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Cockcroft (1982) Notes on the text
Part 1
Part 2 Chapter 5 Mathematics in schools
Part 3 Chapter 12 Facilities for teaching mathematics
Appendices Appendix 1 Statistical information
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The Cockcroft Report (1982)
Mathematics counts Report of the Committee of Inquiry into the teaching of mathematics in schools under the chairmanship of Dr WH Cockcroft London: Her Majesty's Stationery Office 1982
ISBN 0 11 270522 7
Chapter 5 Mathematics in schools
187 In the second part of our report we discuss the teaching and learning of mathematics in schools as well as methods which are used to assess attainment. Before turning to particular aspects such as mathematics in the primary and secondary years we consider some matters which are fundamental to the teaching of mathematics to pupils of all ages, and also certain matters which arise as a consequence of the discussion in earlier chapters, of the submissions which we have received and of our own experience. In order to provide a background we start by drawing attention to the levels of attainment in mathematics which are to be expected of school leavers, so that readers may bear in mind the proportions of the school population to which the different parts of our discussion relate; we also consider the attitudes towards mathematics which pupils develop during their schooldays and the mathematical attainment of girls. 188 In this part of our report we draw on A review of research in mathematical education which summarises the results of the study carried out for our Committee under the direction of Dr A Bell of the University of Nottingham and Dr A Bishop of the University of Cambridge. For the sake of brevity, we shall henceforward refer to it as the Review of research (1).
Attainment in mathematics 189 We believe that there is widespread misunderstanding among the public at large as to the levels of attainment in mathematics which are to be expected among school leavers. At the present time about a quarter of the pupils in each year group achieve O Level grade A, B or C or CSE grade 1; about a further two fifths achieve CSE grade 2,3,4 or 5: the remainder, amounting to almost one third of the year group, leave school without any mathematical qualification in O Level or CSE. These figures are not surprising; they reflect the proportions of the school population for whom O Level and CSE examinations are intended and have been designed. At a higher level, between 5 and 6 per cent of the pupils in each year group achieve an A Level qualification in mathematics; about 1 pupil in 200 reads a degree course in mathematical studies. 190 Figure 5 illustrates in approximate diagrammatic form the 'mathematical attainment profile' of those in England and Wales who left school in 1979, and of those of school age who completed A Level courses in FE or tertiary colleges in that year. It is based on figures collected in the annual 10 per cent survey of school leavers (see Appendix 1, paragraph A3) and on an estimate of the numbers completing A Level courses in mathematics in FE and tertiary colleges, since information available about the examination performance of students in these colleges does not identify separately those who are of school age. Figure 5 'Mathematical attainment profile' for leavers in 1979 191 The number of pupils who have been studying mathematics at A Level in schools and sixth form colleges in England has increased steadily in recent years both in absolute terms, as a percentage of all pupils, and as a percentage of all pupils taking A Level courses. In the school year 1973-74, some 43 per cent of boys and 17 per cent of girls taking A Level courses in the first year of the sixth form were studying A Level mathematics. In the school year 1979-80, these figures had risen to almost 51 per cent of boys (approximately 41,000 and to 23 per cent of girls (approximately 17,000). 192 It is not possible to obtain comparable statistics for those of sixth-form age who are taking A Level courses in FE and tertiary colleges, but we have no reason to believe that the picture would be significantly different. Comparison of examination results in English and mathematics 193 In order to provide more detailed information about mathematical performance in CSE, O Level and A Level, the DES has at our request analysed in a variety of ways the information about results obtained in these examinations which was supplied in 1977, 1978 and 1979 by schools in England and Wales as part of the annual 10 per cent survey of school leavers in both maintained and independent sectors. This has provided more detailed information about examination performance in mathematics than has hitherto been available. Because the survey relates to school leavers in a given year who may be aged 16, 17, 18 or, very occasionally, 19, the information does not relate to a complete year group. Nevertheless, because the patterns for 1977, 1978 and 1979 are very similar, we believe that the picture they provide is unlikely to differ significantly from that which would emerge if it were possible to obtain information relating to a complete year group. 194 Some of those who have written to us have drawn attention to figures published each year in DES Statistics of education Vol 2 which show that the proportion of pupils who achieve O Level grade A, B or C or CSE grade 1 in English at some stage during their school career is much higher than that of those who achieve these grades in mathematics. In some submissions it has been suggested that the standard required in mathematics examinations is therefore too high. In consequence we have also obtained information relating to CSE and O Level results in English for these three years. Figure 6 Proportions of pupils awarded O Level grades A to E and CSE grades 1 to 5 in English and Mathematics/Arithmetic 195 Figure 6 illustrates approximately the O Level and CSE performances in English and mathematics or arithmetic of those leaving school in these three years. The figures on which it is based are set out in detail in Appendix 1, Tables 11 to 12; paragraphs A7 and A9 of this Appendix describe the procedure which has been used to ensure that those who have taken both O Level and CSE in the same subject or, for instance, both mathematics and arithmetic are only included once. The letters in the columns of Figure 6 refer to O Level grades A to E, the numbers to CSE grades 1 to 5. It is reasonable to assume that these figures provide a valid comparison between English and Mathematics for the school population as a whole because, as the report of the National Secondary Survey (2) shows in respect of the maintained sector, virtually all pupils study English and mathematics up to the age of 16 and enter for O Level or CSE in these subjects if they have the ability to do so. Among those who leave school at Easter in any year there will presumably be a small number who could have achieved a graded result at O Level or CSE if they had made the attempt, but we believe this number to be so small that it does not affect the overall picture significantly. 196 These figures show that from 1977 to 1979 there was a small increase in the proportion of school leavers obtaining a graded result in mathematics of some kind. CSE regulations state that in any subject 'a 16 year old pupil of average ability who has applied himself to a course of study regarded by teachers of the subject as appropriate to his age, ability and aptitude may reasonably expect to secure grade 4'. It is clear that the position of grade 4 in mathematics is as nearly in accordance with this definition as could reasonably be expected. The proportion of O Level and CSE grade 1 passes has remained constant at about a quarter, and also reflects the proportion of the school population which is expected to obtain these grades. In this sense it cannot be maintained that the standard required in mathematics examinations at this level is too high. 197 It is, however, the case that the positions of the grade boundaries in English are very different from those in mathematics and it is therefore to be expected that there will be many pupils who will achieve significantly better grades in English than they will achieve in mathematics.
Attitudes towards mathematics 198 As we stated in Chapter 2, the research study into the mathematical needs of adult life revealed the extent to which the need to make use of mathematics could induce feelings of anxiety and helplessness in some people. It also revealed that many people had far from happy recollections of their study of mathematics at school. 199 In the course of their work, those involved in the Bath and Nottingham studies (see paragraph 59) gathered reactions from a large number of young employees, and from some who were older, to the mathematics teaching which they had received at school. The views expressed are, of course, likely to have been influenced by subsequent experience in employment, particularly in respect of what was seen as the usefulness or otherwise of particular topics. Nevertheless, the remarks do record the attitudes towards mathematics of these young people and their experience at school as it has been remembered at a later date. 200 The report of the Bath study states that 'we met many young people who had liked mathematics at school. Most of these, but not all, seemed to be among the mathematically more able as measured by school results. We did come across some who got low grades, or were ungraded or not entered at CSE level, who said they enjoyed mathematics; and one or two of these who thought they were quite good at it, which we found encouraging. There were many who were non-committal. "Okay" was a frequent response. Mathematics was there, you had to do it and that was that'. However, the studies found that there were also many, mainly but not exclusively among the less able, who had disliked mathematics and who had seen no point in it. Their criticism was mainly of two kinds, one concerning the content of the mathematics course and the other the way in which mathematics had been taught. 201 Formal algebra seems to have been the topic within mathematics which attracted most comment. Those engaged in the Bath study were 'left with a very strong impression that algebra is a source of considerable confusion and negative attitudes among pupils'. In some cases this was because the work had been found difficult to understand; in other cases it was felt that exercises in algebraic manipulation and topics such as sets and matrices had had little point. It is, however, of interest to note that some of those who found algebra difficult at school were finding it easier at college; 'formulae make sense now'. Many other topics including fractions, percentages, graphs, trigonometry, Pythagoras' theorem ('the name being remembered but not much else about it') were referred to unfavourably by some, but comments of this kind seemed often to be related to the usefulness or otherwise of these topics in the job with which the employee was concerned. For example, some clerical workers who had not understood trigonometry, and did not make use of it in their work, felt in retrospect that they should pave done more about percentages. 202 Adverse comments on the teaching of mathematics often concentrated on an alleged inability on the part of some teachers to explain clearly, on a tendency to ignore some of those in the class, on an unwillingness to answer questions and on moving through the course too quickly. There was criticism also of teachers who had not required their pupils to do sufficient work and of teachers who had been unable to state the purpose of the work which was being done - 'do it to pass your exams'. Many of those interviewed said that they had been given little or no practice in mental calculation at secondary school, especially after the second year. The report of the Bath study comments that 'on a number of occasions the feeling came through to us very strongly that the sort of teacher appreciated by many pupils is one who can control the class, expects pupils to work hard, explains things briefly and clearly, gives plenty of practice and is prepared to help pupils individually'. Although this description does not, as we explain in paragraph 243 of this chapter, include all the elements which should be present in good mathematics teaching, we nevertheless believe that the qualities to which it draws attention are essential in those who teach mathematics, if pupils are to develop good attitudes. We believe that a somewhat different criterion, stated by one young employee - 'any teacher who can make maths interesting must be good' - also provides food for thought. 203 The comments of young employees showed that very often, but not always, inability to do mathematics and not liking it seemed to go together. Equally, success more often than not seemed to lead to a favourable attitude; 'you enjoy it when you can do it'. Nevertheless, the belief that mathematics was useful was shared by many of those who disliked mathematics as well as by those who liked it. Most of those interviewed, whether young or old, saw mathematics in vocational utilitarian terms. Very few saw it as serving any wider purpose, especially of what might be termed a cultural or general education kind. 204 Very few who had taken mathematics to A Level or beyond were included among those who were interviewed, nor were there very many who had taken any subjects at A Level. The attitudes of such people might well be very different, but those who take A Level form only a small minority of the school population. 205 It is to be expected that most teachers will attach considerable importance to the development of good attitudes among the pupils whom they teach and so it is interesting to compare the comments made by young employees with the findings of research studies. The Review of research discusses the considerable amount of research on pupils' attitudes which has been carried out in recent years, including that which is being undertaken in England and Wales as part of the work of the Assessment of Performance Unit. It states that 'it is not easy to pick out points which summarise all the research on attitudes to mathematics. Strongly polarised attitudes can be established, even amongst primary school children, and about 11 seems to be a critical age for this establishment. Attitudes are derived from teachers' attitudes (though this affects more intelligent pupils rather than the less able) and to an extent from parents' attitudes (though the correlation is fairly low). Attitude to mathematics is correlated with attitude to school as a whole (which is fairly consistent across subjects) and with the peer group's attitude (a group attitude tends to become established). These things do not seem to be related to type or size of school or to subject content. Throughout school, a decline in attitudes to mathematics appears to go on, but this is part of a decline in attitudes to all school subjects and may be merely part of an increasingly critical approach to many aspects of life'. 206 The Review of research points out that research studies also show that there is a strong tendency among pupils of all ages to believe mathematics to be useful but not necessarily interesting or enjoyable. 'There appears to be an identifiable (although small) correlation between attitude and achievement: it is not clear, however, in what way attitude and achievement affect one another. This does not necessarily contradict the teacher's perception that more interesting and enjoyable work will lead to greater attainment. For one thing, the research does not deal with changes in achievement which might result within particular individuals or classes from improvements directed towards attitude; it rather shows that, broadly speaking, the set of people who like mathematics has only a relatively small overlap with the set of those who are good at it. However, research certainly suggests caution against over-optimism in assuming a very direct relation between attitude and achievement.' We have already noted the interest displayed by the Assessment of Performance Unit in respect of attitudes toward mathematics. It is clear that there is need for continuing investigation in this field. Parents and schools 207 Parents can exercise, even if unknowingly, a considerable influence on their children's attitudes towards mathematics. Encouragement to make use of mathematics during normal family activities, for example to weigh and measure, to use money for shopping and to play games which involve the use of dice or the keeping of scores, can assist children to develop familiarity with numbers and confidence in making use of them. However, in some cases parents can expect too little; 'don't worry, dear, I could never understand mathematics at school either'. In other cases parents can expect too much of their children and so, as we have noted in paragraph 24, exert pressure which can lead to failure and consequent dislike of mathematics. It can happen, too, that parents fail to understand the purpose of the mathematics which their children are doing and so make critical remarks which can also encourage the development of poor attitudes towards mathematics in their children. We believe that it is therefore important that schools should make active efforts to enlist the help of parents by explaining the approaches to mathematics which they are using and the purposes of mathematical activities which parents themselves may not have undertaken while at school. Schools should also encourage parents to discuss with teachers their children's progress in mathematics, so that both parents and schools may work together to assist the mathematical development of pupils.
The mathematical performance of girls 208 In recent years increasing attention has been paid to the fact that, as measured by the results of public examinations, the overall level of performance of girls in mathematics is significantly lower than that of boys. In this section we discuss this matter in general terms; a more detailed discussion is contained in Appendix 2, which includes relevant statistics and also reference to some of the research studies into the mathematical performance of girls which have been carried out in recent years both in this country and in other parts of the world. 209 Concern about differences in the performance of boys and girls is not confined to the United Kingdom. This was demonstrated clearly at the Fourth International Congress on Mathematical Education, held at the University of California at Berkeley in August 1980, at which papers about the mathematical performance of girls were presented by participants from many countries. We include in this section some of the ideas which were put forward in these papers and in the discussions which followed them. 210. It is not easy to establish why girls should perform less well than boys at mathematics and many possible reasons have been suggested. Some of these relate to biological factors, some to child-rearing and social factors, some to factors within schools and some to career expectation. It has been suggested that among the child-rearing practices which may influence mathematical attainment is the fact that, whereas girls are usually given dolls and 'domestic' toys, boys are given significantly more scientific and constructional toys, which encourage the development of spatial concepts and problem solving activities. It has also been suggested that boys are encouraged to be more independent than girls, which may again encourage experiment and problem solving; and that adults respond to boys as if they find them more interesting and more attention-provoking than girls. Among the social factors are the perception of mathematics as a 'male' activity (though more recent studies suggest that the sex-typing of mathematics is decreasing), and peer group pressures which cause girls to fear that high attainment in mathematics will inhibit the development of their relationships with boys. The Committee for Girls and Mathematics has told us that 'schools, the careers service and industry appear to have shown little initiative in encouraging or attracting girls with ability in mathematics into some of the fields where there are shortfalls of good applicants; ... whereas a boy is usually counselled to work towards a qualification in mathematics as a career essential, this is not always the case with girls'. 211 Factors related to school may also help to reinforce the impression that mathematics is a male domain. Men teachers in primary schools often take the older classes; at secondary level more men than women teach mathematics. The applications of mathematics which are found in many textbooks and examination questions reflect activities associated with men more often than they reflect activities associated with women. In primary schools it can be the case that boys engage in craft activities of a three-dimensional kind while girls do needlework, though we believe that differences of this kind exist in fewer schools now than was the case a few years ago. It also seems likely that there is, even if unconsciously, an expectation among teachers that girls will perform less well at mathematics than boys. Research studies, to which we refer in Appendix 2, suggest that during mathematics lessons teachers in secondary schools interact more with boys than they do with girls, give more serious consideration to boys' ideas than to those of girls and give boys more opportunity than girls to respond to higher cognitive level questions. It has also been found that high achieving girls may receive considerably less attention in mathematics lessons than do high achieving boys. In such circumstances, girls are likely to receive the message that they are not expected to perform as well as boys and to react accordingly. 212 We believe that the question of classroom interaction and expectation is of considerable importance. It was suggested at the Fourth International Congress that girls may have a greater need than boys to develop understanding through discussion and that many girls consider that mathematics teachers do not listen sufficiently to what girls say in response to questions. We shall argue later in this chapter that discussion should be an essential part of all mathematics teaching; it seems likely that in classrooms where it is not practised sufficiently girls are liable to be especially disadvantaged. Even in classrooms in which there is plenty of discussion teachers may need to take steps to ensure that boys do not dominate the discussion and that girls are given opportunity to play their full part. We may note at this point that, whereas girls perform less well than boys at mathematics, they perform better overall than boys at English. We believe that this gives further support to the suggestion that girls need to undertake extended discussion in order to clarify their ideas and understanding and that plenty of verbal interaction is essential if they are to learn mathematics satisfactorily. 213 Research studies suggest that, whereas boys more often attribute their successes in mathematics to ability and their failures to lack of effort or bad luck, girls more often attribute their success to hard work or good luck and their failures to lack of ability. If this is the case, it underlines the need to do all that is possible to encourage girls to develop confidence in their mathematical powers. It is sometimes suggested that girls succeed better at mathematics when they are taught in single-sex groups but we are not aware of any studies which establish this fact. Although it is possible to identify some girls' schools in which levels of mathematical attainment are high, it is often the case that there are other factors, such as the fact that the school is selective, which may provide the explanation. We are aware of a small number of instances in which a secondary school has arranged to teach mathematics to girls and boys in separate classes but the scale on which grouping of this kind has been carried out is as yet too small for it to be possible to draw conclusions. 214 The HMI booklet Girls and science (3) discusses the attitudes of girls to science and various matters relating to the approaches to teaching which are helpful in encouraging more girls to take an interest in science. We believe that much of this discussion is also relevant to the teaching of mathematics to girls and we commend the booklet, which also contains an appendix on 'girls and engineering', for study by those who teach mathematics. 215 In Appendix 2 we suggest a number of strategies which we believe may contribute to improvement in the mathematical performance of girls. We wish here to draw attention to two of these. The first is the need for appropriate careers guidance to be available to girls at an early stage so that their attention can be drawn to the fact that lack of an appropriate mathematical qualification can exclude entry to many fields of employment. The second is the need for teachers to be aware of the differences in mathematical performance which at present exist between boys and girls. If teachers hold this fact in mind and make a conscious effort to ensure that they do not make use of teaching methods which are liable to put girls at a disadvantage, we believe that an improvement in the overall performance of girls could be achieved. 216 Some programmes have recently been set up in the United States of America which are designed to improve the motivation and attitudes of girls towards mathematics. One of these, the Math/Science Network, operates mainly in California and works to encourage greater participation by girls in 'math-based fields of work and study' by holding conferences for teachers, providing careers information and developing appropriate activities in schools. A second, Women and Mathematics, is sponsored by the Mathematical Association of America and operates more widely. It attempts to change attitudes by arranging for women who are using mathematics in interesting careers to speak at high school conferences. The group also organises conferences for careers advisers. We have been told that these initiatives are meeting with some success. It may be that a similar initiative would be helpful in the United Kingdom.
The teaching of mathematics through the medium of Welsh 217 The number of children who are taught mathematics through the medium of Welsh has increased considerably in recent years. Some of these children come from homes in which Welsh is the family language but increasing numbers are entering Welsh-medium primary schools from homes in which the family language is English. In our meetings with teachers in Welsh-medium schools it has been made clear to us that the problems of teaching mathematics through the medium of Welsh to children who are Welsh speaking are fundamentally no different from the problems of teaching mathematics in English to children who are English speaking. 218 There is, however, one major difficulty which exists for those who teach mathematics through the medium of Welsh. This is the great shortage of mathematics textbooks which are written in Welsh. There is at present no complete primary mathematics scheme in Welsh. Such materials as are available are confined mainly to infant level; other materials are fragmentary and deal only with parts of the mathematics curriculum. This means that teachers have to prepare most of their own classroom materials. This not only takes a great deal of time but also results in a standard of presentation which is less good than that of a published text. For this reason there are some classrooms in which although the teaching and discussion are in Welsh, children work from books which are written in English. 219 Work is in progress on the production of a secondary mathematics course written in Welsh. Books covering the first three years of the course have now been published and material for the fourth year is to appear shortly. However, even when the whole course has been published, it will be the only one which is available and will have to serve the needs of all pupils, whatever their level of attainment. This cannot be considered to be a satisfactory situation. 220 If it is government policy to support the provision of Welsh-medium schools, it is essential that a suitable supply of teaching materials is made available. It has been suggested to us that, because these materials are needed urgently, a first step should be to commission the translation into Welsh of at least two mathematics courses which have already been published in English. This is a proposal which we support. We believe that it will be necessary for special funding of some kind to be made available for this purpose because of the inevitably small market for teaching materials written in Welsh and the need for some kind of subsidy if they are to be produced for sale at a price which is not unreasonably high.
The teaching of mathematics to those for whom English is not the first language 221 In recent years there has been a considerable increase in the number of children in schools for whom English is not the first language. Even when, as is increasingly the case, these children, who are mainly from European, African and Asian family backgrounds, have been born in Britain, they very often start their schooling as only partial English speakers; sometimes they are not able to speak English at all. In the early stages, therefore, it is necessary to ensure that the language demands of discussion and of written materials such as work cards take account of the vocabulary and language structures available to these children. It is, however, important that they should take part in oral work both to assist the development of their general language skills and also to enable them to become familiar with the language which is used in mathematics. 222 Especial care is likely to be needed with young children in the early stages of naming numbers and counting. Almost all European languages show irregularities in the naming of some or all of the numbers between 10 and 20. In English, for example, numbers from 13 to 19 are spoken 'back to front' compared with numbers from 20 onwards, so that we say 'twenty-four' but 'fourteen'; the words 'eleven' and 'twelve' are even more irregular. In the major Asian languages, each number up to 40 has its own name. Those who teach children whose first language is not English therefore need to take steps to find out about the number system which is used in the countries from which the children's families originate so they can be aware of the kind of difficulties which may arise. 223 With older pupils it is necessary to ensure that difficulty in speaking and understanding English does not lead to placement in a mathematics teaching group whose level is too low. The notation which is used in mathematics, and also the way in which numerals and other mathematical symbols are printed, is the same in very many countries, including some in which not only the language but also the written script is different. For this reason these pupils may well be able to work at a higher level in mathematics than is, for the time being, possible for them in some other subjects in which lack of fluency in English leads to greater difficulties. 224 It is possible to make positive use of mathematical ideas drawn from other cultures, especially when discussing shape and space. For example, many of the Rangoli patterns which are used by Hindu and Sikh families to decorate their homes on important occasions have a geometrical basis in which symmetry plays a major part. Practice in drawing patterns of this kind can help to develop geometrical concepts. Again, the intricate patterns which decorate many Islamic buildings are formed by fitting together various geometrical shapes. Patterns of this kind can be examined and discussed and children can then create patterns of their own. As children grow older, it is possible to discuss the ways in which the numerals which we now use have developed from those which were originally used in eastern countries, and the contributions to the development of mathematics which have come from different countries and different cultures.
The teaching and learning of mathematics 225 In the preceding chapters we have shown that, in broad terms, it is possible to sum up much of the mathematical requirement for adult life as 'a feeling for number' and much of the mathematical need for employment as 'a feeling for measurement'. Underlying both of these, and essential to their development, is the need to establish confidence in the use of mathematics while at school. 226 However, we do not believe that mathematical activity in schools is to be judged worthwhile only in so far as it has clear practical usefulness. The widespread appeal of mathematical puzzles and problems to which we have already referred shows that the capacity for appreciating mathematics for its own sake is present in many people. It follows that mathematics should be presented as a subject both to use and to enjoy. 227 The study of shape and space, of graphical methods of presenting information and of the properties of number such as those of even, odd, prime and square numbers, provides opportunity to develop the powers of 'abstraction' and 'generalisation' and their expression in algebraic form on which higher level mathematics depends. Although many pupils will not attain to work at this level, nevertheless we believe that all should have opportunity to gain some insight, however slight, into the generalised nature of mathematics and the logical processes on which it depends. At all ages, pupils should be encouraged to look for 'pattern' in the results they obtain and to explain this in words even though they may not be able to express in algebraic terms what they have observed. Whatever the level of attainment of pupils, carefully planned use of mathematical puzzles and 'games' can clarify the ideas in a syllabus and assist the development of logical thinking; 'if I move this piece to that position, then I shall be able ...', 'because those two numbers are even, this one cannot be odd'. The determination to command a computer to do one's bidding can be a further potent force in encouraging pupils to think logically and mathematically. Activities of all these kinds require pupils to think about numbers and the processes of mathematics in ways which are different from those encountered within the more usual applications of mathematics and so enable confidence and understanding to increase. 228 Mathematics is a difficult subject both to teach and to learn. One of the reasons why this is so is that mathematics is a hierarchical subject. This does not mean that there is an absolute order in which it is necessary to study the subject but that ability to proceed to new work is very often dependent on a sufficient understanding of one or more pieces of work which have gone before. Whether or not it is true, as is sometimes suggested, that each person has a 'mathematical ceiling' (and so far as we are aware no research has been undertaken to establish whether or not this is the case), it is certainly true that children, and adults, learn mathematics at greatly differing speeds. A concept which some may comprehend in a single lesson may require days or even weeks of work by others, and be inaccessible, at least for the time being, to those who lack understanding of the concepts on which it depends. This means that there are very great differences in attainment between children of the same age. A small number reach a standard which enables them to study mathematics at degree level but many others have time to advance only a very short distance along the mathematical road during their years at school. Because of the hierarchical nature of mathematics these pupils do not reach a position from which they are able to tackle the more abstract branches of the subject with understanding or hope of success, though some can and do continue their advance after they have left school. 229 Mathematics is also a subject which requires hard work and much practice, whatever one's level of attainment may be. It can be straightforward to understand the solution of a problem which someone else has worked out; it is usually very much more difficult to discover a solution by oneself. Indeed, it is 'getting started' which is often the most difficult part of solving a mathematical problem and it is easy to underestimate the qualities both of determination and of imagination which can be required. 230 One of the reasons why it is difficult to teach mathematics is the fact that attainment and rate of learning vary so greatly from pupil to pupil. If the pace of the teaching is too fast, understanding is not able to develop; on the other hand, if the pace is too slow pupils can become bored and disenchanted. The amount of ground which it is appropriate to cover in anyone period of work on the same topic also varies with the attainment of the pupils. Those whose attainment is high are often able to advance a considerable distance at one time but those whose attainment is low need to advance by smaller stages and to return to the topic more frequently. The achievement of a correct balance in these matters requires skilled professional judgement and presents problems to the teacher which should not be underestimated. Whatever their level of attainment, pupils should not be allowed to experience repeated failure. If this shows signs of occurring, it is an indication that the advance has continued too far and that a change of topic is needed. Understanding 231 In recent years there has been considerable discussion of the nature of mathematical understanding. There is general agreement that understanding in mathematics implies an ability to recognise and to make use of a mathematical concept in a variety of settings, including some which are not immediately familiar. A distinction (4) is sometimes made between 'relational understanding' (in brief, both knowing what to do in particular cases and the relating of these procedures to more general mathematical knowledge) and 'instrumental understanding' (the rate memorising of rules for particular classes of examples without knowing why they work). However, mathematical understanding is not 'all or nothing'. It develops as knowledge of mathematics develops and needs to exist at a level which is sufficient for the work which is being done at the time. Thus, the level of understanding which is required for the study of mathematics in higher education is very different from that required by pupils at school. Indeed it is a common, and sometimes somewhat disconcerting, experience to those embarking on degree courses in mathematics to find that their understanding of topics which they have tackled with apparent success at school is questioned and shown to be insufficient. For this reason the distinction between relational and instrumental understanding can never be clear cut, and in this sense it over-simplifies a complex situation, but the distinction can be a helpful starting point for discussion of the nature of understanding. 232 Because understanding is an internal state of mind which has to be achieved individually by each pupil, it cannot be observed directly by the teacher. The fact that a pupil is able to solve a particular problem correctly does not necessarily indicate that understanding of the relevant concepts is present. A much better indication of the depth of understanding which exists can be obtained in the course of discussion, by means of appropriate practical work or through more general problem-solving activities. As understanding develops the teacher may need from time to time to challenge more deeply the understanding which already exists so as to make a pupil aware of a need to think more deeply and more critically. 233 Because, too, understanding develops gradually over a period of time as a result of successful experience, especially of a problem-solving kind, it is important that teachers should be aware of the fact that it can be counter-productive to continue work on the same mathematical topic for too long a period. Very often work on some other topic can provide insight from a different direction which will assist in the process of consolidating understanding and enabling ideas to mature. Memory 234 The Review of Research discusses at length the question of memory and distinguishes between 'short term' or 'working' memory and 'long term' memory. Material held in short term memory fades in a matter of seconds unless it is deliberately held in consciousness; the amount of material which can be retained in this way is also limited. Once an item has been stored in long term memory it tends to be forgotten very slowly or not at all, though it may not necessarily be easy to retrieve. 235 Short term memory plays an important role in all tasks in which several attributes or items of information have to be considered simultaneously, for example in mental calculations, problem solving, the understanding of complex concepts and the construction or following of an explanation or argument; in other words, in most learning tasks. In order to carry out these tasks it is necessary to draw on information stored in long term memory. Research evidence makes it clear that information is stored better in long term memory if it is assimilated in such a way that it becomes part of a network of associated and related items which support one another. An everyday example of this is provided by the fact that some children whose ability to remember number facts appears to be weak are often able, because of their interest in and knowledge of sport, to remember without difficulty the scores in football or cricket matches which have been played weeks or even months earlier. 236 It follows that a child's long term memory tends to improve as he grows older and develops a network of items which contains more interconnections into which new information can be fitted. Information which has been remembered in this way is also easier to retrieve because association between the purpose for which it is required and the information which is stored in the memory provides a cue which triggers recall. Although it is possible to store in long term memory material which has not been integrated in this way, it is likely to be less well retained and, perhaps more importantly, to be more difficult to retrieve because fewer associations have been formed to act as retrieval cues. Furthermore, if relationship with other material has not been established, it is more difficult to use powers of reasoning to make complete material which has been remembered only in part. 237 For long term memory to be effective, 'rehearsal' is necessary; that is, recall of the material and the strengthening of its relationship with other facts which are already known. The more that such rehearsal can increase links with the existing network, the more effective it is likely to be. Thus retention of number facts is likely to be improved if practice in recall is associated with some kind of explanation or checking procedure; for example, that the sum of two odd numbers must be even or that any number in the '5 times table' ends in either 0 or 5. Rote learning 238 We have received several submissions which have urged that more emphasis should be placed on 'rote learning'. The Oxford English Dictionary defines 'by rote' as 'in a mechanical manner, by routine; especially by the mere exercise of memory without proper understanding of, or reflection upon, the matter in question; also, with precision, or by heart'. There are certainly some things in mathematics which need to be learned by heart but we do not believe that it should ever be necessary in the teaching of mathematics to commit things to memory without at the same time seeking to develop a proper understanding of the mathematics to which they relate. As our discussion of memory shows, such an approach is unlikely to meet with long term success. 239 However, the need to teach in a way which will help to develop long term memory and understanding need in no way be in opposition to, or at the expense of, the development of skills in computation and algebraic manipulation. It is important that children should practise routine manipulations until they can be done with an appropriate degree of fluency; this applies all the way from routines such as addition and subtraction to those required for A Level mathematics and beyond. Well-mastered routines are necessary in order to free conscious attention as much as possible so that it can focus on aspects of a task which are novel or problematic. Here again, we need to distinguish between 'fluent' performance and 'mechanical' performance. Fluent performance is based on understanding of the routine which is being carried out; mechanical performance is performance by rote in which the necessary understanding is not present. Although mechanical performance may be successful in the short term, any routine which is carried out in this way is much less likely either to be capable of use in other situations or to be retained in long term memory. Teaching methods 240 The Review of research points out that in the teaching of mathematics it is possible to distinguish between three elements - facts and skills, conceptual structures, and general strategies and appreciation. Facts are items of information which are essentially unconnected or arbitrary. They include notational conventions - for example that 34 means three tens plus four and not four tens plus three - conversion factors such as that '2.54 centimetres equals 1 inch' and the names allotted to particular concepts, for example trigonometrical ratios. The so-called 'number facts', for example 4 + 6 = 10, do not fit into this category since they are not unconnected or arbitrary but follow logically from an understanding of the number system. Skills include not only the use of the number facts and the standard computational procedures of arithmetic and algebra, but also of any well established procedures which it is possible to carry out by the use of a routine. They need not only to be understood and embedded in the conceptual structure but also to be brought up to the level of immediate recall or fluency of performance by regular practice. Conceptual structures are richly interconnected bodies of knowledge, including the routines required for the exercise of skills. It is these which make up the substance of mathematical knowledge stored in the long term memory. They underpin the performance of skills and their presence is shown by the ability to remedy a memory failure or to adapt a procedure to a new situation. General strategies are procedures which guide the choice of which skills to use or what knowledge to draw upon at each stage in the course of solving a problem or carrying out an investigation. They enable a problem to be approached with confidence and with the expectation that a solution will be possible. With them is associated appreciation which involves awareness of the nature of mathematics and attitudes towards it. 241 Research shows that these three elements - facts and skills, conceptual structures, general strategies and appreciation - involve distinct aspects of teaching and require separate attention. It follows that effective mathematics teaching must pay attention to all three. Classroom practice 242 We wish now to discuss the implications of the previous sections for work in the classroom. We are aware that there are some teachers who would wish us to indicate a definitive style for the teaching of mathematics, but we do not believe that this is either desirable or possible. Approaches to the teaching of a particular piece of mathematics need to be related to the topic itself and to the abilities and experience of both teachers and pupils. Because of differences of personality and circumstance, methods which may be extremely successful with one teacher and one group of pupils will not necessarily be suitable for use by another teacher or with a different group of pupils. Nevertheless, we believe that there are certain elements which need to be present in successful mathematics teaching to pupils of all ages. 243 Mathematics teaching at all levels should include opportunities for
244 We believe that one of the reasons for this may be that a brief statement such as 'mathematics teaching should include opportunities for investigational work' does not explain sufficiently what is intended. We wish, therefore, to consider more fully each of the elements which we have listed. Exposition 245 Exposition by the teacher has always been a fundamental ingredient of work in the classroom and we believe that this continues to be the case. We wish, though, to stress one aspect of it which seems often to be insufficiently appreciated. Questions and answers should constitute a dialogue. There is a need to take account of, and to respond to, the answers which pupils give to questions asked by the teacher as the exposition develops. Even if an answer is incorrect, or is not the one which the teacher was expecting or hoping to receive, it should not be ignored; exploration of a pupil's incorrect or unexpected response can lead to worthwhile discussion and increased awareness for both teacher and pupil of specific misunderstandings or misinterpretations. Discussion 246 By the term 'discussion' we mean more than the short questions and answers which arise during exposition by the teacher. In the National Primary Survey report (5) we read 'In some cases, particularly in the older classes, more attention could usefully have been given to more precise and unambiguous use of ordinary language to describe the properties of number, size, shape or position'. The National Secondary Survey report (6) noted that 'the potential of mathematics for developing precision and sensitivity in the use of language was underused'. The ability to 'say what you mean and mean what you say' should be one of the outcomes of good mathematics teaching. This ability develops as a result of opportunities to talk about mathematics, to explain and discuss results which have been obtained, and to test hypotheses. Moreover, the many different topics which exist within mathematics at both primary and secondary level should be presented and developed in such a way that they are seen to be interrelated. Pupils need the explicit help, which can only be given by extended discussion, to establish these relationships; even pupils whose mathematical attainment is high do not easily do this for themselves. Practical work 247 Practical work is fundamental to the development of mathematics at the primary stage; we discuss this in detail in the following chapter. It is too often assumed that the need for practical activity ceases at the secondary stage but this is not the case. Nor is it the case that practical activity is needed only by pupils whose attainment is low; pupils of all levels of attainment can benefit from the opportunity for appropriate practical experience. The type of activity, the amount of time which is spent on it and the amount of repetition which is required will, of course, vary according to the needs and attainment of pupils. The results of the practical testing carried out by the Assessment of Performance Unit and described in the reports of both primary and secondary tests (7) illustrate clearly the need to provide opportunities for practical experience and experiment for pupils of all ages. Practice 248 All pupils need opportunities to practise skills and routines which have been acquired recently, and to consolidate those which they already possess, so that these may be available for use in problem solving and investigational work. The amount of practice which is required varies from pupil to pupil, as does the level of fluency which is appropriate at any given stage. However, as we have pointed out already, practice of fundamental skills is not by itself sufficient to develop the ability to solve problems or to investigate - these are matters which need separate attention. Problem solving 249 The ability to solve problems is at the heart of mathematics. Mathematics is only 'useful' to the extent to which it can be applied to a particular situation and it is the ability to apply mathematics to a variety of situations to which we give the name 'problem solving'. However, the solution of a mathematical problem cannot begin until the problem has been translated into the appropriate mathematical terms. This first and essential step presents very great difficulties to many pupils - a fact which is often too little appreciated. At each stage of the mathematics course the teacher needs to help pupils to understand how to apply the concepts and skills which are being learned and how to make use of them to solve problems. These problems should relate both to the application of mathematics to everyday situations within the pupils' experience, and also to situations which are unfamiliar. For many pupils this will require a great deal of discussion and oral work before even very simple problems can be tackled in written form. Investigational work 250 The idea of investigation is fundamental both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to extend knowledge and to solve problems in very many fields. We suspect that there are many teachers who think of 'mathematical investigations' as being in some way similar to the 'projects' which in recent years have become common as a way of working in many areas of the curriculum; in other words, that a mathematical investigation is an extensive piece of work which will take quite a long time to complete and will probably be undertaken individually or as a member of a small group. But although this is one of the forms which mathematical investigation can take, it is by no means the only form nor need it be the most common. Investigations need be neither lengthy nor difficult. At the most fundamental level, and perhaps most frequently, they should start in response to pupils' questions, perhaps during exposition by the teacher or as a result of a piece of work which is in progress or has just been completed. The essential condition for work of this kind is that the teacher must be willing to pursue the matter when a pupil asks 'could we have done the same thing with three other numbers?' or 'what would happen if ...?' Very often the question can be resolved by a few minutes of discussion either with the pupil or with a group of pupils; sometimes it may be appropriate to suggest that the pupil or a group of pupils, or even the whole class, should try to find the answer for themselves; sometimes it will be necessary to find time on another occasion to discuss the matter. The essential requirement is that pupils should be encouraged to think in this way and that the teacher takes the opportunities which are presented by the members of the class. There should be willingness on the part of the teacher to follow some false trails and not to say at the outset that the trail leads nowhere. Nor should an interesting line of thought be curtailed because 'there is no time' or because 'it is not in the syllabus'. 251 Many investigations lead to a result which will be the same for all pupils. On the other hand, there are many investigations which will produce a variety of results and pupils need to appreciate this. For example, the answer to the question 'In how many different ways can you carry out this calculation on your calculator; which way requires the least number of steps?' depends on the particular model of calculator which is used, and pupils who undertake an investigation of this kind will produce a variety of answers, all of which may be equally valid. Mathematical puzzles of various kinds also offer valuable opportunities for investigational work. Even practical routine skills can sometimes, with benefit, be carried out in investigational form; for example, 'make up three subtraction sums which have 473 as their answer'. The successful completion of a task of this kind may well assist understanding of the fact that subtraction can be checked by means of addition. 252 It is necessary to realise that much of the value of an investigation can be lost unless the outcome of the investigation is discussed. Such discussion should include consideration not only of the method which has been used and the results which have been obtained but also of false trails which have been followed and mistakes which may have been made in the course of the investigation. Some specific aspects 25 3 We turn now to certain more detailed aspects of the teaching and learning of mathematics which are relevant to work in both primary and secondary schools and which we consider to be of sufficient importance to consider further at this stage. Mental calculation 254 We have already referred several times to the need to be able to carry out straightforward calculations mentally. 'Mental arithmetic' was once a regular part of the mathematics taught in both primary and secondary schools; very often it occurred as a separate heading in school reports. It is clear that it now occupies a far less prominent position within most mathematics teaching; reports which have come to us confirm the comments of young employees, to which we referred in paragraph 202, that in some classrooms it is no longer practised at all. We believe that one reason for this change is the increasing use of individual learning programmes in which a pupil works for much of the time on his own using prepared materials, very often in the form of work cards or work sheets. This method of working reduces opportunity for discussion and oral work generally. Again, most primary classes and many secondary classes, especially in the earlier years, contain pupils of a very wide range of ability. It is difficult to find mental questions which are suitable for all the pupils in such classes and so, because of an increasing awareness of the effects which a sense of failure can produce, teachers have tended to avoid methods of working which are liable to draw attention to a child's lack of success. Many of our readers will no doubt recall the 'ten quick questions' at the beginning of a mathematics lesson. For the mathematically able they could be a source of enjoyment and challenge but for those whose mathematical ability was limited they were much more likely to lead to increasing loss of confidence, increasing antipathy to mathematics and sometimes even to feelings of humiliation which would long be remembered. 255 We believe that the decline of mental and oral work within mathematics classrooms represents a failure to recognise the central place which working 'done in the head' occupies throughout mathematics. Even when using traditional methods of recording calculations on paper, the written record is usually based on steps which are done mentally. For instance, the written calculation 27 + 65, carried out by the method which is traditionally taught, requires the mental calculations 7 + 5 = 12 and 1 + 2 + 6 = 9; a relatively simple division sum can involve mental experiment with various multiples of the divisor before the correct one is chosen. 256 However, a more important reason for including the practice of mental calculation is the now well established fact that those who are mathematically effective in daily life seldom make use 'in their heads' of the standard written methods which are taught in the classroom, but either adapt them in a personal way or make use of methods which are highly idiosyncratic. It is, for example, common when carrying out calculations mentally to deal with the hundreds or the tens first and then the units afterwards; for instance, in the example we quoted in the previous paragraph, to use the sequence '20 + 60 = 80, 5 + 7 = 12, 80 + 12 = 92'; we would stress however that this is only one of several possible methods which can be used to carry out this calculation mentally. Again, when adding sums of money mentally, it is common to add the pounds first and then the pence, rather than deal with the pence first and then the pounds as is usually done when working with pencil and paper. Although many pupils come to realise by themselves that methods which may be convenient on paper are often not well suited to use 'in their heads', we believe that in the case of many other pupils it is necessary for the teacher to point this out explicitly and to discuss at length the variety of methods which it is possible to use. However, no attempt should be made to force a single 'proper method' of performing mental calculations; pupils should be encouraged to make use of whatever method suits them best. Teachers should also encourage pupils to reflect upon the methods which they develop for themselves so that facility in mental computation can be consolidated and extended. Estimation 257 LEA guidelines and schemes of work in both primary and secondary schools almost always include a reference to the need for pupils to be able to estimate and sometimes state more specifically that pupils should be encouraged or required to write down an approximate answer before carrying out a calculation. The earlier chapters of this report make it clear that ability to estimate is important not only in many kinds of employment but in the ordinary activities of adult life. 258 However, from our own observations and from what we have been told by others, we believe that, even though it may be advocated in guidelines and schemes of work, estimation is not practised in very many classrooms. In our view this is not because of unwillingness on the part of teachers to encourage and develop this skill in their pupils but because of a failure to realise how much is implied by the words 'ability to estimate' and how long it takes to develop this ability. 259 There are several aspects of estimation. One is that of obtaining, before a calculation is carried out, a 'rough answer'; in other words, an answer in rounded terms which will enable a check to be made that the result of the calculation is of the correct 'order of magnitude'. Estimation of this kind is probably most commonly applied to the operations of multiplication and division; it concentrates on ensuring that the result of the calculation is not, for example, ten or a hundred times too large or too small. The method usually involves working 'to one significant figure', for example replacing 26 x 52 by 30 x 50 in order to obtain an approximate answer. However, for very many children this replacement is conceptually far more difficult than might be expected and the ability to work in this way takes a long time to develop. A much more elementary application is that the sum of two numbers, each less than 50, must be less than 100. We may, perhaps, describe this as 'awareness' rather than estimation but it is awareness of this kind which is needed if a 'feel' for number and the ability to estimate are to develop. In our view they are most likely to develop as a result of a great deal of discussion and 'thinking aloud' by both teacher and pupils, coupled with appropriate practical work on measurement. 260 A second aspect of estimation may be described as 'realising whether the answer is reasonable'. Many pupils find it difficult to develop this ability. In some instances it is concerned, as is the previous aspect, merely with computation but, as is frequently the case when using a calculator, after the calculation rather than before. Again, most mathematics teachers will recall instances in which, for example, a pupil has not only given as the area of a table top a measurement which more nearly corresponds to the area either of a postcard or of a football pitch, but has also failed to realise the incongruity. This type of mistake shows a lack of appreciation of the size of units and of their relation to everyday objects. There is need for pupils to establish everyday equivalences - for instance, to know that in most living rooms the door is about 2 metres high - and for the teacher to discuss and ask questions; 'would it go through the doorway?', 'could you lift it?'. It is to be hoped that, at a later stage, this kind of estimation will become part of 'common sense', but we believe that its development needs conscious attention. 261. A related aspect, that of estimating measurements of various kinds, is most evidently the aspect in which success is achieved by continuing usage and practical experience. Very specific skills of estimation, such as those of the decorator who is able to gauge by eye the quantity of paint or wallpaper he will require for a given task, have in the end to develop on the job. Nevertheless, throughout their time at school pupils should be encouraged to practise the estimation of lengths, areas, capacities and weights. 262 As we pointed out at the beginning of this section, work of the kind we have described in this section does not at present figure prominently in most school mathematics courses. There is therefore little experience available on which to draw or evidence of approaches which are known to have been successful. We believe that this is an area in which further study is required. Approaches to calculation 263 We have already drawn attention to the fact that in adult life written calculations are often carried out in ways which are different from those traditionally taught in the classroom. Although there are occasions on which it can be both quick and convenient to carry out written calculation in the traditional way, 'back of an envelope' methods are often not only quicker but also more straightforward. This raises important questions concerning the approach to computation which should be adopted in the classroom and the way in which calculations should be recorded on paper. 264 At the primary stage, computation is commonly introduced through the use of counting or structural materials and methods of recording develop in the first instance from the use of such apparatus; we discuss this further in the following chapter. By the age of 11 many children are able to carry out with confidence the operations of addition, subtraction, multiplication and division on whole numbers and to record their working in the traditional way. However, many other children do not reach this stage by the age of 11, especially in respect of multiplication and division. 265 It is common in secondary schools to insist that all pupils make use of the traditional methods of recording; these are often taught as standard routines to be followed. Nevertheless, even those pupils for whom this does not present undue difficulty need to be encouraged to develop alternative methods as well. These will frequently be similar to methods which are used to carry out mental calculations (see paragraph 256) or will make use of 'short cuts' of various kinds. Discussion of such alternative methods with a group or a class provides valuable opportunity for developing confidence and 'feel' when doing number work. 266 In the same way, pupils should be encouraged to approach other calculations, such as those involving percentages, 'on their merits' and not be expected always to use a single standard routine. For example, in order to calculate value added tax (VAT) at 15 per cent on a given sum of money, many people are likely to find it easier to write down 10 per cent of the sum; halve it and add the two results together than to multiply by 15/100. This method takes advantage of the particular relationship between 15 and 10 and also demonstrates understanding of the operation which is being carried out; if the rate of VAT were 8 per cent, most would probably find it easier to carry out the calculation in a single step by multiplying by 8/100. 267 Some low-attaining pupils have great difficulty in carrying out and recording computation using standard routines. However, after discussion and practical work with appropriate counting materials, they are often able to carry out calculations successfully by making use of methods of their own devising. They should be encouraged to do this and should not be dissuaded from using their own 'best method' provided that they can do this reliably. It can be counter-productive and inappropriate to drill these pupils in standard routines from the time they enter secondary school. 268 The availability of the electronic calculator is another factor which needs to be taken into account when considering approaches to calculation, especially in the case of those who have difficulty in using the standard routines. We discuss this further in Chapter 7. Measurement 269 Measurement is fundamental to the teaching and learning of mathematics because it provides a natural 'way in' both to the development of number concepts and also to the application of mathematics over a very wide field. Practice in ordering lengths, capacities and weights enables a young child to develop understanding of concepts such as 'more than', 'less than', 'longer than', 'shorter than'. After this the child learns first of all to use non-standard units such as handspans and cupfuls and then standard units to measure continuous quantities such as length and capacity. 270 All such forms of measurement are inexact and exist only within limits which can be specified. These limits may be chosen in accordance with the use which is to be made of the measurement or may be imposed by the limitations of the measuring instrument which is being used. Mathematics: 5-11 (8) states: Children should be led to master the concept of 'betweenness'. If the child's answer to the question 'What is the time?' is 'Between five and ten past two', then the answer is absolutely true.We wish to stress that in our view this concept is fundamental and children should from the earliest stages be encouraged, where appropriate, to record measurement in this way. 'The jug holds more than 5 cupfuls but less than 6'; 'this pencil is more than 12cm long but less than 13cm long'. This approach to measurement leads to the necessity for the sub-division of units. When this has been understood it is possible to go on to discuss degrees of accuracy and the idea of measurement within a given tolerance. 271 Measurement of area, capacity and angle involve geometrical concepts of shape and space; the recording of measurement gives opportunity for the introduction of a wide variety of graphical methods. At later stages study of measurement can be used to introduce concepts such as those of scale, rate, and ratio, and the use of compound measures of the kind used for speed, density and pressure. 272 At all stages the teaching of measurement should be based on extensive practical work, including the use of a variety of measuring instruments, the use of compasses and other drawing instruments, and the construction of geometrical shapes and models. All children at all levels need experience of this kind. Some children will acquire facility in measurement and in the use of drawing instruments without difficulty but others will require a long time and much practice to achieve a reasonable competence. This time needs to be provided over a period of years. Metrication 273 The change from imperial to metric units of measurement has not proceeded as quickly as had at one time been expected and, although considerable progress has been made, it is clear that both metric and imperial units will continue in use in England and Wales for some years to come. We have discussed some of the reasons for this in paragraph 82. 274 The continuing existence of the two systems of units has led to some confusion in schools in recent years. The guidance issued in 1974 by the DES and Welsh Office (9) drew attention to the 'growing familiarity within the schools, particularly the primary schools, with the use of metric quantities'. It acknowledged that 'the greatest difficulty for schools is not metrication itself, but the fact that, for some purposes, the imperial system appears likely to remain in everyday use for some time to come'. It suggested that schools should teach children to carry out calculations in metric units but should also enable them to maintain a general familiarity with imperial units. It also encouraged a policy of 'thinking metric'. 275 We believe that this advice continues to be sound and that the metric measures of length, weight and capacity should be used in both primary and secondary schools. Pupils should also learn to estimate in metric units. However, it is clear that it remains necessary for school leavers to have some knowledge of imperial units. We therefore consider that pupils in secondary schools should become familiar with the more common imperial units such as feet, inches, pounds, ounces, pints and gallons and should be able to use them for purposes of direct measurement. This means that they should be able to measure length in feet, inches and fractions of an inch (1/2, 1/4, 1/8, 1/16) to measure weight in pounds and ounces, and to measure capacity in gallons and pints. Pupils should become aware of the approximate equivalences between these imperial units and the appropriate metric units, but should not normally be required to calculate in imperial units.
'The basics' 276 The word 'basic' appears in very many of the submissions which we have received. Various expressions are used: 'basic skills', 'basic computational skills', 'basic mathematics', 'basic numeracy', 'the basics'. The contexts in which these expressions are used suggest that, while not exactly synonymous, they are effectively different ways of describing what is thought to be much the same thing. Where a definition is given, it is almost always in terms of purely arithmetical skills, with stress on the operations of addition, subtraction, multiplication and division treated in isolation from application to real situations. Many submissions assume that the meaning of whichever expression is used is self-evident and that there is no need to go into details. 277 Knowledge or skills which are 'basic' are presumably needed as a basis either for the mathematics required in employment or in adult life or for further study. We have considered these requirements in the preceding chapters. Although many of the requirements may be considered to be 'elementary' in terms of their position within the hierarchy of learning mathematics and the stage of schooling at which they are first introduced, it does not follow that they are necessarily either simple or straightforward for most pupils to learn and, more importantly, to apply. 278 There is evidence that the public focusing of attention on standards in schools which has occurred in recent years has created pressure in some quarters for a 'back to basics' movement. This has encouraged some primary teachers and some teachers of low-attaining pupils in secondary schools to restrict their teaching largely to the attainment of computational skills. Some of the submissions which we have received advocate a 'back to basics' approach of this kind. However, we hope that the argument of this chapter makes it clear that the ability to carry out a particular numerical operation and the ability to know when to make use of it are not the same; both are needed. The mathematics of employment and of everyday life is always mathematics in context and is based largely on measurements of many kinds made in many different situations. Arithmetical skills are therefore a tool for use in situations which require an understanding of other areas of mathematics, for example the geometry of shape and space and graphical representation of various kinds. An excessive concentration on the purely mechanical skills of arithmetic for their own sake will not assist the development of understanding in these other areas. It follows that the results of a 'back to basics' approach (as we understand the words) are most unlikely to be those which its proponents wish to see, and we can in no way support or recommend an approach of this kind.
Modern mathematics 279 In Britain the beginning of 'modern mathematics' is usually associated with three conferences held in Oxford in 1957, in Liverpool in 1959 and in Southampton in 1961. These were among a number of conferences, held in Europe and the United States of America towards the end of the 1950s, which discussed the teaching of mathematics in schools and resulted in the setting up of a variety of mathematics curriculum development projects. It is interesting to note that the conferences held in Britain were financed by industry and paid considerable attention to modern industrial applications of mathematics. A direct outcome of the conference held in Southampton was the setting up of the School Mathematics Project (SMP) whose syllabuses and teaching materials are prepared by groups of practising teachers and which was again funded by industry in its early stages. 280 The Director of SMP wrote in his report for 1962-63 that 'a major aim of the syllabus is to make school mathematics more exciting and more enjoyable, and to impart a knowledge of the nature of mathematics and its uses in the modern world. In this way, it is hoped to encourage more pupils to pursue further the study of mathematics, to bridge the gulf which at the moment separates university from school mathematics - both in content and in outlook - and also to reflect the change brought about in the world by increased automation and the introduction of electronic computers'. However, this statement related to an O Level course designed for pupils whose mathematical attainment was in the top quarter of their age group and was written at a time at which the provision of 'modern' courses for pupils of lower attainment had not been planned or expected. 281 The intention of those who set out to develop SMP and other modern courses was to introduce changes both in the content of the mathematics syllabus for these higher attaining pupils and also in the teaching methods and approaches which were used. Teaching approaches were designed to encourage investigation, to emphasise the applications of mathematics, and to draw attention to the unified nature of mathematics rather than to its traditional division at school level into arithmetic, algebra and geometry; and the classroom materials which were produced took for granted that the teachers who used them would possess sufficient mathematical insight and experience to enable them to work in these ways. At first most teachers who wished to introduce a modern syllabus were able to attend in-service training courses at which the aims of modern mathematics courses were explained and teaching approaches discussed. However, the unexpectedly rapid expansion of modern mathematics courses meant that it was not long before many teachers were required to teach these courses without the benefit of introductory training. A further, and also very rapid, development was the extension of modern mathematics courses to pupils whose attainment was lower and the introduction of modern syllabuses in CSE mathematics examinations. Not all teachers possessed a sufficient mathematical background to enable them to appreciate the intentions underlying the new courses they were teaching. In consequence the material which was included in modern courses was often not presented as part of a unified structure but as a collection of disconnected topics whose relevance to the mathematics course as a whole did not become apparent to pupils. 282 However, in our view the introduction of certain topics which had not previously been included in most mathematics courses has had a beneficial effect. We may cite as examples the increased emphasis on graphical work, the introduction of work based on the geometrical ideas of symmetry, reflection and rotation, the use of coordinates and the study of elementary statistics. Work on these topics has lent itself to more practical approaches to the teaching of mathematics and has proved to be within the capability of very many pupils. The same has not, however, been true of certain algebraic topics, which have proved difficult for many pupils to understand and whose purpose and use have not been evident to them. It is these topics, notably the algebra of sets and matrices, which have attracted considerable public attention and criticism and which have come, in the eyes of many people, to exemplify modern mathematics even though they form only a relatively small part of many courses. 283 During the last few years, a number of 'modern' O Level and CSE syllabuses have been modified so as to exclude some of the more abstract algebraic topics. At the same time many 'traditional' courses have also been modified to include such topics as elementary statistics and a greater emphasis on graphical work; as a result the differences between 'modern' and 'traditional' mathematics have become much less marked. In our own discussions we have not thought in terms of traditional or modern mathematics nor has the evidence which we have received suggested that it is any longer profitable to do so. Our discussion of mathematics teaching in the primary and secondary years which follows makes no distinction between the two because, in our view, it is no longer appropriate to make such a distinction; we believe that very many people now share this view.
Footnotes (1) A brief summary of the Review may be purchased from the Shell Centre for Mathematical Education, University of Nottingham; see also paragraph 756. (2) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979. (3) HMI Series: Matters for discussion 13. Girls and science HMSO 1980. (4) cf RR Skemp The psychology of learning mathematics Penguin 1971. (5) Primary education in England A survey by HM Inspectors of Schools. HMSO 1978. (6) Aspects of secondary education in England A survey by HM Inspectors of Schools. HMSO 1979. (7) Assessment of Performance Unit. Mathematical development: Primary survey report No. 1 and No. 2. HMSO 1980 and 1981; and Secondary survey report No. 1. HMSO 1980. (8) HMI Series: Matters for discussion 9. Mathematics 5-11. A handbook of suggestions. HMSO 1979. (9) Department of Education and Science Administrative Memorandum 9/74. Welsh Office Administrative Memorandum 4/74. 
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