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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
Preliminary pages Foreword
Few subjects in the school curriculum are as important to the future of the nation as mathematics; and few have been the subject of more comment and criticism in recent years. This report tackles that criticism head on. It offers constructive and original proposals for change. It should be read by those responsible for school mathematics at all levels. The main message is for the education service. The report identifies six agencies whose active response is required. The contribution of all will be necessary if we are to make headway. To the extent that the report calls for extra resources, progress is bound to be conditioned by the continuing need to restrain public expenditure; but many recommendations involve no such call. We hope that there will be widespread discussion of the report's conclusions and that action will follow. The Committee's terms of reference invited it to consider the teaching of mathematics with particular regard to the mathematics required in further and higher education, employment and adult life generally. The early chapters of the report are concerned with these aspects. They will be of interest to many both within and outside the education service. They reveal the hesitant grasp many adults have of even quite simple mathematical skills. They are particularly valuable in examining closely the mathematics which is in fact needed in different kinds of employment and in everyday life, and relating it to what is taught in the schools. The Committee's findings point to the need for teachers to devote more time to the use of mathematics in applications taken from real life. This is a first-class report. We are greatly indebted to Dr Cockcroft and the members of his Committee, and commend their work to all those concerned about the quality of mathematics taught in our schools.
Membership of the Committee
Dr WH Cockcroft (Chairman) Vice Chancellor, New University of Ulster, Coleraine
The styles, decorations and appointments shown are those held by members at the time of their appointment to the Committee.
[page vi] 10 November 1981 Dear Secretaries of State On behalf of the Committee of Inquiry into the teaching of mathematics in primary and secondary schools in England and Wales, I have the honour to submit our report to you. Yours sincerely
WH COCKCROFT The Rt Hon Sir Keith Joseph Bt MP
The Rt Hon Nicholas Edwards MP
Contents
Introduction Explanatory note Part one 1 Why teach mathematics?
Part two 5 Mathematics in schools
Part three 12 Facilities for teaching mathematics
Appendices 1 Statistical information
Index
Introduction
In its report published in July 1977, the Education, Arts and Home Office Sub-Committee of the Parliamentary Expenditure Committee stated that 'it is clear from the points which were made over and over again by witnesses that there is a large number of questions about the mathematical attainments of children which need much more careful analysis than we have been able to give during our enquiry. These concern the apparent lack of basic computation skills in many children, the increasing mathematical demands made on adults, the lack of qualified maths teachers, the multiplicity of syllabuses for old, new and mixed maths, the lack of communication between further and higher education, employers and schools about each group's needs and viewpoints, the inadequacy of information on job content or test results over a period of time, and the responsibility of teachers of mathematics and other subjects to equip children with the skills of numeracy'. The Committee recommended as 'possibly the most important of our recommendations' that the Secretary of State for Education and Science should set up an enquiry into the teaching of mathematics. In their reply presented to Parliament in March 1978, the Government agreed 'that issues of the kind listed in the Committee's report need thorough examination' and announced their decision to 'establish an Inquiry to consider the teaching of mathematics in primary and secondary schools in England and Wales, with particular regard to its effectiveness and intelligibility and to the match between the mathematical curriculum and the skills required in further education, employment and adult life generally'. They further undertook that the Inquiry would examine the suggestion that there should be a full analysis of the mathematical skills required in employment and the problem of the proliferation of mathematics syllabuses at A Level and at 16+. Terms of reference Our Committee met for the first time on 25 September 1978 with the following terms of reference: To consider the teaching of mathematics in primary and secondary schools in England and Wales, with particular regard to the mathematics required in further and higher education, employment and adult life generally, and to make recommendations.Meetings and visits The full Committee has met on 64 days, which have included three residential meetings. Its Working Groups have met on 143 days in all, and there have been less formal discussions on many occasions. 54 schools and 26 companies of various kinds in England and Wales have been visited by members of the Committee and there have been six meetings with groups of teachers in different parts of the country. Small groups of members have visited the Scottish Education Department in Edinburgh, the Institute for the Development of Mathematics Education (IOWO) at Utrecht, Holland, the Institute for the Teaching of Mathematics at the University of Bielefeld, West Germany and the Royal Danish School of Educational Studies in Copenhagen; two members have visited industrial companies in Nuremberg, West Germany. Several members of the Committee were present at the Fourth International Congress on Mathematical Education held at the University of California at Berkeley in August 1980. Individual members of the Committee have been invited to attend the conferences and meetings of a number of professional bodies. Submissions of evidence Throughout our work we have been greatly encouraged by the welcome which many people have given to the setting up of the Inquiry and by the helpful response which we have received to our requests for information and written evidence. We have received written submissions, many of them of considerable length, from 930 individuals and bodies of many kinds. 73 individuals and groups have met members of the Committee for discussion. A list of those who have submitted evidence and who have met members of the Committee for discussion is given in Appendix 3. Research studies When we started to consider how best we might respond to our terms of reference, we became aware that we needed more detailed information about the mathematical needs of employment and of adult life generally than we were likely either to receive in written evidence or to be able to obtain by our own efforts. We therefore requested the Department of Education and Science (DES) to commission two complementary studies into the mathematical needs of employment and also a small study into the mathematical needs of adult life. One of the studies into the mathematical needs of employment was based at the University of Bath under the direction of Professor DE Bailey, assisted by Mr A Fitzgerald of the University of Birmingham, and the other at the Shell Centre for Mathematical Education, University of Nottingham, under the direction of Mr RL Lindsay. The study into the mathematical needs of adult life was carried out by Mrs B Sewell on behalf both of the Committee and of the Advisory Council for Adult and Continuing Education. The DES also agreed to commission a review of existing research on the teaching and learning of mathematics which was carried out by Dr A Bell of the University of Nottingham and Dr A Bishop of the University of Cambridge. The Steering Groups for all these studies have included members of the Committee, and relevant evidence which has been received has been made available on a confidential basis to those engaged in the studies. The reports which have been produced have proved to be of very considerable help to us; we refer to them and draw on their conclusions in a number of the chapters which follow. At a later stage the DES commissioned a small survey of mathematics teachers in secondary schools who were in their first three years of teaching; this was carried out for us by the National Foundation for Educational Research. Publications and announcements Since we started our work a considerable number of official reports and other publications have been issued which relate wholly or in part to the teaching of mathematics in schools. These include from the DES: Mathematical development. Primary survey reports Nos 1 and 2 (The APU Primary Surveys)
from HM Inspectorate: Primary education in England (Report of the National Primary Survey)
from the Schools Council: Mathematics and the 10 year old (Schools Council Working Paper 61)
from other sources: Engineering our future (Report of the Finniston Committee)
We have studied all these documents and make reference to several of them in the course of this report. Since we started work, the Government have announced that the present O Level and CSE examinations are to be replaced by a single system of examining at 16+ , that GCE A Levels are to be retained, that the Certificate of Extended Education will not be introduced but that there will be a prevocational examination at 17+, and that consideration is being given to the introduction of Intermediate levels (I Levels). We have considered the implications of these announcements so far as mathematics is concerned. Statistical information As a result of the work which has been carried out for us by the Statistics Branch of the DES and by the Universities Statistical Record we have been able to obtain a considerable amount of information which has not hitherto been available. We refer to this, and to other existing information, from time to time throughout the report. In general we quote this information in rounded terms or present it in diagrammatic form. The detailed tables from which the information is taken are set out in Appendix 1, which also gives in each case the source from which the information has been obtained. This Appendix also contains some tables to which no direct reference is made in the text but which we believe to be of interest. The Appendix discusses, where appropriate, any assumptions which it has been necessary to make in order to prepare the tables and includes brief comments on some of them. Views from the past In the light of present day criticism of standards, it is interesting to assemble a collection of quotations from documents of various kinds, some of which date back to the last century, which draw attention to the allegedly poor mathematical standards of the day. We content ourselves with examples from approximately a century, a half century and a quarter century ago. In arithmetic, I regret to say worse results than ever before have been obtained - this is partly attributable, no doubt, to my having so framed my sums as to require rather more intelligence than before: the failures are almost invariably traceable to radically imperfect teaching.Those comments are taken from reports by HM Inspectors written in 1876. Many who are in a position to criticise the capacity of young people who have passed through the public elementary schools have experienced some uneasiness about the condition of arithmetical knowledge and teaching at the present time. It has been said, for instance, that accuracy in the manipulation of figures does not reach the same standard which was reached twenty years ago. Some employers express surprise and concern at the inability of young persons to perform simple numerical operations involved in business. Some evening school teachers complain that the knowledge of arithmetic shown by their pupils does not reach their expectations. It is sometimes alleged in consequence, though not as a rule with the support of definite evidence, that the teacher no longer prosecutes his attack on this subject with the energy or purposefulness for which his predecessors are given credit.That extract comes from a Board of Education Report of 1925. The standard of mathematical ability of entrants to trade courses is often very low ... Experience shows that a large proportion of entrants have forgotten how to deal with simple vulgar and decimal fractions, have very hazy ideas on some easy arithmetical processes, and retain no trace of knowledge of algebra, graphs or geometry, if, in fact, they ever did possess any. Some improvements in this position may be expected as a result of the raising of the school leaving age, but there is as yet no evidence of any marked change.Our final quotation comes from a Mathematical Association Report of 1954; the school leaving age was raised to 15 in 1947. It is therefore clear that criticism of mathematical education is not new. Indeed, throughout the time for which we have been working we have been conscious that for many years a great deal of advice to teachers about good practice in mathematics teaching has been available in published form from a variety of sources. These include the publications of the DES, of HM Inspectorate, of the Schools Council and of the professional mathematical associations; there have also been references to mathematics teaching in the reports of Committees of Inquiry, for example that of the Newsom Committee. Much of this advice is still relevant today and serves as a background to our own work. General approach In writing our report we have tried so far as is possible to avoid the use of technical language and to put forward our views in a way which we hope will be intelligible to mathematician and non-mathematician alike. For this reason we have at times omitted detail which, had we been writing only for those engaged in mathematical education, we would have included. We hope that those who would have wished us to discuss certain matters in greater detail will understand the reason why we have in certain places used a somewhat 'broad brush'. We hope that our attempt to draw attention to those aspects of the teaching of mathematics which we believe to be of fundamental importance will be of use both inside and outside the classroom. We wish to stress that many of the chapters in our report, and especially those in Part 2, are interrelated. For example, in Chapters 5 and 6 we discuss the elements of mathematics teaching at some length; the fact that we do not repeat this discussion in Chapter 9 (Mathematics in the secondary years), but deal mainly with matters of syllabus content and organisation, does not mean that the teaching approaches we have recommended in earlier chapters are not equally applicable at the secondary stage. We therefore hope that those who read our report will view it as a whole. In our report we have not considered the needs of pupils with severe learning difficulties. We hope, however, that those who teach pupils of this kind will find that our discussion of mathematics teaching in general, and of the needs of low-attaining pupils in particular, as well as our discussion of the mathematical needs of adult life, will be of assistance. Acknowledgements We would like to express our thanks to all those who have written to us and with whom we have talked both formally and informally. We are grateful for the help we have received from the heads, staff and pupils of the schools we have visited, from the teachers whom we have met at meetings in different parts of the country and from those at all levels whom we have met during our visits to commerce and industry. We are grateful, too, to those who were kind enough to arrange these visits and meetings. We wish to thank the Scottish Education Department for arranging our visit to Edinburgh and those whom we visited in Denmark, Holland and West Germany for the help they gave us and the arrangements they made on our behalf. Our thanks are due to all those who have carried out the various research studies and also to those in DES Statistics Branch and the Universities Statistical Record who have undertaken a great deal of work for us and found ways of answering our questions. We are grateful for the help we have received from many officers in the DES, in particular our Assessor, Mr PH Halsey; and from members of HM Inspectorate, especially Mr TJ Fletcher who has, at our invitation, attended many of our meetings. We wish also to express our thanks for the help and support which we have received from the members of the Committee's secretariat. Mr WM White has taken major responsibility for obtaining, putting in order and interpreting a very great deal of statistical information. Mr EL Basire, our Assistant Secretary, Miss E Kirszberg and Mr RW Le Cheminant, in addition to their other duties, have given much personal help to members of the Committee. Among those to whom we are indebted, our Secretary, Mr WJA Mann HMI, stands out. We are conscious of the burden we have placed upon his shoulders and of the conscientious way in which he has accepted the load. Most of all, we are grateful to him for the able and efficient way in which he has taken the outcome of our many diffuse and varied deliberations and moulded it into a coherent whole.
Explanatory note
Throughout the report there are certain passages which are printed in heavier type. In selecting these passages, we have chosen those which either relate to matters we consider to be of significance for all our readers or which call for action by those who are outside the classroom. This means that, especially in Chapters 5 to 11 which are concerned in particular with the teaching of mathematics, we have not picked out many of the passages in which we make suggestions relating to classroom practice. As we have pointed out in the Introduction, Chapters 5 to 11 are interrelated and we have not wished to draw attention only to certain passages in them, except in so far as these passages fulfil the purposes we have already set out. We do not regard the passages printed in heavier type as in any way constituting a summary of the report. 
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