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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 4 The mathematical needs of further and higher education
Further education 149 A wide range of both full-time and part-time courses is available within further education establishments for those who leave school at ages from 16 to 18. Many of the vocational courses which have a mathematical component operate under the regulations of the Business Education Council (BEC), the Technician Education Council (TEC), the City and Guilds of London Institute (CGLI) and the Royal Society of Arts (RSA). CGLI and RSA have been in existence for many years; BEC was set up in 1974 and TEC in 1973. The role of BEC and TEC is to plan, administer and keep under review the establishment of a national system of non-degree courses. BEC is concerned with those whose occupations fall, or will fall, within the broad areas of business and public administration. TEC caters for those in, or about to enter, all levels of technician occupation in industry and elsewhere. BEC and TEC courses have been designed to subsume the National Certificate and Diploma Courses (ONC, OND, HNC, HND). Types of course TEC courses 150 Courses for technicians at all levels in industry and elsewhere are provided by means of the TEC programmes. These are arranged within three sectors - engineering, construction and science-based occupations. The 16 year old school leaver will follow the course for the TEC Certificate or Diploma. The Diploma is an extension of the Certificate; in general it provides broader coverage at the same level and not study in greater depth. Both programmes may be taken by either full-time or part-time study. The certificate and diploma courses are made up of units of study offered at different levels. The Level I mathematics unit is common to programmes in all three sectors and is designed to be appropriate for those who have a minimum of CSE grade 3 or O Level grade E in relevant subjects. Level II units follow on from Level I but in some cases, including mathematics, may be entered directly by those who have CSE grade 1 or O Level grade C or above. 151 The mathematical content of the TEC Level I unit and the associated technological units usually extends beyond the mathematics which technician apprentices will require during their first year at work. Some companies insist that technician apprentices who are qualified to enter the Level II course in mathematics must nevertheless take the Level I unit in order to consolidate their understanding of topics which are specifically relevant to their work. In some cases this insistence stems from complaints by company training officers about lack of competence among some O Level entrants in certain areas of mathematics which are considered to be of particular importance. As we explained in paragraph 127, some firms make no distinction between craft and technician apprentices in the first year of training and CSE or O Level qualifications will usually determine whether an apprentice follows a TEC or CGLI course. However, it does not follow that all who start a TEC programme will eventually be classified as technicians; similarly, some of those who start a CGLI course may advance to become technicians. 152 TEC has published a mathematical 'Bank of Objectives' as a resource to be used in the preparation of mathematics units; it has also issued a set of 'standard units' based on the materials in the Bank. Colleges can make use of these standard units if they wish to do so, but are at liberty, subject to validation, to modify these units or to devise their own units to meet local needs. In the latter case they are able, in consultation with local firms, to make use of the Bank of Objectives. However, in practice many colleges include the standard units within the certificate and diploma courses which they offer. 153 The TEC units and Bank of Objectives are kept under review in order to attempt to match school syllabuses and to meet industrial requirements. Both units and Bank were revised in 1980 after extensive discussion with colleges and industry and in the light of comments from mathematics panels of GCE and CSE boards and from the mathematics committee of Schools Council. Recent changes in the standard units for Level I and Level II include the removal of slide rule and logarithm tables for purposes of calculation and greater emphasis on the correct use of calculators. BEC courses 154 BEC courses are designed to provide a foundation of vocational education for a range of related careers and to meet the needs of one or more of four Boards - Business Studies, Financial Sector Studies, Distribution Studies, Public Administration and Public Sector Studies. 155 In written evidence to us, BEC stated: The Council has been saddened and concerned to receive consistent reports of the inability of secondary school leavers to cope with basic arithmetical tasks requiring manipulative skills or to show that they have a 'feel for the order of magnitude of a quantity', or to apply basic quantitative concepts and techniques to business problems. Accordingly the Council has included compulsory quantitative studies in all of the courses leading to its awards.All BEC courses incorporate four 'central themes' of which one is 'a logical and numerate approach to business problems'. The courses are designed on a modular basis and 'cross-modular assignments' form part of each course. 156 BEC General Level and National Level courses are open to 16 year olds. The lower of these is the General Level course which is designed on a 'fresh start' policy and requires no formal examination qualification on entry. However, many entrants to this course possess examination qualifications at a level which is below the minimum of four O Level grade C or equivalent required for entry to the National Level course. 157 The General Level course contains a compulsory module entitled Business calculations which is examined separately. The emphasis in this module is on accurate and efficient performance of routine business calculations, ability to interpret numerical data and information, and ability to make use of numerical skills in the solution of business problems. The syllabus attempts to reflect the requirements of junior employees in clerical-type jobs and includes the operations of addition, subtraction, multiplication and division applied to whole numbers, decimals and fractions; metric and imperial units used in business; the purpose and use of approximations; the nature and function of percentages; the use of the average in business; the use of appropriate visual presentation of data. The learning objectives include 'appropriate and effective use of calculators'. 158 The entry requirement of four O Levels or equivalent for the BEC National Level course is not subject specific and so the mathematical attainment of those who enter this course from school varies widely. Those who proceed to the course as a result of having achieved credit standard at General Level will already have taken the Business calculations module. The National Level course includes a compulsory module entitled Numeracy and accounting. The 'numeracy' part of this module includes some, but not all, of the Business calculations module at General Level; notable omissions are metric and imperial units and the purpose and use of approximation. It also includes simple algebraic operations; construction and interpretation of graphs, representation of data in tabular and diagrammatic form; calculation and interpretation of averages; calculation of weighted averages; method used to construct selected index numbers, for example retail price index. CGLI and RSA courses 159 Courses at craft level for those employed in technical jobs are offered by CGLI. The mathematics within these courses is not usually treated as a separate topic, nor is it usually examined separately. Instead, it is regarded essentially as a tool for vocational studies and is commonly taught as part of these studies as and when necessary. 160 In order to provide for the needs of less academic students, CGLI has recently introduced a number of Foundation Courses, each focused in a broad vocational area. The main aims of these courses are 'to improve basic educational skills such as literacy and numeracy; to ease the transition from full-time education into the world of work; and to provide students with the basis on which they can make a more informed choice of career'. These courses are full-time and may be taken at school, at FE college or as 'link courses', in which the course is partly at school and partly at college. Each course contains a numeracy component which is developed in a context relevant to the vocational focus of the course. During the time we have been working, CGLI has introduced a course entitled Numeracy, which may be taken at school or at college. 161 In 1980 RSA introduced a Vocational Preparation (Clerical) course which is aimed at those initially seeking employment at operative or equivalent levels. Successful completion of the course secures a profile certificate which attests, as part of Communication, competence in various areas of arithmetic. Among the single-subject examinations offered by RSA are arithmetic, mathematics and machine calculating. These latter examinations may be taken from school or college. The match between school, employment and further education 162 Ideally a school leaver should experience continuity of learning in mathematics after moving from school to further education and employment. However, this is by no means a simple process. Further education courses are designed for broad categories of employees and entry requirements are framed in such a way as to allow admission to young people of as wide a range of educational attainment as possible. Furthermore, the variety which exists in the content of O Level and CSE syllabuses and the fact that attainment of a given grade does not indicate competence in any specific part of the syllabus make it difficult to identify with any degree of certainty topics with which most of the entrants to a particular course will be familiar. It is probably for this reason that we have been told that many courses, especially those which do not require an O Level or equivalent qualification in mathematics, often start at a very elementary level but then move so quickly that weaker students find great difficulty in keeping up. 163 We have already drawn attention in paragraph 130 to the fact that the mathematical demands of FE courses are likely to be considerably more than the demands of the job itself. One reason can be that in some cases it is necessary to go beyond immediate requirements in order to develop confidence and familiarity with essential topics. It is also the case that many courses are intended to provide not only the specific skills which are needed in the early years of employment but also a base for forty or more years of working life. Nevertheless, mathematical skills which are not used regularly can very easily atrophy, especially if they have proved difficult to comprehend, and so may not prove to be available when they are needed. It seems also to be the case that promotion can often lead to the use of less mathematics rather than more, because time is spent on supervisory and other duties. 164 We noted in paragraph 130 that craft and technician apprentices are sometimes required to follow the same initial training course. In some cases, apprentices who will eventually be designated as craftsmen are required in FE colleges to undertake the technician courses which are academically more demanding and for which they may not be adequately prepared. We believe that it is with this type of entrant that the mismatch between the mathematics content of FE courses and the future demands of the job gives most cause for concern. A comparable problem can arise in the case of entrants to BEC National Level courses whose four or more O Levels do not include mathematics; however, we have no evidence to indicate that this is so great a problem. The diversity of school syllabuses can also lead to problems of mismatch between college courses and courses which have been followed at school. 165 Although the range of mathematical ability can be particularly wide among those on craft courses, the fact that mathematics is not usually examined separately on CGLI courses makes it possible for students to avoid the more mathematical questions in examinations and still obtain good grades. This would seem to underline the fact that many craftsmen need to use only a very limited range of mathematical skills. 166 The Bath and Nottingham studies found that attitudes towards mathematics among students at FE colleges were very often more favourable than had been the case when they were at school. In some instances this seemed to arise from the fact that the applications of mathematics were more immediately apparent. Even when this was not the case there were some who persevered because they felt that the mathematics they were having to learn was certain to be needed at some stage or it would not have been included in the college Course.
The mathematical requirements of higher education Non-university sector 167 Within the non-university sector of higher education in England and Wales there are thirty Polytechnics and more than sixty Colleges of Higher Education, which offer a wide range of courses at degree level. Many of these courses are validated by the Council for National Academic Awards but some universities validate Bachelor of Education and other degrees offered at colleges within their locality. Polytechnics and colleges also offer a considerable range of full-time and sandwich courses at sub-degree level. The majority of these have a specific vocational slant, which is further reinforced by work experience when the course includes 'sandwich' placement in an appropriate firm. Some courses are aimed directly at the membership grades of various professional bodies (see also paragraphs 184 to 187). 168 Entry requirements for both degree and sub-degree courses are usually stated in terms of success at O and A Level in appropriate subjects, which may include mathematics, or the successful completion of other relevant courses such as the BEC National Certificate or Diploma or an appropriate TEC Certificate or Diploma. 169 There is at present no single body which is concerned with the administration of, or entry to, the complete range of courses in the non-university sector of higher education. We have sought information about students in this sector but the information which we have been able to obtain is not classified and analysed in a way which makes it possible to identify the mathematical qualifications of students on either degree or sub-degree courses. University sector 170 However, within the university sector of higher education, detailed information about the A Level qualifications of undergraduates entering degree courses at universities in the United Kingdom is collected by the Universities Statistical Record (USR). It has therefore been possible to establish the number of undergraduates with an A Level qualification in mathematics who have entered universities in England and Wales in recent years and also the degree courses which they have chosen. In the following paragraphs we discuss the general information which these figures provide before considering certain degree courses for which a specific mathematical entry qualification is normally required. The figures which we quote refer to undergraduates at universities in England and Wales with home fee-paying status; students from overseas are not included. We estimate that the statistics we give for 1979 refer to about three quarters of all those with home fee-paying status who started degree courses at institutions in England and Wales. 171 In 1979 almost 92 per cent of all new entrants with home fee-paying status to first degree or first degree and diploma courses at universities in England and Wales had entry qualifications based on A Levels. About 2½ per cent, of whom just over half were entrants to courses in engineering and technology, had qualifications based on National Certificates or Diplomas; the remainder had a variety of other qualifications, including qualifications gained at other universities or in other countries. For mathematical studies and most science subjects some 95 percent of entrants had an A Level qualification; for engineering and technology the figure was about 85 per cent, reflecting the greater numbers who enter these courses on the basis of National Certificates or Diplomas. 172 These proportions have remained substantially the same since 1973, the first year for which we quote figures. Between 1973 and 1979 the total number of undergraduates increased by just over 28 per cent, though there were considerable variations from subject to subject. The number reading mathematical studies increased by about 19 per cent, physical sciences by about 17 per cent (but physics itself by about 28 per cent) and engineering and technology by about 34 per cent. Undergraduates with A Level mathematics 173 Mathematics is the only subject other than music which it is possible to offer as either one or two subjects at A Level. This is commonly referred to as taking 'single-subject' or 'double-subject' mathematics and we shall use these terms. For the present it is sufficient to be aware that both single-subject and double-subject mathematics are available at A Level; we discuss the differences which exist in the structure of the double-subject examinations and in the syllabuses for both single- and double-subject examinations in Chapter 11. Those who take the double subject cover a more extended syllabus than those who take the single subject. It does not follow that they are necessarily more able mathematically than those who take the single subject; however, the fact that they will almost certainly spend more time on mathematics during the A Level years enables them not only to cover more ground but also to develop increased confidence and competence in the content of the single-subject syllabus. For many years almost all of those who took double-subject mathematics combined it with A Level physics; some also took A Level chemistry to give a total of four subjects in all. Many still take these traditional combinations but in recent years it has become increasingly common to combine both single- and double-subject mathematics with a wide variety of other subjects. 174 The numbers of men and women entering universities in England and Wales from 1973 to 1979 who had an A Level qualification in mathematics (1) are shown approximately in Figure 2 (see also Appendix 1, Table 27). We may note that throughout these years the percentage of men with an A Level qualification in mathematics has been about twice that of women. 175 We now examine the degree subjects studied by entrants to degree courses who had one or more A Levels in mathematics. The information for 1973 and 1979 is given in Table A (see also Appendix 1, Tables 28 and 29). Table A Distribution between subject groups of entrants to degree courses at universities in England and Wales who had one or more A Levels in mathematics 176 We present this information in a different way in Figure 3 (below) which draws attention to the proportion of entrants to each group of courses who had an A Level qualification in mathematics. The most significant changes between 1973 and 1979 have been in respect of those starting medical and dental studies, in which the proportion has increased from 33 per cent in 1973 to 44 per cent in 1979, and those starting courses in business and management studies, economics and accountancy, in which there has been an increase from 42 per cent to 54 per cent. Undergraduates with double-subject mathematics 177 There has been a marked change in the proportion of entrants to universities in England and Wales with a double-subject qualification in mathematics (2). In 1973 almost 32 per cent of those entering with an A Level qualification in mathematics had a double-subject qualification; in 1979 the proportion had fallen to little more than 21 per cent. In absolute terms, although there were over 6000 more entrants with A Level mathematics in 1973 than in 1979, the number of entrants with a double- subject qualification had dropped by about 650. The number and percentage change over this period is shown in Table B below (see also Appendix 1, Table 28). Figure 2 Numbers of men and women entering universities in England and Wales from 1973 to 1979 who had an A Level qualification in mathematics Figure 3 Proportion of entrants to each group of courses who had one or more A Levels in mathematics: Universities in England and Wales 1973 and 1979 Table B Levels of mathematical qualification of entrants to degree courses at universities in England and Wales 178 There have also been changes in the way in which those with double-subject mathematics have been distributed among the various degree courses. This is shown in Table C. (also Appendix 1, Tables 28 and 29). Table C Distribution between subject groups of entrants to degree courses at universities in England and Wales who had double-subject A Levels in mathematics 179 About 80 per cent of entrants with a double-subject qualification in mathematics at A Level read engineering and technology, physical sciences or mathematical studies. Figure 4 illustrates the way in which entrants with a double-subject qualification were distributed between these three areas of study in 1973 and 1979. (See also Appendix 1, Table 30). Figure 4 Numbers of university entrants in England and Wales admitted on the basis of A Levels. Subject choices and numbers with double-subject A Level qualifications in mathematics: 1973 and 1979 Degree courses in mathematical studies 180 We wish to draw attention to the drop in the proportion of those reading mathematical studies (3) at universities in England and Wales who have a double-subject qualification and to its implications. In 1973 the proportion was almost 80 per cent; in 1979 this had dropped to 55 per cent. Although there are some universities at which it is still the case that a very high proportion of those reading degrees in mathematics have taken the double subject at A Level, our own enquiries have established that there are others at which substantially less than half of those reading mathematics have a double-subject qualification. We believe that this information may come as a surprise to many people in both universities and schools. It is not within our terms of reference to comment on its implications for those who teach mathematical studies in universities; the implications for those who teach in schools are very great. 181 It is very commonly supposed that it is almost essential to have taken double-subject mathematics at A Level in order to read mathematics successfully at university. However, it is very important that those who teach mathematics in sixth forms and those who advise pupils about their choice of degree course should realise that there are now universities in which more than half of those reading mathematics are doing so from a basis of single-subject A Level. It follows that they should not dissuade pupils who have taken only the single subject at A Level from applying to read degree courses in mathematics. We see no likelihood that the demand for mathematics graduates will decrease - indeed, we believe that the demand will continue to grow - and those whose interests and abilities lie in this field need every encouragement to study mathematics at degree level. As we pointed out in paragraph 173, it does not follow that those who have taken only the single subject are necessarily less able at mathematics than those who have taken the double subject and so less fitted to embark on a mathematics degree course. There seems no doubt that at most universities they will be increasingly likely to find themselves in the company of others who are similarly qualified. Degree courses in engineering and technology 182 A knowledge of mathematics is essential for the study of engineering and of most other technological subjects. We drew attention in paragraph 172 to the fact that the number of entrants to courses in engineering and technology increased by 34 per cent between 1973 and 1979 whereas the overall university entry increased by 28 per cent. This increase has been much greater than the increase in total entry to all other mathematics and science courses, which has risen by only 20 per cent. Furthermore, despite an overall drop during this period in the total number of entrants to universities in England and Wales with a double-subject qualification in mathematics, there has been an increase in the proportion of these entrants who have chosen to read engineering and technology, and also a small absolute increase in their numbers. Tables A and C show that degree courses in engineering and technology are attracting an increasing proportion of university entrants with an A Level qualification in mathematics and, in particular, of those with double-subject mathematics. The proportion of entrants with double-subject mathematics is, however, decreasing both within engineering and technology courses and overall.
The mathematical requirements of professional bodies 183 Although not directly within our terms of reference, we have given some attention to the mathematical requirements of professional bodies. Many of those engaged in professional activities seek membership of the appropriate professional institution or association. In some cases membership of such a body is a necessary qualification for professional advancement; in other cases membership, although not essential for career purposes, provides opportunity to keep abreast of current developments by reading publications, attending meetings and taking part in the work of committees. Most institutions conduct their own examinations, commonly in two or three parts, for admission to membership, which is usually offered at more than one grade. The possession of an appropriate academic qualification often secures exemption from some or all of these examinations but admission to higher grades of membership normally requires evidence of relevant professional experience. 184 A number of the professional bodies who have written to us have stated the mathematical requirements for direct entry to their various grades and have also supplied details of their own examinations. When entry is at graduate level, it is usually assumed that any necessary mathematics will have been covered either at school or during the degree course and no further mathematical requirement is stipulated. However, one exception to this is the Institute of Actuaries whose final examinations require a considerable extension of mathematical and statistical knowledge and its application. When entry to a professional body is at lower levels, any mathematical requirement is normally stated in terms of success at A or O Level; any further mathematics which is required is then included within subsequent professional study. 185 Almost all the professional bodies who submitted evidence stressed the importance of being able to apply computational skills confidently in a variety of ways. These include accuracy and speed in mental calculation and ability to check the reasonableness of answers; in some cases extended and complex calculations are necessary. Specific calculations identified by bodies whose members are concerned with commerce include interest, discount and value added tax, cash flow, costing and pricing, and budgetary control; it is frequently necessary to be able to deal with both metric and imperial units. There were also many references to the need to be able to interpret data with understanding. 186 Most institutions take for granted the mathematical foundation provided by an entrant's previous study. Any mathematics included within professional examinations is usually limited either to topics of a specialist nature which are unlikely to have been studied before entry to the profession or to applications of mathematics in unfamiliar contexts. This is especially true of professional bodies whose members are concerned with business and commerce. The examinations of these bodies frequently include applications of statistics, and to a lesser extent techniques of operational research, which are used within the particular profession. The collection, classification, presentation and analysis of data, use of probability distributions, hypothesis testing, correlation and regression analysis, survey methods and sampling techniques all occur frequently within the syllabuses of professional examinations. This emphasis on statistics no doubt reflects the fact that at the present time few school leavers will have studied the subject to any depth.
Footnotes (1) A Level Statistics and A Level Computer Science are not included as mathematical qualifications for the purposes of the figures and tables in this chapter. (2) This category is restricted to those whose A Levels include Applied Mathematics or Further Mathematics. (3) Mathematical studies includes degree courses in mathematics, statistics, computer science, combinations of these and also a variety of courses which combine mathematics with subjects other than these. Analysis of information provided by USR about first degrees awarded in 1979 within the field of mathematical studies shows that about 65 per cent were in mathematics only, about 25 per cent in computer science, either wholly or in part, and about 8 per cent in statistics. 
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