that all parts of their initial training are relevant to their needs. The relevance of courses on methods of teaching and classroom organisation is clear, as is the relevance of school experience, but the relevance of education theory may be less apparent. It has been suggested to us that this may be because there are few education tutors who have a background of mathematics teaching. In consequence, for instance when illustrating education theory with examples from the classroom, they may seldom make use of examples from mathematics but instead choose examples from other curricular areas of which they have greater experience. There is need for close cooperation between education tutors and mathematics tutors so that education theory can be seen by mathematics students to be firmly grounded on practice in schools and in this way to make a valid contribution to their initial training.
Induction 702 Initial training prepares a student for entry to the profession but much of that preparation is likely to be less effective than it should be if it is not followed up and developed during the first year of teaching. 'There is no major profession to which a new entrant, however thorough his initial training, can be expected immediately to make a full contribution. The government share the view of the James Committee that a teacher on first employment needs, and should be released part-time to profit from, a systematic programme of professional initiation, guided experience and further study.' (8) The aim set out in the paper from which these words are taken was for the introduction of a national 'induction' scheme in the school year 1975-76. This was to provide both enhanced support and a lightened teaching load for teachers during their first year of service. Sadly, this intention has not yet been realised. 703 Most LEAs make provision of some kind for the induction of probationary teachers. A survey carried out in 1979 (9) on the induction of teachers serving in maintained schools suggested that a high proportion of new entrants to teaching were involved in some kind of induction programme but that relatively few induction programmes included provision for significant release from normal classroom duties and/or a lightened teaching load. It is possible that recent expenditure reductions may have caused a reduction in the modest provision which existed during the year 1978-79. 704 The results of the NFER survey which we described in paragraph 671 showed clearly that the provision made for the teachers included in the sample had been uneven and often inadequate. Of those teachers who had been teaching for at least a year, only a quarter said that they had attended more than two meetings arranged by the LEA during their probationary year; one third had attended no meeting at all. On the other hand, eighteen teachers who had attended at least ten meetings had evidently taken part in substantial induction programmes. Within schools, eleven teachers seem to have taken part in formal induction programmes but almost half of the teachers said that they had not attended any formal meetings for probationers in their schools. This does not necessarily mean that probationers lacked guidance in their schools; they may, for example, have been given help and support by the head of the mathematics department which they did not classify under the heading 'meetings for probationers'. Indeed, over 40 per cent of those who had completed their probation said that they were very satisfied with the guidance and help which they had received from their schools during their first year of teaching and a further 50 per cent said that they were satisfied or fairly satisfied; only 8 per cent expressed dissatisfaction. On the other hand, only 7 per cent expressed themselves as very satisfied with the support given by the LEA and 30 per cent expressed dissatisfaction; 40 per cent said that they were fairly satisfied - perhaps being unaware of the level of support which can, and in our view should, be provided. Average teaching loads during the probationary year were very little lighter than subsequently and some teachers who had been given a somewhat lightened teaching timetable felt that they had been imposed upon by having to cover more frequently than their colleagues for teachers who were absent. We do not have such specific evidence in respect of primary teachers but the general indications which are available to us do not suggest that provision for induction is significantly better. 705 It has been pointed out to us that revised patterns of initial training which have been adopted in many institutions in recent years have been planned on the assumption that college courses will be followed by a properly structured induction year. If this is not provided, the training of all who have taken these courses will be incomplete. In our view a proper programme of induction, which should include some lightening of the teaching load, is necessary in the first year, as is good support from the mathematics coordinator or head of department, the head teacher and LEA staff. 706 We appreciate the problems which schools can face in providing support and a lightened teaching load during the first year of teaching. In some cases probationary teachers are appointed at short notice to replace experienced teachers and at a stage at which it is very difficult to change arrangements which have already been made for the following school year. Nevertheless, it is essential that schools and LEAs recognise the need for suitable induction procedures and make every effort to provide them. We have already pointed out that mathematics is not an easy subject to teach; it follows that adequate support is of vital importance during the first year of teaching as well as subsequently. Ideally, probationers should be appointed only to schools in which suitable support is assured. When an appointment of this kind is not possible, LEAs should make every effort to supplement the support which the school is able to give. Teachers in secondary schools for whom mathematics is the second teaching subject can be especially in need of support, particularly in schools in which the standard of mathematics teaching is not high. In the absence of such support, these teachers are liable to find themselves falling back on a narrow style of teaching aimed mainly at containing the class and keeping order, rather than working in the ways which we recommend in this report and which, we hope, will also have been advocated during their training.
Future developments 707 In the course of the chapter we have quoted from Developments in the BEd degree course, from Teacher training and the secondary school, and from other evidence we have received. All these sources highlight the complexity of existing initial training courses, draw attention to problems of the kind which we have discussed in this chapter and pose questions which we believe should be answered. 708 It is not our task to prepare a report on the initial training of teachers nor is our Committee suitably constituted to do so. We are, however, concerned that there seems to be so little knowledge in the training institutions about the relative effectiveness of, for example, the different forms of curriculum and method courses in mathematics, their length and their positioning within the course as a whole. We are also concerned that little systematic attempt appears to be being made to evaluate the different forms of provision which exist and to distil or discover good practice. PGCE in the public sector (10) refers to the 'bewildering variety of practice that exists'. The submissions which we have received make it clear that this is indeed the case and we find it hard to believe that such variety can be justified. We believe that there is an urgent need to review and evaluate the courses provided for the initial training of all those who will teach mathematics. 709 We are disturbed to read that 'the problem for teacher training is to know what the newly qualified teacher should be equipped with on emerging from his course and what could be left to induction and in-service training. There is no consensus on this, and even if there were it would produce no ready formula which would fit the circumstances of every institution' (11). We believe that it is essential that efforts should be made to achieve such a consensus. 710 We have noted that the recent report from the Education, Science and Arts Committee of the House of Commons (12) recommends that HMI should no longer be concerned with courses in the higher education sector. It is not for us to comment on the general issue but we feel it to be essential that HMI should continue to be involved with teacher training courses. Indeed, we find it anomalous that, as part of their duties, HMI appraise initial training courses in institutions in the public sector but have not been given a similar duty in respect of university departments of education which are also responsible for the initial training of teachers. We are aware that HMI maintain informal links with university departments of education; we recommend that they should be given the duty of appraising the initial training courses which these departments provide. 711 We have received several suggestions that PGCE courses and three-year BEd courses should be made longer in order to provide a better preparation for the intending teacher. Although we can appreciate the reasons for these suggestions, and although we have been told that the number of three-year BEd courses is decreasing, we doubt whether any overall extension of initial training courses would be practicable in the foreseeable future. However, this is not to say that there should not be changes. We believe, for example, that consideration should be given to the extent to which it would be possible to make use of the weeks which lie outside the existing university and college terms, and especially of those which lie within school terms. 712 We have already stressed the need for good induction programmes; we believe that consideration should also be given to ways in which initial training and induction might be more firmly linked. 713 Our report has many implications for the initial training of teachers of mathematics in both primary and secondary schools. We have suggested emphases on the teaching of mathematics which are different from those found in many schools at present. These include a greater differentiation in the curriculum of secondary schools, the use of a greater variety of teaching styles, and a greater emphasis on discussion, practical experience, applications of mathematics, problem solving and investigation. It will be the responsibility of those who train teachers to ensure that students experience a variety of forms of learning for themselves, are aware of the need for variety in teaching methods, and start teaching with an ability, at a beginner's level, to use these methods in their teaching. Many students will not yet be able to observe our suggestions in practice in the classrooms in which they find themselves; but young teachers can influence their schools, as well as be influenced by their schools, and we hope that the staff of initial training institutions will respond to our report and take account of its recommendations in their work with students. Perhaps the most important characteristics with which a new teacher of mathematics can enter teaching in the 1980s is a determination continually to monitor and reappraise his teaching and his curriculum. It must be for those who train teachers to seek to develop a flexible attitude in their students which will enable them to respond positively to curriculum change in mathematics.
Footnotes (1) HMI Series. Matters for discussion 8 Developments in the BEd degree course HMSO 1979. (2) Teacher training and the secondary school An HMI discussion paper. DES 1981. (3) In 1979 rather more than 25,000 teachers in primary schools (ie about 13 per cent of all primary teachers) had a graduate qualification; of these only about 300 had mathematics as the first subject of their degree. (4) As defined in Appendix 1, paragraph A12. (5) Teacher training and the secondary school An HMI discussion paper. DES 1981. (6) Teacher training and the secondary school An HMI discussion paper. DES 1981. (7) HMI Series. Matters for discussion 8 Developments in the BEd degree course HMSO 1979. (8) Education. A framework for expansion HMSO 1972. (9) See DES Statistical Bulletin 9/80. (10) PGCE in the public sector An HMI discussion paper. DES 1980. (11) Teacher training and the secondary school An HMI discussion paper. DES 1981. (12) House of Commons. Fifth Report from the Education, Science and Arts Committee The funding and organisation of courses in higher education HMSO 1980. 
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