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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 16 Some other matters
767 In this chapter we discuss four matters for which we have not found place elsewhere in our report; they are not related to each other. These matters are mathematics in middle schools, the teaching of statistics, mathematical education in other countries and statistical information relating to mathematical education.
Mathematics in middle schools 768 The mathematical needs of children are related to their ages, their levels of attainment and their rates of progress, and not to the type of school in which they are being taught. It is for this reason that, in the titles of Chapters 6 and 9, we have referred to mathematics in the primary and secondary years and not to mathematics in primary and secondary schools. Pupils in middle schools span the ages traditionally associated with the words primary and secondary but the mathematical needs of children in these schools are not different from those of children of the same age in either primary or secondary schools. For this reason we have not included a separate chapter on mathematics in middle schools; had we done so we would of necessity have repeated a great deal of what we have written elsewhere in the report. 769 We have received very few submissions which make specific reference to middle schools. However, we have been able to visit middle schools and, in the course of our series of meetings with teachers in different parts of the country, to meet teachers who work in middle schools. In addition, we have been able to discuss the teaching of mathematics in middle schools with a small group of mathematics advisers whose areas contain schools of this kind. We have also known that HM Inspectors have carried out a survey of middle schools for pupils aged 9 to 13 during the school year 1979-80 and a survey of middle schools for pupils aged 8 to 12 during the year 1980-81. We understand that a report on each of these surveys is to be published and that each report will contain information about the teaching of mathematics. Staffing 770 Mathematics is most often taught by the class teacher for at least the first one or two years of the middle school course, but setting for mathematics is introduced in many middle schools in the later years. The results of the 1977 survey of staffing in maintained secondary schools, to which we referred in Chapter 14 and which included middle schools 'deemed secondary' (1), show that there is a very considerable shortage of mathematically qualified teachers in middle schools. In the middle schools included in the survey, 62 per cent of the teaching of mathematics to pupils aged 11 to 13 in these schools was in the hands of teachers who had no qualification in mathematics (see Appendix 1, paragraph A15). We have no comparable information about the qualifications of those who teach mathematics to pupils aged 11 to 12 in middle schools deemed primary but we have no reason to suppose that these teachers are any better qualified; indeed, it seems possible that the reverse is the case. 771 Most middle schools have been formed from either primary or secondary modern schools. Staff from the original schools often remained as part of the staff of the new middle schools and so were in some cases required to teach pupils of an age of which they did not have experience or for which they had not been trained. We were, for example, told by the group of mathematics advisers whom we met that former primary teachers who are now teaching in middle schools for pupils aged 9 to 13 can find difficulty in meeting the needs of 13 year olds, especially those whose attainment is high; equally, former secondary teachers can very often find it difficult to work with 9 year olds. As existing middle schools recruit new staff the situation may change, but the advisers reported difficulty in attracting mathematically qualified teachers to work in middle schools. One reason for this would seem to be the present shortage of mathematics teachers in secondary schools, in which the promotion prospects are also likely to be better. Another may be the fact that very few teachers in middle schools are able to work as specialist teachers of mathematics; they are likely instead to have to teach one or more other subjects or take responsibility for the general teaching of a class. Furthermore, because of the uneven distribution of middle schools throughout the country, it is difficult for teacher training institutions to provide initial training courses which focus specifically on middle schools. The mathematics coordinator 772 Because, as we pointed out in paragraph 770, a high proportion of mathematics teaching in middle schools is undertaken by teachers who lack suitable qualifications, the post of mathematics coordinator (or head of department) is clearly of particular importance. We have, however, been told that there are some mathematics coordinators in middle schools who are not themselves mathematically qualified; we find this disturbing. Furthermore, it is often the case that the mathematics coordinator has to combine responsibility for mathematics either with responsibility for another subject or with administrative or pastoral responsibility of some kind. This clearly limits the time which is available for the oversight of mathematics teaching within the school. Often, too, the mathematics coordinator teaches only the older, and perhaps also the higher attaining, pupils. It is desirable that he should have experience of teaching pupils in each year group but, if it is not possible to arrange this, the mathematics coordinator should at least be enabled to visit the classrooms of all those who teach mathematics and work with these teachers from time to time. 773 When preparing a scheme of work the mathematics coordinator will need to pay special attention to maintaining continuity of progress in mathematics both when pupils transfer from first school and when they move on to upper school. This may not be easy, especially if pupils move to any one of a number of upper schools; we have been told that difficulties can sometimes arise because of requests from upper schools that, by the time they leave middle schools, pupils should be working from a particular textbook. Because pupils from several middle schools very often proceed to the same upper school, liaison is necessary not only with first and upper schools but also with other middle schools in the area. We consider that mathematics advisory staff are likely to have an important role to play in encouraging and facilitating liaison of this kind.
The teaching of statistics 774 Surprisingly few of the submissions which we have received have made direct reference to the teaching of statistics. In the course of our report we have ourselves made several references to it, either directly or by implication, for instance in the foundation list (paragraph 458 [in chapter 9]) and in Chapter 11. We now consider briefly certain aspects in greater detail. 775 Statistics forms part of the mathematics course in the majority of schools at all age levels, ranging from the collection and representation of data in primary schools to the mathematically oriented material in A Level syllabuses. A survey carried out in 1976 by the Schools Council Project on Statistical Education 11-16 (POSE) indicated that over three quarters of all secondary schools were teaching some statistics as part of the mathematics courses followed by 11-16 year olds. However, the submissions which we have received from POSE and from the Joint Education Committee of the Royal Statistical Society and the Institute of Statisticians have stressed that, although statistics is commonly taught within mathematics courses, it should not be regarded solely as part of mathematics. The submission from POSE states that 'statistics is not just a set of techniques, it is an attitude of mind in approaching data. In particular it acknowledges the fact of uncertainty and variability in data and data collection. It enables people to make decisions in the face of this uncertainty'. 776 Statistics is essentially a practical subject and its study should be based on the collection of data, wherever possible by pupils themselves. It should consider the kinds of data which it is appropriate to collect, the reasons for collecting the data and the problems of doing so, the ways in which the data may legitimately be manipulated and the kinds of inference which may be drawn. Work in subjects such as biological science, geography and economics can therefore contribute to the learning and understanding of statistics. When statistics is taught within secondary mathematics courses too much emphasis is very often placed on the application of statistical techniques, rather than on discussion of the results of ordering and examining the data and on the inferences which should be drawn in the light of the context in which the data have been collected. The work can therefore become dry and technique-oriented and fail to show the power and nature of statistics. 777 Many of the ideas of which statistics makes use need time and exposure in order to mature. This means that some of the more elementary ideas should be introduced at an early stage so that understanding can develop and deepen over a period of time. We have been told that in the preparation of many CSE, O Level and A Level syllabuses the difficulty of some of the statistical concepts which they contain appears to have been underestimated, with the result that the statistical content of these syllabuses is too extensive. This is said to be even more true of the syllabuses of other subjects which make use of statistical techniques. 778 Few teachers, including those whose degree or other courses have included the study of statistics, have received training in how to teach statistics in schools. There is therefore a considerable need for in-service training courses on the teaching of statistics not only for mathematics teachers but also for teachers of other subjects. It is possible to identify three levels at which such training is required. The first is training for those who will teach statistics as a subject in its own right or as part of a mathematics course which includes statistics. The second is training for those who teach subjects which make use of statistics and who may themselves have to introduce their pupils to the statistical methods which are used. The third is training to enable other teachers to develop in their pupils a numerically critical approach to data and an awareness of the forms of misrepresentation which are very often to be found in published materials of various kinds. It is desirable that attention at the appropriate level should also be paid to these matters within initial training courses, preferably with groups of students whose main subjects span a variety of disciplines. We were pleased to hear of one PGCE course in which such an inter-disciplinary curriculum option on the teaching of statistics in schools has been introduced; we would support the provision of a similar option within other initial training courses. 779 Because work in subjects other than mathematics can contribute to the learning and understanding of statistics, efforts should be made to ensure that there is cooperation between all those in a school who make use of statistics in their teaching. Such cooperation could be assisted in secondary schools by the nomination of a member of staff who would identify the use which was made of statistics in the teaching of a variety of subjects and act as coordinator for the teaching of statistics; such a coordinator need not necessarily be a teacher of mathematics. 780 We have been told that there is at present a shortage of suitable books about statistics for both pupils and teachers, and that such textbooks as are available concentrate on theory rather than practice; it seems clear that there is need for the provision of further teaching materials which will emphasise a practical approach to the teaching of statistics. The introduction of electronic calculators has eased problems of calculation and so has made it possible for pupils to make use of 'real-life' data rather than of data which has been artificially contrived in order to avoid heavy calculation. This provides opportunity to emphasise the interpretation of data rather than techniques of calculation. In our view the increasing availability of microcomputers and the visual display which they provide should also offer opportunities to illuminate statistical ideas and techniques; we believe that development work in this field is also required. 781 To conclude our discussion of the teaching of statistics we quote from the submission we have received from POSE. Statistical numeracy requires a feel for numbers, an appreciation of appropriate levels of accuracy, the making of sensible estimates, a commonsense approach to the use of data in supporting an argument, the awareness of the variety of interpretation of figures, and a judicious understanding of widely used concepts such as means and percentages. All these are part of everyday living. Good statistical teaching can encourage pupils to think in these ways.We endorse this statement and draw attention to the way in which it complements our discussion in Chapter 2 of the mathematical needs of adult life.
Mathematical education in other countries 782 Our terms of reference have not required us to study the teaching of mathematics in other countries. Nevertheless, among the earliest submissions which we received were some which drew comparisons with other countries and it was clear that both similarities and differences existed in the approaches to mathematics teaching which were used in these countries. We therefore decided to seek information about mathematics teaching in other English speaking countries as well as in some comparable European countries, while realising that any effort which we were able to make would of necessity provide only very limited information which we could not assume would be representative of the country from which it had come. We were aware that different cultural and social attitudes influenced education at all levels, from central legislation to classroom practice, even in other English speaking countries, but we wished to see whether our own debate could be enhanced by some knowledge of methods of working in other countries. 783 With the help of the League for the Exchange of Commonwealth Teachers and the Central Bureau for Educational Visits and Exchanges we were able to write to a number of teachers from England and Wales who had recently been teaching on exchange in Australia, New Zealand, Canada and the United States of America as well as to teachers from these countries who were at the time working in schools in England and Wales; we received many helpful replies from these teachers. Through the good offices of the Secretariat of the European Community we received official statements about mathematics teaching in several of the countries within the community, and the British Council provided us with information relating to countries in other parts of the world. Some of our members were able to pay short visits to Denmark, Holland and West Germany and, nearer home, to Scotland. During these visits we were able to talk with teachers and industrialists, as well as with those concerned with curriculum development and teacher training, and to visit several schools. 784 We cannot overemphasise the diversity of approach which we ourselves have observed and of which we have been told, a diversity which we have also found in the schools in this country which we have visited. This diversity has, on the one hand, provided a valuable stimulus to our discussion but, on the other, has made it clear that, especially on the basis of the very limited information which we possess, we can in no way generalise about mathematical education in any other country. It is clear that there are no model solutions; in all countries of which we have some knowledge we have been aware of concern to improve the quality of mathematical education. 785 Notable differences exist between countries in the extent to which there is central control of the curriculum. In Scotland, the existence of a single examination board exercises a unifying effect on the secondary curriculum, as does the use of centrally developed materials at both primary and secondary levels. In Holland and the various Lander in West Germany, curriculum control is exercised by national or regional government. In France, there is centralised control of the syllabus and of the associated textbooks. However, the existence of nationally or regionally determined curricula in no way removes the requirement for teachers to exercise professional judgement in their mathematics teaching. 786 The part played by public examinations also varies considerably from country to country. In some countries, only those pupils who wish to proceed to higher education are required to attempt a public examination; other pupils are assessed by their teachers, often according to criteria which are specified in detail. In some other countries pupils attempt an examination only at the time at which they leave school. On the other hand, in some countries internal assessment procedures within schools can mean that pupils can be required to repeat a year if their progress is adjudged to have been insufficient. These practices are in marked contrast to the situation in England and Wales in which public examinations exert a strong influence throughout the secondary years, but in which it is unusual for pupils not to be promoted with their fellows. 787 In other countries it is often the case that secondary teachers are required to have a higher academic qualification than those who teach in primary schools. Furthermore, the subjects which secondary teachers are permitted to teach are often restricted to those in which they have a graduate qualification or to subjects which are closely related to these. The restriction which is often placed on the level of qualification of those who teach in secondary schools as well as on the subjects which they can teach is also in marked contrast to the situation in England and Wales. It is clear that at the present time it would not be possible to require all those who teach mathematics at secondary level to hold a minimum mathematical qualification. However, just as we have urged that all who teach mathematics should be professionally qualified, so we believe that they should possess suitable academic qualifications. We believe, therefore, that there should be an investigation into ways in which such a requirement might be introduced over a period of years and that the first steps towards introducing such a requirement should be taken as soon as possible. 788 Shortage of mathematics teachers did not seem to be thought a problem in Denmark, Holland or West Germany and in all of these countries teachers appeared to enjoy a high status. In contrast we were told that, in some parts of the United States of America, there is an extreme shortage of mathematics teachers above elementary level. In West Germany salaries of teachers in secondary schools were said to be comparable with those of 'middle management' in industry and commerce. In Bavaria we were told that teachers were assessed every four years by the head teacher and that the results of this assessment could materially affect the time which was spent by a teacher at a given grade. 789 Our discussions with teachers in other countries, as well as the letters from the exchange teachers to whom we wrote, revealed a very wide range both of teaching styles and syllabus content. It is clear, for example, that both modern courses and traditional courses are sometimes taught in very formal ways. We have received criticism of modern courses taught in this way from those who see such courses as too abstract and difficult for the average pupil. Equally, we have received criticism of formal approaches to traditional courses because such approaches can lead to teaching of a kind which emphasises routines at the expense of understanding. On the other hand, in the course of our visits abroad, as well as from other sources, we have become aware of the high regard which is often accorded in other countries to some of the teaching materials which have been produced in this country and some of the educational research and curriculum development which has been undertaken. Both are seen as more firmly based on classroom practice and less theoretical in approach than is often the case abroad. 790 We have been impressed by the Danish tradition of in-service training for teachers which goes back for very many years. We were told that each year about one third of all teachers in Folkeskoler take in-service training courses organised by the Royal Danish School of Educational Studies. In the United States of America many teachers follow Master's degree programmes in education during the summer vacations and in some areas there is an expectation that teachers will undertake training of this kind. In West Germany we were told that, in some Lander, head teachers had a responsibility for staff development which could include a requirement that teachers should undertake appropriate in-service courses. We have also been given to understand that, in some Canadian provinces, the conditions of service of teachers require them to undertake some in-service training each year. 791 The diversity of practice in schools which we have outlined in the previous paragraphs is complemented by the diversity of employers' views of which we have become aware, both through the comments of those engaged in education in other countries and also in the course of the small number of discussions which we have been able to have with employers themselves. No uniform picture has emerged and the mixture of satisfaction, reservation and criticism has been not unlike that which we have obtained from employers in England and Wales. In Denmark and Holland we gained the impression from educationalists that there was little criticism of the mathematical attainment of school leavers. In West Germany we were made aware of the advantages which were seen by those employers to whom we talked to result from the Berufsschule system of vocational education, which requires all those who have left full-time education before the age of 18 to continue part-time vocational education up to this age for the equivalent of about one day a week. These employers expressed enthusiasm for the courses provided by the Berufsschulen and words such as 'meaning' and 'relevance' were used frequently in connection with the courses which were provided. Another advantage was seen to lie in the fruitful liaison which developed between industry and schools. We were told that courses for some 15 year olds in parts of West Germany were very much more vocationally orientated than is the case for pupils of the same age in this country and one UK industrialist has made clear to us his regret that similar provision is not made in schools in England and Wales. We have been told that in the United States of America some employers have complained of a lack of ability among school leavers to apply mathematics; on the other hand one distinguished industrialist expressed the view that the differences to be found in educational provision in the United States 'reflected to a considerable extent the ability of the different regions to support through direct taxation establishments of education that have the quality in their teaching staff essential to produce the best results'. 792 For the reasons which we explained at the beginning of this section, our references to mathematical education in other countries can only be fragmentary and incomplete. However, we believe that the information which we have been able to obtain has been of use in our work. All the points to which we have drawn attention in the preceding paragraphs have been reflected in our own discussions and so have contributed to our thinking and our conclusions; some of them raise more general issues which we believe merit further investigation. We are most grateful to those whom we met in the course of our visits for the help which they afforded us so willingly.
Statistical information relating to mathematical education 793 It has proved very much more difficult and has taken very much longer than we had expected to obtain statistical information about the various aspects of mathematical education. This has not been because of any unwillingness on the part of the DES or of other bodies to provide information but because, for example, much of the information which is collected each year from schools and universities has not hitherto been analysed in ways which provide the kind of information which we have been seeking. Accordingly, the computer programs which were needed to do this did not exist and have had to be specially written. A less important, though not insignificant, problem is that it is not, in general, possible to obtain separate figures relating to computer studies and statistics because they have traditionally been included as part of mathematics for the purposes of statistical analysis. 794 We have tried to obtain information relating to four main areas. The first of these relates to pupils in schools. In addition to making use of information published in the DES Volumes of Statistics and by the various examination boards, we have made extensive use of information provided by the annual 10 per cent survey of school leavers carried out by the DES. We understand that, as a result of the work which has been carried out for our Committee, computer programs have been developed which will in future enable the information provided by this survey to be analysed more readily. We welcome this move because we believe it is important that this information should be readily available for analysis in ways which are subject specific. We recommend that analyses of the kind which have, at our request, been produced for the years 1977-1979 should continue to be produced for subsequent years. We wish also to recommend, that, so far as is possible, the DES and examination boards should cease to include computer studies and statistics under the general heading of mathematics and treat them as separate subjects for the purpose of statistical analysis. We draw attention to the fact that we have not been able to obtain information about the CSE and GCE results of students aged 16 to 19 who are studying in tertiary and FE colleges. We believe that this information should be available. 795 The second area about which we have sought information relates to the entrance qualifications on a subject specific basis of those entering courses in higher education. As we explained in paragraph 170 [in chapter 4], detailed information about students at universities is collected by the Universities Statistical Record but comparable information about students on courses at other institutions of higher education is not at present available. To our regret we have not therefore been able to obtain information about the mathematical qualifications of students entering courses at these institutions nor have we been able to reconcile information about the numbers of students on degree courses in the non-university sector provided by the DES and the Council for National Academic Awards. We have noted with interest that the recent report of the Education, Science and Arts Committee of the House of Commons (2) recommends that 'the sort of information collected for the Universities Statistical Record should be collected for college and polytechnic personnel' and that 'the databases maintained by the Universities Statistical Record at Cheltenham and the Further Education Statistical Record at Darlington should be made compatible'. We support this recommendation strongly. 796 The third area relates to the mathematical qualifications of those who enter initial training courses for intending teachers. We have not, for example, been able to identify the proportion of students enrolling for BEd degrees, other than at universities, who have an A Level qualification in mathematics. This problem would be overcome if the recommendation of the Parliamentary Committee to which we referred in the previous paragraph were implemented. We believe that there is a need to identify the kind of information which should be available about the academic qualifications of entrants to BEd and PGCE courses and the way in which this is collected and analysed; for example, it should be possible to discover the extent of the mathematical qualifications of those who choose mathematics as a first or second method course in PGCE or as a main course in BEd. 797 The fourth area about which we have sought information relates to the qualifications of teachers and the subjects which they are teaching. A major difficulty in identifying teachers who are mathematics specialists arises from the fact that education is recorded as the first subject of qualification of those who hold the BEd degree. This means that, for BEd graduates who are mathematics specialists, mathematics is recorded as the second subject rather than the first, whereas for mathematics specialists with other kinds of graduate qualification, mathematics is recorded as the first or only subject of qualification. We recommend that education should not be recorded as the first subject of qualification of graduates who hold the BEd degree since the title of the degree itself provides this information; instead, the subject of the main course should be recorded as the first subject of qualification. The survey of secondary staffing carried out in 1977 provided a great deal of valuable information. We believe that surveys of a similar kind should be carried out at least once every five years and that agreement should be reached on the kinds of information which it is necessary to obtain so that any LEA which wishes from time to time to collect information about its own teaching force will be enabled to do so in a way which it knows will be compatible, and therefore comparable, with any information which becomes available as a result of national surveys.
Footnotes (1) Middle schools with pupils aged up to 12 are normally deemed primary; those with pupils aged up to 13 are normally deemed secondary. (2) 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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