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 2 The mathematical needs of adult life
[pages 5 - 11]

13 There are indeed many adults in Britain who have the greatest difficulty with even such apparently simple matters as adding up money, checking their change in shops or working out the cost of five gallons of petrol. Yet these adults are not just the unintelligent or the uneducated. They come from many walks of life and some are very highly educated indeed, but they are hopeless at arithmetic and they want to do something about it.

During this investigation the firm impression has built up - in the investigator's mind, at least - that functional innumeracy is far more widespread than anyone has cared to believe.

15 Since there appeared to have been very little research carried out to identify the mathematical needs of adults, we decided, as a result of the Make it count study and our discussions with ACACE, that we would ask the DES to commission a small study to be carried out on behalf both of ACACE and of our Committee. We suggested that those involved should investigate the mathematical needs of adults in daily life, and, in particular, try to identify the strategies which were used by those whose mathematical skills and understanding were limited. We felt that such an investigation would be of use to both bodies because, although the problems and methods of teaching adults are different from those of teaching children, an understanding of the goals to be achieved should be of value both to those who teach adults and to teachers in schools. The outcome of this investigation has drawn attention to a number of matters which we believe to be worthy of note, not only by teachers but by many others as well.

The research study

16 The study (2) was carried out in two stages. The first stage consisted of interviews designed to cover four areas:

• a discussion of selected situations, related to shopping and household matters generally, in which mathematics might be involved;
• brief questions on other matters such as the reading of timetables and the use of calculators;
• attitudes to mathematics;
• background information.

Both direct and indirect approaches were tried, the word 'mathematics' was replaced by 'arithmetic' or 'everyday use of numbers' but it was clear that the reason for people's refusal to be interviewed was simply that the subject was mathematics ... Several personal contacts pursued by the enquiry officer were also adamant in their refusals. Evidently there were some painful associations which they feared might be uncovered. This apparently widespread perception amongst adults of mathematics as a daunting subject pervaded a great deal of the sample selection; half of the people approached as being appropriate for inclusion in the sample refused to take part.

18 Because the sample of adults had been small, ACACE decided that it would be desirable to try to validate the findings of the study in some way. The Advisory Council therefore made arrangements for a selection of questions, of the kind which had been used in the study, to be included as part of a national enquiry undertaken by the Gallup Poll. This enquiry covered a representative sample of the population of Great Britain aged 16 or over; almost 3000 people were interviewed. The results of this enquiry, which are included in the ACACE booklet Adults' mathematical ability and performance, suggest that the findings of the original small study are in no way untypical.

Findings of the research study

19 There are, of course, many people who are able to cope confidently and competently with any situation which they may meet in the course of their everyday life which requires them to make use of mathematics. However, the results of the study suggest that there are many others of whom quite the reverse is true.

20 The extent to which the need to undertake even an apparently simple and straightforward piece of mathematics could induce feelings of anxiety, helplessness, fear and even guilt in some of those interviewed was, perhaps, the most striking feature of the study. No connection was found between the extent to which those interviewed used mathematics and the level of their educational qualifications; there were science graduates who claimed to use no arithmetic and others with no qualifications who displayed a high level of arithmetical competence. Nor did there appear to be any connection between mathematical competence and occupational group; people of widely varied mathematical competence were found in each of the five occupational groups. The estimates which those who were interviewed gave of their own mathematical competence did not relate closely to the extent to which they made use of mathematics. There were some who said that they managed very well but who appeared to avoid numbers and others who, although apparently highly competent in the conduct of their everyday affairs, were very hesitant about claiming mathematical skill. There were also some who, while apparently able to perform adequately in the situations which they normally encountered, admitted that they were working at the limit of their mathematical competence and were anxious lest anything more complicated should be required of them.

21 The feelings of guilt to which we referred earlier appeared to be especially marked among those whose academic qualifications were high and who, in consequence of this, felt that they 'ought' to have a confident understanding of mathematics, even though this was not the case. Furthermore, they were aware that others, to whom it was evident that they were well-qualified in general terms, took it for granted that they would be mathematically competent. 'People assume you're good at maths if you're good at other things.' Those who were not academically well-qualified did not appear to feel guilty in the same way. Some arts graduates who had gained O Level passes in mathematics were nevertheless so aware of a lack of confident understanding of the subject that their career choices were seriously reduced as a result of their determination to avoid mathematics.

22 There was another group consisting of those who, although able to perform the calculations which they normally required, felt a sense of inadequacy because they were aware that they did not use what they considered to be the 'proper' method; in other words, they did not make use of the standard methods for setting out written calculations which are normally taught in the classroom. In fact, the study, whilst revealing a very wide variety of approaches to the questions which were asked, also found that many individuals appeared to have only one method of tackling a given problem. If this failed, or if the calculation involved became too cumbersome, they lacked the ability and confidence to attempt a different approach. Nor, in some cases, were they even aware that there might be alternative and possibly more straightforward methods which could be used.

23 Again, just as some felt that there was always a 'proper' method, some felt that there should always be an exact answer to questions involving mathematics and so found themselves in difficulties when it became necessary to approximate or to round off a result. 'I get lost on long sums and never know what to do with the "leftovers".' 'My mind boggles at the arithmetic in estimation.'

24 Failure and consequent dislike of mathematics was often ascribed to a specific cause when young. Such causes included change of teacher or of school, absence through illness, being promoted to a higher class and becoming left behind, having an irascible or unsympathetic teacher who failed to resolve difficulties, or even over-expectation on the part of parents, usually fathers. Criticism by husbands or wives or by other members of the family, especially comment about slowness or the need to use pencil and paper instead of performing a calculation mentally, also eroded confidence and contributed to decreasing use of mathematics. 'I'm afraid I have to write it down. My brother can do it in his head.' 'My husband says I'm stupid'

25 The report also refers to 'those who dreaded what they saw as the innate characteristics of learning mathematics such as accuracy and speed, as well as the traditional requirement to show all working neatly. This recalled the long buried anxieties caused by the pupil's arriving at an answer by a mental method and being required to produce a written solution demonstrating a method which had not been used'. This perception of mathematics, and especially arithmetic, as something which is supposed to lead to exact answers by the use of proper methods seemed to be quite common despite the fact that the numbers which arise in everyday life very often need to be rounded or approximated in some way.

26 Another feature revealed by the study was a widespread inability to understand percentages. Many of those interviewed said that they did not understand them or never used them. 'I'm hopeless at percentages really.' Others who said that they were able to calculate 10 per cent and perhaps, but with greater difficulty, 15 per cent indicated that they would not be able to cope with 8 per cent or 12 per cent. Nor did they seem to have realised that the introduction of a decimalised currency had made it easier to evaluate percentages of sums of money than had previously been the case. It is clear that politicians, administrators, businessmen, journalists and advertisers all assume that the public at large will understand the many statements which are made which express comparisons in percentage terms. The study suggests that this is very far from being the case and the results of the Gallup Poll enquiry confirm this. Even though those who say that they do not understand percentages probably realise that, for example, a 12 per cent increase is greater than one of 10 per cent, it seems that they would certainly not be able to work out the actual size of the difference in relation to their own salary or wage.

27 A further question revealed that one statistic which is normally expressed in percentage terms, that of rate of inflation, is even more widely misunderstood, with many thinking that a fall in the rate of inflation ought to be associated with an overall drop in the level of prices (even though they often did not think that this was likely to happen) rather than a lessening of the rate at which prices were increasing.

28 The reading of charts and timetables was another area which presented difficulty to many of those interviewed but there was a much higher rate of success on a question which tested ability to read a map and to estimate the distance between two points on it. Understanding of the relative sizes of imperial and metric measures in common use was not widespread.

29 About 70 per cent of those interviewed in the first sample had access to a calculator if they required it but one third of them said that they never used one. Some of the latter admitted that they did not know how to use a calculator and others expressed doubt and distrust. 'I never use it because of the risk of major mistakes.' There were also those who maintained that 'brains are better' or that 'they make you lazy'. Some who had tried to use a calculator had been discouraged by the large number of figures which had appeared after the decimal point, for instance when dividing by 3, and had lacked confidence to persevere and to discover how to interpret the answers they had obtained. On the other hand there were some, whose computational skills were weak, for whom the use of a calculator made all the difference. 'I know the theory but without the calculator I couldn't do it.'

30 Many strategies were encountered for coping with the mathematical demands of everyday life. These included always buying £10 worth of petrol, always paying by cheque, always taking far more money than was likely to be needed when going shopping so as to be certain of being able to pay bills without embarrassment. There was frequent reliance on husbands, wives or children to check and pay bills, to measure or to read timetables; and also reliance on past experience. Sadly, it was also clear that lack of mathematical ability had prevented some people from applying for jobs or from following courses of training which they would otherwise have wished to undertake. In this sense, they had been unable to cope.

The mathematical needs of adult life

31 What, then, are the mathematical needs of adult life? In the first place, it is clear that there is hardly any piece of mathematics which everyone uses. For example, those who do not travel by bus or train probably have no need to consult timetables; those who do not drive a car have no need to buy petrol; those who do not have meals in hotels or restaurants have no need to be able to calculate a service charge. The study shows that some people appear to use practically no mathematics because they have organised their lives so as to avoid its use or so as to make use of the mathematical skills of others. There are, however, very few people who do not at some time need to be able to read numbers, to count, to tell the time or to undertake a minimal amount of shopping. This, perhaps, represents a minimum list but it is apparent that many of those who possess only this minimum of mathematical skill, as well as some whose attainment is a good deal greater, frequently experience feelings of stress, inadequacy or helplessness, even though they may have found methods of coping with their everyday needs.

32 Therefore, whilst realising that there are some who will not achieve all of them, we would include among the mathematical needs of adult life the ability to read numbers and to count, to tell the time, to pay for purchases and to give change, to weigh and measure, to understand straightforward timetables and simple graphs and charts, and to carry out any necessary calculations associated with these. There are many who, because of the requirements of their employment, their hobbies or their own interest in mathematics, are able to achieve a great deal more than this. Some develop very specialised skills; for example, of the kind which are frequently exhibited by those who play darts or make use of betting shops. However, we believe that those who teach mathematics in schools should do all that is possible to enable their pupils to include as part of their mathematical knowledge those abilities which we have listed.

33 We believe too that, as a necessary accompaniment to the list which we have given, it is important to have the feeling for number which permits sensible estimation and approximation - of the kind, for instance, which makes it possible to realise that the cost of 3 items at 95p each will be a little less than £3 - and which enables straightforward mental calculation to be accomplished.

34 Most important of all is the need to have sufficient confidence to make effective use of whatever mathematical skill and understanding is possessed, whether this be little or much.

Numeracy

35 The words 'numeracy' and 'numerate' occur in many of the written submissions which we have received. In the light of our discussion in the preceding paragraphs we believe that it is appropriate to ask whether or not an ability to cope confidently with the mathematical needs of adult life, as we have described them, should be thought to be sufficient to constitute 'numeracy'.

36 The concept of numeracy and the word itself were introduced in the Crowther Report (3) published in 1959. In a section devoted to the curriculum of the sixth form, 'numerate' is defined as 'a word to represent the mirror image of literacy'. Later paragraphs in the report make clear that this definition is intended to imply a quite sophisticated level of mathematical understanding. 'On the one hand ... an understanding of the scientific approach to the study of phenomena - observation, hypothesis, experiment, verification. On the other hand ... the need in the modern world to think quantitatively, to realise how far our problems are problems of degree even when they appear as problems of kind. Statistical ignorance and statistical fallacies are quite as widespread and quite as dangerous as the logical fallacies which come under the heading of illiteracy.' 'However able a boy may be ... if his numeracy has stopped short at the usual fifth form level, he is in danger of relapsing into innumeracy.'

37 In none of the submissions which we have received are the words 'numeracy' or 'numerate' used in the sense in which the Crowther Report defines them. Indeed, we are in no doubt that the words, as commonly used, have changed their meaning considerably in the last twenty years. The association with science is no longer present and the level of mathematical understanding to which the words refer is much lower. This change is reflected in the various dictionary definitions of these words. Whereas the Oxford English Dictionary defines 'numerate' to mean 'acquainted with the basic principles of mathematics and science', Collins Concise Dictionary gives 'able to perform basic arithmetic operations'.

38 The second of these definitions reflects the meaning which seems to be intended by most of those who have used the word in submissions to us. However, if we are to equate numeracy with an ability to cope confidently with the mathematical demands of adult life, this definition is too restricted because it refers only to ability to perform basic arithmetic operations and not to ability to make use of them with confidence in practical everyday situations.

39 We would wish the word 'numerate' to imply the possession of two attributes. The first of these is an 'at-homeness' with numbers and an ability to make use of mathematical skills which enables an individual to cope with the practical mathematical demands of his everyday life. The second is an ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease. Taken together, these imply that a numerate person should be expected to be able to appreciate and understand some of the ways in which mathematics can be used as a means of communication, as we have described in the previous chapter. We are, in fact, asking for more than is included in the definition in Collins but not as much as is implied by that in the Oxford dictionary - though it will, of course, be the case that anyone who fulfils the latter criteria will be numerate. Our concern is that those who set out to make their pupils 'numerate' should pay attention to the wider aspects of numeracy and not be content merely to develop the skills of computation.

Footnotes

(1) Make it count A study by David Stringer. Independent Broadcasting Authority 1979.

(2) The results of the study are reported in detail in Use of mathematics by adults in daily life: Bridgid Sewell, which may be purchased from the Advisory Council for Adult and Continuing Education. The Advisory Council has also published a summary of the report, together with a summary of the results of a Gallup Poll national survey, in Adults' mathematical ability and performance.

(3) 15 to 18 A report of the Central Advisory Council for Education (England). HMSO 1959.