Cockcroft (1982)

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

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

Part 2

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

Part 3

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

Appendices

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

Index

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

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

ISBN 0 11 270522 7

Chapter 15 In-service support for teachers of mathematics
[pages 217 - 231]

714 In the course of our report we have drawn attention to a number of ways in which we believe that the mathematics curriculum for some pupils should be changed, as well as to ways in which we believe it is necessary for mathematics teaching to develop in the coming years. Whatever the reaction of central or local government to our recommendations, the extent to which those recommendations are implemented in the classroom will depend upon the response of those who teach mathematics and upon their ability to work in the ways which we have suggested. We wish now to consider the various forms of support which can and should be provided to enable those who teach mathematics to develop and extend their professional skill.

The need for in-service support

715 In our view the need for such support is self-evident. Even if greatly increased numbers of teachers who are well equipped to teach mathematics were to enter teaching in primary and secondary schools in the next few years, it would take many years for this to have a significant effect on the overall quality of the mathematics teaching force. It follows that any improvement in the standards of mathematics in schools must come largely as a result of the efforts of those teachers who are already in post; they must, therefore, receive all possible support to enable them to improve the effectiveness of their teaching.

716 The analysis of the present teaching force which we have given in Chapter 13 shows that there are many teaching mathematics in primary, middle and secondary schools whose qualifications for this task are weak or non-existent. It should not, however, be supposed that in-service support is needed only to remedy the deficiencies of those who lack suitable qualifications. However good their initial training and induction may have been, all those who teach mathematics need continuing support throughout their careers in order to be able to develop their professional skills and so maintain and enhance the quality of their work.

717 The need to provide adequate support for teachers is stressed in very many of the submissions which we have received, including submissions from LEAs, training institutions, professional bodies and teachers themselves. The need for support is not, of course, confined to teachers of mathematics alone and a number of the matters which we discuss in this chapter apply equally to teachers of other subjects. We are, however, concerned with the specific needs of mathematics and believe that there are a number of reasons which justify support for teachers of mathematics on a scale which may not, on financial grounds, be possible for teachers of all subjects at the present time.

718 In the first place, there is the general acceptance that mathematics is an essential part of the curriculum for all pupils in both primary and secondary schools. Secondly, there is the public concern about the teaching of mathematics which has led to the setting up of our Inquiry. Thirdly, there is the lack of suitable qualification, to which we have already drawn attention, of many of those who teach mathematics. Fourthly, the curricular changes which we have recommended in this report will require many teachers to work in ways which are different from those which they use at present. Fifthly, the increasing availability of calculators and computers requires all who teach mathematics to reconsider the content of the mathematics curriculum and the methods which they use in the classroom. Sixthly, other areas of the curriculum which make use of mathematics are at a disadvantage if mathematics is not well taught. Finally, as we have noted in Chapter 13, it is possible that falling rolls in secondary schools in the coming years, and consequent reductions in staffing, will increase the pressure for mathematics teaching to be undertaken by teachers for whom mathematics did not form a significant part of their initial training.

Types of in-service support

719 In-service support for teachers needs to be provided in a variety of ways. We may distinguish in general terms between the support and training which is provided within the school itself, the support which comes from meetings with other teachers, visits to other schools and membership of professional subject associations, the support provided by the staff of local authorities and training institutions, and the support which comes from training courses of various kinds; these categories are not, of course, wholly distinct. There is one other kind of support which, although perhaps not so immediately apparent to many teachers, is provided by research into mathematical education and by the centres which exist to undertake and disseminate work of this kind. We wish to discuss all of these.

School-based support

720 We are in no doubt that school-based in-service support for teachers is of fundamental importance. It can be directed specifically to the needs of the school and its pupils, so that those who teach mathematics develop professionally as a result of working together to improve the work of the school. Above all, it can and should be a continuing process which is not limited to the length of a lecture, a discussion or a course.

Every staff group has within it the ingredients of a kind of continuous educational workshop. For it is in the staff group itself that meeting points can be found between student and practitioner, between the young and the middle-aged, between the inexperienced and the experienced, between the enthusiasts and the cynics, the optimists and the pessimists, between the so-called 'pupil oriented' and the so-called 'subject oriented' teachers. (1)

722 It is important that school-based activities do not become too inward-looking. It is therefore helpful from time to time to invite someone from outside the school to join in meetings and discussion in order to offer new ideas and additional expertise. Such a person might perhaps be a teacher from another school, a mathematics adviser or advisory teacher, the warden of a teachers' centre or a member of the staff of a teacher training institution. Unless this person knows the school well and is well known to those who teach mathematics, a 'one-off' session may be of little value; it is usually more profitable to arrange a series of meetings so that mutual understanding can develop and discussion be based on the perceived needs of the school.

723 Because the effectiveness of school-based support depends upon the leadership of mathematics coordinators or heads of department, it is essential that they themselves should receive support and training. We believe that such training must be the responsibility of local education authorities and that it is necessary for it to be provided not only for newly appointed coordinators and heads of department but also, on a continuing basis, for those who are already in post. Training should emphasise the leadership and organisational role of the coordinator or head of department as well as the need to be aware of current developments in mathematical education. In our view the training of coordinators and heads of department is likely to contribute most quickly and effectively to the overall improvement of mathematics teaching and should be given top priority. Training provided by local education authorities will need to be accompanied by support for the work of the coordinator or head of department within each school. This must be the responsibility of the head. We realise that it is not easy in either primary or secondary schools to provide opportunity within the school day for the coordinator or head of department to work with other teachers but we believe that it is essential that this problem should be overcome, if necessary by the use of part-time staff to provide cover.

724 Except, perhaps, in small schools the coordinator or head of department will need assistance with some of the tasks for which he is responsible. This will need to come from other teachers of mathematics who should be encouraged, singly or in groups, to undertake specific jobs within the school or department; help of this kind will not only assist the coordinator or head of department but will also contribute to the professional development of the teachers concerned. The delegation of responsibility in this way also encourages cooperation, both inside and outside the classroom, between those who teach mathematics. There are sufficient examples of cooperative working of this kind to be found in both primary and secondary schools to show that such working is both possible and effective. However, it is necessary to realise that such methods of working have to be developed gradually; perseverance is required over many terms throughout which support for the coordinator or head of department must continue both inside the school and from outside.

Meetings with other teachers

725 However good the support which is provided within a school, it needs to be supplemented by provision of other kinds. In particular, it is necessary for those who teach mathematics to have opportunity to meet other teachers. Local teachers' centres can play an important part in providing facilities for teachers to meet each other. We have been interested to note that very considerable impetus to the establishment of teachers' centres in their present form was provided by the centres set up as part of the Nuffield Mathematics Project in the latter part of the I 960s. These centres were seen as places in which teachers could meet and where the ideas and activities put forward in the Nuffield Teachers' Guides could be 'discussed, elaborated and modified'. We have no doubt that there is need for provision of this kind and regret that, according to reports which we have received, several teachers' centres have recently been closed. We hope that they will be reopened as soon as circumstances permit and that in the meantime LEAs will endeavour to provide in other ways the kind of support which can come from teachers' centres.

726 Teachers' centres can not only facilitate formal and informal discussion and arrange courses of various kinds but can also act as resource centres. A few authorities maintain one or more teachers' centres which are devoted entirely to mathematics. We have received evidence of the excellent work which is carried out by these mathematics centres and of the support which they provide for the teachers whom they serve. The staff of these centres are mathematicians who are able to arrange mathematical activities in their centres for groups of teachers, for teachers with their classes, and sometimes also for children. They also give help to individual schools and teachers and generally provide a focal point for mathematical activity in their area. In some cases mathematics centres also produce a magazine or newsletter for teachers. We believe that, provided they are adequately resourced, centres of this kind can playa most valuable part in improving the teaching of mathematics; we strongly support their continuation.

727 In areas where mathematics centres do not exist we recommend that efforts should be made to provide at least one resource centre for mathematics in each authority. This should have a library of reference books on the teaching of mathematics, a selection of journals, a range of published textbooks and a collection of mathematical equipment; these should be available both for use in the centre and for loan to teachers. Arrangements should be made for a mathematics specialist to be available in the mathematics resource centre at stated times so that he can discuss with teachers the classroom use of the materials which are in stock.

Visits to other schools

728 Teachers of mathematics - as, indeed, teachers of most other subjects - very often have little idea of what goes on in other schools in their neighbourhood. Time spent observing and joining in with the teaching in another school can provide valuable insight into different forms of organisation and different teaching methods. For this reason we consider that all teachers should be enabled to visit other schools from time to time. These visits must of necessity take place during school hours and cover of some kind will have to be provided for the teachers involved. In secondary schools the fact that many fifth form pupils are no longer in school during the second half of the summer term may make it easier to release staff for visits to other schools during this period. Some authorities arrange for a school to be closed for a single day from time to time so that the whole staff may engage in in-service work. An arrangement of this kind can enable all the teachers in a school to spend the day visiting other schools in their own or a neighbouring LEA.

729 Further opportunities of meeting and working with other teachers are provided by membership of examination panels and especially of local working parties. These may be concerned with such matters as the preparation of guidelines in mathematics, the preparation of teaching and assessment materials, arrangements for continuity and transfer between schools or liaison with local employers. Work of this kind is not only of benefit to the mathematics teaching of an authority or group of schools but can also provide a valuable means of advancing the professional development of those teachers who take part.

The professional associations for teachers of mathematics

730 The professional mathematical associations provide yet another means of enabling those who teach mathematics to meet other teachers, both through local branch meetings and national conferences. They also assist professional development through their journals and other publications. However, a disappointingly small proportion of those who teach mathematics in schools belongs to one or other of these associations. We understand that the joint membership of the two associations which are most directly concerned with mathematics teaching in schools - the Mathematical Association and the Association of Teachers of Mathematics - amounts to some 12,000 but that, if those who belong to both associations are counted only once, the total is nearer to 9,000; nor do all of these teach in schools. In comparison with some 30,000 teachers who teach mathematics in secondary schools and the very many who teach mathematics in primary and middle schools, this is a very small number. Both the Mathematical Association and the Association of Teachers of Mathematics publish journals which are expressly intended for those who teach mathematics in schools and which contain many articles which offer help and suggestions at both primary and secondary levels. We believe that every effort should be made to encourage membership of the professional mathematical associations and that the associations themselves should do as much as possible to develop their local activities.

731 The professional mathematical associations are already playing an important part in the development of mathematical education. We consider that it is vital that they should continue to be able to present informed and independent opinions. If they are to be able to do this effectively it is necessary that serving teachers should play a full part in the working groups and committees of these associations. We hope that LEAs will give such support as may be necessary so that those of their teachers who are invited to take part in the work of these groups and committees may be able to do so.

Mathematics advisory staff

732 We believe that mathematics advisory staff (2) have an essential part to play if the kinds of in-service support which we have already discussed are to operate effectively. The first task of a mathematics adviser must be to monitor the quality of the mathematics teaching in the schools for which he is responsible; in LEAs which do not have an adviser specifically for primary mathematics this must entail close contact with the advisers who work in primary schools. He has then to take such steps as are open to him to improve the quality of mathematics teaching throughout the LEA and to encourage and disseminate good practice.

733 We have already stressed the central role of the mathematics coordinator or head of department in providing school-based support for those who teach mathematics. It follows that the mathematics adviser must be aware of the strengths and weaknesses of the mathematics coordinators and heads of department in the schools which he visits and be prepared to offer them such help as is possible. He will need, in particular, to make sure that the necessary in-service training is provided for them; in some cases it may be helpful to arrange this on a shared basis with other authorities.

734 In addition to monitoring the work of schools he will also need to maintain regular contact with wardens of teachers' centres and staff of local training institutions, as well as with local employers' organisations. He may need to set up local working parties and should be able to advise on visits of teachers to other schools. He will be concerned with arrangements for the provision of in-service work within the authority and will need to identify teachers and others who are able to help with local courses. He is likely also to be called upon to give advice about the appointment of mathematics staff in schools.

735 Just as it is necessary for teachers in schools to be aware of what is going on in other schools, so it is necessary for the mathematics adviser to be aware of what is going on in other authorities. He must also be aware of current developments in mathematical education both regionally and nationally. In some parts of the country it has become the custom for mathematics advisers from a group of neighbouring authorities to meet together regularly, perhaps once in each term, to exchange information and ideas, and sometimes also to arrange in-service activities which are shared between two or more authorities. We believe that consultation of this kind is to be encouraged and that it provides a valuable method of sharing experience and assisting the development of good mathematics teaching.

736 The duties to which we have already referred constitute a formidable list, and we do not suggest that it is complete. If the adviser is to be able to do his job effectively and offer the necessary support to others, it is essential that he himself receives support. We do not believe that all advisers receive adequate training, either on appointment or subsequently. On appointment they should be provided with a proper programme of induction which includes opportunity to see advisers at work in other authorities; there should also be opportunity for further training from time to time.

737 There is considerable variation between authorities in the level of provision of mathematics advisory staff. Some authorities have two or more mathematics advisers and a team of advisory teachers; some have a single mathematics adviser; some have an adviser who has responsibility for mathematics as well as for one or more other subjects; some have no mathematics adviser. It is the responsibility of LEAs to ensure that the teachers whom they employ are sufficient in quality and quantity. We do not believe that an LEA can ensure that the quality of mathematics teaching in its schools is adequate unless it has within its advisory staff adequate mathematical expertise to carry out the necessary assessment and identify schools which are in need of assistance. The money required for the appointment of a mathematics adviser is exceedingly small compared with the cost of the education service as a whole or the cost of that part of the teaching force which deals with mathematics. We therefore believe that more expenditure on mathematics advisers is essential, since without it there are bound to be wide and unacceptable variations in quality and inadequate resources to improve it. If mathematics advisers are to spend their time to maximum advantage, they need to be provided with adequate secretarial and other administrative support and, where appropriate, with support from mathematics advisory teachers.

738 Some advisers devote their whole time to mathematics while others combine a specialism in mathematics with more general duties. Whichever arrangement obtains within an authority, we believe it to be important that mathematics advisers should be aware of the place of mathematics within the whole curriculum and be able to view the organisation for teaching mathematics in the context of the organisation of the school as a whole. It is to be expected that advisers who are seen to be effective in their own field are likely to be asked for comment and help on other matters and so they need to be aware of developments and sources of help outside their specialist field. We do not therefore support the suggestion which has been made to us in some submissions that mathematics advisers should not also be required to work in more general ways. However, it must be for each LEA to ensure that sufficient time overall is available for advisory work in mathematics and to appoint further advisory staff if necessary.

Mathematics advisory teachers

739 Although many LEAs do not make use of advisory teachers for mathematics, some have teams of advisory teachers with well defined tasks. In some LEAs there are teachers who work part-time in a school and part-time as mathematics advisory teachers. Advisory teachers most commonly work alongside teachers in the classroom. They may spend a period of several days in the same school or visit each of a group of schools on a regular basis. They may perhaps help with drawing up or revising a scheme of work or assist with the in-service programme of the LEA or a school. We believe that advisory teachers can play a valuable role in providing in-service support in a school and in assisting with the introduction of new approaches to teaching. This is perhaps especially the case in primary schools which lack a member of staff who has mathematical expertise; it is likely that regular visits from an advisory teacher who is able to work alongside teachers in the classroom could be a very effective method of improving the level of mathematics teaching within such a school. An advisory teacher may be able to make an especially valuable contribution in rural areas where the distance between schools and problems of transport can make it difficult for teachers to take part in the activities of a teachers' centre or to meet teachers in other schools.

Establishments of higher education

740 Establishments of higher education of all kinds have for many years made a major contribution to the in-service support of teachers. In addition to providing full-time and part-time courses for serving teachers both on a regular basis and in response to particular needs, their staff engage in many other kinds of in-service support. One form of support which we believe to be of great value, even though it is on a relatively small scale, is the provision of 'school-teacher fellowships'. Teachers who are appointed to these fellowships, which are sometimes funded by the institutions themselves and sometimes by outside sources including industry, are enabled to undertake a period of study or research away from the classroom.

741 Those who work in establishments of higher education, and especially in training institutions, often assist with courses organised by local authorities or teachers' centres and serve on local in-service committees. Increasingly, too, staff of training institutions assist with school based in-service work, especially in schools in which they have become known as a result of their visits to supervise students on teaching practice. In some cases, too, their institutions serve as resource centres for local teachers, who are encouraged to make use of libraries and other facilities. Furthermore, training institutions are often able to work across LEA boundaries, which enables teachers from different LEAs to meet and work together.

742 Our attention has been drawn to a matter which is said to operate against the involvement of members of staff of training institutions in in-service work with teachers. Such work is clearly of direct relevance to those who are concerned with the initial training of teachers but, if undertaken, reduces the time available for academic research. However, there can be no doubt that publication of the results of academic research is considered to be essential for promotion to higher posts in institutions of higher education, especially universities. We have been told that some of those who work in training institutions therefore feel that time spent on in-service work can be to the detriment of their academic careers, because those who have responsibility for making appointments do not value evidence of experience gained during in-service work as highly as evidence of published work. If this is the case, we greatly regret it.

743 'Consultancy' work in schools undertaken by the staff of training institutions is not only of benefit to schools but also enables those who work in training institutions to gain up to date and first-hand knowledge of the work which is going on in primary and secondary classrooms. We have been told that the way in which time given to work of this kind is 'costed' can in some cases limit the amount of time which training institutions feel able to give to it, particularly if a significant amount of travelling is involved; nor do the methods which it is intended should be used to 'cost' time given to work in schools appear to be entirely clear. We understand that the Pooling Committee is aware of this problem and is at present seeking ways of clarifying the situation. We believe it to be very important that any arrangements should be such as to encourage and facilitate consultancy work undertaken by the staff of training institutions.

In-service courses

744 Although it is not our view that courses should be regarded as the most important form of in-service support for teachers, we nevertheless consider that they have an essential role to play because not all the forms of support which are required by teachers can be provided within a school, by meetings with other teachers, by advisory staff or by the professional mathematical associations. Courses provide a means whereby teachers from several schools can come together for a special purpose, either to consider a particular aspect of mathematics teaching or to add to their own knowledge or qualifications. We have, for example, already referred to the need to provide training for mathematics coordinators and heads of department; this will almost certainly have to be done, at least in part, by the provision of suitable courses on a local or regional basis.

745 Most short courses, and also some longer courses, operate on a part-time basis, very often with one session each week or fortnight. It is common for these sessions to take place after the end of the school day. However, many have argued strongly in submissions to us that courses held at this time are often less effective than they should be because teachers are tired after a full day in school; in addition, preparation for the next day's work may suffer. They have therefore urged that more courses should be held during school hours than is the case at present; we believe that there is substance in this argument. We know that in some LEAs it has been found helpful to start each session in mid-afternoon, so that half of the session takes place during school hours.

746 An advantage of courses which operate on a part-time basis can be that teachers are more easily able to make use in the classroom of ideas and activities which are suggested during the course and, if necessary, to discuss these further or obtain any help which they require during later sessions. On the other hand, the interval of a week or a fortnight between sessions may make it difficult to maintain continuity in certain types of course. Full-time courses, whatever their length, take teachers completely out of the classroom and enable them to devote their full attention to the course without the distractions of daily school life. If such courses are residential, there are additional opportunities for reflection and discussion. Some longer courses, for example DES regional courses, often combine regular weekly or fortnightly sessions with occasional periods of full-time work for two or three days. In our view there is need for both part-time and full-time courses.

747 The range of mathematical knowledge and experience which exists among teachers on the same course can often be very wide and it is not easy to ensure that the content and level of a course is suited to all those who are taking part in it. For this reason it is important that, when a course is advertised, there should be a clear explanation of its purpose and of the level of mathematical knowledge and experience which will be expected of those for whom it is intended. In this way teachers, and those who will support their attendance, will be able to judge in advance whether a course is relevant to their needs.

748 The long term effectiveness of in-service courses, especially those which last only a short time, can be greatly diminished unless there is suitable follow-up. In the case of courses which are locally based, efforts should therefore be made to arrange one or more follow-up sessions at intervals after the course has been completed. The cost of organising such sessions is likely to be small in comparison with the cost of the original course. It has been pointed out to us that, if several teachers from the same school have attended a course, either together or on successive occasions on which the course has been offered, it may be more effective to organise follow-up work within the school itself. There was strong support for this method of working from teachers and others whom we met during our visit to the Scottish Education Department. In any case, teachers who have been on courses should be encouraged to share their experiences with their colleagues on the staff and to discuss them at staff or departmental meetings. It is all too easy for in-service training courses to result in no long-term improvement because of lack of interest or support when a teacher returns to his school.

749 Full-time courses for teachers lasting either a year or a term are offered by a number of training institutions but recently a considerable number of both primary and secondary mathematics courses have had to be cancelled because they have been insufficiently subscribed. Although we have not been able to obtain complete information, it is clear that some LEAs have reduced the number of teachers whom they have seconded to these full-time courses because of the effects of current financial stringency. If a sustained effort is to be made to improve the qualifications of those who at present teach mathematics, the number of teachers seconded to full-time courses in mathematics will need to increase substantially.

Diploma in Mathematical Education

750 A substantial contribution to the in-service education of teachers in primary and middle schools has been made by the introduction of the Mathematical Association's Diploma in Mathematical Education which is intended for teachers of children in the 5-13 age range who have at least two years of teaching experience. Two year part-time courses for this diploma started in 1978 and are now offered at some fifty centres throughout the country. About 600 teachers enrolled for the diploma course in 1978 and much the same number have done so in each following year. Of those who enrolled in 1978, about 400 obtained the diploma in 1980; a similar number are expected to do so in 1981. Teachers who obtain the diploma are entitled to one increment on the Burnham salary scale. We believe that every support should be given by LEAs to teachers who wish to enrol for a diploma course since it provides an effective means of increasing the mathematical qualification of teachers at a significant rate.

751 Because diploma courses normally take place in teacher training establishments, there are parts of the country in which it is not possible for teachers to take the course. We believe that energetic efforts should be made to explore ways in which diploma courses can be made available in areas which are remote from training institutions.

Open University

752 In recent years many serving teachers have gained mathematics degrees awarded by the Open University or degrees which include some mathematics. We have been pleased to note the extension of the University's work to include in-service support for teachers. A course on Mathematics across the curriculum is already being offered and a complementary course on Developing mathematical thinking is in preparation. There are also provisional plans to include these courses as part of a mathematics diploma course.

753 We believe that the Open University, which has now developed considerable experience and expertise in the development of 'distance learning' courses in mathematics, could play a major part in improving the quality of the mathematics teaching force. At the present time the course fees for courses such as Mathematics across the curriculum are high because courses of this kind which the University offers are required to be financially self-supporting. We believe that means should be explored of providing financial support for in-service work in mathematics provided by the Open University, since the provision of suitable courses which could be followed by groups of teachers within LEAs might prove to be an effective method of providing inservice support of good quality on a wide scale.

Radio and television

754 Mathematics programmes designed for use in the classroom are broadcast by both BBC and IBA [Independent Broadcasting Authority], who consult to prevent duplication. These provide material which can be used to form the basis of a mathematics course or supplement other work in the classroom; different series of programmes cater for the needs of pupils of different ages and levels of attainment. The programmes are usually accompanied by notes for teachers and often by work books for pupils. Programmes of this kind provide another form of in-service support for teachers which can not only be used in the classroom but can also serve as a basis for discussion by groups of teachers at a teachers' centre or within a school.

755 In preparation for the raising of the school leaving age in 1973, the BBC broadcast a series of television and radio programmes for teachers which were followed and discussed by large numbers of teachers either at teachers' centres or within individual schools. We believe that programmes of this kind on different aspects of mathematical education, designed to provide a basis for discussion among groups of teachers and perhaps considering some of the issues raised in this report, could make yet another valuable contribution to in-service support and training in mathematics. We have been told by the BBC that, in their view, an advantage of broadcast programmes is that they are impersonal, so that the ideas contained within them can be criticised frankly without fear of giving offence; this is not always possible following input from a visiting lecturer or adviser.

Research into mathematical education

756 A considerable number of research studies have been carried out which relate to aspects of mathematical education. These include work in such fields as concept formation and development, mathematics learning, the classroom behaviour of teachers and pupils, and curriculum development and evaluation. Many of these studies are summarised in the Review of research to which we have already referred several times. We are pleased to learn that plans are being made for a book based on this review to be published shortly (3) because we believe that much of the research which has been undertaken remains largely unknown to those who teach mathematics in schools and also to many who produce textbooks and other teaching materials. Even when teachers become aware of the existence of a research study on a topic, they very often find it difficult to appreciate its relevance to their own classroom. One of the reasons for this is that research reports are usually written in a technical style which is not always easy to follow and are very often published in journals which most teachers do not see.

757 More recently some research projects have published their work by means of articles, which are written in a non-technical style, in the journals of the professional mathematical associations. The project Concepts in secondary mathematics and science, to which we have already referred, is an example of this. We suggest that more use should be made of the educational press both to disseminate the results of specific research projects and also to review and interpret for teachers the state of research on different topics. Advisers and others who have responsibility for in-service support for teachers should be aware of the need to disseminate and interpret the results of research studies so that teachers can be helped to find ways of making use of these in their own thinking and classroom practice.

Centres for mathematical education

758 There are a few centres whose reputation is international, such as those at the University of Nottingham and at Chelsea College, University of London, which foster curriculum development and research into mathematical education and which also provide in-service support for teachers of mathematics. The work which these centres undertake is closely related to mathematics in the classroom and we believe it to be of the greatest value. We would welcome the establishment of a few further centres of comparable quality; we believe that the creation of such centres could lead to a significant improvement in the quality of mathematical education. However, a paramount consideration in the establishment of any further centres must be the appointment of staff of suitable quality. In our view any further centres should be suitably situated geographically and be based on existing institutions, whether in universities or elsewhere, which are known to be strong in the field of mathematical education. In order to ensure that the work done by any new centre would be able to be strongly classroom-related, it would be important that the LEAs, whose teachers were likely to be most closely involved, should take part in the discussions leading to the setting up of that centre.

Financial support

759 There are two matters which are fundamental to the provision of almost all kinds of in-service support - finance and time. Unless the will exists to make the necessary financial provision and to provide the necessary time, no appropriate plans for in-service support can be formulated, either within a school or an authority, or on a regional or national basis. It is clear from national surveys, as well as from information which some LEAs have provided for us, that a great deal of in-service work is undertaken by mathematics teachers outside school hours and sometimes at their own expense. This voluntary effort is praiseworthy and to be encouraged. However, the problem of providing support and training for all those who teach mathematics cannot be solved entirely by schemes which depend on the voluntary attendance of teachers. The fact that many teachers undertake in-service work outside school hours shows that they already accept in-service training to be part of their professional commitment; the fact that LEAs provide in-service training courses shows that they accept the provision of training and support for teachers to be part of their duty. We are aware that in some countries it is part of the conditions of service of every teacher that a certain number of days each year should be spent on in-service education and we believe that many local authorities and teachers would wish to see a similar provision in this country. Provision of this kind has been recommended in many of the submissions which we have received. We realise that there are difficulties, not only of a financial kind, in such a proposal but without some such arrangement we do not believe that there can be sufficient opportunity to influence and improve the quality of mathematics teaching. We are convinced that the curricular changes which we are proposing and the changes in attitudes and perceptions on the part of teachers which they will require would have a far better chance of implementation if such an arrangement were to exist.

760 There are two elements in the cost of providing in-service training for teachers: one is the cost of providing the training courses themselves and the other is the cost of releasing from their schools the teachers who take part in them. These costs are met from a variety of sources. Part of the cost is borne by local authorities, either individually or jointly; voluntary colleges and universities contribute through the courses which they provide. Central government contribute through the rate support grant to local authorities and through their funding of voluntary colleges and universities. However, no part of the rate support grant is specifically earmarked for in-service training; nor is any part of the grant to voluntary colleges or universities earmarked in this way. Teachers themselves contribute both in terms of the expenses which they incur and also by giving up their own time.

761 There are arrangements whereby LEAs can recover a proportion of their expenditure on certain in-service training costs for teachers, including tuition fees, travel and salaries, from a national 'pool' which is financed by contributions from all LEAs according to an agreed formula. Pooling was at one time restricted to full-time courses included in a national programme and lasting either a year or a term, but the present position is that, in general, full-time courses lasting for more than four weeks and part-time courses lasting for at least sixty hours are eligible for pooling. In addition, the costs of providing courses of in-service training in departments of education in local authority maintained colleges are redistributed between authorities through the Advanced Further Education (AFE) Pool. Many maintained sector departments of education provide direct in-service support in schools; a proportion of the lecturer costs involved in giving this support is also chargeable to the AFE Pool.

762 We have noted two types of training which are not at present eligible for pool support. The first is the cost to authorities of releasing mathematics teachers to gain experience of industry and commerce. The second is the cost of appropriate courses from the Open University Associated Student Programme which are taken by teachers. We recommend that both of these should be eligible for support.

763 Most LEAs pay, in full or in part, the expenses of teachers who undertake in-service work, either locally or further afield. We have, though, been told of cases in which unhelpful limitations have been placed on teachers, for example in terms of travel or of refusal to permit attendance at a course which, although suitable and near at hand, is taking place in the area of another LEA. We regret restrictions of this kind and also failure to recompense in full expenses which have been incurred; in our view, such disincentives to take part in in-service work should not be placed in the way of teachers.

764 The methods by which in-service support is provided by local authorities are likely to vary considerably. It will be appropriate for LEAs to pay for their own facilities, whether in schools, teachers' centres or elsewhere and to make recompense for facilities provided for their teachers by other LEAs. In some cases joint resource centres may be established, often in conjunction with training institutions, and joint funding will be appropriate. We believe that in any such arrangement there should be an incentive for the training institution to provide what is required. Payment, at least in part, for services rendered would help to ensure that good provision not only survived but was developed, whereas less good provision would cease.

765 The Secretaries of State bear considerable responsibility for the quality of the initial training of teachers; we believe it is also appropriate that they should influence the quality of teachers during their subsequent careers by the provision of direct financial support for programmes of in-service training. We understand that under Section 3(a) of the Education Act 1962 they possess powers to do this which, until the introduction of the National Scholarship scheme (see paragraph 652 [in chapter 13]), have not hitherto been used. Although we have been told that local authorities do not, in general, like the concept of funding for specific purposes, they nevertheless have experience of it and accept it in a number of other fields. The evidence which is available to us suggests that at the present time sufficient money is not being spent on the provision of in-service training and that in some areas the position is worsening. Unless the Secretaries of State take effective action in this field we do not believe that sufficient resources to improve the quality of mathematics teaching will be made available. In this matter we believe that the Secretaries of State should work in close association with local authorities as the employers of teachers and also, as appropriate, with teacher training institutions.

766 In the words of the government paper Education: a framework for expansion published in 1972, 'expenditure to achieve an expansion of in-service training ... is a necessary investment in the future quality of the teaching force'. The scale on which it was at that time proposed to increase the provision of in-service training has not been achieved. We note that the government's expenditure plans presented to Parliament in March 1981 state that 'provision for the release of teachers to in-service training and induction has been held at the current level'. However, we believe that in-service support and training for those who teach mathematics needs to be increased and that this need is pressing if the developments and changes in mathematics teaching which we are advocating are to be realised. Not all of our recommendations will cost money but many of them will. It will be for those who control finance to determine whether additional money can be made available or whether money must be found from within the public expenditure limits already determined. In his address to the 68th North of England Conference in January 1981, the Secretary of State for Education and Science said that 'Even within our restricted resources, I believe it to be crucial that a high priority should continue to be given to the right in-service training ... In asking authorities and teachers to give a high priority to carefully-managed in-service programmes ... I am in effect asking that, in their very proper concern to do their utmost for today's children, they should not lose sight of the need to do equally well by tomorrow's'. Unless additional money is spent on in-service support for those who teach mathematics, the improved mathematical education which we believe could and should be provided for children in the future is unlikely to be available to them.

Footnotes

(1) Richardson E The teacher, the school and the task of management Heinemann Educational Books 1973.

(2) There are differences in the titles used by LEAs. Some have 'advisers', some have 'inspectors' and some have both. We use the terms 'mathematics adviser' and 'mathematics advisory staff' to refer to all those who have responsibility for assessing the quality of the mathematics teaching within an LEA and providing support for those who teach mathematics.

(3) Information is available from the Shell Centre for Mathematical Education, University of Nottingham; see also paragraph 188 [in chapter 5].