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 6 Mathematics in the primary years
[pages 83 - 108]

284 In this chapter we discuss the mathematical work of children between the ages of 5 and 11, whether they are in infant, junior, infant and junior, first or middle schools. We do not discuss the mathematical development of children of pre-school age.

285 We have received much evidence which is supportive of the work of primary schools and believe that the great majority of primary teachers are aware of their responsibility to provide a sound mathematical foundation for the children in their care. Some teachers have told us that they would welcome guidance in this task and we hope that the matters we discuss in this chapter will provide a basis for study and discussion in staffrooms and elsewhere. We have also received some comments which are critical of mathematics teaching in primary schools. Some of these refer to matters which we discuss in this chapter; others reflect a failure on the part of some of those who have written to us to appreciate the detailed and careful approach to the teaching of mathematics which is necessary, especially in the early years, in order that children may develop confidence and understanding.

The primary mathematics curriculum

286 The use of practical methods in the primary classroom is sometimes thought to be of relatively recent introduction but this is not the case; work of this kind has been advocated for very many years. For example, in the Handbook of suggestions for teachers (1), published in 1937 by the Board of Education, we read:

First, by way of introduction, should come practical and oral work designed to give meaning to, and create interest in, the new arithmetical conception - through deriving it from the child's own experience - and to give him confidence in dealing with it by first establishing in his mind correct notions of the numerical and quantitative relations involved in the operation.

Before we discuss these areas of work in more detail, we wish to emphasise, as we shall continue to emphasise, that work of this kind needs to be carefully structured and followed up by the teacher. Much of its value will be lost unless the work which has been done, and the results which have been obtained, are discussed with the children so as to establish the necessary concepts and make links with other pieces of work which have been undertaken.

287 The primary mathematics curriculum should enrich children's aesthetic and linguistic experience, provide them with the means of exploring their environment and develop their powers of logical thought, in addition to equipping them with the numerical skills which will be a powerful tool for later work and study. The practical and intuitive experience which should be the result of a course of this kind provides an invaluable base for further work in the secondary years. However, we do not believe that mathematics in the primary years should be seen solely as a preparation for the next stage of education. The primary years ought also to be seen as worthwhile in themselves - a time during which doors are opened onto a wide range of experience.

288 We believe that the public criticism of recent years, to which we referred in paragraph 278, has caused some teachers in primary schools to wonder whether they have been right to adopt the broader approach which we have described. We hope that our discussion in Chapter 5 of the fundamentals of mathematics teaching and learning will have convinced them, and also the critics of primary mathematics, that emphasis on arithmetical skills does not of itself lead to ability to make use of these skills in practical situations. It is only within a broadly based curriculum that the ability to apply mathematics is enabled to develop.

Practical work

289 Practical work is essential throughout the primary years if the mathematics curriculum is to be developed in the way which we have advocated in paragraph 287. It is, though, necessary to realise at the outset that such work requires a considerable amount of time. However, provided that the practical work is properly structured with a wide variety of experience and clear stages of progression, and is followed up by the teacher by means of questions and discussion, this time is well spent. For most children practical work provides the most effective means by which understanding of mathematics can develop. It enables them to think out the mathematical ideas which are contained within the various activities they undertake at the same time as they are carrying out these activities; and so to progress within each topic from the handling of actual objects to a stage in which pictures or diagrams can be used to represent these objects and then to a final stage at which symbols are used which can be manipulated in abstract ways. (We give an example of this in paragraph 305.)

290 Children vary greatly in the amount of time which they take to move through these stages. It is as harmful to insist that one child should continue to use practical materials for a process which he understands and can carry out by using symbols as to insist that another should proceed to diagrammatic or symbolic representation before he is able to carry out the process by using practical materials. It is therefore a mistake to suppose that there is any particular age at which children no longer need to use practical materials or that such materials are needed only by those whose attainment is low. It is not 'babyish' to work with practical materials while the need exists and we believe that many children would derive benefit from a much greater use of these materials in the later primary years than occurs in many classrooms.

Measurement

291 In paragraph 269 we emphasised the importance of practising measurement of all kinds and explained the approach to measurement which we believe to be necessary. The measurement of length, capacity, weight, area and time should be part of every child's experience; some children will extend their work to the measurement of angle and of speed, volume and density. Practice in measurement needs to be associated with practice in estimation so that children gain an appreciation of the size of units and of their relation to everyday objects. Measurement should be linked with number work and attention drawn, for example, to the fact that the mathematical structure of metres, decimetres and centimetres is identical to that of hundreds, tens and units. This enables a ruler or tape measure to be used as a portable 'number-line'; this can be helpful where numbers have to be added or subtracted.

Shape and space

292 All children should have experience of work with a variety of plane shapes and solids. Here again, progression is essential. After the early stages of drawing round, cutting out, folding and colouring a variety of shapes, children should form them into patterns and then experiment to discover which shapes will 'tessellate' (that is, fit together without leaving any spaces between adjoining shapes) and which will not. They can also explore the ideas of symmetry, rotation and reflection. Geometrical knowledge grows through investigations of this kind and the properties of geometrical shapes become apparent as different patterns are constructed. From time to time the teacher should draw on the knowledge and experience which the children have gained in order to discuss these properties explicitly. For some children the idea of proof can start to develop as they seek to discover, for example, why triangles and quadrilaterals of all kinds will tessellate but many other shapes will not. The construction of plane shapes and solids helps to develop skill and accuracy in the use of instruments for measuring and drawing and also the ability to visualise three-dimensional figures. Almost all children find pleasure in working with shapes, and work of this kind can encourage the development of positive attitudes towards mathematics in those who are finding difficulty with number work. The rich variety of practical work which is possible in the primary years provides a foundation on which the more formal geometry of the secondary years can be based.

Graphical work

293 Throughout the primary years attention should be paid to methods of presenting mathematical information in pictorial and graphical form, and also to interpreting information which is presented in this way. It can often be the case that graphical work lacks variety and progression, so that older children are limited to drawing graphs which differ little from those which are to be found in infant classrooms. Children need experience of a wide variety of graphical work; the mere drawing of graphs should not be over-emphasised. It is essential to discuss and interpret the information which is displayed both in graphs which children have themselves drawn and also in graphs which they have not. Children should be encouraged to collect examples of graphs and charts from newspapers, magazines and books, to discuss in detail what they depict and to make deductions from them. Work of this kind often enables children to link their work in mathematics with their work in other areas of the curriculum.

294 As well as drawing graphs which provide factual information, some children should be able to construct graphs which display mathematical relationships such as those of the multiplication tables or of the growth of squares and cubes. Games such as 'Battleships' can be used to introduce the idea of coordinates to identify spaces and, later, single points; it then becomes possible to record graphically relationships such as 'pairs of numbers which add up to 10'. Many of these activities provide a basis from which later work in algebra can develop. When discussing graphs which have been constructed, attention can be drawn to the relationships which they display; for instance, 'every time we added another 20 grams the length of the spring increased by 3 centimetres'.

Logic

295 'To speak of logic in connection with young children may surprise some people, but no highly theoretical notions are involved. It is rather a matter of describing things accurately, noticing their resemblances and their differences, and saying how they are related to one another. In games and puzzles moves often have to be made according to rules, and finding the best moves involves logical thought'. (3) In its most straightforward forms, the activity of sorting objects and of recording the results in diagrammatic form is practised in most infant classes and forms the basis on which the concept of number is built. As children become older, it can develop into the more sophisticated activity of sorting shapes which vary in colour, size and thickness according to their attributes such as, for example, 'large and blue', 'thin and square'. A wide variety of work with shapes of this kind can be undertaken in order to encourage precision of language and the development of logical thinking. Games such as noughts and crosses, dominoes, draughts etc can also encourage logical thinking; 'if I go there ..., then he will ..., then I shall have to ...'

General activities

296 In addition to practical activities related to specific areas of the mathematics curriculum, of the kind which we have discussed in the preceding paragraphs, all children need experience of practical work which is directly related to the activities of everyday life, including shopping, travel, model making and the planning of school activities. Children cannot be expected to be able to make use of their mathematics in everyday situations unless they have opportunity to experience these situations for themselves. For most children a very great deal of practical exploration and experience is needed before the underlying mathematical ideas become assimilated into their thinking. We emphasise again that discussion both with the teacher and with other pupils is a necessary part of this process. A few children pass through the various stages of mathematical development in rapid succession and need to advance to more challenging and more abstract work before they leave primary school. For the majority, however, the transition from the use of concrete materials to abstract thinking takes place slowly and gradually; and even those children to whom abstract thinking appears to come easily often need to undertake practical exploration at the beginning of a new topic.

297 There is another aspect of practical work which we have not yet discussed. This is the use of number apparatus as an aid to the understanding of the number system and of methods of computation. We deal with this in the following paragraphs.

Number and computation

298 The skills of mental and written computation are founded on basic concepts which need to be developed through measurement, shopping, the use of structural apparatus and many other activities. These concepts include the meaning of the operations of addition, subtraction, multiplication and division and the very important concept of place value (that is, for example, that the 2 stands for 2 units in the number 52, for 2 tens in the number 127 and for 2 hundreds in the number 263). Understanding of place value enables number facts stored in long term memory to be used as a means of carrying out calculations involving larger numbers; for example, the knowledge that 14 - 8 = 6 can be used to work out 140 - 80 or 54 - 8. It is therefore essential that children should be helped to attain a secure and rapid recall of addition facts up to 10 + 10 and the related subtraction facts, and of multiplication facts up to 10 x 10 and the related division facts. This knowledge, together with understanding of place value, provides a basis for calculation involving small or large numbers. The learning of the number facts to which we have referred needs to be based on understanding, but understanding does not necessarily result in remembering. A time comes, therefore, when most children need to make a conscious effort to commit these number facts to memory. We have, though, to be aware that there are some children who have not attained secure and rapid recall of addition and multiplication facts by the age of 11.

299 Understanding of place value needs to be developed not only by means of structural apparatus and the abacus but also by using as examples the structure of hundreds, tens and units which underlies both measurement (metres, decimetres and centimetres) and money (pounds, ten pences and pence). It should not be assumed that a child who understands the structure of hundreds, tens and units will necessarily be able with ease to make the generalisation to thousands and higher powers of 10. Many children need further practical experience with structural apparatus so that they can work out for themselves the meaning of large numbers and be able to carry out operations with them. Other steps in understanding of place value are the understanding of tenths and hundredths, and of multiplication and division of both whole numbers and decimals by 10 or 100. These lead to more advanced computational skills such as long multiplication. They also provide a basis for developing the ability to approximate and to estimate the size of the answer which is to be expected as the result of carrying out a given calculation.

300 The skills of computation involving fractions are based on an understanding of the concept of equivalent fractions; that is, for example, that 5/10 and 1/2 have the same value. For many children understanding of this concept and of the notation for fractions are only just beginning to develop during the primary years. Furthermore, it is difficult to find everyday situations which require fractions to be added or multiplied and there seems to be little justification for teaching the routines for adding, subtracting, multiplying or dividing fractions to the majority of children during the primary years. Children should, however, become familiar in practical situations with terms such as 'one half of', 'one quarter of' and it is entirely appropriate, for example, to work out the number of children in 'half the class'. However, some children whose attainment is high are fully capable of understanding the equivalence of fractions and of applying this understanding to the solution of problems.

301 There is one aspect of computation which needs specific attention with most children towards the end of the primary years. This is the computation of time. Unlike other measures with which children become familiar, the relation between hours and minutes is based on 60 and not on 10 or 100. This means that children have to remember that routines which they normally use for addition and subtraction need to be modified if they are, for example, to be able to calculate the time taken for a journey which starts at 10.45am and ends at 1.30pm. They have also to be able to understand times expressed in terms of the 24 hour clock. It is not sufficient for computation of time to be practised in the abstract; it should be related to practical situations involving the planning of journeys and the use of timetables.

302 Some of the submissions which we have received have suggested that many primary schools do not pay sufficient attention to developing computational skill in their pupils. We do not believe this to be the case, nor is the view supported by the report of the National Primary Survey (4) which says that

suitable calculations involving the four rules with whole numbers were practised in all classes at all ages. It also states that in about a third of the classes, at all ages, children were spending too much time undertaking repetitive practice of processes which they had already mastered ... the efforts made to teach children to calculate are not rewarded by high scores in ... examples concerned with the handling of everyday situations. Learning to operate with numbers may need to be more closely linked with learning to use them in a variety of situations than is now common.

The beginnings of calculation

304 Young children should not be expected to move too quickly to written recording in mathematics. The forming of figures correctly on paper is a skill which needs to be learned. Until it has been mastered, attempts to carry out written calculation can inhibit the development of mathematical knowledge and understanding. In the early stages, therefore, mental and oral work should play a major part in learning. It has been pointed out to us that, albeit with the best of intentions, some parents can exert undesirable pressure on teachers to introduce written recording of mathematics, and especially 'sums', at too early a stage, because they believe that the written record is a necessary sign of a child's progress.

305 We believe that there are many who do not appreciate the number of stages through which a child must pass before even such an apparently simple 'sum' as 3 + 2 can be carried out with understanding. First of all the child must be able to recognise and form groups of three objects and two objects and to combine these to form a group of five. At the same time he will talk about what he is doing: 'There are three toy cars - fetch two more - now there are five'. When he carries out similar actions in different situations he needs to realise that essentially the same mathematics is contained in 'There were three children in the room; two more came in; now there are five.' and in 'I had three crayons; Mary gave me two; now I have five'. The next stage is to illustrate these in pictorial form; for instance, the crayons may be drawn as:

It is only when these experiences have all been assimilated and it has been realised that they all lead to the mathematical symbolisations:

that this sum, and similar ones, can be properly understood in the abstract. Only then is it appropriate for the sum to be carried out without concrete materials. Children should then be encouraged to make up their own stories for some of the sums they are asked to do. Some children are able to pass through these different stages quite quickly but for others it can take a long time. A premature start on formal written arithmetic is likely to delay progress rather than hasten it.

Language

306 Language plays an essential part in the formulation and expression of mathematical ideas. In paragraph 246 we drew attention to the need to extend and refine the use of mathematical language in the classroom. Development of this kind can only take place by means of continuing practice; from their earliest days at school children should be encouraged to discuss and explain the mathematics which they are doing. In the words of a submission we have received from the head of an infant school, 'there is a need for more talking time ... ideas and findings are passed on through language and developed through discussion, for it is this discussion after the activity that finally sees the point home'.

307 Children vary greatly in the level of the language skills which they possess at the age of 5. Some are already familiar with words and expressions such as 'heavy', 'light', 'larger than', 'shortest' and with the concepts to which these relate, but many are not. All children need, as a first stage in their learning of mathematics, to develop their understanding of words and expressions of this kind by means of activities and discussion in the classroom, and this development of mathematical language should continue throughout the primary years.

308 It is important to be aware of the great variety of language which is used in connection with many of the mathematical operations which children will meet. For example, the instructions 'add 5 and 3', 'add 3 to 5', 'find the sum of 5 and 3', 'find the number which is 3 more than 5' all require the same mathematical operation to be carried out. These are only four of many ways in which it is possible to frame this instruction; the particular form of words which is used is most often the one which arises naturally in the context of the moment. Children need to be able to interpret these apparently different instructions, to use them in their own speech and thought, and eventually to be able to attach them all to the symbolic form 5 + 3. Unless children become familiar with the many different ways in which the same mathematical idea can be expressed and are able to recognise the same idea within different forms of words, they will not only have difficulty in dealing with computational examples of the kind which we have quoted but also in dealing with problems expressed in words.

309 Children whose grasp of language is not secure often try to overcome their difficulty by looking out for words such as 'more' or 'less' and using them as 'verbal cues' which they believe will indicate the operation they are required to carry out. However, this does not resolve the difficulty. For example, the two problems

Janet has 5p and John has 3p more than Janet; how much money has John?

and

Janet has 5p and John has 3p; how much more money has Janet than John?

310 Children need also to learn that certain words are used in mathematics in ways which are not the same as those in which they are used in everyday speech. We have been told of a visitor to a junior classroom who, in response to the question 'what is the difference between 10 and 7?' was surprised to receive the answer '10 is even and 7 is odd' instead of the answer '3' which had been expected. 'Difference' is only one of many words with whose mathematical meaning children have to become familiar.

311 Reading skills in mathematics should be built up alongside other reading skills so that children can understand the explanations and instructions which occur in the mathematics books which they use. If the skills of reading mathematics are not developed, many children will evolve their own strategies for avoiding such reading. We have already referred to reliance on verbal cues. Another strategy is to avoid reading any explanatory passages which come at the beginning of a work card or textbook exercise and to start on the questions in the hope that they can be done without first studying the explanation or instruction which precedes them. Yet another is to seek help from a friend or the teacher. The policy of trying to avoid reading difficulties by preparing work cards in which the use of language is minimised or avoided altogether should not be adopted. Instead the necessary language skills should be developed through discussion and explanation and by encouraging children to talk and write about the investigations which they have undertaken. Children should also be encouraged to suggest their own problems and to express them in written form.

The use of books

312 Even although a child may without difficulty be able to read what is written in a mathematics textbook or on a work card, he may well find great difficulty in learning an unfamiliar piece of mathematics from the written word. This is likely to be the case however careful has been the choice of the language which is used. The ability to learn mathematics from the printed page is one which develops very slowly, so that even at the age of 16 there are few pupils who are able to learn satisfactorily from a textbook by themselves. At the primary stage new topics and the concepts should always be introduced by appropriate oral and practical work and the necessary links with what has gone before established by discussion.

313 Nevertheless, textbooks provide valuable support for teachers in the day by day work of the classroom. They can provide a structure within which work in mathematics can develop and provide ideas for alternative approaches. They can be a source of exercises which have been carefully graded and are likely to provide revision exercises at suitable intervals. Accompanying teachers' manuals may suggest other kinds of work which should be undertaken alongside the exercises in the textbook and indicate ways in which the topic can be developed further for some pupils. However, it is always necessary to use any textbook with discrimination, and selections should be made to suit the varying needs of different children. It may be better, too, to tackle some parts of the work in an order which is different from that in the book or to omit certain sections for some or all children. It should not be expected that any textbook, however good, can provide a complete course or meet the needs of all children; additional activities of various kinds need to be provided.

314 By the middle junior years some children are skilled readers and have become accustomed to acquiring information from books. Although the printed word is seldom a satisfactory means of introducing new mathematical concepts, the same limitation does not apply to the use of mathematical problems, puzzle and topic books, and books of this kind should be available in the classroom or school library. Their use can enable children to realise that mathematics is a living subject which is full of interest and of use outside the classroom, and can also contribute to the children's overall mathematical development. Although some books of this kind are available, more are needed; suitable topics would be the mathematics used in everyday life, the exploration of shape, communication by means of graphs and diagrams, the history and development of counting, calculation and measurement, and links between mathematics and science or art. More books of puzzles, problems and suggestions for investigations are also required.

Mental mathematics

315 We refer in this section to 'mental mathematics' rather than 'mental calculation' because we wish to include within our discussion both mental calculation and also the oral work which should play an important part in the teaching of primary mathematics. For the reasons to which we drew attention in paragraph 254 there has been a decrease in the use of mental mathematics in schools of all kinds in recent years and we believe that this trend should be reversed.

316 We have already explained that young children should not be allowed to move too quickly to written work in mathematics. It follows that, in the early stages, mental and oral work should form a major part of the mathematics which is done. As a child grows older, he needs to begin to develop the methods of mental calculation which he will use throughout his life; as we pointed out in paragraph 256 these will not necessarily be the same as the methods used on paper. Practice in the handling of money, the giving of change by 'counting on' in the way which is commonly used in shops, calculation of journey times and mental calculations involving measurement of various kinds should all start during the primary years. We may note that, although it is possible to practise written methods of computation as routines with little understanding of the underlying method, good mental methods have to be based on understanding of place value accompanied by recall of addition and multiplication facts; it follows that the practice of mental methods of computation will also assist in the understanding and development of written methods. Mental mathematics may also be used in the primary years to build up speed and confidence in the recall of basic number facts and to extend mathematical insights without the added complication of formal recording.

317 While one aspect of mental mathematics is work 'in the head', another is the promotion of mathematical discussion in the classroom. Exchanges between child and teacher, and between different children, should be encouraged. Even in a class in which an individual learning scheme is used for much of the time, there are some skills, puzzles and problems which are appropriate for every child no matter what stage of learning he may have reached and short class sessions can be arranged for work of this kind. If answers are recorded on paper, difficulties and weaknesses can be dealt with on an individual basis later so that the limited success of certain children is not drawn to the attention of the whole class as might be the case with oral answers. However, this does not preclude general discussion of certain problems; on the contrary some problems should be posed with general discussion in mind. Both children and their teachers learn from the different strategies and methods which other members of the class use and explain in answer to questions. It is valuable experience for children, and something which many children do not find easy at first, to explain the approach which has been used; even a wrong answer or a false start, if carefully handled by the teacher, can be illuminating when discussed. Different points of view offer considerable opportunities for exploring and increasing the depth of understanding of all members of the class. Sometimes, too, children can be asked to pose their own problems.

318 Mental mathematics can also be used as a means of introducing informally mathematical ideas which will later be developed in greater depth and perhaps in different ways by different children. Questions such as 'if 2 x [square] + [triangle] = 17, what numbers can we write in the box and the triangle?' can be used to introduce algebraic ideas. The ability to visualise shapes in two and three dimensions can also be developed through mental work.

319 Even though an individual learning system may be in use the teacher will often assemble a small group to begin a new topic or to draw together common strands in work which is going on. On such occasions mental mathematics is easily and naturally introduced, both in the form of mental calculations and of questions which develop new ideas or bring together the work of the various children. Since such small groups will often consist of children of similar attainment, there is less chance of dispiriting failure and the size of the group makes it easier for the teacher to handle difficulties sensitively. With a small group of children the teacher can appropriately press them, according to their ability, to increase their speed of response and calculation. Repeated failure should be avoided; nevertheless children respond to questions which extend them and by careful questioning, either of the whole group or of individual children, confidence can be built up.

320 Whatever textbooks or work cards are used, the level of difficulty can never be matched exactly to every child's needs. Questioning and mental mathematics have their part to play in improving the match, in helping the weaker over difficulties and in increasing the challenge to those whose attainment is high. Mental mathematics also provides a means of developing the skills of estimation to which we referred in paragraphs 257 to 261.

Using mathematics to solve problems

321 All children need experience of applying the mathematics they are learning both to familiar everyday situations and also to the solution of problems which are not exact repetitions of exercises which have already been practised. When young children first come to school, much of their mathematics is 'doing'. They explore the mathematical situations which they encounter - perhaps sorting objects into different categories or fitting shapes together - and come to their own conclusions. At this stage their mathematical thinking may reach a high level of independence. As they grow older this independent thinking needs to continue; it should not give way to a method of learning which is based wholly on the assimilation of received mathematical knowledge and whose test of truth is 'this is the way I was told to do it'.

322 Mathematical explorations and investigations are of value even when they are not directed specifically to the learning of new concepts. Children should therefore be encouraged, for example, to work out the best way of arranging the seating for the audience at the school concert or to compare the cost of various packet sizes and brands of food for the classroom pets. The extent to which children are enabled to work in this way will depend a great deal on the teacher's own awareness of the ways in which mathematics can be used in the classroom and in everyday life.

323 The development of general strategies directed towards problem solving and investigations can start during the primary years. Children should therefore be given opportunity to become familiar with the processes which can be used in work of this kind. One of these is to make a graphical or diagrammatic representation of the situation which is being investigated; for example, if two dice are being thrown, the scores obtained can be recorded graphically. There may be a pattern in the results which are being obtained which can lead to the making of a conjecture to forecast later results; for example, 2 points on a circle can be joined by one line, 3 points can be joined in pairs by 3 lines, 4 points by 6 lines and so on. Efforts can then be made to discover whether, and explain why, the conjecture is or is not correct. It is sometimes appropriate to set up an experiment, for example to discover the length of a seconds pendulum, or to employ the strategy of looking at a simpler related problem; an example of this latter strategy is that the number of squares (of any size) on a full-sized chessboard may be too many to count, but a 2 x 2 and a 3 x 3 board are more manageable, and a pattern begins to emerge. It is necessary to develop persistence in exploring a problem, for example the number of different shapes which can be made from a given number of squares of the same size, and the ability to record the possibilities which have been tried. Finally, it is important to develop the ability to work with others in the discussion of possible approaches and to be able to communicate progress which has been made by means of words, diagrams and symbols.

324 Not a great deal is yet known about the, ways in which these processes develop nor are suitable materials for teachers readily available. There is need for more study of children's spontaneous problem solving activities and of the extent to which strategies and processes for problem solving can be taught. Present knowledge suggests that, if children are not enabled to tackle problems which are at the right level for them to achieve success as the result of concentrated effort, their problem solving abilities do not develop satisfactorily.

Links with other curricular areas

325 The experiences of young children do not come in separate packages with 'subject labels'; as children explore the world around them, mathematical experiences present themselves alongside others. The teacher needs therefore to seek opportunities for drawing mathematical experience out of a wide range of children's activities. Very many curricular areas give rise to mathematics. Measurement and symmetry arise frequently in art and craft; many patterns have a geometrical basis and designs may need enlarging or reducing. Environmental education makes use of measurement of many kinds and the study of maps introduces ideas of direction, scale and ratio. The patterns of the days of the week, of the calendar and of the recurring annual festivals all have a mathematical basis; for older children historical ideas require understanding of the passage of time, which can be illustrated on a 'time-line' which is analogous to the 'number-line' with which they will already be familiar. A great deal of measurement can arise in the course of simple cookery, including the calculation of cost; this may not always be straightforward if only part of a packet of ingredients has been used. Many athletic activities require measurement of distance and time. At the infant stage many stories and rhymes rely for their appeal on the pleasure of counting.

326 It would be easy to compile a much longer list of areas of the primary curriculum which provide opportunities for the use of mathematical skills; pressure of work in the classroom makes it much less easy for the teacher to make sure that advantage is taken of these opportunities when they arise. When planning the activities of the classroom, and especially any extended topic or project work, it is therefore necessary for the teacher to try to identify at the outset the mathematical possibilities which exist within the work which is planned. Not all of these will necessarily be realised but by planning in this way it becomes easier to make the most of whatever opportunities present themselves and perhaps also, by appropriate discussion, to draw attention to others.

327 We have not yet referred to one area of the curriculum which has clear and direct links with mathematics, that of science. Almost every investigation which is likely to be undertaken will require the use of one or more of the mathematical skills of classifying, counting, measuring, calculating, estimating, recording in tabular or graphical form, making hypotheses or generalising, and will provide opportunity for making use of mathematics in practical situations. Indeed, there is a great deal of overlap between practical mathematics and science in the primary years and many activities such as recording the growth of a plant or animal, measuring temperature and rainfall or investigating the chain-wheels of a bicycle could take place under either heading. The report of the National Primary Survey (5) draws attention to the fact that too few of the schools visited had effective programmes for the teaching of science. The government paper The school curriculum (6) states that it is intended to take further action in relation to science in schools. The development of science teaching in primary schools will provide valuable opportunities for developing the use of mathematics in practical ways; we hope these will be exploited.

328 There is one matter to which attention will need to be paid. If, because of lack of suitable expertise among other teachers in a school, the teaching of science is undertaken on a specialist basis, it will be essential for those who teach mathematics and those who teach science to work closely together so that full advantage can be taken of the overlap between science and mathematics and links established between them.

329 The overall aim must be to develop in children an attitude to mathematics and an awareness of its power to communicate and explain which will result in mathematics being used wherever it can illuminate or make more precise an argument or enable the results of an investigation to be presented in a way which will assist clarity and understanding.

Attainment in mathematics

Children whose attainment is high

330 In this section we discuss the teaching of children who are within about the top 10 per cent of their age group in terms of their attainment in mathematics. High attainment in mathematics is very often associated with high attainment in other areas of the curriculum but it is important to be aware that it can also exist in children whose performance in the rest of their work is no more than average. The capacity for high attainment in mathematics can sometimes become apparent at a very early age. A fascination with numbers and a self-developed capability in the use of large numbers may be signs of such capacity. However, some children do not display such ability until much later. During the primary years a teacher may notice that a child learns and grasps new ideas with great speed, that he shows energy and perseverance in pursuing ideas, that he understands abstract concepts easily and is able to make use of them in a variety of situations. All of these characteristics can be signs of capacity for high attainment but it is possible for such capacity to go unrecognised. This may be because poor linguistic or reading skills can conceal capacity for mathematical attainment or, especially in the case of gifted children (those whose attainment is within the top 2 to 3 per cent), because of behaviour problems which are the result of boredom and frustration arising from work which is insufficiently demanding. It can also be the case that gifted children seek to hide their powers so as not to appear different from their fellows.

331 The Russian psychologist Krutetskii has listed some characteristics which are often found during the primary years in children who are very highly gifted mathematically. (7) These include ability to perceive and use mathematical information and grasp the inner structure of a problem; ability to think with clarity and economy when solving problems; ability to use symbols easily and flexibly and to reverse a mathematical process with ease; ability to remember generalised mathematical information, methods of problem solving and principles of approach. Although few children will display gifts of such a high order, this list indicates the directions in which high attainment in mathematics develops.

332 The problems which primary teachers can encounter in making suitable provision for high-attaining children are not always appreciated. It is not sufficient for such children to be left to work through a textbook or a set of work cards; nor should they be given repetitive practice of processes which they have already mastered. 'The statement that able children can take care of themselves is misleading; it may be true that mathematically such children can take care of themselves better than the less able, but this does not mean that they should be entirely responsible for their own programming; they need guidance, encouragement and the right kind of opportunities and challenges to fulfil their promise.' (8) High attaining children should combine more rapid progress through the mathematics syllabus with more demanding work related to topics which have already been encountered. In particular, they should be given opportunity to undertake activities and investigations which encourage the development of powers of generalisation and abstraction; older juniors may, for example, become able to express in terms of algebraic symbols the relationships which arise from graphical work or the investigation of number patterns. Geometrical work should also be encouraged. 'Abler top juniors ... are capable of a considerable amount of geometry. Through the strength of their own intuition, they can cover the greater part of the geometry demanded by an O Level syllabus. They do, however, need suitable resource material including topic books and reference books, but not secondary school texts.' (9)

333 There is undoubtedly a need to make specific provision for primary children whose mathematical attainment is high. However, suitable resource material is not easily accessible for the use of teachers who are not mathematics specialists and more is needed. We believe that in some parts of the country arrangements have been made for a peripatetic teacher who is a mathematics specialist to teach groups of children in different schools from time to time. In some areas clubs which meet on Saturdays have been organised by LEAs, colleges or other agencies with considerable success; such clubs enable children to share their enthusiasm for mathematics with others who are like-minded. If these children's teachers are also able to take part, they are likely to be able to make use of some of the ideas which arise to enrich the curriculum of other children. Within a particular school it may be possible to regroup children for mathematics in order to provide more effectively for the higher attainers, and we believe that careful consideration should be given to this possibility. We return to this point in paragraph 350.

Children whose attainment is low

334 Low attainment in mathematics can occur in children whose general ability is not low. Among the reasons for this can be inappropriate teaching, lack of confidence, lack of continuity, especially because of change of school, and frequent or prolonged illness; poor reading skills can also hinder progress in mathematics. In such cases it is essential to try to diagnose the reason for the 'mathematical blockage' and to remedy the lack of understanding which exists. Because the problem is likely to be individual to the child the diagnostic process should be very largely oral and practical so that by observation and discussion the teacher may establish which concepts are understood and which are not. The use of suitable diagnostic tests may also be helpful. Failure can only be compounded if efforts are made to build further upon a foundation which does not exist. Such efforts are likely only to result in confusion and lack of confidence because of continuing lack of success, and so lead to dislike of mathematics and further failure.

335 In the case of children whose low attainment in mathematics is associated with low general ability, the mathematics course needs to be specifically designed to build up a network of simple related ideas and their applications, so that the children can feel confident in their ability to make use in their daily lives of the mathematics which they know. Advance should be by very small stages with frequent opportunity for repetition and reinforcement. Use should be made of extensive and varied practical and oral work related to everyday situations such as measurement, shopping and the use of money. In this way efforts can be made to develop the necessary network of associations which is essential to long term memory.

336 Low-attaining children need extensive experience of counting objects of many kinds by grouping them in tens so that they become aware of the relative size of numbers. Work of this kind needs to be accompanied by the use of number apparatus to develop the idea of place value and an understanding of the different number operations. Familiarity with number and counting can also be developed by means of board and other games played with dice. Practice is also needed in counting money and using it for shopping. However, stress should not be placed on the development of number skills to the exclusion of other activities. Low-attaining children need to join with their fellows in experiencing the pleasure of simple work with shapes and the discovery of pattern. They should also undertake straightforward work of a graphical nature and learn to use a ruler and other simple geometrical instruments.

337 Children whose attainment is very low are often withdrawn from their normal class for part or all of the time to receive special attention from a remedial teacher. Such teachers usually concentrate largely on language work and are not always skilled in the teaching of mathematics or in the diagnosis of associated learning difficulties. Remedial teachers should nevertheless seek to develop the understanding and use of mathematical language alongside other language skills. It is of the utmost importance that the mathematical work of children in remedial classes should not consist of the practice of arithmetical skills in isolation but should be accompanied by discussion of the concepts on which these skills rest and of the ways in which they can be used in the children's everyday lives. More time should be spent on oral and practical work than on written work.

338 The increasing availability of electronic calculators has made it all the more important that, in the teaching of low-attaining children, attention should be given to the development of concepts and applications. Once these are understood, it becomes possible to make use of a calculator to overcome lack of computational skill, but a calculator can be of no assistance until a child knows which arithmetical operation it is necessary to carry out.

Attainment at the age of 11

339 The mathematical course which a child will follow is normally set out in the mathematics syllabus of his school, which is often based on guidelines prepared by the LEA. In some cases syllabuses and guidelines indicate only a progression of topics while in other cases an attempt is made to suggest levels of attainment which are thought to be appropriate to a majority of children of a given age. The amount of guidance which is given as to the way in which different topics should be approached can also vary considerably. However, even when detailed guidance is given, it may not be heeded by a particular teacher. The report of the National Primary Survey (10) draws attention to the fact that 'individual schools or teachers are making markedly individual decisions about what is to be taught based on their own perceptions and choices'. We need therefore to be aware of the differences which are likely to exist between the mathematics which the syllabus intends should be taught, the mathematics which has actually been taught by a particular teacher and that part of what has been taught which a child has learned and understood. Furthermore, any test which is given to a child can measure performance only on that part of what he has learned to which the test questions relate; even then, he may not always be able to demonstrate his knowledge under test conditions.

340 During recent years large-scale testing of children's performance in mathematics has been introduced in many countries. In addition to studying the reports of the Assessment of Performance Unit (11), which provide information about children in England and Wales, and the results of some other tests carried out in England, we have been able to examine the results of large-scale tests carried out in Scotland and the United States of America. We have also studied the results of a survey of basic numeracy carried out in Australia. It is clear that the differences in attainment which exist between children of the same age in any one country are very much greater than the small variations which exist between the performance of an 'average' pupil from each country. It is also apparent that the general pattern of the development of number knowledge is very similar in all the countries whose tests we have studied. We do not therefore believe that there are any grounds for thinking that the overall performance of children in England and Wales is markedly different from that of children in these other countries.

341 Because different countries test pupils on a large scale at different ages, it has been possible to build up a composite picture, as demonstrated by the results of written tests, of the performance in some numerical topics of an English speaking child whose attainment is at about the fiftieth percentile and also of one whose attainment is much below average (at about the fifteenth percentile from the bottom). For the child whose attainment is at the fiftieth percentile, understanding of the topic of place value probably develops along these lines. At age 9 he can identify the largest or smallest of a set of numbers which range from units to thousands and he can write down the total number of postage stamps from a picture showing the stamps in blocks of 100, strips of 10 and single stamps. By age 11 he is clarifying his concept of place value in the range from thousands to tenths. He understands that the 2 in 12, 205 and 40.2 represents units, hundreds and tenths respectively but he cannot yet arrange the numbers 0.07, 0.02, 0.1 in ascending order of size. He can write down the number which is 1 less than 2010 and probably also the number which is 1 more than 6399. However, test results suggest that not until the age of 15 are at least half the children in a year group able to read a scale to two decimal places or to state that the 1 in the number 2.31 represents 1 hundredth. Although about half of the children in our schools will know more about place value than this 'average' child, the other half will know less and will be progressing through these stages more slowly. For example, the Concepts in Secondary Mathematics and Science study (12) found that pupils whose attainment was at the fifteenth percentile from the bottom had reached the third year of secondary school before they were able to give the number which is 1 more than 6399. On the other hand, there are a few 7 and 8 year olds who are able to answer this question and a few 9 year oIds who are able to use two decimal places with understanding.

342 It therefore seems that there is a 'seven year difference' in achieving an understanding of place value which is sufficient to write down the number which is 1 more than 6399. By this we mean that, whereas an 'average' child can perform this task at age 11 but not at age 10, there are some 14 year oIds who cannot do it and some 7 year olds who can. Similar comparisons can be made in respect of other topics. For example, the top 15 per cent of 10 year oIds in England are able to answer the question 'There are 40 children in a class and three fifths of them are girls. How many boys are there in the class?'. By contrast, the bottom 15 per cent of 14-15 year old pupils in Scotland find difficulty in working out 3/4 of £24; the bottom 15 per cent of 14 year olds in Australia find difficulty with the comparable question 'Mixed concrete costs $24 per cubic metre. What would 3/4 of a cubic metre cost?'. There is little evidence to show the attainment of the most capable 11 year olds, because large scale tests do not usually include very many items which will extend these children and so provide the necessary evidence. However, one American study found that there were a number of 12 and 13 year old pupils in Baltimore who performed at the same level as the top 10 per cent of 17 year olds on a mathematics test designed to reveal potential for college study.

343 We believe it is clear from the preceding paragraphs that it is not possible to make any overall statement about the mathematical knowledge and understanding which children in general should be expected to possess at the end of the primary years. However, the test results which we have quoted, and others which we have studied, indicate that even in the primary years the curriculum provided for pupils needs to take into account the wide gap in understanding and skill which can exist between children of the same age.

344 The study of test results provides evidence only of the achievement of children under test conditions as a result of the curricula and the teaching methods which have been used. They cannot indicate what might be the results given different curricula or different teaching methods. However, the fact that the overall picture is similar in different English-speaking countries suggests that any improvements in teaching are likely to produce slow change rather than rapid results. Even if the average level of attainment can be raised, the range of attainment is likely to remain as great as it is at present, or perhaps become still greater, because any measures which enable all pupils to learn mathematics more successfully will benefit high attainers as much as, and perhaps more than, those whose attainment is lower.

Attitudes

345 During every mathematics lesson a child is not only learning, or failing to learn, mathematics as a result of the work he is doing but is also developing his attitude towards mathematics. In every mathematics lesson his teacher is conveying, even if unconsciously, a message about mathematics which will influence this attitude. Once attitudes have been formed, they can be very persistent and difficult to change. Positive attitudes assist the learning of mathematics; negative attitudes not only inhibit learning but, as we discussed in Chapter 2, very often persist into adult life and affect choice of job.

346 By the end of the primary years a child's attitude to mathematics is often becoming fixed and will determine the way in which he will approach mathematics at the secondary stage. He may thoroughly enjoy his work in mathematics, or he may be counting the days until he can stop attending mathematics lessons. He may have learned that mathematics provides a means of understanding, explaining and controlling his environment, or he may have failed to realise that it has any relevance outside the classroom. He may have learned the importance of exploration and perseverance when tackling a problem and have experienced the pleasure which comes from finding its solution, or he may regard mathematics as a series of arbitrary routines to be carried out at the teacher's behest, with no opportunity for initiative or independent thought. He may be well on the way to mastering some of the mathematician's skills, or he may already see mathematics as an area of work which he cannot understand and in which he always experiences failure.

347 In the previous paragraph we have set out extremes of attitude in order to stress the importance of doing all that is possible to develop positive attitudes towards mathematics from the earliest days at school. At the age of 5, children usually show an uninhibited enthusiasm and curiosity; school is enjoyable and they learn rapidly and with interest as they encounter a great variety of new experience. The challenge for the teacher is to present mathematics in a way which continues to be interesting and enjoyable and so allows understanding to develop. In the course of our visits to primary schools we met a number of teachers who were succeeding in presenting mathematics in this way and whose pupils were clearly enjoying their work. The notes which we made after the visits contain comments such as 'the level of work and presentation were most impressive ... pupils went about their work in a quiet, business-like yet enthusiastic manner', 'a most enjoyable visit ... a happy and conscientious staff working together to achieve common aims'. However, not all the classes we visited were achieving these levels of involvement and the importance of creating positive attitudes to mathematics did not always seem to have been realised. Even in schools in which the general atmosphere was lively and supportive, the need of children to work at mathematics in practical ways had not always been realised, so that attitudes to mathematics in some classrooms contrasted strongly with attitudes towards other work. In some classrooms children were working at abstract calculations with numbers beyond their experience; their need to measure, weigh and pour water, to count real things and to learn about hundreds, tens and units with apparatus and with money had gone largely unrecognised.

The organisation of teaching groups for mathematics

348 In most primary schools children work in mixed ability classes which usually contain children of only one year group. The teacher in charge of the class is normally responsible for the greater part of the work of the class, including mathematics. This arrangement allows flexibility in the organisation of work in mathematics, as in other areas of the curriculum. It is not, for example, necessary for children to do mathematics at a fixed time or for a given length of time, nor for all children to do mathematics at the same time. It is therefore possible for the teacher to work with part of the class while the remaining children are engaged in activities which require less immediate attention from the teacher. Because the same teacher is with the class for most of the week, there is also maximum opportunity to relate work in mathematics to work in other curricular areas (see paragraph 325). However, the quality of the mathematics teaching inevitably depends largely on the strength and interest of the class teacher. If this teacher lacks enthusiasm for mathematics and confidence in teaching it, the children in the class will be disadvantaged. Even though the membership of a class may not change very much from one year to the next it is common for there to be a change of class teacher at the beginning of each year. It is therefore possible for some children to be taught mathematics by seven different teachers during the primary years, with consequent problems of ensuring continuity. An arrangement whereby the same teacher remains with a class for more than one year is likely to improve continuity but may not improve the teaching of mathematics if the teacher lacks the necessary expertise.

349 'Vertical grouping', in which children of two or more year groups are placed in the same class, is quite often used in the infant years and is a necessity in some small primary schools. Although an arrangement of this kind may lessen problems of continuity, there will almost certainly be a greater spread of attainment in mathematics among the children in the class and a consequent increase of difficulty for the teacher in matching levels of work to the needs of the children. (13) We do not therefore consider that this form of grouping offers any advantages for the teaching of mathematics.

350 Although most primary teachers group the children in their classes according to attainment for some part at least of their work in mathematics, there are some junior and middle schools in which this practice is extended further so that children from several classes are rearranged for mathematics into groups based on attainment. This enables the range of attainment in any one group to be reduced and so makes it easier for teachers to match the levels of the work appropriately. However, it is necessary to realise that even when children are grouped in this way, considerable differences will exist within each group. This is illustrated by a submission which we have received from a teacher in a middle school: 'The middle group itself could have been divided into three, so great was the disparity of understanding'. Furthermore, even when grouping of this kind is used, the teaching still has to be shared among the teachers of the classes from which the groups have been formed and so some groups may be taught mathematics by teachers who lack interest in the subject or confidence in teaching it. Nevertheless, rearrangement of this kind is likely to make it easier to provide appropriately for higher-attaining pupils.

351 Some of our members who visited Denmark were able to observe mathematics teaching in a Folkeskole, a comprehensive school for children aged 7 to 16. From the age of 7, children in this school were in classes whose teaching was shared by three teachers, and were usually taught by the same small team of teachers for several years. This not only ensured continuity of mathematics teaching but also enabled teachers to make use of their particular strengths. We do not suggest that this form of organisation would necessarily be appropriate in this country but we believe that it suggests the need to examine the advantages of some form of team teaching, perhaps by means of two or three classes working in association with each other, so that continuity of mathematics teaching could be maintained for two or three years. If a teaching team contains a teacher with enthusiasm for mathematics and with some specialist knowledge, this teacher is able to lead the work in mathematics of a much larger group of children than would otherwise be the case and also to assist the other members of the team. A few schools already work in this way with teams of two to four teachers working together. Each teacher takes a major responsibility for one area of the curriculum and follows the lead of the other teachers in the remaining areas. An arrangement of this kind enables children to experience greater continuity of teaching and still to be well known to their class teacher. It also provides for teachers the opportunity of observing the mathematical development of children over a longer period than a single year.

352 The report of the National Primary Survey (14) draws attention to the need for schools to consider how best to deploy staff in order to make the best use of the strengths of individual teachers. We believe that all schools should examine the extent to which the form of organisation which they are using enables the best use to be made of the mathematical strengths of their staff and provides continuity of teaching in mathematics for the children in the school.

Time allocation for mathematics

353 On average, junior classes devote about five hours a week to mathematics, but this figure conceals wide variations from less than three hours to more than six. If, as we believe should be the case, mathematics plays a part in many areas of the curriculum, children may do quite a lot of mathematics outside the time specifically allocated to it. We do not therefore believe that such specific allocation should exceed five hours per week. On the other hand, we do not consider that the time allocation should fall substantially below four hours. So far as is possible, time for mathematics should be flexible, so that an interesting discussion can be followed to its conclusion or a piece of practical work completed; mental mathematics is usually best done for a fairly short period of time. Even older children should not normally work at mathematics for more than one hour at a stretch.

The mathematics coordinator

354 The effectiveness of the mathematics teaching in a primary school can be considerably enhanced if one teacher is given responsibility for the planning, coordination and oversight of work in mathematics throughout the school. We shall refer to such a teacher as the 'mathematics coordinator'.

355 In our view it should be part of the duties of the mathematics coordinator to:

• prepare a scheme of work for the school in consultation with the head teacher and staff and, where possible, with schools from which the children come and to which they go (we discuss this further in paragraph 363);
• provide guidance and support to other members of staff in implementing the scheme of work, both by means of meetings and by working alongside individual teachers;
• organise and be responsible for procuring, within the funds made available, the necessary teaching resources for mathematics, maintain an up to date inventory and ensure that members of staff are aware of how to use the resources which are available;
• monitor work in mathematics throughout the school, including methods of assessment and record keeping;
• assist with the diagnosis of children's learning difficulties and with their remediation;
• arrange school based in-service training for members of staff as appropriate;
• maintain liaison with schools from which children come and to which they go, and also with LEA advisory staff.

357 Good support from the head teacher is essential if the mathematics coordinator is to be able to work effectively, and some modification of the coordinator's teaching timetable is likely to be necessary in order to make it possible to work alongside other teachers. Appropriate in-service training for the mathematics coordinator will also be required; we discuss this further in paragraph 723.

358 There is at present a great shortage of teachers who are suitably qualified to become coordinators but we believe that every effort should be made to train and appoint suitably qualified teachers in as many schools as possible. We consider that, in all but the smallest schools, the responsibility should be recognised by appointment to a Scale 2 or Scale 3 post, or by the award of additional salary increments; we discuss this further in paragraph 662.

Mathematics guidelines

359 About half of the LEAs in England and Wales have issued mathematics guidelines. These are documents which provide guidance to teachers about the content of the mathematics curriculum and sometimes also about teaching method. The majority of guidelines relate to mathematics for pupils up to the age of 11 or 13, though some relate to mathematics for infants or for middle schools. Many LEAs have sent us copies of their guidelines. We have noted that those which have been produced most recently reflect an increasing concern with assessment. Some LEAs have produced record sheets and assessment materials related to their guidelines which can be used by teachers.

360 Most LEA guidelines have been produced by groups of teachers working under the leadership of an LEA adviser. In many cases the adviser and some of those who have helped to prepare the guidelines have then introduced them to groups of teachers at local meetings. We believe that some procedure of this kind should always be followed, so that the thinking which underlies the guidelines can be explained and discussed. If the guidelines are issued without explanation and discussion, we consider that they are likely to lose much of their effectiveness. Such initial introduction should be followed by discussion of the guidelines in each school.

361 It is essential that guidelines are kept under review and revised regularly. At present, for example, few which we have seen offer guidance about the use of calculators within primary mathematics. We believe also that many guidelines may need adjustment in the light of the information which is now available in the reports of the Assessment of Performance Unit (15) so as to place greater emphasis on the differences in attainment which exist between children of the same age.

362 One of the purposes of issuing guidelines is to ease transfer to secondary schools or other primary schools within the LEA. Therefore, where LEA guidelines exist, they should be taken into account when preparing a school's scheme of work; we discuss this further in the following paragraph.

Schemes of work

363 A scheme of work is essential as a basis for the teaching of mathematics in a school. The responsibility for its preparation lies with the head teacher but is likely to be delegated to the mathematics coordinator. A carefully planned scheme of work can assist greatly in maintaining continuity both of syllabus content and of approach as children move from class to class. In addition to setting out the progression of work in mathematics which should be followed, the scheme of work should provide guidance about the resources, including both practical equipment and books or work cards, which are available; it should also make suggestions as to the ways in which they can be used for the different topics in the syllabus. It should outline the approaches to be used in the teaching of particular topics and give guidance about such matters as assessment and record keeping.

364 The preparation of a scheme of work takes a considerable time. Wherever possible all members of staff should collaborate in the task. This not only provides an excellent form of in-service training but makes it easier for teachers to implement the scheme in their classrooms because they will be aware of the intentions which underlie it and of the discussion which has taken place during its preparation. As we have already pointed out, the scheme should take account of LEA guidelines where they exist; it is desirable, too, that there should be consultation with the mathematics adviser or advisory teacher and with the schools from which the children come and to which they transfer. It is essential that the scheme of work should be appraised and revised regularly in the light of the experience of the teachers who have been using it in their classrooms.

Small schools

365 The teaching of mathematics in small schools can give rise to a number of problems. The small number of teachers makes it less likely that there will be a member of staff with mathematical expertise; the head teacher may therefore have to act as mathematics coordinator. The task of preparing a scheme of work is as great in a small school as in a larger school but there are fewer teachers to share in the work. Because the head teacher is likely to have responsibility for a class, the operation of the scheme of work may be difficult to monitor. The age range of the children in each class is likely to be wide and, because small schools are often in somewhat isolated situations, the teachers in them may lack the support which can come from professional contact with other teachers.

366 For these reasons, some of the suggestions which we have put forward are likely to be harder to carry out in a small school than in one which is larger and an increased degree of support from outside the school is therefore likely to be required. We have been told that in some areas an additional teacher has been assigned to a group of small schools. This teacher is a part-time member of the staff of each school and acts as mathematics coordinator for all of them, working in each school in turn on a regular basis. We commend such initiatives. Advisory teachers can also give help of a similar kind and enable those who teach in small schools to become aware, for example, of the different kinds of resources which are available to assist in the teaching of mathematics and of the ways in which they are used.

Aims and objectives

367 It is common for schemes of work in mathematics to begin with a statement of aims. Such a beginning should not be regarded merely as a necessary formality but should be a statement of intent which has been discussed, developed and accepted by those who teach mathematics in a school. The aims of mathematics teaching in primary schools should be closely related to the general aims of primary education. The primary years are a time when children are not only acquiring the skills of language and number but are also experiencing a variety of methods of learning; they are learning to think, to feel and to do, to explore and to discover.

368 Aims are conceived at different levels and are of necessity expressed in general terms; they are realised as much through the methods of teaching which are used as through the topics which are being taught. Amongst the most general can be aims such as developing a good attitude to mathematics and 'opening doors' onto a wide variety of mathematical experience. Others which are rather more specific can relate to the development of spatial awareness, of ability to solve problems, of ability to use and apply mathematics, of ability to think and reason logically, of ability to use mathematical language and to the acquisition of certain skills. We believe that few would dissent from these aims, but the emphasis they receive may vary. An excellent set of aims is set out in Mathematics 5-11 (16). This is developed from the statement 'We teach mathematics in order to help people to understand things better - perhaps to understand the jobs on which they might later be employed, or to understand the creative achievements of the human mind or the behaviour of the natural world'. It concludes 'Finally, there is the overriding aim to maintain and increase confidence in mathematics ...'.

369 For classroom purposes some aims can be translated into more explicit and precise objectives but others, such as those concerned with the development of positive attitudes and appreciation of the creative aspects of mathematics are less tangible and so not easily expressed in terms of objectives. For this reason they can sometimes fail to receive their due attention. The way in which a particular topic is to be taught and a particular objective achieved needs to be considered in relation to the aims of the course as a whole and especially of those aims to which it can make a direct contribution. Some topics, for example, offer particular opportunities to develop appreciation of space or pattern, others to help in the development of logical thinking, others to develop persistence in sustained work. It is therefore necessary when working at a particular topic in the classroom to have in mind both the need to relate it to other work which has been covered and also to consider ways in which it can contribute to broader aims. Unless both of these needs are held in mind, it is possible for certain topics to become ends in themselves rather than means through which wider mathematical understanding can develop.

370 It is necessary, too, to ensure that approaches which are used in the classroom do not conflict with the aims which have been agreed for teaching mathematics in the school. For example, one of the aims set out in Mathematics 5-11 - 'to develop an understanding of mathematics through a process of enquiry and experiment' - will not be achieved if the methods of teaching which are used do not allow or encourage children to work in this way. Because it is easy for long-term aims to become overlooked as a result of the day by day pressures of the classroom, all teachers need to review the aims of their teaching regularly in order to discover whether these aims are being fulfilled within the classroom or whether they are giving way to other more limited and unintended aims.

371 It must be for each school to develop its own aims and objectives for teaching mathematics in the light of its approach to the curriculum as a whole; it must be for each class teacher to seek to achieve these aims and objectives by the provision of suitable activities for the children in the class. It is essential that support is available to help teachers in their task; in the provision of such support the head teacher and the mathematics coordinator should play a major part.

Footnotes

(1) Board of Education Handbook of suggestions for teachers. HMSO 1937.

(2) Mathematics in primary schools Schools Council Curriculum BulIetin No 1. HMSO 1965.

(3) HMI Series: Matters for discussion 9. Mathematics 5-11 A handbook of suggestions. HMSO 1979.

(4) Primary education in England A survey by HM Inspectors of Schools. HMSO 1978.

(5) Primary education in England A survey by HM Inspectors of schools. HMSO 1978.

(6) Department of Education and Science and Welsh Office. The school curriculum. HMSO 1981.

(7) VA Krutetskii, The psychology of mathematical abilities in schoolchildren: survey of recent East European mathematical literature. Trans J Teller; ed. J Kilpatrick and I Wirszup. University of Chicago Press 1976.

(8) E Ogilvie. Gifted children in primary schools. The report of the Schools Council enquiry into the teaching of gifted children of primary age 1970-71. Macmillan Education, for the Schools Council 1973.

(9) HMI Series. Matters for discussion 9. Mathematics 5-11 A handbook of suggestions. HMSO 1979.

(10) Primary education in England. A survey by HM Inspectors of Schools. HMSO 1978.

(11) Assessment of Performance Unit. Mathematical development, Primary survey reports No 1 and No 2. HMSO 1980 and 1981.

(12) KM Hart (Editor), Children's understanding of mathematics: 11-16. John Murray 1981.

(13) 'There is clear evidence from the survey that the performance of children in classes of mixed age can suffer.' Primary education in England. A survey by HM Inspectors of Schools. HMSO 1978.

(14) Primary education in England. A survey by HM Inspectors of Schools. HMSO 1978.

(15) Assessment of Performance Unit. Mathematical development, Primary survey reports No 1 and No 2. HMSO 1980 and 1981.

(16) HMI Series: Matters for discussion 9. Mathematics 5-11 A handbook of suggestions. HMSO 1979.