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 3 The mathematical needs of employment
[pages 12 - 41]

Views expressed before the Committee was set up

40 It is clear from the report of the Parliamentary Expenditure Committee (1) to which we referred in the introduction that the volume of complaints which seemed to be coming from employers about lack of mathematical competence on the part of some school leavers was one of the principal reasons for its recommendation that our Inquiry should be set up.

41 We believe that these complaints started to come to the fore in 1973 and 1974 when a number of articles and letters which were highly critical of mathematics teaching in schools, and of 'modern mathematics' in particular, appeared in Skill, a news-sheet published by the Engineering Industry Training Board for group training schemes in the engineering industry. These complaints were followed in 1975 and 1976 by articles and letters in Blueprint, the newspaper of the Engineering Industry Training Board, which also expressed dissatisfaction with the mathematical attainment of some entrants to the industry. More widespread criticism appeared in newspaper articles and correspondence columns during these years.

42 In his speech made at Ruskin College, Oxford in October 1976, Mr James Callaghan, at that time Prime Minister, said:

I am concerned on my journeys to find complaints from industry that new recruits from the schools sometimes do not have the basic tools to do the job that is required. ... There is concern about the standards of numeracy of school leavers. Is there not a case for a professional review of the mathematics needed by industry at different levels? To what extent are these deficiencies the result of insufficient coordination between schools and industry? Indeed how much of the criticism about basic skills and attitudes is due to industry's own shortcomings rather than to the educational system?

Employers are becoming increasingly concerned that many school leavers, particularly those leaving at the statutory age have not acquired a minimum acceptable standard in the fundamental skills involved in reading, writing, arithmetic and communication. This shows up in the results of nearly every educational enquiry made amongst the CBI membership, and is backed up by continuing evidence from training officers in industry and further education lecturers that young people at 16+ cannot pass simple tests in mathematics and require remedial tuition before training and further education courses can be started.

Mathematics, I think - or arithmetic, which is really the primary concern rather than mathematics themselves - is the one area which is really brought up every time as a problem. It seems that industry's needs are greater in this respect than almost any other. This is the way, certainly, in which shortfall in the education of children makes itself most manifest immediately to an employer.

The Engineering Industry Training Board, over the last two years, received from its industry increasing criticism, with supporting evidence, of the level of attainment, particularly in arithmetical skills, of school leavers offering themselves for craft and technician training ... In the view of the Engineering Industry Training Board the industry needs a higher level of attainment in basic mathematics among recruits than it is now getting and believes that, with closer cooperation between school and industry, children can while still at school be motivated to achieve this ... Mathematics is, however, not simply a question of basic manipulative skills. An understanding of the concepts is also needed and these are better taught by 'innovative methods', which also appear to enhance the ability to acquire planning and diagnostic skills, of great importance to craft and technician employees.

Employers' views expressed to the Committee

45 We have received written evidence, amounting to some 200 submissions in all, from major companies, industry training boards, the Confederation of British Industry, the Trades Union Congress, the Association of British Chambers of Commerce and, with the help of Rotary International in Great Britain and Ireland, from many smaller employers. We have studied the reports of the research studies into the mathematical needs of various types of employment carried out at the Universities of Bath and Nottingham and have also had access to the detailed notes of visits to more than 100 companies on which these reports have been based. Members of the Committee have themselves visited 26 firms chosen to span a wide range of types of employment. During these visits it was possible to talk with young employees as well as with management and training staff, and so to receive their views directly. We have also taken oral evidence from the CBI, from two major training boards and from groups of small employers.

46 The overall picture which has emerged is much more encouraging than the earlier complaints had led us to expect. We have found little real dissatisfaction amongst employers with the mathematical capabilities of those whom they recruit from schools except in respect of entrants to the retail trade and to engineering apprenticeships, both of which involve significant numbers of young people; we discuss these two groups in paragraphs 53 and 54. We have also found little evidence that employers find difficulty in recruiting young people whose mathematical capabilities are adequate.

47 Because we were concerned that there might nevertheless be a substantial amount of criticism which had not reached us, we made special efforts to invite further evidence. However, amongst other initiatives, an article published in the autumn 1979 issue of Skill, which drew attention to the existence of our Committee and invited its readers to write to us, resulted in only one letter. A further article published in the Financial Times in July 1980 produced three letters, only one of which contained major criticisms.

48 Employers have, in general, expressed no more than mild reservation about the fluency in arithmetical skills possessed by the young people whom they recruit; lack of fluency in mental arithmetic has attracted the most comment. However, employers have also told us that, under the motivation which employment can provide, these problems can soon disappear. This situation is consistent with the preliminary findings, published in the spring of 1980, of the study Young people and employment, at present being carried out at Lancaster University under the direction of Professor Gareth Williams. These show that, of a sample of 300 'employing establishments' in England and Wales, only 14 per cent criticised the educational standards of their recruits. More common criticisms found in the Lancaster study were of inability to take work seriously, lack of interest, unwillingness to work, bad time keeping and poor attendance.

49 There appear to be several reasons for this apparent change in the views expressed by employers. In part it may reflect the efforts which we believe have been made by many schools to meet criticisms of the kind which were voiced during the 'Great Debate'. It may also reflect the increasing number of local initiatives which have led to the setting - up of school/industry liaison groups of various kinds. We discuss these groups in paragraph 103.

50 We believe that there is increased understanding on the part of employers of the changes which have occurred in the education system in the post-war years and, in particular, of the fact that, because more young people are going from school to further and higher education, the recruitment of 16-18 year olds to jobs at a given level is taking place from further down the ability spectrum than was the case in earlier years. We may add here that we were much encouraged to hear one of the CBI representatives draw attention in oral evidence to his own company's experience. Although some of the craft apprentices who were currently being recruited were less well qualified academically than some of their predecessors, they were nevertheless being trained successfully to carry out more complex work than their predecessors had been required to undertake. The other CBI representatives agreed with this view.

51 It is possible that a further reason for the present absence of criticism is the current high level of unemployment among young people which allows employers to pick and choose to an extent which was not possible in, for instance, the mid-sixties and early seventies. It is, of course, not possible to quantify the extent to which those young people who are at present unemployed would, in a time of full employment, exhibit mathematical shortcomings in their work. However, it may be of relevance to consider the pattern of unemployment and notified vacancies among school leavers over the last twenty years which is shown in Figure 1.

Figure 1 Vacancies for young people and numbers registered as unemployed

52 The raising of the school leaving age to 16 in 1973 undoubtedly had some effect on recruitment patterns in 1973-74, but this was also the time at which complaints about lack of numeracy among school leavers came sharply into the public notice, typified by letters and articles in Skill. It does seem possible that high levels of employment, and so a smaller pool of applicants from which to choose, may result in greater criticism from employers about the mathematical abilities of their young recruits. We are not aware of the same expression of concern in the mid-sixties, so the connection between high employment and complaint must be treated with caution. Furthermore, industry training boards were only just coming into existence at that time and so there were perhaps not as many vehicles for complaint by employers. However, since it may perhaps be the ease that reduction in criticism is in part a result of high levels of unemployment among young people, we are very conscious that this fact must not be allowed to result in complacency about the state of mathematical education at the present time.

53 We have already referred to the fact that most of the criticism of young employees which we have received refers to entrants to the retail trade and to engineering apprenticeships. The former attract the more serious criticism. Those who enter the retail trade on leaving school at 16 commonly have very modest or no mathematical qualifications but are often required from the outset to give change, count stock, fill in stock sheets and calculate discounts. One large employer has complained to us that it is necessary to spend a substantial amount of time teaching newly recruited shop assistants to carry out tasks of this kind.

54 The complaints which we have received relating to engineering apprentices seem to stem largely from the performance of applicants in company selection tests, which are very often tests of computational ability only. However, most of the criticism relates to those applicants who are rejected. Employers have expressed comparatively little dissatisfaction with the mathematical performance of those whom they have taken on as apprentices. Moreover, where difficulties, mainly in arithmetic, do arise, we have been told that they can almost always be overcome relatively quickly during the initial apprenticeship year as apprentices gain practical experience in the workshop and so realise why it is necessary to be able to carry out certain specific calculations.

55 It is important at this stage to note that, while agreeing with the importance of arithmetical attainment, the EITB has stressed to us its view that tests of arithmetical skill play too dominant a role in selection for engineering apprenticeships and that conceptual skills, such as spatial awareness, the understanding of orders of magnitude, approximation and optimisation, are of equal or greater importance. In the course of oral evidence, one of the EITB representatives expressed concern that some schools, in over-reacting to the perceived criticisms of industry and concentrating excessively on computational work, were failing to give to their pupils the wider mathematical experience which was of value for the intending craftsman.

56 A number of submissions have drawn attention to the need which is experienced by many employers to provide so-called 'remedial' training for young employees; we have referred to these in earlier paragraphs. It is important at the outset to distinguish between three kinds of training for which this term may be used. The first is the need to revise skills which may have become rusty through lack of practice - and all those who work within classrooms know not only how much can apparently be 'forgotten' during the summer holiday, but also how quickly memory can be refreshed. The second is the need to provide opportunity to become familiar with applications, possibly of quite elementary mathematics, which may not have been encountered in the classroom even though they are of frequent use in a particular job, or to teach certain specialised techniques. The third is the need to spend time on training which is remedial in the true sense of the word; in other words, to spend time teaching mathematics which, although it has been included in the school course, has not been understood or is being used incorrectly.

57 We believe that the need to undertake a certain amount of training of the first two kinds, which should only take a short time, is to be expected; though we appreciate that for the small employer or the business which has a rapid staff turnover it will represent some drain on resources. On the other hand, we accept that employers may with reason expect not to have to undertake training of the third kind, since this may properly be considered to come within the responsibility of schools. Inevitably, the line between the two categories is not always easy to draw. In the course of our work we have become aware that a degree of over-expectation exists in many quarters as to the level of attainment in mathematics which some school leavers are able to reach; we discuss this in paragraph 189. This is one of many reasons why good liaison between employers and schools, which we discuss later in this chapter, is of such great importance.

58 There is a further point to which we wish to draw attention. Because of the high level of unemployment which exists among young people at the present time, lack of use of mathematics on their part may well lead to a degree of 'rustiness' which will require sympathetic consideration on the part of employers to whom these young people apply for jobs, and perhaps also the provision of special assistance.

The research studies

59 We turn now to the findings of the two research studies into the mathematical needs of various types of employment which were carried out at the University of Bath and the Shell Centre for Mathematical Education at the University of Nottingham. For the sake of brevity we shall henceforward refer to them as the Bath study and the Nottingham study.

60 The reports which these studies have produced are extensive and wide-ranging. We have space in our own report to draw attention only to the points which we believe to be most significant. We hope that means will be found to make the detailed findings of the studies available widely (2); we believe that they will be of assistance to those who teach in schools and colleges and to those who employ and train young people. We believe, too, that they may well be of considerable interest to a much wider section of the public.

61 Both studies paid particular attention to the types of employment open to those leaving school at ages from 16 to 18. However, both also included some jobs which, because of legal or other restrictions, were open only to employees over the age of 18 if it was the case that these jobs required no greater experience or qualifications than had been obtained at school.

62 The Bath study was designed to cover as wide and as representative a sample of the various categories of employment as possible. More than 90 companies and other establishments were visited and a considerable amount of time was spent both in observation of work which was going on and in discussion with employees themselves and with managerial and training staff.

63 The Nottingham study was designed to complement the work undertaken at Bath by looking in greater depth at the more specific areas of clerical and retail work, agriculture, and the medical and paramedical occupations within the Health Service which recruit from those who leave school at ages from 16 to 18. It was also able to draw on similar work which had already been undertaken in some other areas, notably the training of engineering apprentices. In both studies stress was laid on obtaining first-hand information. A report was written on each visit and agreed with the company or other establishment concerned.

64 The report of the Bath study draws attention to the fact that it is possible when observing work in progress to describe certain aspects of it in mathematical terms. For instance, the mathematical concept of geometrical symmetry is present within many manufacturing processes; it is possible to describe different methods of stacking and packing in geometrical terms or in terms of sorting and classifying. However, even if the mathematical concepts involved have at some time been encountered in the classroom, the employee will probably not consciously analyse in such terms the operations which are being performed; nor, if he were to do so, would he necessarily be able to do his job any better.

65 On the other hand, many jobs require the employee to make explicit use of mathematics - for instance, to measure, to calculate dimensions from a drawing, to work out costs and discounts. In these cases the job cannot be carried out without recourse to the necessary mathematics and it is this latter use of mathematics to which we refer in the paragraphs which follow. However, even when mathematics is being used, frequent repetition and increasing familiarity with a task may mean that it may cease to be thought of as mathematics and become an almost automatic part of the job. A remark which was overhead - 'that's not mathematics, it's common sense' - is an illustration of this and, indeed, indicates an 'at-homeness' with that particular piece of mathematics which we would wish to commend.

66 Both studies draw attention to the diversity of types of employment which exists, to the variety of mathematical demands within each and to the considerable differences which were found to exist even within occupations which might be assumed from their titles to be similar. It is therefore not possible to produce definitive lists of the mathematical topics of which a knowledge will be needed in order to carry out jobs with a particular title. It is, however, possible to describe in general terms the types and levels of mathematics which are likely to be encountered by certain broad categories of employees. These descriptions may be found in paragraphs 120 to 148 at the end of this chapter; those of our readers who have a detailed interest in the mathematical needs of employment may wish to study them before proceeding further.

Some common findings

67 A number of findings emerged from the studies with sufficient regularity to suggest that they are likely to have general validity; it is these which we discuss in the paragraphs which follow.

Range of mathematics required

68 Both studies found that almost all of the mathematics which young people need to use, whatever their job, is included within all the existing O Level and CSE Mode 1 (3) syllabuses. Even the restricted content of the limited grade CSE syllabuses which are offered by some boards includes the mathematics required in most of the jobs which those who take these examinations are likely to have the opportunity to enter. Since most pupils do not know in advance the type of employment which they will follow in later life, it is important that this should be the case.

69 Nevertheless, the studies also identified certain important differences between the ways in which mathematics is used in employment and the ways in which the same mathematics is often encountered in the classroom. We believe that these differences may be among the factors which have contributed to the criticisms which have been voiced in recent years.

Calculation

70 The need to be able to carry out artithmetical calculations of various kinds appears among the mathematical requirements of almost all the types of employment which we discuss in paragraphs 120 to 148. These calculations are sometimes carried out mentally, sometimes with pencil and paper and sometimes with a calculator. Some jobs specifically require an ability to carry out mental calculations of various kinds. In almost all jobs the ability to carry out some calculations mentally is of value and lack of ability to do this is a frequent cause of complaint by employers. It was also found to be a cause for complaint by some young employees who maintained that an ability to carry out mental calculations which they had possessed in their primary and early secondary years had been allowed to atrophy as a result of lack of practice in the classroom.

71 Both the Bath and the Nottingham studies found that the methods which are used when at work to carry out calculations with pencil and paper are frequently not those which are traditionally taught in the classroom. Employees use a variety of idiosyncratic and 'back of an envelope' methods, especially for long multiplication and division. Sometimes these methods have been devised by the employees themselves but frequently they have been passed on by their fellow workers. The methods which are used depend very much on the user's confidence in his own mathematical ability. However, at all levels there appears to be a preference for carrying out a calculation in a succession of relatively short stages rather than for making use of a single calculation which is mathematically more sophisticated, and perhaps quicker, but which may be more difficult to use with confidence. For example, we read in the report of the Bath study:

We met a number of situations where people by-passed the more traditional school methods, eg a boy who had left school before examinations and at work performed quite a complex check (which he was handling with confidence) on whether he was up to quota. He required 7 x 96, which we might classify as use of the seven times table. However, he proceeded as follows:

First 3 x 96 = 288

then	288
	288
	576
	96
	672

Although this employee did not know his seven times table, his method shows that he knew what 'seven times' something meant. This is one of the many examples we found where people resorted to longer methods with which they evidently felt confident, rather than using what a mathematics teacher might have thought to be the obvious direct method.

Use of calculators

73 Our own enquiries, and the evidence which we have received, lead us to believe that there is an ambivalent attitude to the use of calculators in industry and commerce at the present time. The Bath and Nottingham studies found their use to be widespread in many types of employment. These include a wide range of clerical and administrative jobs such as accounts departments, banks, insurance offices and related areas of employment. Calculators are also used widely by those who work in laboratories and engineering design offices and by those concerned with quality and production control; these are all jobs which require a considerable amount of calculation and analysis of data. In all of these situations calculators are regarded as desirable aids to speed and accuracy.

74 Calculators are also used increasingly by many who work on the shop floor but their use is still viewed with suspicion by some managers and supervisors who were themselves trained to use slide rules or logarithm tables. This seems to be especially true of those who supervise engineering and other technical apprentices and craftsmen of various kinds. We are aware of situations in which new apprentices, who had been issued with logarithm tables by their training supervisor, preferred instead to make use of the personal calculators which they were encouraged to use on the college courses that they attended. There are, of course, many straightforward calculations which a craftsman needs to be able to carry out mentally but this does not seem to be a reason for denying the use of a calculator when it is sensible and time-saving to use one. However, the majority of young employees who were seen to be using calculators at work had not been trained in their use either at school or on the job. In consequence, calculators were frequently not being used in the most effective way.

Fractions

75 Although fractions are still widely used within engineering and some other craft work these are almost always fractions whose denominators are included in the sequence 2, 4, 8, ... , 64. This sequence is visibly present on rules and other measuring instruments and equivalences (4) are apparent. Addition or subtraction of lengths which involve fractions of this kind can be done directly by making use of the gradations on the rule. When the calculation is carried out with pencil and paper it is never necessary to work out the common denominator which will be required because it is always present already; for example 2 1/4 + 3 5/16 has the necessary denominator, 16, already visible. Even so, the methods used are not always those which might be expected. The report of the Bath study comments that

Frequency of use promotes assimilation of equivalents. This is evident from the following method that a craftsman used to add fractions:

3/16 + 5/64	=	3/16 + 1/16 + 1/64
	=	4/16 + 1/64
	=	16/64 + 1/64
	=	17/64

This is another instance of someone rewriting numbers in a more convenient form. It could be described as very mathematical but it is not necessarily the method which would be reinforced in schools.

76 The notation of fractions appears in some clerical and retail jobs, for instance 4 3/7 to represent 4 weeks and 3 days or 2 5/12 to represent 2 dozens and 5 singles. However, school-type manipulation is rarely found and then only in very simple cases; for instance, the calculation required to find the charge for 3 days based on a weekly rate is division by 7 followed by multiplication by 3.

Algebra

77 One of the more surprising results of the studies is the little explicit use which is made of algebra. Formulae, sometimes using single letters for variables but more often expressed in words or abbreviations, are widely used by technicians, craftsmen, clerical workers and some operatives but all that is usually required is the substitution of numbers in these formulae and perhaps the use of a calculator. The report of the Bath study quotes an example of one such formula which was not even written down, but was remembered by the employee concerned:

A wages clerk explained: 'To get the rate of pay per hour, we add together the gross pay, the employer's National Insurance, and holiday stamp money, subtract any bonus, and then divide by the hours worked'.

It is not normally necessary to transform a formula; any form which is likely to be required will be available or can be looked up. Nor is it necessary to remove brackets, simplify expressions or solve simultaneous or quadratic equations, although algebra of this kind is sometimes encountered on courses at further education colleges. Solution of linear equations is required very occasionally.

Estimation

78 Industry and commerce rely extensively on the ability to estimate. Two aspects of this are important. The first is the ability to judge whether the result of a calculation which has been carried out or a measurement which has been taken seems to be reasonable. This enables mistakes to be detected or avoided; examples are the monthly account which is markedly different from its predecessors or the measured dose of medicine which appears unexpectedly large or small. The second is the ability to make subjective judgements about a variety of measures. This is of use in situations in which measurement is difficult or awkward, in which trial and error is possible or in which tolerances are large. Skills of estimation develop on the job but employers often complain that young entrants to industry and commerce lack a 'feel' for both number and measurement, even in terms of the units, whether metric or imperial, which they can be expected to have encountered at school or in everyday life.

Measurement

79 Although estimation is important, counting and measurement are paramount. A very great deal of the mathematics which is used in employment is concerned with measurement in one or other of a wide variety of forms, by no means all of which are directly concerned with the use of measuring instruments. Measurements are specified in a variety of ways, for instance in terms of number of items or total of money; of length, weight or volume; of ratio, percentage or rate. Use is made of metric and imperial units as well as of units which are peculiar to a particular industry.

80 There are two different aspects of measurement - the first in which an existing measurement such as length, weight or number in stock has to be determined, the second in which it is necessary to create a desired measurement. In every case it is necessary to be aware of how accurate the measurement needs to be; in some cases it is necessary to be able to choose and use an appropriate measuring instrument. The labourer may well measure in terms of buckets or shovels full. The skilled engineering craftsman may work to different tolerances for different parts of a job and will need in each case to choose the appropriate tool or instrument for creating or checking the measurement he requires. The cashier will need to count the money in the till exactly.

81 There are also those who have to be keenly aware of the meaning of measurements, even though they seldom measure anything for themselves. These include staff who are involved in ordering and costing, in calculating turnover and profit margins. Here again, measurement is seldom exact but needs to operate within specified degrees of accuracy. This concept of measurement as something which is sometimes exact but more often needs to lie within stated limits is one which is very different from that which is commonly encountered in the classroom; it is also one which requires a conceptual understanding which takes time to develop. We discuss this important point in more detail in paragraphs 269 to 272.

Metric and Imperial units of measurement

82 We have received a number of submissions which draw attention to the fact that imperial units are still used in many parts of industry. Although there are many companies which use only metric units, others still operate mainly with imperial units and many companies use a mixture of both metric and imperial. There are also certain companies which make use of units which are peculiar to a particular industry. There are a number of reasons for the continuing use of imperial units. Among the most significant are the expense of re-equipping workshops with new machinery and instrumentation, the need to maintain a supply of spare parts which conform to standard imperial sizes and the needs of customers overseas who still make use of imperial measures. However, where imperial units are still in use, only a limited range is normally encountered on any particular job, for example yards or feet and inches, pounds and ounces. The use of fractions of an inch and of the 'thou' (thousandth of an inch) is also common. Where conversion from imperial to metric units, or vice versa, is necessary it is usual to make use of a conversion table.

Implications for the classroom

83 It is of fundamental importance - and, we believe, not as self-evident as some might suppose - to appreciate the fact that all the mathematics which is used at work is related directly to specific and often limited tasks which soon become familiar. The different aspects of mathematics which are used are related to each other by their context so that their application is immediately evident. Increasing experience of carrying out calculations or measurements helps to develop skills of estimation and approximation and an awareness of whether or not a result is sensible.

84 However great the effort which is made, illustrations of the practical applications of mathematics within employment which are given to a group of pupils, whose members will enter many different types of job, cannot provide the immediacy of the actual job itself. Nevertheless, it is important that the mathematical foundation which has been provided in the classroom should be such as to enable competence in particular applications to develop within a reasonably short time once the necessary employment situation is encountered.

85 The preceding paragraphs give some indication of the kinds of mathematical skill and understanding which are needed. We believe that it is possible to summarise a very large part of the mathematical needs of employment as 'a feeling for measurement'. This implies very much more than an ability to calculate, to estimate and to use measuring instruments, although all of these are part of it. It implies an understanding of the nature and purposes of measurement, of the many different methods of measurement which are used and of the situations in which each is found; it also implies an ability to interpret measurements expressed in a variety of ways.

86 If boys and girls are to leave school equipped in this way, they will need, in the mathematics classroom and elsewhere, to have taken part in a wide variety of appropriate mathematical activities and to have discussed these at length with their teachers and with each other. We consider ways in which this may be achieved in the second part of our report.

Employers' selection tests

87 Some firms, especially those in the engineering industry and some branches of the retail trade, require applicants for jobs to undergo written selection tests in mathematics. We have received many comments about these tests both in submissions from teachers and from employers themselves. We are grateful to the many employers who have provided us with copies of the selection tests which they use and also, in some cases, with photocopies of scripts completed by applicants for employment.

88 Employers who make use of written selection tests appear to do so for one or more of the following reasons:

• because recruitment takes place before the results of O Level and CSE examinations are available;
• because employers wish to satisfy themselves that those whom they engage are able to carry out specific types of calculation;
• because, even when the results of O Level and CSE examinations are available, there is a feeling that success in the examination, especially if it is at a low grade, does not guarantee competence in the particular mathematical skills which the employer requires;
• because of unwillingness to rely solely on reports provided by schools since, as some employers have explained to us, these reports are often written in terms which are too general and sometimes include predictions of likely examination results which do not prove to be accurate;
• because it is necessary to choose from a large number of applicants or to sort them in some other way.

91 The kinds of tests which are used and the level at which they are set vary widely. We are concerned that testing procedures are often in the hands of people who have neither training nor appropriate experience in testing procedures, including the setting and marking of papers. We have been surprised to find that this can be the case even within major companies.

92 Although some tests are specifically related to tasks which the employee will be required to undertake as soon as he starts work, many others are set at a level which is higher than this. In some cases it is claimed that this is because a higher standard of mathematics than the job requires will be needed for success in, and perhaps entry to, some form of training at a college of further education either immediately or in the future. It may also be because a higher standard of mathematics will be required for jobs to which the employee may be promoted in due course.

93 We have also been told that some tests are set at a higher level than will be required on the job because it is considered that success in a harder test will indicate an ability to perform well at more elementary tasks or will indicate a more 'flexible' employee whom it will be possible to employ on a variety of jobs. Some also take the view that ability in mathematics is a good indicator of a more general ability. We are aware, too, of some cases in which a test is set at a higher level merely because a commercially produced test is being used which does not reflect the specific needs of the job.

94 Even in cases in which the requirements of a job have been carefully identified and a test devised which is matched to these, the test may not serve the purpose which is intended and so lead to unjustified criticism of the performance of those who are required to attempt it. There can be many reasons for this. One of the most common is that questions are very often worded ambiguously or set out on the page in a confusing manner. The latter can apply especially to question papers which have been typed and stencilled; fractions, for example, are difficult to set out clearly on a typewriter and the fact that the same key is used for both a decimal point and a full stop can lead to confusion unless great care is taken. Some tests which we have seen use language which is mathematically incorrect; some also use the jargon of the job - 'cast these figures across and down' - which those taking the test may not have met before and so do not understand.

95 We are also concerned at the rigid way in which test papers are very often marked and at the number of errors in marking which we have identified on scripts which have been sent to us. Marking schemes are usually kept very simple, often with just one mark for each answer and frequently no credit is given if, as a result of a single mistake in one part of the question, several subsequent answers are incorrect, even though the mathematical operations involved have been carried out correctly. We have also found instances in which candidates have dealt successfully with one possible interpretation of an ambiguously worded question but have been given no credit for it. In our view, too, the time allowed for completing some tests is far too short so that candidates do not have the opportunity of showing properly what they can do. Furthermore, the stress of working is increased when it is necessary to work against the clock; provided that the mathematics is understood, speed of working will develop with continuing use on the job.

96 We wish to draw attention to one further aspect of the testing undertaken by employers. There seems to be a very general unwillingness to provide applicants with a specimen of the test paper which will be set. Indeed, we have to record with regret that a few employers have not even been willing to make available to the Committee a copy of the test which they use and have devised, despite an assurance that its confidentiality would be preserved. We find it difficult to appreciate the reason for this unwillingness to provide a specimen test paper. Candidates for public examinations of all kinds, whether they are pupils at school, students in further or higher education establishments, or applicants for the membership of professional bodies, are provided both with an examination syllabus and also access to papers set in earlier years. If, as we believe should be the case, one of the purposes of tests set by employers is to discover whether the applicant is able to carry out certain mathematical tasks satisfactorily, it must surely be of value to make clear what these tasks are. This can apply particularly to certain clerical and accounting operations which, although making use only of relatively elementary mathematics which will certainly have been studied at school, often require it to be applied in ways which may not have become familiar in the classroom. If, as a result of being provided with a specimen test paper, the candidate wishes to practise these tasks and improve his ability at them, this can only be of advantage to all concerned. Nor need the provision of a specimen test paper in any way invalidate the confidentiality of the actual test which is used or inhibit its use over a period of years if this is felt to be desirable. We believe the same to be true of the commercially produced tests used by some employers; in our view specimens of these tests should also be made available.

97 In some areas local groups of teachers and employers have discussed the recruitment tests used by the employers, often as part of the work of liaison groups to which we refer in paragraph 103. As a result of such discussions, tests have often been modified so as to enable them to fulfil their purpose better. In a few cases, a further development has been that several employers in the same locality have agreed to use the same test. This avoids the necessity for boys and girls who have applied to several firms to take a succession of tests. On the other hand, a system of this kind may act to an applicant's disadvantage unless it provides opportunity for a second attempt to redeem a poor, and perhaps uncharacteristic, performance at the first attempt.

98 We are aware of an increasing number of schemes which set out to provide evidence of achievement for pupils whose ability is below that for which the CSE examination is intended. Some of these schemes have been developed by a single school or a group of schools, and some by local education authorities. There is also the national SLAPONS test (School Leaver's Attainment Profile of Numerical Skills) which was developed by the Schools and Industry Committee of the Mathematical Association and initially administered by the Shell Centre for Mathematical Education; it is now administered by the Royal Society of Arts. Although these schemes are not, in the strict sense, employers' tests, the intention of all of them is to provide information for employers about the mathematical attainment of school leavers and so render separate recruitment tests unnecessary. We discuss schemes of this kind in Chapter 10.

Liaison

99 Many of those who have submitted evidence to us have drawn attention to the need for better liaison between schools and employers. There is very general agreement that liaison activities are crucial and that greater efforts need to be made to encourage, develop and extend them. We have received encouraging evidence of the way in which liaison of this kind is developing in many areas.

100 It must be a primary aim of liaison to enable schools and employers to obtain a better understanding of each other's needs and problems and of the way in which each operates. Such an improved understanding will both assist pupils to make a more informed choice of employment and also ease their transition to working life, whether this takes place immediately on leaving school or after a period of further or higher eduation. Heads and other teachers in schools, local careers officers, staff in colleges of further and higher education and local employers all have a contribution to make. Many organisations participate in careers activities in individual schools and in careers conventions organised by LEA Careers Services. Many companies hold careers evenings on their own premises for pupils and parents. In some cases pupils visit firms to see at first hand the kind of work which is going on and to learn about the career opportunities which are available. In other cases, some pupils may have the opportunity of gaining work experience by spending a period oftime with a firm. Joint activities of this kind can provide valuable and we believe necessary opportunities for employers to keep abreast of current educational developments and for schools to be aware of changes in patterns of employment and of opportunities which are available.

101 In the following paragraphs we consider only liaison which contains a mathematical dimension, although it is likely to be the case that such liaison will also contribute to a more general understanding of the school/employment interface for both teachers and pupils, including the development of a better understanding of the place of industry in the economy.

102 Moves to develop school/employment liaison relating to the mathematics curriculum are by no means new. The Schools and Industry Committee of the Mathematical Association has existed for some twenty years and consists of representatives of industry, commerce and mathematical education. Through its meetings and other related activities it has provided opportunity for much valuable discussion. This discussion has also revealed some of the problems which have to be overcome if liaison is to be effective. These include the difficulties of making both teachers and employers aware of the work which is being done and of persuading teachers who lack direct knowledge of industry that they should nevertheless look for opportunities of introducing examples of the industrial and commercial uses of mathematics into their work in the classroom.

103 A few local groups were in existence before the Schools and Industry Committee was set up. Since that time other groups and projects have come into being, a few of them with national coverage and, especially since the mid-1970s, many others at local level. The latter have usually consisted of teachers, local employers and the local careers officer. The LEA mathematics adviser and staff from a nearby further education college or college of education have often been involved as well. In 1978 Professor DE Bailey of the University of Bath published a list of some seventy such local groups (5) and subsequent work has identified some twenty more. In the main, each of these has been concerned with one or more of three broad areas:

• the identification of the mathematical needs of various types of employment;
• the preparation of materials for use in the classroom which provide examples of ways in which mathematics is used in industry;
• the discussion of recruitment tests used by employers.

Identification of mathematical needs

104 Many local groups and a number of research projects have set out to identify the mathematical needs of various types of employment. The scope of these investigations has varied considerably and, because most of the groups and projects have worked in isolation from each other, it has perhaps been inevitable that certain investigations have been repeated many times. In drawing attention to this fact, we in no way wish to undervalue the work which has been done, the very considerable expenditure of time and effort which it has involved, the valuable experience which has been gained by those who have carried out the work and the increased understanding between employers and schools which has resulted. However we hope that the results of the work which has been carried out at our request at Bath and Nottingham, together with the results of the Schools Council Project Mathematics and the young entrant to employment (6), will provide a fIrm base from which more detailed, and perhaps local, studies can develop without the necessity of going yet again over the preliminary ground.

105 A not infrequent outcome of the studies which have been made has been some modification of the mathematics courses in local schools. In some cases, too, CSE Mode 3 (7) examination syllabuses have been adopted which have reflected these revised courses. Another outcome has been that the performance of pupils in certain areas of mathematics has been certificated separately by the school or sometimes by the local education authority in cases where several schools have been involved. We are also aware of some local schemes in which experiments are being made with proflle reporting, which gives more detailed information about a pupil's mathematical strengths and weaknesses. We suspect, however, that there are relatively few employers for whom such detailed information is of real value.

106 Where modifications to mathematics courses have resulted from local initiatives, these seem to have been directed mainly towards pupils of average or below average attainment. However, an exception to this has been the Mathematics in Education and Industry Project (MEI) which was set up in 1963 specifically to develop new school syllabuses, initially at A Level, which were more in line with the mathematics to be found in industry and in higher education than were those which existed at the time. These syllabuses were based on the experience of teachers who had spent time on secondment to various large firms. One difficulty which arose was that, while there were plenty of suggestions for new topics which could be included in the curriculum, it was not easy to decide what should be omitted to make room for them. The MEl work has been influential, both through its own A Level (although the size of the entry has been small) and through its influence on other A Level syllabuses. However, its work has not yet had very much influence on the 11-16 curriculum. We have been told that attempts by local MEl groups to work with local employers on ideas suitable for those in the 11-16 age range have not hitherto met with very much success, although there have been recent indications of improvement.

Classroom materials

107 A number of local groups have set out to produce materials for use in the classroom based on the mathematics used in various types of employment. Such materials have most usually been produced as a result of a teacher or teachers visiting local firms to observe the different kinds of work which were going on. Following the visits, worksheets have been produced which are based on examples of the uses of mathematics which have been observed and which explain the situations in which these occur. The preparation of materials of this kind seems to have been most effective when it has been carried out in consultation with a member of staff of the firm concerned, who has sometimes also been able to visit the school to talk with pupils or demonstrate equipment. However, the evidence which is available to us suggests that, although the benefits of such materials can be considerable both to the teacher who has prepared them and to his pupils, it should not be assumed that these benefits are necessarily easily transferable to other teachers, whether in the same school or in other schools, who have not had the same direct involvement with the firm concerned. It is therefore necessary to provide appropriate in-service training for teachers in the latter category so that they may be able to use the materials effectively.

108 Preparation of materials for the classroom has also been carried out on a national scale. The School Mathematics Project (SMP), which has from the outset stressed the use of mathematics in application, supported an Industrial Fellow for two years in 1964-66. His task was to gather examples of the uses of mathematics in various kinds of employment so that these could be incorporated into the SMP texts wherever possible. The fact that this initiative achieved only limited success again draws attention to the fact that the preparation of classroom materials related to the world of work is more difficult than might be expected.

109 A more recent initiative on a national scale has been the setting up in 1975 of the Working Mathematics Group, which developed from the Continuing Mathematics Project and receives non-financial sponsorship from the Council for Educational Technology. The Group is a working liaison of industrialists and teachers which produces instructional materials in modular form designed to show mathematics in action in the world of work. The members of the Group work in pairs, one from industry and one from education, and give time voluntarily to the preparation of materials. The Group published its first ten instructional units in the autumn of 1980, so it is as yet too early to assess the contribution which these materials will be able to make in the classroom. In its written evidence to us, the Group drew attention to the long time which is required to produce classroom materials - a fact which once again underlines the problems attendant on work of this kind and the time scale which is involved.

Visits

110 Although visits by groups of pupils to companies and other employers can be of considerable value, we have to accept that the number of such visits which can be made by any one school, and by any one group of pupils within the school, will inevitably be small. When it is possible to arrange visits of this kind, it is essential that there is both adequate preparation of pupils before the visit and also follow-up when the visit has taken place, otherwise much of the value is likely to be lost. Nor should the opportunity for visits be restricted, as sometimes happens, to the academically less able; pupils of high ability can also benefit from visits of this kind and should have the opportunity to take part.

Attachments for teachers

111 A number of the submissions which we have received urge the provision of greater opportunities for mathematics teachers and mathematics staff in training institutions to undertake a period of attachment to industry. We believe that such attachments can be of considerable value to those concerned and we commend schemes such as those offered by the Understanding British Industry project despite the fact that it has been pointed out to us, somewhat ruefully, by the mathematics adviser of one LEA that an unforeseen outcome of three such attachments to industry was the loss to teaching of three good mathematics teachers. We have been told that at least one LEA has appointed full-time mathematics teachers to its supply staff in order to permit mathematics teachers in schools to be released for attachment to industry. Whatever the method adopted, suitable arrangements need to be made to make replacement staffing available so that industrial attachments can take place. However. we have to accept that the number for whom it will be possible to arrange attachment to industry will remain small in comparison with the total number of mathematics teachers. It is therefore of the utmost importance that every effort is made to use to the full the experience which teachers gain during such attachment; and that efforts are made to share with other members of staff the experience which has been gained and to prepare teaching materials based on this experience for use in the classroom, notwithstanding the difficulties to which we referred in paragraph 107.

112 It is essential that, in any scheme for liaison between schools and employers, the two-way nature of the relationship is accepted by both sides. Not only is it necessary for teachers to have a better understanding of the needs of industry, but also for those in industry to maintain an up to date knowledge of present day practice in schools. We believe that representatives of industry and commerce should visit classrooms to observe teaching in progress and be encouraged to give a talk or series of talks to pupils.

Liaison with further education

113 We end this section by drawing attention to the need for liaison between schools and local further education (FE) establishments. Liaison between further education and industry has traditionally been relatively strong. A major reason for this is that the part-time and block release courses which FE establishments provide for young employees provide opportunity to establish links between company training officers and college staff; furthermore, many lecturers in FE colleges, especially at craft level, have traditionally been recruited from industry. Training officers and managers are also often members of college governing bodies and advisory committees; and the Business Education Council (BEC) and Technician Education Council (TEC), as part of their course approval and monitoring procedures, require evidence that discussions have taken place between colleges and employers to identify the needs of the young employees concerned.

114 Direct and regular contact between schools and further education establishments seems often to be less effective. 'Link courses', in which staff of both school and college are involved, can provide valuable opportunity for cooperation and also considerable motivation for pupils. However, these courses are almost invariably more expensive than those provided solely in schools. At a time of pressure on resources and cash limits we can understand that LEAs may wish to limit the provision of courses of this kind. We believe, however, that such limitation is short-sighted and we would wish to see a development of link courses wherever practicable. In any case, whether link courses are arranged or not, general liaison between schools and colleges needs to be developed.

115 Links between the school and the further education sectors at regional and national levels are also tenuous. For example, further education is generally only weakly represented on GCE and CSE subject panels. Similarly schools are poorly represented on City and Guilds of London Institute (CGLI), BEC and TEC committees and similar bodies. We believe that much more three-way cooperation between the school, further education and industry sectors should take place in a variety of ways.

116 It is clear that over the last few years a good deal of effort has been put into liaison activities of various kinds. However, if the improvements achieved by the work of local groups and by other local initiatives are to be maintained, continuing discussion is needed; resources, too, must be maintained and where at all possible improved. It is not the case that discussion followed by the publication of a report, syllabus or test paper will set things right once and for all. All parties must continue to learn from each other. To maintain a continuing discussion is perhaps more demanding than the initial effort required to produce a document of some kind. It also makes a continuing demand on the time of the teachers, industrialists and others who are involved. Local education authorities, careers advisers, teachers and employers will all need to take responsibility and initiative for the various aspects of liaison.

The future

117 At a time of rapid technological and social change such as the present, it is particularly difficult to assess the extent to which the mathematical needs of those entering employment are likely to change. The dramatic reduction in the cost and size of computing and automation equipment made possible by the advent of the silicon chip is already beginning to have a profound effect on our working and social lives and it is evident that this effect will increase in the coming years. Automation has already been introduced into many industries engaged in mass production; leaders in this field have been the process and electronic industries. More recently there have been substantial advances in the automotive and some other industries. However the introduction of automation has not been limited to production only; computer-aided techniques are increasingly being used in design and in various aspects of management information processing. Within manufacturing industry a growing number of examples can be quoted of the integration within one system of many different but interrelated facets of the same business - for instance, design, manufacture, sales and management control. In the wider fields of commerce and of the retail and distribution industries, a similar move towards automation and integration is apparent and it seems likely that few, if any, fields of employment will be unaffected by the time that those children now entering primary school are ready to take up employment.

118 In this situation, and with the possibility of the introduction of factors at present unknown, it is impossible to anticipate further needs in detail. However, we believe that it is possible to put forward some general conclusions. The principal effect of increasing automation will be to reduce the amount of human effort involved in producing a product or providing a service. Thus, for the vast majority of those in employment, it would appear that there could be some reduction in the use of mathematics, at least at the level of arithmetical calculation. This may be considered as an extension of the effect of the introduction of the electronic calculator which, as we have already pointed out, is being used more and more in employment. However, for a limited number of people we expect that there will be an increased need to use mathematics; these will be those who are involved in the maintenance of automated equipment and in the development of computer-aided engineering systems. At still more sophisticated levels we also expect that there will be an increased use of mathematics in conjunction with computers for modelling purposes. At all levels of employment we believe that an understanding of simple mathematical concepts will enable those in employment to take an intelligent interest in their work as it changes under the impact of automation. The need for understanding will apply not only to those operations with which the employee is directly concerned in his work; it will also be important if the employee is to understand the financial performance of his company and participate intelligently in those aspects of its management which he is in a position to influence. The laying of adequate mathematical foundations at school will, moreover, remain of central importance so as to provide a basis for any further training which career development or change of employment may require.

119 Bearing in mind the pace of technological development and the unexpected directions which history shows such development may take, we see no reason for supposing that the levels of mathematical need of those who will enter employment in ten or twenty years time will be less than they are today. Although the specific needs may well be different, a secure grasp of simple mathematical ideas and the ability to apply them will remain as important as ever. For a limited number of people there will be a call for more mathematics than is needed by their counterparts currently in employment; there will also be a need for a certain number of people who are very highly qualified mathematically to be responsible for the development work which very rapidly changing technology will require. In any event it will be of the utmost importance to maintain and develop liaison between schools and industry.

The mathematical needs of some areas of employment

120 In the remainder of this chapter we discuss in very general terms the types and levels of mathematics which are likely to be encountered by certain broad categories of employees. Although these categories do not cover all types of employment available to those who leave school at 16, we believe that they include a wide cross-section. In some cases we indicate the recruitment procedures which are most commonly used and the qualifications required. We wish to emphasise that even the general descriptions which we give must not be taken to apply to all those who may seem to be in the categories which we have chosen; it is likely that only some of the mathematics which we mention will be needed by any particular person. The requirements are also likely to vary with the size of company in which an employee works. Those who work in small companies may well undertake a wider range of work, and so make use of a wider range of mathematics, than those who work in a specialised section of a large company.

Manufacturing industry

121 Within manufacturing industry, three levels are commonly identified - operatives, craftsmen and technicians - though the boundaries of each category are by no means clearly defined and the category to which an employee may be assigned varies from industry to industry, and also from firm to firm within the same industry.

Operatives

122 Among those identified as operatives there are very many whose jobs do not appear to require any formal application of mathematics. Their jobs include feeding and removing articles from machines, assembling small components, trimming off surplus material and operating pre-set production machinery. It seems to be extremely rare for selection for these jobs to be based on school qualifications; personal qualities and the way in which applicants present themselves at interview are usually considered to be of greater importance. Selection tests are rarely used, though there may in some cases be tests for specific requirements such as manual dexterity or freedom from colour blindness; training is usually undertaken on the job. However, in cases where operatives in this group obtain promotion, for instance to senior operator or chargehand, arithmetical work may be required and so they will move into a second category, also large in number, consisting of operatives who use a limited range of mathematical skills.

123 Operatives in this second category may need to count articles which are being stacked and to record the result; to recognise, copy and interpret numerals, for instance code numbers; to add, subtract, multiply and sometimes divide whole numbers, perhaps with the help of a calculator; to carry out straightforward mental calculations; to read dials and guages, though this may merely involve making sure that a needle or other indicator stays within specified limits; to weigh and measure in both metric and imperial units, including measurement involving fractions of an inch; to be familiar with the idea of gross and net weights. Recruitment procedures are usually similar to those for the previous group.

124 A third category is that of operatives who make use either of a wider range of elementary skills or of certain more advanced mathematical techniques; some of the tasks which these operatives carry out are also undertaken by those designated as craftsmen. These tasks include checking dimensions using a micrometer, vernier or other type of gauge; calculating or estimating area; understanding tolerances expressed as +/- or in other ways; reading engineering drawings, often in the form of freehand sketches with some dimensions marked but sometimes also of a more complicated kind. Dimensions may be given in millimetres or inches; the latter may involve both fractions of an inch in the sequence 1/2, 1/4, ... 1/64 and multiples of these, and also decimals which are frequently expressed in 'thou' (thousandth of an inch). Work of this kind requires the operative to possess the necessary geometrical awareness to interpret a two-dimensional drawing in three-dimensional terms. Associated with the reading of gauges and dials may be a need to understand rates of many kinds, for instance rotational speed, fluid flow or pressure. Some operators will be required to use a calculator-type keyboard, to read pen-trace graphs and to use a variety of reference tables. Others may need to mix substances in a given ratio, perhaps expressed in percentage terms. It may occasionally be necessary to substitute numbers into a simple formula, which will usually be expressed in words. However, it is important to remember that any one operative will hardly ever need to undertake more than a small number of these tasks.

Craftsmen

125 Many industries involve craft-type work and in consequence the mathematical needs vary widely. In many cases the needs will be similar to those listed in the previous paragraph, though within a given industry the craftsman will usually make use of a wider range of mathematical skills than will the operative. He may also need to understand a broader range of geometrical concepts so as to be able, for instance, to estimate or calculate areas and volumes of non-rectilinear shapes in two and three dimensions, measure angles and carry out simple geometrical constructions.

126 Engineering craft apprentices frequently need a still broader mathematical base. Most still need in their work to be fluent with fractions in the sequence 1/2, 1/4, ... 1/64 and their multiples and to be familiar with the decimal equivalents of these; to be able to select the nearest fractional equivalent in this sequence to a given decimal; to be able to make conversions between millimetres and inches, probably with the help of a reference table. They need to be able to work within given tolerances and to understand the significance of figures in digital read-out meters. The reading of meters may also require understanding of prefixes such as micro-, milli-, kilo-, mega- and their associated symbols. The need to be able to use a wide range of drawings, for example in plan view, first angle projection and cross-section, requires familiarity with the conventions of technical drawing and a good grasp of geometrical concepts; ability to make use of geometrical instruments may also be necessary. Further geometrical requirements can include the understanding of similar triangles and Pythagoras' theorem and the use of sine and tangent of an angle in right-angled triangles. Craft apprentices have little need of formal algebra but may be required to substitute numbers into a simple formula, very often given in words, to solve simple linear equations and to draw graphs of experimental results with suitable choice of scale and axes. The need to transpose algebraic equations and formulae is usually avoided.

127 Although few young employees are likely to need the breadth of mathematical knowledge implied by the total list in the preceding paragraphs, many will meet most of the topics we have listed during their first year of training. In some cases, one reason for this is that, within the engineering industry, off-the-job training in the first year is very often the same for both craft and technician apprentices (see paragraph 129). Firms adopt widely differing strategies for recruiting craftsmen. A primary requirement is that the applicant should show the motivation and other interests which indicate likely success as a craftsman. In the event, many so recruited come from the CSE grade 4 level (the grade awarded to the average candidate) in appropriate subjects. However, school leavers with qualifications at this level will rarely have confidence in the whole range of mathematical skills and concepts which we have outlined; we believe that it is with this group of young entrants to employment that most concern has been expressed by manufacturing industry.

Technicians

128 Almost all of those who are recruited as engineering technician apprentices also undertake an associated course of further education, commonly a TEC course at certificate or diploma level. These courses are designed for those with at least CSE grade 3 in appropriate subjects, including mathematics. For this reason engineering technician apprentices are normally required to have at least this level of qualification; many have O Level grade C or higher. Many will also have been required to take a company selection test in mathematics.

129 The relationship between craft and technician apprentices is by no means precise and we have already drawn attention to the fact that both may foilow the same off-the-job training in the first year. Thereafter, technician apprentices are likely to encounter a wider variety of tasks. However, some firms make little or no distinction in the first two or three years in so far as in-company work is concerned.

130 The mathematical requirements will be those which we have already listed for craft apprentices. Although by no means all this mathematics will be needed at work during the first year, all is likely to be covered in the associated TEC course. Indeed, the demands of the further education course are likely to be considerably more than the demands of the job itself, a point to which we return when we discuss these courses in the next chapter. We may however note at this point that, because the initial training of craft and technician apprentices is frequently the same, some craft apprentices in particular can be confronted by an unnecessarily taxing programme either in the training centre or at college, or both.

131 Many training centres for craft and technician engineering apprentices include a period of training in 'basic mathematical skills' in the early weeks of training. This training may take from a few hours up to two or three weeks. We have been given evidence of the improvement which is achieved during this period of intensive concentration on number skills and simple mathematical topics. However, we believe that the two main factors at work here are that revision refreshes topics which may not have been used for several months immediately before the start of the training period and that apprentices have a strong motivation to succeed, since it is at this stage that they can begin to see the uses which will be made of mathematics within their own job. There is little evidence either from industrial training centres or from the performance of technicians on TEC programmes to suggest that weakness in mathematics is a general source of problems during the first year of training.

132 The specific mathematical requirements of other groups of technician apprentices vary with the nature of the employment. In general the needs are less broad than for engineering, though a deeper understanding of certain topics may be required. For example, a science laboratory technician may have to plot and interpret graphs of various kinds arising from the results of experiments, the calibration of instruments or the output of computerised testing machines; he may need to determine points of inflexion, to recognise linear and non-linear relationships and to use graphs with logarithmic scales. Some technicians need a sound understanding of numerical processes and orders of magnitude so that they can, for example, weigh and measure very small quantities on digital read-out machines in a variety of units. They may also need to make use of both direct and inverse proportion to scale results obtained from tests which have been carried out on substances diluted for test purposes. Such work also needs a good understanding of the degrees of accuracy which are required in the appropriate practical situation. For work of this kind the use of calculators is widespread.

Clerical work

133 Clerical work covers a wide range of jobs such as accounts, sales, wages and records; the work of many typists, receptionists and secretaries also includes a clerical element. The mathematics required by this large body of employees, most of whom are female, is predominantly arithmetic, although some tasks may include substitution of numbers into a formula, usually expressed in words, and drawing graphs of sales or production. The total list of possible arithmetical skills which may be needed is large, though any one job is likely to require the use of only a limited range of them. The skills include counting, recording numbers and arranging them in order or in tabulated form; addition, subtraction, multiplication and division with whole numbers, decimals and money; a limited use of fractions, usually in situations involving division by a whole number; percentages applied in a variety of ways; ratio and proportion; rates, for example in the form of wages per hour or cost per tonne. It may be necessary to make use of reference tables and to round up or down, for instance to the nearest penny. The use of calculators for almost all of this work is now widely accepted. Appropriate checking procedures, although perhaps of a different kind from those used formerly, continue to be of crucial importance.

The retail trade

134 Young employees in the retail trade, usually employed as sales assistants or trainee managers, normally need only a limited range of arithmetical skills for their work. Many of their tasks are similar to those listed for clerical workers in the preceding paragraph. The work may entail using a till, ready-reckoner or table of goods and prices. It may be necessary to count goods in dozens and singles and to record the result on stock sheets; to prepare bills, perhaps allowing for discount or value added tax; to work with averages and percentages. Tables of numerical data may require adding both by rows and columns. Although calculators may be used for the more complicated tasks, sales assistants will often be required to carry out simple calculations mentally.

Agriculture

135 The mathematical requirements of those who work in agriculture may appear to be modest, the main skills needed being the ability to count and measure. However, the present high cost of basic materials and advancing technology mean that quantities of such things as animal feed, fertiliser and weedkiller must be calculated and measured with considerable accuracy. Precision drilling requires the farm worker to carry out accurate calibration and provide precise quantities of seed. The feeding of dairy cattle is now finely balanced in respect of such things as starch and protein requirements and these need to be calculated with reference to the characteristics of a range of feeding stuffs. Milk yields have to be recorded.

136 Farm machinery, too, is complex. It is necessary for the farm worker to be able to read a range of dials and gauges and, for instance, to be able to insert the appropriate discs to control the spray rate of a crop sprayer or make the appropriate settings for different types of seed. Although metric quantities are increasingly being used, a knowledge of imperial units is also still required, not least because much of the machinery at present in use has been built to non-metric standards.

137 In addition to being familiar with the aspects which we have already listed, the farm manager needs to be able to deal with farm accounts and to have planning skills so as to make efficient use of both manpower and machinery. An ability to estimate, especially in respect of irregularly shaped areas, is also desirable.

The construction industry

138 As with other sectors of industry, it is difficult to identify precise mathematical needs for all employees. The size of firms within the industry varies greatly. It is estimated that there are over 40,000 firms employing less than ten people; but 50 large firms account for almost 10 per cent of the construction work which is carried out each year. Within a large organisation, the work of an individual may become very stereotyped and involve much the same routines for most of the time. On the other hand, the self-employed craftsman is likely to have to deal with a wide range of different tasks. For him, as for the small firm in general, estimation of materials is crucial. An overestimate may result in a residue of materials which cannot be used up elsewhere; an underestimate can lead to delays and loss of time and profit.

139 Many of those who work in the construction industry will need to interpret plans and drawings and to be capable of accurate measurement. The bricklayer, the plasterer, the painter and decorator must become adept at estimating the amount of material they will require for a variety of tasks. Employers in the construction industry often complain that young entrants have initial difficulties in this field and lack any 'feel' for orders of magnitude, especially in respect of length, area and volume.

140 Some tradesmen require specific mathematical techniques. For example, the electrician may need to use the appropriate formula for combining resistances or calculating the current in a circuit. The plumber requires a sound appreciation of the quantities and shapes involved when working in three dimensions, for instance in order to shape the flashings around chimneys or to erect pipework. The joiner may have to carry out quite detailed calculations when constructing stairways or erecting roofing timbers. The foreman who is responsible for laying out a building site will need good planning and measurement skills.

Hotels and catering

141 The main mathematical requirement for work in the hotel and catering industry is an ability to carry out arithmetical calculations accurately. Many workers are concerned with calculations involving money, with weighing and measuring and with counting stock.

142 Those concerned with reception and accounts have to keep records and account books and to calculate service charges and value added tax. They may also be called upon to help visitors with bus or train timetables. Those who prepare food are concerned with weighing and measuring and with calculating the time required to prepare dishes. They require a knowledge of proportion in order to be able to adapt recipes for larger or smaller quantities and may also need to work out the cost per portion. Some of those employed in institutional catering must also be able to calculate specific nutritional requirements and take account of factors such as the provision of a balanced diet; accurate costing is likely to be essential.

143 Waiters and bar staff have to add up bills, handle money and memorise prices. Bar staff often have to deal with orders for a large number of items, each with its associated price, and by custom they are expected to add up the cost mentally.

Work with computers

144 Relatively few school leavers are likely to work directly with a computer. Their work will usually be at clerical or operator level dealing with the input and output of data, though some leavers with A Level qualifications obtain posts as junior programmers. The preparation of data for input to a computer entails the strict discipline of presenting data accurately in the required format; the handling of computer output often involves extracting data from tables which contain more information or more figures than are needed at that moment. These tasks demand little in the way of mathematical expertise apart from the need to feel 'at home' with the handling of numerical information. In some cases it is also necessary to be able to carry out straightforward artithmetical calculations which may involve the use of decimals and percentages.

Nursing

145 Except in a very few specialised cases, it is not possible to start training as a nurse until the age of 18 but many of those who start their training at this age will have left school at 16. The minimum entry qualifications for training as a State Registered Nurse (SRN) are stated in terms of passes at O Level or CSE grade 1, mathematics not being a compulsory subject. An alternative method of entry is by passing the educational test of the General Nursing Council; this examination contains a section on mathematics. To become a State Enrolled Nurse (SEN) the minimum requirement is a good general education. Selection is on the basis of interview and sometimes also of an educational test. However, individual hospitals are free to set qualifications for entry to training as SRN or SEN which are higher than these minima and in some cases a qualification in mathematics is specified. The fact that there can be a two year gap between leaving school and starting to train as a nurse can mean that, in some instances, computational skills have deteriorated through lack of use.

146 Almost all the mathematics which is required in nursing is concerned with measurement and recording, often in graphical form. This can include the measurement and recording of temperature, pulse rate, respiration, fluid input and output, gain or loss of weight. Measurements which have been recorded need to be interpreted. Measurement is now almost entirely in metric (SI) units and nurses need to be familiar with prefixes such as mega-, kilo-, milli- and micro- and to understand that in the SI system gradation of units is in multiples of a thousand. An understanding of ratio, proportion and percentage is essential.

147 Various formulae exist which have to be used and manipulated with ease and accuracy. These include, for example, the formula which a nurse uses to work out the amount of food to be given to a baby and the formula which is used to work out the dose to be given if a drug which has been prescribed is not available in the specified strength. Because nurses frequently work under pressure, an ability to carry out calculations of this kind with speed, accuracy and confidence is essential.

148 However, although accuracy is important, so too is the ability to estimate and approximate which helps to ensure accuracy, especially if a calculator has been used. Errors of the order of a multiple of 10 can occur easily when working constantly with metric units and it is vital to detect them before harm results. Skills of visual estimation are also required of the kind which will lead to a realisation that a drip-feed bottle is emptying too quickly or that a syringe of unusual size appears to be required.

Footnotes

(1) House of Commons, Tenth Report from the Expenditure Committee The attainments of the school leaver HMSO 1977.

(2) Copies of the Bath Study Mathematics in employment 16-18 may be purchased from the School of Mathematics, University of Bath.

A series of reports based on the Nottingham Study will be available for purchase from the Shell Centre for Mathematical Education, University of Nottingham.

(3) Modes of examining vary according to the degree of involvement on the part of the candidate's school. The principal forms are:

Mode 1 - examinations conducted by the examining board on syllabuses set and published by the board;

Mode 2 - examinations conducted by the examining board on syllabuses devised by individual schools or groups of schools and approved by the board;

Mode 3 - examinations set and marked internally by the individual school or groups of schools but moderated by the board.

(5) A survey of mathematics projects involving education and employment. University of Bath; also supplement, May 1980.

(6) D Bird and M Hiscox, Mathematics in schools and employment: a study of liaison activities Schools Council Working Paper 68, Methuen Educational 1981.

(7) See footnote (3) above.