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Preface

The National Survey of Primary Education carried out by HM Inspectorate over a two-year period, ending in 1977, showed that mathematics was given high priority by teachers. The subject was supported by a scheme of work or guidelines in 88 per cent of the schools sampled, a higher percentage than for any other subject. Children in about three-quarters of the classes were grouped by ability when taught mathematics. For the purposes of the survey, teachers were asked to indicate which children were above average, average or below average within each class. The work was found to be reasonably well matched to the average and below average children's abilities in rather more than three-quarters of the classes, though this fell to about a half for the more able children. HM Inspectors expressed some disappointment that the efforts of the children and teachers - and those involved with initial and in-service training - did not lead to better results in the mathematics test, E2, administered for the purposes of the survey by the National Foundation for Educational Research, and reported in Primary education in England (HMSO £3.50).

This booklet has been produced against that background and with knowledge of primary schools that have proved particularly successful in mathematics. It does not call for more effort or greater priority for the subject. Rather, it sets out to identify those parts of the subject which, in HMI's view, should be taught to every child, and gives some guidance on approaches which teachers have shown to be effective. It also indicates additional aspects of mathematics which are being taught successfully in a substantial number of schools and which provide opportunities that experienced teachers can explore with their children, particularly those more able in mathematics.

The book is addressed in the first instance to heads and teachers with a special responsibility for mathematics in primary schools, as well as advisers and those responsible for teacher training. The greater part should also be of direct use to all teachers concerned with the teaching of mathematics between the ages of 5 and 11. There are, however, some matters of importance about which little is said here. For example, in this country, as in many other parts of the world, exciting developments are taking place in broadening the range of mathematical experience and knowledge which can be made available to primary children. The relevance of these developments should be given careful thought, but this is not the place to discuss such matters since they require a fuller background knowledge of mathematics than we assume in this booklet. They can, of course, be followed up in specialist publications.

In speaking of 'the mathematics which should he taught to every child', it is necessary to remember that schools contain children ranging from the very able to those who find simple mathematical ideas difficult. There is no point on the scale at which there is some sudden, abrupt change in abilities. Consequently, no attempt is made to identify minimum standards which can apply to all without exception. Though teachers aim to help all children to achieve mastery in certain areas of mathematics, it is not always possible to succeed, and the limitations of pupils as well as their capabilities need to be recognised.

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1 Introduction

During the last fifteen years many reforms have been introduced at all levels of mathematical education. At their best, new syllabuses introduce broader content and methods of teaching which involve more exploratory investigation by the pupils; at their worst, they bring only a fresh unthinking conformity without conviction. Recently, there has been a measure of reaction against innovation, and there have been protests from some critics of the educational system that certain traditional standards are not being adequately maintained.

In a minority of primary schools, mathematics is taught much as it was twenty years ago. In others a more practical and informal approach has been adopted and new content has been introduced. The new content may have been introduced piecemeal, or in the form of a coherent course of study combining traditional material with new ideas. The majority of schools have made some change of both method and content and may be claiming to 'teach modern mathematics'; but this claim needs to be interpreted with care. It means so many different things to different people that only by enquiring more deeply can we know what it means in any particular case. There is room for improvement in the teaching of mathematics, but practices vary so much from school to school that there are no universal remedies. A booklet of general guidance like this one does not absolve schools from the need to appraise their individual problems.

The belief that children should be enabled to discover important mathematical ideas for themselves has been developed over many years in numerous publications; among these Schools Council Curriculum Bulletin No 1, Mathematics in primary schools, has been particularly influential. Discovery methods are a sound approach when they are used to lead children to acquire a deeper understanding of the processes involved and more enthusiasm for the subject. Of course, neither teachers nor children have the time or skill to ensure that children discover everything, but, if their future attitudes to the subject are to be positive, it is important that each should have sufficient experience of personal discovery.

There is a certain amount of misunderstanding about investigation in mathematics: it is sometimes thought to be synonymous with practical work. In fact, while young pupils' investigations usually involve actual material and practical experimentation, it should be remembered that for much of the investigation by older children in the primary school only pen and paper are required. The practical and the abstract are interdependent at every stage, but with increasing maturity the theoretical component should increase for most children.

It used to be thought that school mathematics should consist only of learning a particular sequence of specific skills. Fortunately this idea is now less commonly held, and there is therefore less risk of skills being developed in such a way that only the able are successful, while the less able are frustrated by slowness and inaccuracy. The sequence most generally adopted is first to engage children in work that demonstrates

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the value of skills and some insight into the way skills function, and then to ensure that these same skills are acquired and consolidated by practice.

What range of understanding and which skills should be acquired by pupils by the time they are 11? The freedom which schools have to decide their own curricula means that in many areas children from different primary schools have gone on to secondary education with very different mathematical backgrounds. In many parts of the country recent initiatives by local education authorities, by schools and by individual teachers have resulted in the production of guidelines which should ensure a more uniform core of knowledge for children moving from class to class, and from one phase of education to the next.

Agreement about the levels to be achieved by pupils at different stages of transfer is more difficult to reach than it might seem at first sight. Targets that are too high may result in the slower children being taught excessively by rote methods which, though they may sometimes improve speed and accuracy with figures, can also lead to such a poor notion of the working principles that absurd answers arising from the misapplication of rules are never questioned by the children. On the other hand, targets that are too low can lead to able children being presented with an insufficient challenge, as the inspectorate has found in too many schools. Another difficulty is that targets which are easy to define may be given undue priority over those which are more difficult to specify, but which may be of greater importance in the long term.

There is a clear need to reach some agreement on these issues within any one local education authority area. In an educational system which does not seek to achieve a consistently developing programme of work for children by central direction, there is a responsibility on the various partners to achieve it by consultation.

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2 The place of mathematics in primary schools

Mathematics, at least in the form of arithmetic, has had an important place in the education of young children since well before the Education Act of 1870. The obvious reason for this is that numbers are a part of everyday life and adults find it useful to calculate at work, while shopping and when engaged in sports and pastimes. Nearly all primary school children are capable of learning what is necessary for most of these purposes.

For many years, speed and accuracy in working out sums on paper helped some people to obtain clerical jobs that were more secure and physically less arduous than many others. Today, with a widespread use of mechanical and electronic aids, speed is governed by the machine operator, but it is all the more important in offices and factories to know whether the answer arrived at is reasonable, or whether it has been distorted by a slip on the part of the operator or a fault in the device used. Consequently, there is a special need for understanding of the processes used, and for the ability to estimate what might reasonably be expected as an answer. Both can be improved by working at written calculations in which the method is displayed. Moreover, it would be foolish to rely completely on mechanical or electronic devices. Facility with mental arithmetic is as important as ever.

Children should also be able to weigh and measure with a degree of accuracy appropriate to the task in hand, and given suitable opportunities the majority of primary school children can learn to do so.

In recent years it has become more important to understand the various ways in which mathematical information is presented. Graphs of various kinds and statistical tables are commonplace in the newspapers and on television. They are often introduced into primary schools now and quite young children can both get pleasure and acquire insight, if the work is suitable and its implications are sufficiently probed by the teacher.

The learning of the necessary skills in mathematics is certainly part of the basic course in primary schools. Not all the children who might acquire them do so and, as later chapters of this booklet suggest, care has to be taken in pacing the work and in providing the right preparation and range of activities for it.

But there is a good deal more that mathematics can do for children if the teacher is sufficiently knowledgeable to be confident in the subject. Clearly, it is not essential that every man in the street should know that the ratio between the circumference and diameter of a circle is about three and one-seventh and is represented by the Greek letter π. Most people manage to get through life reasonably well without realising that the shell of a nautilus, the florets in the head of a daisy and a spider's web are all arranged, nearly enough, in the form of equiangular spirals. Still more often, they get by without knowing that the number of left turning spirals is different from the number of right turning spirals in the head of a daisy, and, what is more, that the two numbers appear side by side in an interesting mathematical sequence named after the greatest of

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medieval mathematicians, Leonardo of Pisa (nicknamed Fibonacci). People who have been made aware of these curious but orderly aspects of nature may gain more pleasure in life as a result. There is a whole range of relationships and patterns of a mathematical kind that can catch children's imagination and, though they are not essential, it is a pity if some of them are not included. They cannot however he allowed to oust the essential, and there is no sense in introducing them if the teacher feels uncertain of the material, or under strain. Certainly there should be some way of providing such aspects of mathematics for children who have a flair for them - not always those children who are quick and accurate when doing sums.

In the course of learning mathematics children have to learn a good many other things. For example, they have to learn to be neat and tidy, for muddled working may produce a wrong answer. They need to learn to be careful and they ought to learn to be discriminating. They should learn to check what they have done; they should also learn that understanding and knowing how to do something is as important as getting the right answer.

The purpose of teaching mathematics

The need for schools, both primary and secondary, to provide their pupils with a good foundation of mathematics has been argued on three main grounds. Mathematics is useful, mathematics is part of our culture, and mathematics trains the mind. There are many other arguments but they nearly always reduce to variations of these three; and the three interrelate with one another.

The utilitarian argument can be presented in both narrow and broad versions. At one extreme each person needs to know enough arithmetic to make simple purchases, count change, check wages and understand a popular newspaper, but it can be fairly claimed that he has no basic need for very much else. The broad utilitarian case goes much beyond this. It can be argued that mathematics is of fundamental importance for the understanding of the physical sciences and technology, that it is increasingly employed in the social sciences, business management and social administration, and that the person with a knowledge of mathematics has the key to many other areas of knowledge as well.

The arguments for mathematics as a cultural subject can also be presented in a variety of forms. Different people may define culture in different and unnecessarily divisive ways; our problem is not so much to define utility or culture, but to establish common objectives towards which teachers with differing outlooks can work. If a sufficiently broad view is taken of utility on the one hand and culture on the other, then these two arguments for teaching mathematics reinforce each other. 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 is the particular power of mathematics that its central ideas help us to do all of these things.

The third argument, that mathematics is a training of the mind (quite apart from any use of the actual material studied), has some validity, but it is open to abuse. Mathematics can provide a valuable mental training, but many other things can do this just as well; and mathematics cannot be justified on this ground alone if it deteriorates into stereotyped working to rule, with the higher functions of the mind

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neglected. Mathematics can be justified as a training for the mind, but the training also needs to serve other purposes which can be understood by the pupil at the time. There is something wrong with the teaching if the reply to 'Why are we learning this?' has to be 'You will understand later on!'

The mathematical aims

Aims are essentially declarations of intent that give direction and shape to a scheme of work or teaching programme. Although they may lack detail or precision, they should remain in the forefront of the teacher's mind at all times when he or she is considering the provision of experiences which lead to the mathematical development of the child. The formulating of aims and then of objectives is necessary before attempting to draw up a scheme of work.

Listed below are some of the aims within the primary school from which objectives will later be derived. It is not an exhaustive list and the order is to some extent arbitrary. It will however allow the teacher to consider their relative importance and to encourage discussion with colleagues who share in the planning of the work throughout the school.

The general aims should be to develop:

i. a positive attitude to mathematics as an interesting and attractive subject;
ii. an appreciation of the creative aspects of the subject and an awareness of its aesthetic appeal;
iii. an ability to think clearly and logically in mathematics with confidence, independence of thought and flexibility of mind;
iv. an understanding of mathematics through a process of enquiry and experiment;
v. an appreciation of the nature of numbers and of space, leading to an awareness of the basic structure of mathematics;
vi. an appreciation of mathematical pattern and the ability to identify relationships;
vii. mathematical skills and knowledge accompanied by the quick recall of basic facts;
viii. an awareness of the uses of mathematics in the world beyond the classroom. Children should learn that mathematics will frequently help them to solve problems they meet in everyday life or understand better many of the things they see, and provide opportunities for them to satisfy their curiosity and to use their creative abilities;
ix. persistence through sustained work in mathematics which requires some perseverance over a period of time.

The objectives

Having decided on the general aims, teachers also look for ways of achieving them, and for classroom practices which accord with these aims. It is therefore essential that aims should be converted into more detailed descriptions of performance which teachers can accept as evidence of learning and as 'milestones' along the path of progression and development. These can be regarded as objectives. In forming

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objectives, we are categorising evidence of learning in terms of performance and achievement, ie thinking, remembering and understanding. The objectives of a programme can usually be set out precisely and clearly in terms of what the children are intended to learn.

Listed in Appendix I are the concepts and skills which children are normally expected to acquire before leaving the primary school.

It must be recognised that the objectives listed in the appendix will never be achieved within the primary school by some children, particularly those with special educational difficulties. On the other hand these 'milestones' should have been reached by the age of 11 years, by perhaps 75 per cent of children, and surpassed by some.

In view of the present diversity of practices and aims, and the current criticisms of the standards being achieved, it would have been very attractive to seek improvement by establishing a central core or agreed minimal goals. But a discussion of standards and quality implies pre-established criteria; at the present time primary teachers are not all agreed on common aims and objectives in mathematics. Some stress factual knowledge to the exclusion of other objectives; others emphasise the importance of processes and understanding, and the enjoyment of mathematics. The formulation of minimum requirements could be a helpful step but, unless the formulation is accompanied by continuing attempts by teachers to develop their professional judgement, minimum standards could too easily become an accepted norm. Any statement of minimal goals needs to be accompanied by some non-prescriptive indication of the broader range of work which might be appropriate for most children.

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3 Planning the mathematics programme

Introduction

The primary teacher today is faced with a considerable task, brought about by the changes which have taken place in the teaching of mathematics. These have involved new content, new terms, new concepts and what many regard as a new approach to the teaching of the subject (although this approach has a very long history).

Today, the child is encouraged to make enquiries, investigate, discover and record; learning is not looked upon only as something imposed from without. It is recognised that it is through his own activity that the child is able to form the new concepts which will in turn be the basis of further mathematical ideas and thinking. These early experiences provide the foundation on which future learning is built.

The school which concentrates single-mindedly upon arithmetical skill alone is neglecting the whole range of important logical, geometrical, graphical and statistical ideas which children can meet before the age of 12. The school which stresses 'modern' topics and practical work for its own sake at the expense of consolidating number skills, systematic thought and learning, is equally guilty of neglecting its charges. The challenge is to encourage children to develop their mathematical education along a broad front of experience while ensuring systematic progression and continuity.

The primary teacher, confronted with the task of providing a wide experience for her children, can be bewildered by the wealth of apparatus, material and equipment which are now available. She must be capable of informed choice, bearing in mind the needs of her individual class within the school. Planning is vital; and it cannot be achieved by teachers in isolation. Infant and junior teachers need to plan together within the school and between schools.

Although most schools have a scheme of work for mathematics, in many schools this needs to be revised; what is perhaps a greater problem is that large numbers of teachers experience difficulty in translating the scheme into an effective mathematics programme.

Within some primary schools today, teachers are endeavouring to work for some of the time in such a way that subject barriers are not emphasised. Many 'integrated' studies, resulting from this way of working, lend themselves admirably to the introduction of mathematics. This mode of working requires understanding of the mathematical potential of a wide variety of situations, and this in turn demands more mathematical knowledge than many teachers possess. As a result, the opportunities for developing mathematics from an integrated topic are too often under-developed. The thematic approach is unlikely to motivate all the mathematics which most children need to cover within the age range 5-11 years; it is necessary to provide adequate time for mathematics, to cover a scheme of work systematically, and to include sufficient regular revision of those skills which have been identified as necessary for further progress.

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Organisation

The school or classroom organisation can be critical in determining the effectiveness of mathematical learning. Any decision on organisation needs to take into account the aims and objectives decided by the school or by teachers themselves.

It is essential for the teacher to intervene appropriately and give support and help. not only in the later stages of mathematical learning but also in structured 'play'; otherwise these activities are not fully utilized and can easily become meaningless and result in time wasting and a lack of progression. An activity which does not have the teacher's attention can seem to be less important to the children. In addition, if the only work which draws the teacher's attention is that written in a book, it is quite natural that children should desire to work in this way, however inappropriate it may be, in order to attract the teacher's approval. There are occasions, particularly with the younger children, when the teacher needs to recognise that participating with the children in the activities she arranges might sometimes be the best use of mathematical time.

When the activities of a class are well organised children are able to work with much less direct supervision and teacher support. This allows the teacher to work with smaller groups and individuals within groups. Class organisation which allows children the opportunity to exercise an informed choice needs not override the wish of the teacher to withdraw a group of children in order to teach them. Indeed, professional time can easily be wasted if on one day a teacher finds that she needs to introduce or teach the same skill eight times with eight individuals separately. In general, it is the extremes in classroom organisation which militate most acutely against the effective learning of the subject.

Forms of organisation which require children and their teacher to change their activity after a set period of time inhibit sustained work in mathematics. This is particularly true if the child has been working constructively with material or apparatus and needs an extension of time to complete his task before the equipment is packed away or used again for something else. A child will often work with deep concentration and effort on a task which has interested him, and to ask him to move off quickly on to some other area of experience can be unwise. If it is decided to impose timetabling restrictions these should be interpreted flexibly, bearing in mind the needs of the individual child. Over-fragmentation of the child's day should be avoided.

A further cause for concern is the quality of mathematical education which is available to those children who are able in the subject. Too often, schools present an insufficient challenge to the more able or highly gifted. In primary schools the problem is as important as at the secondary stage.

The efficacy of an organisation for mathematics can be judged by the following criteria.

Does the organisation provide opportunities for:

i. direct teaching of individuals, groups of various sizes and the whole class,
ii. practical work with appropriate material in a range of situations,
iii. children to use mathematics across the curriculum and to see the relevance of mathematics in the different areas of study which mathematics pervades.

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iv. discussion and consolidation of mathematical ideas with individuals, groups and the class,
v. project work or studies,
vi. effective remedial work for a variety of ability levels,
vii. extended experiences for the more able pupils,
viii. children to reflect on their experience and the kind of thinking they are engaged in, so that they are aware that the activities in which they are involved are mathematical,
ix. children to learn relevant work skills:

recording and clear presentation, including an understanding of why this is important,
the use of reference books,
the use of measuring instruments.

If teaching is to be successful, it is essential that the teacher should assess what is happening. Assessment, evaluation, diagnosis and prescription are all important and should feature in the planning of work in a school, in a particular class or for groups or individuals. These forms of assessment are essential if children are to learn mathematics effectively and to make progress that is in accord with their ages and abilities.

Responsibilities at various levels

Responsibility for assessment operates at different levels - national, local authority, school and teacher.

There was not, until recently, a standardised system for judging the attainment of pupils on a national level. The Department of Education and Science (DES) set up an Assessment of Performance Unit (APU) in 1976 to provide for this. Mathematical attainment began to be monitored in 1978 by the National Foundation for Educational Research on behalf of the APU.

Local education authorities (LEAs) have discretion with regard to their own procedures for monitoring attainment and many authorities are taking steps to establish systems of their own. In the future, it is possible that many will use material related to the national system of monitoring as these assessment procedures become available. LEAs have freedom to employ their own methods and this provides opportunity for various systems to be tried and for practice to develop. Nevertheless, there could be disadvantages if the schemes they adopt are too widely disparate.

Assessment in the school

Schools and individual teachers have problems at other levels, and it is with these that this booklet is mainly concerned.

As a first step, the school should decide the purpose of its assessment procedures. The aim may be to grade children in order to assist transition to the next stage of education, and in this case there is little choice. A uniform scheme devised by the LEA would seem almost essential. Where the school has discretion, assessment may be part of a philosophy which embodies a belief in the stimulus of competition, or its purpose may primarily be diagnostic (seeking to reveal the learning problems of individual children), or it may be part of a more general strategy seeking to modify the future teaching planned for a group of children in the light of the collective progress made. All of these

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objectives imply different types of tests and appropriate record-keeping procedures.

It is necessary to evaluate what both individuals and groups are learning; the results may or may not reflect what the teacher believes she has taught. These procedures demand great courage and professionalism. A teacher should not feel a failure if, on occasion, what has been taught has not been learnt, providing that assessment is continually being carried out. Following the evaluation of the work done by pupils, it may be necessary for the teacher to diagnose the difficulties of a group of children or of a single child within a group. When a difficulty has been identified a prescription which gives specific help should follow, if success is to be achieved.

It is all too easy to restrict assessment to those aspects of mathematics teaching which are most easily tested. Efforts should be made to broaden assessment procedures to include as many as possible of the initially planned objectives of the course.

It is essential to know the ability of the child to apply skill and knowledge to problems associated with the world in which he lives. The teacher needs to know the child's attitudes towards mathematics, his perseverance, creativity (elaboration, fluency, flexibility and originality), his understanding, visualisation and psychomotor skills. At the present time, skills which are described as mathematical are applied across the curriculum. It is necessary to assess this - to assess the ability to generalise, to classify and to identify and select the essentials which determine the solution of a practical problem.

Formal examinations, based on syllabus content, frequently limit the teaching of mathematics to that which is to be tested. Objective tests, although they give a wide coverage and facilitate rapid marking, seldom reflect good methods of teaching or satisfactory levels of learning. In addition, the existing tests in no way assess mathematical creativity.

Oral questioning is an important method of checking, particularly for some areas of the curriculum and for some pupils. Certain aspects of the work can be reliably tested particularly well in this way (for example, rapid recall of number facts). For other work, judgement based on this type of testing, unless very carefully prepared, can be unreliable. Oral questioning is usually very time consuming.

There are certain long established standardised tests. The use of a well validated test will not of itself be helpful unless the teacher takes the trouble to learn the purpose of the test, studies the appropriate method of administration, and appreciates the limitations. Some teachers prefer to plan their own assessment tests, believing that such tests can be more closely related to the teaching objectives. Where this is the practice, an attempt should be made to learn something of the expertise laboriously acquired by professional testers over many years, and to apply it appropriately.

It is also necessary for schools and for teachers to evaluate the teaching methods and materials they use. This involves the careful scrutiny of materials, schemes of work, text-books, work-cards, equipment and apparatus, to see if they are providing what is required.

Assessment might be regarded as a procedure which challenges the teacher to define aims and objectives more clearly, and subsequently leads to more effective teaching and learning. It allows the teacher to check if the aims have been achieved and the objectives reached. Most

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learning experiences need to be planned, and it is at the planning stage that the fullest assessment (evaluation, diagnosis and prescription) is important if the experiences are going to meet the real needs of the children. For this there are no standardised tests, and the teachers involved must rely on their professional judgement. This judgement can often be sharpened by collaborative work within the school or at a teachers' centre.

Finally, no methods of assessment are sacrosanct. From time to time, the methods themselves require reappraisal in order to decide whether or not the purposes they are intended to serve are being achieved.

Teacher development

As headteachers, advisers and organisers of in-service training know, a problem almost as large as that of the initial training of teachers is that of helping teachers already in post to acquire the necessary knowledge and skills. In spite of the great efforts which have been made over recent years, it is still the case that too many teachers have to teach mathematics without knowing enough about the subject, or about current ideas of teaching it. Additional provision of in-service training is only part of the larger problem of enhancing the quality of the teacher's professional life. Teaching innovations fail unless the teachers are fully conversant with, and convinced by, the reasons underlying the innovation. In-service training must be directed above all to the development of the teacher's own capacity to make judgements.

The primary teacher today needs all possible help, support and encouragement in teaching mathematics to those for whom she has responsibility. Whereas, for some subjects, teaching groups within which there is a wide range of ability give rise to few difficulties, in some aspects of mathematics they can present major problems for all but those who have substantial mathematical background and are both experienced and skilful teachers.

Individual and group work is essential and there is a need to provide and supervise a range of different tasks each lesson and to keep adequate progress records for each child. These demands make the problem of classroom organisation difficult and support is often needed by those teachers who are insecure or less experienced in the subject. Some success is being achieved in schools which have one teacher on the staff with special knowledge of and interest in mathematics, who has some responsibility to undertake the in-service education of the other members of staff. Support at this level is vital if the primary teacher is to gain the knowledge and confidence she needs to carry out her task.

In schools where the mathematics programme is of some quality, the head supports his colleagues and gives them a positive lead in what is for many a difficult area of the curriculum. It is essential that there is a coherent and systematic policy for mathematics throughout the school. The head also needs to ensure that the appropriate resources are as far as possible made available within the school to serve the teachers' needs. Where a teacher has been given special responsibility for mathematics, it is necessary to provide appropriate opportunities for her to work in a way which makes it possible to carry out her responsibilities - once these have been defined - in giving support and help in the classroom to her colleagues and in arranging that meetings of staff are planned and occur regularly.

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Local Education Authority mathematics advisers, the staffs of colleges of education, institutes of higher education and other providers of in-service training, including the wardens of teachers' centres, are all playing a valuable part in the development of in-service training courses for primary teachers at all levels. It is important that these courses not only offer opportunities to stimulate development and help teachers achieve realistic goals, but also enable them to improve their own understanding of mathematics and of the ways in which children learn. Activity at these levels needs to be encouraged and increased.

The curriculum followed in any school will be found set out in the Head Teachers' scheme and the class teachers' syllabuses of work. It is considered essential nowadays that each teacher should possess a copy of the complete scheme of work of the school, so that he may be able to understand the part which he is called upon to play in the education of pupils.

Handbook of Suggestions for Teachers.
Board of Education (1937)

The planning for the provision of mathematics in the primary school is as essential today as it was forty years ago. The complexity of the problem has increased and the problem will be solved only by the development of a scheme of work drafted by teachers sharing ideas and planning a cohesive framework on which to build a sound foundation of experience and learning for their children.

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4 Communication: language and logic

Language

Language - whether everyday, formalised or symbolic - is an essential tool for the formation and expression of mathematical ideas. Having to express an experience or an idea in everyday language helps even young children to become more clearly aware of it. Moreover, the teacher engaging in conversation with them can appreciate their level of understanding and facilitate the next step, clarifying the idea and extending the control of language at the same time. By these means children also find out what it is that they did not understand, and are enabled to move towards a solution of the immediate problem. Their ideas become more precise as their grasp of language is extended and refined, and it is here that the teacher can do most to help them. But the wider vocabulary and the more complex phrasing towards which they must move will have meaning for them only if the teacher can start from the child's absorbing interest in himself and his environment.

One element of this interest is the urge to touch and handle whatever he sees whether in the natural or the man-made environment. So he is likely to come to school recognising streets, houses, shops, vehicles; probably familiar with trees, plants, birds, stones, steps; and possibly with some knowledge of shells, pebbles, rocks and sand. Already he has experience which is fundamental to many mathematical ideas. Some children have begun to use such prepositions as over, under, on, behind, next to, between; can say whether a stone or a tree is big or little; whether a shell is heavy or a rock light. School provides a child with more opportunities to 'feel the language'. He is encouraged to experiment with large bricks and constructional toys, with water and clay, with all the household articles in the home corner. He is told stories and verses incorporating counting and number names. He paints 'right to the edge', fills the jug to the brim. He may match cups to saucers, find out if a crown 'just fits' a king's head, or take three steps and a jump to match the rhythm of a polka. He can feel flat and curved surfaces, balance a stick across the table - and learn to talk about all of these activities. So he builds up the language of number, order, size, shape and weight, as the teacher seizes her opportunities to introduce the appropriate new word or phrase.

After meeting them often enough in context, the child incorporates them in his own running commentary on his activities; skilful questioning on the part of the teacher can extend his linguistic range even further. At this stage the natural environment and man-made materials, including play equipment, are all that the teacher needs to lay the foundations of mathematical language.

Structuring of experiences is essential if language is to develop over a sufficiently wide range of ideas. For instance, children need to handle objects which are relatively heavy and light, of the same and of different shapes and sizes, and to compare them and arrange them in various ways. When a balance is being used, two objects only can be compared at the same time; the problem of finding which is the heaviest of three objects is much more difficult. The use of language to make

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comparisons is of great importance, and from conversation about such experiences words describing relationships are learned: relations of size: longer, shorter, thicker, fatter, heavier, lighter: relations or position: first, second, last, middle, inside, outside: relations of time and speed: early, later, before, after, faster, slower.

Within this general provision special care needs to be given to making available a range of materials which offer children experiences related to the four operations of arithmetic. Discrete materials (separate objects) and continuous media (eg string, water etc) are both required. Initially this work should be carried on without counting or formal measurement, but later on appropriate activities can be devised to introduce the various processes of arithmetic. (Activities of this kind and the formal processes which they introduce are described in detail on pages 26 and 27.)

So far, children's 'language of mathematics' is oral, and based on everyday words. But the mature language of mathematics has special features of its own, to which children need to be introduced progressively. The correct terminology (eg multiplied by, horizontal, diagonal), the symbols and the formal methods of calculation should be learned when they can be understood and appreciated. It is important that children should be enabled to express their mathematical experiences and ideas in a variety of ways: not only talking and writing, but drawing, making diagrams and models, constructing graphs. One medium of expression is not enough. Having to talk about a diagram often reveals what is misleading about it, and constructing a model from written instructions can lead to correction of the instructions themselves. Once the children become practised operators in a range of media, they can begin to choose the best way of expressing different mathematical ideas.

As time goes on the language of mathematics acquires a momentum of its own; for example, children become able to calculate without having to wonder what the symbols mean. It is because the language of mathematics is so elegantly precise that it gives so much help in analysing complicated problems. But the use of symbolism cannot be hurried. Initially children need to see the meaning of the arithmetic they are doing step by step, and ordinary language is needed all the time to explain and interpret calculations.

The role of language in the learning of mathematics is clearly recognised in many mathematics schemes in use in schools today. Current schemes (as indicated by LEA guidelines, development projects, commercial publications and attainment tests) incorporate a far wider vocabulary than those of twenty years ago. This arises from a determination, on the part or those designing the scheme, to base mathematics on a broader foundation than is provided by counting and number work alone. However, one indirect consequence of some styles of teaching (for example, over-reliance on work-cards) is a regrettable diminution of the role of language in the learning of mathematics. Yet mathematics affords invaluable opportunities for encouraging precision and clarity of language. The teacher has to use her judgment continually, making decisions about the need to repeat experiences, to extend them, to consolidate vocabulary, to question or to answer. Conversation - between her and an individual, or a group, or the class - is essential as a way or assessing progress and planning the next

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step. Whereas answers to 'sums' may be right for the wrong reason or wrong for good reasons misapplied, conversation is a reliable indicator of what has or has not been understood. A child may recognise a pattern or discover a relationship, but often the idea is consolidated only when he has talked about it. It is essential that the teacher should listen to any spontaneous comment from the child, and try to assess the degree of understanding revealed by it. If he is half-way to the discovery, she needs to ask the question that will lead to it. If he has arrived, she needs to find out whether he can generalise the discovery by applying it to other cases. She has to estimate the intellectual leaps of which each child is capable. She has to refrain from 'telling' when he seems to have reached a dead end, and from depressing him by expecting too much and being disappointed. With the appropriate words and gestures, she encourages the slow, coaxes the reluctant and challenges the able to further discoveries. Furthermore, she gradually helps each of them towards a more exact use of language in mathematics as elsewhere, discussing alternative words and phrases, their meaning and ambiguities. and the importance of precision.

English is not the first language of some children in our schools today. There may well be interference with their learning of mathematics from the type of systematisation* of the number names used in their mother tongue, and the mathematical notions tacitly embodied in it which may be distinct from our systematisation. It would help the teacher considerably to know something of the number system of the children's country of origin, so that she can understand their difficulties. More information is required, especially by teachers engaged in remedial work with children from ethnic minority groups.

Logic

Just as practical investigations, thinking about what one has seen and done, and discussions in every day language lead to the ideas of number and space, they also lead to the intuitive ideas of logic. 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. These notions all have mathematical aspects, and they are the foundations on which an understanding of number and methods of calculations is built.

There are schemes of work, sometimes involving structural apparatus, which have been specially devised to teach logical ideas to young children in a systematised way. A certain degree of knowledge and skill is required before the benefits of these schemes can be realised. Before undertaking such work many teachers need experienced guidance, at a local teachers' centre or in some other way. It is good that strong teams of experienced teachers should devise schemes of work which develop children's logical ideas: but no school can teach everything, and it is easy to attempt too much. Without such a specialised approach a school can still develop a satisfactory scheme.

Much the same can be said of the explicit introduction of the terminology of sets into primary school mathematics. The ideas are explained in detail in hooks such as Nuffield Guides (Guide to the Guides. Sets, p 41 ) and Primary Mathematics Today (for details see

*For instance, there is considerable irregularity within the teens in English: eleven and twelve are not constructed like the rest, but even the numbers from thirteen to nineteen present difficulties because the numerals used are not in the same order as are the elements of the words used to refer to them; we write 13 but must say, not 'teen three' but 'thirteen'. For the Italian child there are no totally irregular number names in the teens, but the change from 'backwards' to 'forwards' comes at a different stage, after 16 (sedici) - 17 is written diciasette, and 18 and 19 follow suit. For the Cypriot child there are different discrepancies: endeka 11 and dodeka 12 run 'backwards', and the remainder of the teens all begin with deka, just as they are written in numerals. In the major Asian languages, each numeral up to 40 is a distinct word, not built up in a pattern.

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Appendix III). These ideas can be very helpful, but if the teacher is in doubt as to their value, or if skilled guidance is lacking, a satisfactory scheme can be developed which does not require any special notation or terminology.

Logical thinking should certainly be cultivated in the primary school in some way; and this can be done without any special technical language provided that there are enough suitably chosen materials and previously established ideas available to discuss. Most children display evidence of logical thought but not always in a way which is recognised by an adult. Children's level of thinking can sometimes be in advance of the ideas the teacher expects. It is moreover necessary to listen carefully to children to try to find the underlying reasons for their apparently incorrect replies to questions. In arithmetic, as in the grammatical aspects of language, a child's 'mistake' may arise from continuing to use a previously established pattern in circumstances where it no longer applies.

It may be unwise at the primary stage to think of any ideas labelled 'logic' as constituting part of the essential syllabus for all children: but the associated aspects or language require careful consideration when a scheme of work is planned in mathematics.

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5 Number

The approach to number

An experimental and practical approach should be made to number. At each stage the ideas should arise from investigations carried out by the pupil with the encouragement of the teacher; subsequently the key ideas have to be identified, arranged systematically, and practised by applying them in a variety of ways. There is nothing new in the suggestion that children should approach number skills through practical activity, and that they should achieve conviction and understanding before consolidating their knowledge by deliberately committing certain essential facts and processes to memory. This theme is frequently to be found in the professional advice on the teaching of arithmetic which has been given over many decades.

The achievement of a proper balance between teaching for understanding and teaching for skill is currently the subject of much debate, and complete agreement is very unlikely as fundamental questions of educational aims are involved. But it is unnecessary to set understanding and basic skills against one another since they are complementary: both are needed. Many of the novel ideas which have come into primary schools in recent years were introduced with the intention of establishing basic skills on a surer foundation. Their success in doing this has been mixed: further progress demands a continuing reappraisal of what has been achieved so far. An agreed policy is needed on reasonable levels of skill as targets at various crucial stages of education - especially ages of transfer.

Early stages of number development

Early language and number

Before a child starts to talk, his parents and brothers and sisters begin to introduce him, often without intending to do so, to a variety of mathematical ideas. Counting fingers and toes sets up associations between sounds and sets of objects.

Comments such as 'What a big/strong/heavy/boy you are' introduce ideas of size and measure. Handling him - 'Ups a daisy', 'Down we go', or pushing a child on a swing 'backwards', and 'forwards' - involves notions of direction.

Early school activities, nursery rhymes and stories such as The Grand Old Duke of York, Jack and Jill, Three Little Pigs, Three Billy Goats Gruff, Ten Men Went to Mow, help to consolidate and extend these experiences. Many children's games introduce number in one form or another: these include skipping, hop scotch, conkers, marbles, 'casting out' rhymes.

None of these experiences guarantees methodical progress. The fact that a child can recite the numbers up to fifty does not necessarily mean that he can count. Infant teachers know that counting involves counting 'things' and matching number names to objects one by one and not merely reciting number names. These various activities provide a wealth of matching experiences for the children. Some of them include matching sets of objects and learning their corresponding

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number names and symbols. Others involve matching colours and matching shapes: logic blocks, jigsaws and tiling games. Some of the matching (such as words to pictures) is aimed at language development, some at mathematical growth. Resulting from all these are activity, free and directed conversation, recording of various kinds, reading and writing.

Teachers need to choose words with care and to see that children use words with care; they also need to help children to see what common factors there are in what appear to be very different experiences ('red' covers a large range of colours; 'food' refers to many different objects).

Pre-number and early number experiences

Children need many preliminary experiences in sorting and matching, in comparing objects and materials of all kinds, and in using arbitrary measures such as a child's hand or foot, long before they learn to count systematically and to use standard measures. The following notes assume the rich background of experience with which many children are provided in school.

Although the concepts are noted in a logically progressive order, the order is not intended to be adhered to rigidly in practice; it would be unwise to believe that any one concept can be considered to be 'understood' in any final sense before another one is embarked upon. Throughout, the language employed is appropriate for discussion between teachers; it is not necessarily the language the teacher needs to use with the child.

Sorting
Sorting is a natural activity which children enjoy and from which they learn. It is also a necessary stage towards understanding the meaning of number; the result of sorting is one or more sets of things and number is a property of sets. Discrimination and classification are required in order to sort. In addition, the notions introduced lead to further logical thought and decision making.

Matching experiences leading to the appreciation of one-to-one correspondence
Experience of matching in a variety of ways and in many different situations is helpful in the process of establishing the idea of cardinal number. The cardinal number 'two' is an abstraction from all conceivable pairs: two toys, two cats, two eyes, two blocks, two legs, etc.

Ordering
Putting objects in order of size can assist children to appreciate later on that numbers can be put in an order of magnitude: 1, 2, 3, 4, 5, 6 ...

Learning to count
The ability to count is a skill that should be acquired by almost all children before they are seven years of age. It is not such a simple matter as it seems once the skill has been acquired. It rests upon the ability to see things both as individual items and as members of a set.

Another essential element in learning to count is that the ideas of 'more, equal and less' must be well appreciated. Children need to be able to recite the number names in their proper order as far as

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they are expected to count. The association of those number names, one by one, with the objects being counted is no easy task for some 5 and 6 year-old children. They need a great deal of experience in matching one object to another and one series of objects to another, for instance buttons to button-holes, if they are to absorb the idea of matching and apply it as it must be applied when counting.

Numerals
A numeral is a symbol used to denote a number, although this logical distinction is not usually important. Number symbols are all around us and children must gradually come to associate them correctly with the numbers they have learned about practically and orally. Much experience should occur in the course of reading and writing the numerals used in daily activities.

Conservation of number
This idea is sometimes misunderstood.

Many 4 and 5 year old children, and some older ones, have to learn that the number of objects in a group remains the same no matter how they are spread. Sets of five objects remain sets of five however dispersed or positioned, and there is a one-to-one correspondence between members of these sets. Children at this stage need considerable experience to establish this concept.

The two aspects of number
The two aspects of numbers are the 'ordinal' and the 'cardinal'. When we use numbers to say how many objects there are in a set, we are using numbers 'cardinally'. When we say where an object comes in a sequence, we are using number 'ordinally'. The words first ... second ... third ... etc. introduced to emphasise that we are interested in the order, are sometimes called 'ordinal numbers'. The misuse of vocabulary often leads to confusion: many young children, having counted five objects, will offer the second when asked for two. Nevertheless, teachers should not be over-pedantic and force unnatural usage on children in this matter. But teachers must exercise great care in their own speech to use words correctly.

Counting is an activity that should be extended as soon as possible: on occasions children should be encouraged to go on counting as far as they can.

Children come to school accustomed to solving problems. We should not therefore divorce number from problems as we so often do. Children are not ready to practise 'sums' until they can explain much of what they are doing. They should not use symbols until they can convey orally and pictorially what they mean. When children meet a new type

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of 'sum' they should ordinarily meet it in the form of a problem; when given a 'sum' they should be able to suggest a situation from which it might have arisen. There are some children who are fascinated by numbers and enjoy looking for patterns and learning number facts. They memorise facts far in excess of the minimum normally expected. Children interested in number should be encouraged to memorise facts if they are prepared to do so.

There is only one way in which a teacher can find the extent of a child's number knowledge - that is by observing and questioning him, much as she hears him read. This should take place often with young children and regularly with juniors. Teachers need to keep careful records of each child's number knowledge.

Early number relationships

Between the ages of 6 and 7 and before children move on to the four processes of number and use expressions involving two numbers, teachers need to ensure that they have been given a wide range of opportunities to gain experience of one number and its relationships with others, particularly for the numbers 0 to 10 inclusive.

Children should acquire the confidence at this stage:

i. to count with ease:
ii. to understand that the number of objects in a group does not vary, even though the grouping is rearranged;
iii. to put numbers in order
eg the staircase of numbers 1, 2, 3 ... 10 and to rearrange a sequence of numbers, eg 7, 2, 12, 9, 4, in ascending order:
iv. to appreciate the 'spatial concept' of number
eg possible spatial arrangements of five objects

v . to understand the relationships of one number with others in the range 1 to 10, through the use, for example, of number ladders and strips;
vi. to remember the pairs of numbers which add up to 10;
vii. to be able to 'partition' numbers, ie 'break up' numbers into several parts
eg 5 is: 4 and 1, 3 and 2, 2 and 2 and 1, 1 and 4, 2 and 3, 1 and 2 and 1 and 1, 0 and 5.

Most children, by the age or K, need to be able to:

• add/subtract 1 or 2 to/from other numbers up to 10;
• double numbers up to 10 and halve even numbers up to 20;
• understand and use facts such as 5 + 2 = 7 = 2 + 5, ie the commutative law of addition. (Once a child can use this law, whether or not he knows its name, the number of addition facts he has to learn is almost halved);

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• know all addition and subtraction facts up to 10, for example

It is most important that number pairs whose sum is 10 should be thoroughly known:

• add 10 to numbers 0 to 10 with immediate response. (This takes time, and children often revert to counting on in ones in moments of stress);

• by quick recall (not necessarily immediate recall) add 9 to numbers 0 to 10 eg 9 + 6 = 15 (giving a reason such as 10 + 6 = 16, and 9 is one less than 10);

• by quick recall, know 'near doubles' eg 11 + 7 = 15 (giving reasons such as 'two eights minus one' and 'two sevens plus one');

• understand and use different addition processes: eg 8 + 5 = 8 + 2 + 3 or 5 + 5 + 3 or 10 + 5 - 2 (Children will need special help in learning these facts and will find it helpful to discuss the many different methods used);

• understand place value (Use of commercial structural material and the abacus is very valuable at this stage. It is important that children should be given frequent opportunities to undertake counting of collections they are making, organising these objects in tens. There is a need to concentrate as much on numbers between 10 and 20 as on those up to 10, before developing further work beyond 20, again consolidating place value);

• by quick recall, recognise repeating patterns of numbers,
  eg 7+10= 17, 17 + 10 = 27 etc
  18 + 6 = 24, 28 + 6 = 34 etc
  and 8 + 16 = 24, 8 + 26 = 34 etc and corresponding subtraction patterns:
  94 6, 84 6, 74 6 etc.

A number line is invaluable in the early stages for the discovery and consolidation of many of the above relationships. Addition and subtraction are seen as steps forward and back, and this can he reinforced by superimposing strips of card of the appropriate length.

A NUMBER LINE

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Teachers need to keep a careful record of each child's number knowledge.

A completed square, of the type below, for each child's record is a quick and easy method of recording number knowledge.

ADDITION TABLE

The table displays all the addition facts up to 10 + 10. Children need to consider the different number patterns in the array. The leading diagonal shows the doubles of numbers up to twice ten, ie the sequence 2, 4, 6, 8 .... 20, and the numbers on either side of this line 'balance' one another, giving symmetry to the array, for example:

It is this symmetry which approximately halves the number of addition facts which have to be remembered (see page 20).

Place value

Although children may be familiar with single digit numbers, and even use two-digit numbers with some success, establishing an understanding of the symbol '12' as one ten and two units may require much experience, carefully arranged by the teacher. Children are able to equate 12 with twelve without realising that the digits indicate 10 + 2.

The understanding of place value, with a full working appreciation of the number system and the notation we use, should be a fundamental aim with all children in the primary school. On this aim there is substantial agreement, but there are considerable differences of opinion as to how it may be best achieved.

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Activities with a spike abacus (consisting of a base-board with spikes on which calculations may be performed with washers) are valuable although a 'number board' is easier to make than an abacus, and in some ways easier to manipulate. (It consists of a board ruled in columns on which calculations may be performed with counters, used as on an abacus). This device was in practical use for centuries.

Counting objects in various bases and the use of multibase blocks for the four operations is another possible approach. The study of number bases is, however, only justified if understanding of the principle is being developed: routine work in bases other than ten can result in misplaced effort unless it is clearly directed towards consolidating the understanding of place value and the related notation for numbers, and to reinforcing the four fundamental processes of calculation. Proficiency in the routine skills of multibase calculations should never be regarded as an essential requirement for some subsequent stage of mathematical education.

Number skills

Multiplication and division facts

A sound knowledge and recall of multiplication facts should he acquired if children are to be able to perform anything beyond the simplest calculations involving multiplication and division. Teachers should encourage memorisation and recognise the importance of precision when there is the need. Random recall is vital but this by no means implies rote memorisation: for we read in the Shorter Oxford English Dictionary.

by rote: 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.

At the Infant or First school level, children introduced to the basic concept of multiplication should be able to double (and halve) numbers up to 2 x 10; many should be able to go well beyond. By the age of seven or eight years most children should also have immediate recall of the simplest tables (2, 3, 4, and 5); at the age of ten, a large majority should know all the multiplication and division facts of the 10 x 10 table square.

It is important that children come to understand that, for example, 3 x 7 = 7 x 3 as this almost halves the multiplication facts one needs to memorise; to know squares of numbers up to 10 x 10; to be able to multiply any number by 10 with understanding. From the tables of 2 and 3 children can be asked to find others, such as tables of 4, 8, and 6 (by doubling).

Children should make a multiplication table square (10 x 10) for their own use; making it is often a great help to memorisation and the completed square is a useful aid. They need to recognise the number patterns contained within the table which will help the learning of

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number facts. The leading diagonal shows the squares of numbers up to 10 x 10: this line bisects the table and the symmetry of the table about the diagonal (arising from the commutative law of multiplication) should be appreciated. The table also facilitates the division of numbers contained in the array and gives pairs of factors of these numbers.

MULTIPLICATION TABLE SQUARE

Various activities can be introduced (number patterns on a 1 to 10 square and the 10 x 10 table-square, 'adding on' using a number line, etc) to encourage and help children to acquire competence in their knowledge of tables and form the basis for further learning. Copies of the table square can be coloured in various ways, eg to show where multiples of 2 or 3 occur. Frequently occurring numbers, such as 24, can be ringed. At the appropriate time, children need to learn their addition and multiplication tables and to be able to apply their knowledge to arithmetical problems. From this time on, confidence and immediate access to recall of tables is essential.

These should be the basic aims in the numerical part of mathematics. Some children will not reach this level of attainment and will find learning such facts a great ordeal. Other important experiences should not be withheld from them meanwhile.

Memorising facts

It is vital that children should regard numbers as useful and manageable constructs and that they should use and manage them. The proper wish on the part of both teachers and parents, that children should master

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numbers quickly should not be allowed to lead to over-expectation: this merely discourages them. It is tempting to suppose that a concentrated period of rote-learning will do the trick. The price that has to be paid in misunderstanding and anxiety when that approach is adopted has been recorded in a succession of HMI reports and other documents produced during the first half of this century.

As well as learning to recall the results of combining the smaller numbers children must also learn how to arrange the larger numbers on paper in ways that make it possible to deal with them without memorising every combination; this involves, among other things, an understanding of 'place value'.

Calculations

Children should be able to perform calculations accurately and give and record a rough estimate of the answer before undertaking a calculation.

To do this they need to understand the relative size of numbers: what numbers are appropriate to describe the number of pages in a book, the number of words on a page, the height of a boy or how long is a thousand seconds. This skill is not easily acquired; but a good start is made if calculations asked for are commonly related to realistic problems: notable exceptions are those needed to investigate pattern and structure in the number system.

The standard of mental skill of children varies. It depends on interest and in particular on children's ability to chain processes together and on the speed with which the processes are performed. Speed is less important than accuracy, though it should also be of concern. Current teaching underplays mental arithmetic. It must be stressed that the way a calculation is set down and worked on paper (technically known as an algorithm) may by no means be the best mental method in all circumstances, eg the difference between 87p and £1 might be set down using the subtraction algorithm:

but 'counting on' as when giving change may arrive at 13p more easily.

Children's use of numbers is rarely enough in itself to enable them to reach the stage of recalling the addition, subtraction, multiplication and division facts readily. Specific efforts of memorisation are also necessary. So is activity that enables the children to see how the number bonds form patterns that both illuminate the relationships between numbers and help memorisation. The memorisation and observation of patterns begins very early, and certainly occurs at the point where reciting the numbers and relating them to quantities are combined. As described on pages 20 to 22 and page 23, by the time must children are eight, they should be able to recall the addition and subtraction of numbers up to ten, and many of the multiplication and division facts. A minority of children of that age know the addition and subtraction facts to 20, and are able to add, in their heads, numbers such as 65 and 17.

The very slowest learners in ordinary schools may not reach that stage with certainty by the time they are 11. Most children should by that age be able to recall instantly the number combinations up to 20 and the multiplication and division facts contained in the 10 x 10 table

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square, and to add and subtract numbers up to 100 in their heads. If they cannot, yet have a working understanding of the processes they are being asked to use, it may be that they have spent too little time on memorisation, or else that memorisation has been spaced so that they have been set too much to learn at once, or perhaps that the task was set at a time when they had insufficient understanding to make sense of it

The processes

Children can be helped to devise methods of calculation based on their practical experiences. Teachers should be prepared to accept a variety of methods in order to help children's understanding.

In subtraction, for example, there are at least two methods (complementary addition and decomposition) which are described in detail later in this section. There are three main aspects of subtraction. Considering the example 62 - 27, these are as follows:

i. what must be added to 27 to obtain 62?, as with giving change in money (complementary addition)

ii. what must be subtracted from 62 to obtain 27?

iii. if 27 is taken away from 62, what remains?

In multiplication, children can devise methods based on addition: young children can answer the question 'how many altogether' or 'how much altogether', for example, 'if I give 7 sweets each to 5 children, how many sweets will I need altogether?'

For long multiplication, most children will require careful questioning to help them to refine the method, for example, 'could you take an easy multiple?'

Both aspects of division need to be considered: the repeated subtraction aspect and the sharing aspect (so that division is seen as more than 'goes into').

eg 21 ÷ 3 can be interpreted in two ways:

a How many threes in 21? If you have 21 sweets, how many children can you give 3 sweets to?
This can be answered by the subtraction aspect of division - subtracting successively sets of 3 sweets resulting in 7 sets of 3.

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b If 21 sweets are shared between 3 children, how many does each receive? This is answered by the sharing aspect of division - sharing the twenty one sweets between A, B and C and resulting in 3 sets of 7.

In the first case, the number of sweets has been determined and the question is how many children can receive three. In the second case, the number of children has been determined and the question is how many sweets does each receive. Even with more difficult problems, both aspects of division need to be practised and understood. In long division, subtraction of easy multiples (eg 10) is a method an older junior may invent. In any event, realistic problems provide the best starting points.

Calculators

Mechanical hand-calculating machines can be a valuable asset as a teaching aid. They provide children with real experience of multiplication as repeated addition, and division as repeated subtraction. Children can usually discover for themselves how to move the carriage and therefore how to obtain, very quickly, larger multiples such as 37, 128. The movement of the carriage embodies the principle of place value. However, even the process of long division is so tedious on mechanical calculators that they are completely superseded as devices for serious calculations by electronic calculators.

Any suggestion that children in primary or, indeed, secondary schools should use electronic calculators is often opposed on the ground that their use is likely - some would say certain - to have an adverse effect on the acquisition of sound computational skills. However, to use this argument is to fail to take account of the opportunities that the calculator offers as a teaching aid. This is a point which is at present considered by too few teachers. Such systematic classroom studies as are currently available suggest that, far from undermining skills in basic computation, proper use of the calculator can help and encourage children to develop and improve skills.

To use any calculator sensibly, children need to he able to check whether the user or the calculator has made a mistake and reached a wrong answer.

This checking can take several forms:

• Does the size of the answer make sense?
• Repeat the calculation in a different order
• Roughly approximate - eg is the decimal point in the right place?
• Use of pattern (32 x 17 should end in a 4)

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• Check that the input data are reasonable
• Use the inverse process - for example, check a subtraction by adding.

When a child has become confident in his use of the calculator the teacher can point out that when a calculator is used for the purpose of carrying out a computation there can be no excuse for obtaining an answer that is not 'correct', and much useful teaching can follow from this. For instance, children need to realise that each calculation should be done twice and that whenever possible the numbers should be entered in a different order. The fact that this is possible when adding or multiplying but not when subtracting or dividing will reinforce their understanding of the commutative nature of addition and multiplication as distinct from subtraction and division. Children need, too, to be aware of the approximate size of the answer they expect to obtain as a further safeguard against any failure of the calculator or error in pressing the keys. The skill of approximation is needed even when a calculator is not being used and it requires considerable practice. This practice is made a great deal easier if a calculator is available.

Current investigations suggest that for some children the calculator can play a useful part in developing an understanding of the equivalence between repeated addition and multiplication. It can also help in the introduction and understanding of decimals and, at a later stage, of negative numbers. Interesting investigations, for example of number patterns, can also be carried out, especially by those who are mathematically able.

Finally, it seems essential to consider one further point. The electronic calculator is with us for good, and it is hard to believe that the children now in our schools will, throughout their adult lives, be without a means of calculation at least as powerful as that which is now available in the electronic calculator. Many children, too, will have access to a calculator at home even if not in the classroom. It therefore seems essential to make sure that our pupils learn to use a calculator correctly and sensibly; and if they do not learn to do this at school, where else will they learn it? It is not a task which can be accomplished in one quick lesson, and the foundations need to be laid in good time.

Practice

Children do require practice in written calculations, but excessive practice is a waste of time. Only the teacher can judge how much practice each child requires to achieve and maintain efficiency.

When children are having practice in written calculations they should be asked, every now and then, to make up a problem (story sum)

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to fit one of the practice examples. This will ensure that the children know the kind of situations which give rise to the written calculations they are doing.

There is a mathematical as well as a computational aspect of arithmetic. Children should be encouraged to look for mathematical pattern at every stage. Frequently a child's genuine desire to investigate a pattern gives valuable computational practice.

Written practice

Written work in number calls for abilities akin to those used in reading. It also calls for the ability to associate values with the position as well as the shape of the numerals, and an even greater ability to set out the work precisely on the page if the layout is to be used as an aid to calculating and not only as an aid to memory. That is, if the following prices are to be added up, and not just recorded individually, it is an enormous advantage if they are set down in an orderly way:

The great majority of children of seven should be able to write down the numbers they use in the practical work they undertake and certainly the numbers up to 99. Some of this age group are likely to have made a start in writing the numbers down and using the written display in order to calculate. But it is children between eight and ten who are most likely to be engaged in mastering this skill, at least as far as addition, subtraction, multiplication and short division are concerned.

The operations; written calculations

In this country, there is a long tradition that set methods of calculation are not prescribed for teachers. The standard algorithms (procedures for calculation) can be understood, not just memorised; the way in which they are taught sometimes suggests that these procedures are merely something to be memorised which cannot be understood in any deeper sense.

Rather, there should be a gradual approach to the standard algorithms by extending, step by step, skills which have previously been established.

It is important that individual children should be helped both to devise their own methods and to make these as efficient as is possible for them. In the end, however, all should he equipped with and practise adequately some reliable method.

As an example consider the addition:

This can be set out:

The working can be shortened considerably but this setting out illustrates the process effectively.

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Subtraction methods

• Subtraction by addition (called complementary addition).

More complex subtractions might be recorded in these ways:

This sum could also, of course, be worked on a number line.

• Subtraction by decomposition (by expanded notation)

Some children may require more steps than this. Others may well require fewer.

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• Subtraction by equal addition

As a written method in which 10 is added to both numbers, it has serious disadvantages which can result in the use of jargon such as 'borrow and pay back'. This method is, however, useful in mental calculations and the principle is important. Adding the same number to, or subtracting the same number from both numbers given leaves the difference unaltered: ie equal additions do not affect differences.

Before this method is used it should be experienced in many contexts.

The method can be recorded:

Again, the process can be shortened considerably by manipulating the numbers mentally, but the steps involved should be clear to the child.

It is most important that children should be able to multiply (and divide) any number by 10 (and by 100) at sight, giving the correct reason. Children may build tables such as

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A gradual development of multiplication will lead to an example requiring the following type of long multiplication.

In a rectangular classroom there were 36 square tiles one way and 37 tiles the other way. How many tiles are there?

Diagrams can be used in the following ways:

Some children can proceed quickly to the traditional method of written multiplication: this can be developed from both diagrams i and ii.

Division

Through a progressive development of division, a question which might eventually be solved is:

A lift can take a maximum load of 2000 kg. Mr Jones weighs 85 kg. How many men of his weight will the lift carry safely?

Children usually begin by subtracting 85 again and again. If they are asked to speed up the method by subtracting easy multiples of 85, provided they know how to multiply by 10 at sight, they can usually be led to suggest subtracting the product of 85 and multiples of 10.

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It will be safe to carry 23 men of Mr Jones' weight.

If, in another context, the calculation is to be extended to decimals it could be continued:

leading to a more formal setting out:

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The extra time (if any) taken by children in setting out calculations like this, until the ideas are grasped, is more usefully spent than in doing 'by the rule' an extra example of the same sum.

Areas of particular difficulty

Decimals

Decimals are encountered initially when dealing with money and, later on, decimal notation is used when recording metric measurement. Teachers must be aware of the difficulties children experience as a result of the different conventions of notation and language used in money. The fact that we should not write £2.5 but are allowed to say 'two-pounds fifty' makes it more difficult when we progress to other decimals. Decimals can also be appreciated as an extension of the pattern of recording the number system. A thorough understanding of tenths and hundredths is essential to the development of work with decimals; experience with base ten structural apparatus is very useful at this stage, taking the 'block' as one unit.

The frequent use of decimals in measurement and money should help children to recognise the relative size of decimal fractions of a unit, eg that 0.9 is greater than 0.12. This experience enables children to become fluent in decimal notation through the recording of money and measurement in a variety of ways; for example, in decimal whole numbers, we think of 89 as 8 tens and 9 units, and in decimal fractions, of 0.89 as 8 tenths and 9 hundredths

and in fractions as

or

Operations on decimals can be introduced side by side with calculations and recording in money and/or measures, for example

and

Note: The deliberate use of the decimetre for teaching is referred to on page 46.

The association of the four operations with actual situations should make it possible to avoid unrealistic calculations and encourage checking.

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Since all calculations are performed in the decimal system there is no need for special teaching in decimals as well as in the metric measures or money calculations. At some time it is important to make children aware of the connection between decimals and percentages (in which 100 acts as the unit). 0.75 is therefore equivalent to 75% (or 75 in 100), and 0.05 is equivalent to 5%.

Averages

The idea of an average can be built up with young children. An infant teacher handed out 'dolly mixtures' to each of three children in turn. She then gave the remainder of the bag to the fourth child in the group. When the children compared their shares, (2, 3, 5 and 6) they said this was not fair: 'Let's put them all back and share them out', they said. In this experience the average was a fair share. Children require frequent experience of real situations which require the finding of an average as in measurement. For example when finding pace lengths or when counting the number of beats of a pendulum in one minute, it makes sense to find the average of, say, ten observations.

Fractions

Fractions: similarly, axioms, definitions, formulations of rules and the formal expression or argument, should always follow and proceed from the children's own expression or individual cases and their own solution or individual problems. Thus, for example, the addition or fractions should be reached by the child as a comprehensive and final statement of the various methods or procedure discovered by himself, guided where necessary by the teacher, in cases of gradually increasing complexity.

Report of a conference on the Teaching of Arithmetic
in London Elementary Schools (LCC 1909).

The amount of time which should be given to the teaching of fractions in schools now that the metric system is being increasingly adopted in this country is likely to be a matter of dispute for some time; but we may be reasonably certain about some matters.

The idea of ratio, which has always been difficult for some children, will continue to be important. Also, there will be very little advantage in being able to manipulate fractions without appreciating the underlying ideas.

Children certainly need to become familiar with fractions at the level of ordinary conversation and still need to use 1/2, 1/4, 3/4, 1/10, 4/10 in practical situations. It is essential that children should understand the notation of fractions and be confident of their meaning. They should be introduced to practical experiences such as cutting things up into equal parts, naming the parts properly and coming to understand the principle of equivalent fractions. This important stage of development will lead children to consider the comparison of fractions, equality, inequality, sum and difference of two fractions.

Before considering how much more should be attempted, let us recognise certain genuine difficulties which are involved. Unless the teacher is confident about these further developments and can present them to children in a way which they fully understand, these extensions are best omitted.

Addition and subtraction and some aspects of the division of fractions can be explained in commonsense terms. The first two operations can only be performed on quantities which are of the same

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kind, and the rules derive from the simple principle that one must reduce the problem to working with the same sort of thing. We can add quarters to quarters or tenths to tenths without trouble, but if we have different kinds of fractions, we must first express them in some common measure. This involves recognising equivalent fractions, and provides opportunities for work on choosing the right unit. In many cases, we can avoid fractions altogether by choosing a smaller unit (for example, a quarter of a metre is 25 cm). The rule for addition can be seen as essentially the same idea; to add thirds to halves, we have to take a sixth as a new unit.

Before teachers consider how to teach multiplication and division the difficulties should be faced. There are genuine complications if we try to teach these operations with fractions with reference to practical examples - and the teaching of the ideas without reference to the practical examples is very hard to justify. Whereas we can add only quantities of the same kind, when we multiply and divide numbers, including fractions, we may be concerned with physical quantities of different types.

Before we discuss this problem, there is another aspect of division which is easier to understand. We frequently want to divide two quantities of the same kind to get an answer which is a number. We may start division by making up relatively simple questions, to which the answer may be obtained without procedural rules by carefully looking at what is involved. How many lengths of wire 2m long may be cut from a piece 6m long? We know the answer is 3 and we learn to write: 6 ÷ 2 = 3

How many lengths 1/2m long may be cut from a piece 6m long? We can see that the answer is 12, and we learn to write: 6 ÷ 1/2 = 12

This we write in order to describe what we see - it is not a question of turning anything upside down and multiplying, a trick that may be justified later if it assists our memories, but the fundamental need is to provide the experience that is worth remembering in the first place. Before resorting to such aids to memory we might lead children on to the idea that, in cases of greater difficulty, we may do this kind of division by our previous method of working in a suitable unit. We may find out how many lengths of 2/3m may be cut from a length of 6m by working entirely in thirds. This means dividing 18/3 by 2/3 and the answer is 9.

A similar approach shows that we can divide some number of quarters by some number of thirds if we have the enterprise to work in twelfths. This leads to a comprehensible general method for the division of fractions which is eminently usable, although this is not the method traditionally taught. We are not suggesting that work of this kind ought to go on in every primary school, but we are saying that if some calculations with fractions are taught, they could proceed by this kind of method.

The practical applications of multiplication very frequently involve multiplying physical quantities of different types. Thus, we multiply a speed and a time to get a distance. If the multiplication of fractions is taught in close connection with practical problems, these complications cannot be avoided. There is, however, another way of introducing the multiplication of fractions which avoids the complications with physical quantities to a great extent. We may ask, for example, what is

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2/3 of 1/2 of something? The first 'of' denotes a mathematical multiplication of two fractions. The second 'of' denotes something different, and it is a matter of opinion whether what it denotes can properly be called multiplication at all. (If I take half of an apple do I multiply an apple by 1/2?).

These difficulties are small if we aim to do no more than state rules; the difficulties are greater if we are concerned about meaning, and the problem of teaching mathematics is the problem of teaching it with meaning.

With the present type of question we can see that 1/2 of 2/3 of something is 1/3 of something. It is not immediately obvious that 2/3 of 1/2 of something comes to the same. In the first case, it is clear that we have divided a quantity of something into 3 parts and taken 2 of them; subsequently we have to take one of these 2 parts. In the second case, we have divided a quantity of something into 2 parts and taken one; we have then to sub-divide this part into 3 parts and take 2. To make it clear that we have 1/3 of the quantity we started with it is necessary to reconsider the problem, and to start by dividing our original quantity into 6 parts by taking 3 (3/6 is equivalent to 1/2). At the second stage, we are then able to take 2 of the 3 parts we have. This leaves us with 2 of the original 6 parts, so the answer is 2/6 and we feel free to use the equivalent fraction 2/3.

With proper care, this takes us to the usual rules of multiplication. On this approach, division has to be introduced as an inverse operation and the appropriate didactic literature must be consulted for details. [See Appendix III.] Once again, we are not suggesting that this should be done in all primary school classes. We are indicating some of the difficulties in adopting an approach to the multiplication of fractions in which the mathematical symbols have meaning; and suggesting that any approach must in some way face up to the genuine conceptual difficulties to which we draw attention. These recommendations could result in some primary schools attempting less with fractions than before. Such a policy requires coordination with the work of the schools to which the children subsequently move.

Positive and negative numbers

The need for negative numbers may arise in measuring winter temperatures or from graphs when children ask, 'What happens next?' The extension of the positive number line back through zero can be dealt with by counting backwards or by the use of number strips to find positions of points on the number line representing negative numbers.

For a very few children, the operations of addition of positive and negative numbers (as one step followed by another step) and subsequently subtraction (as the inverse of addition) might be included in the primary school, but these stages should not be considered essential within a programme of work for the majority of children.

The multiplication of negative numbers is quite a complicated matter if it is to be done properly, and it is unlikely to be of any benefit in the primary school: at this stage, the memorisation of rules such as 'two minuses make a plus' is certainly to be deprecated.

Ratio

Almost all children find ratio and the related idea of proportion very

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difficult to handle. The less precise meanings accepted in ordinary conversation may contribute to the problem. Teachers should approach ratio with exceptional care, patience and sympathy, seizing every opportunity to bring this aspect into discussion at a simple level. There are two ways of comparing things, one of which leads to the idea of difference and the other to the idea of ratio. There is a generous amount of very familiar terminology for the former, but for the latter, the language speedily becomes unfamiliar and too technical for most primary school children.

There is a further difficulty. The difference between any two whole numbers can always be expressed as another whole number, whereas the ratio of two whole numbers can not. To express ratios we use pairs of numbers; in some ways these pairs are like fractions (which the child may well have met in a rather different setting) but in other ways they differ. Children need time to acquire the ideas, with practice in a variety of contexts.

Ratio can arise through the work in measurement and shape in mathematics lessons or within geography lessons (scale) or home economics (mixtures in cooking): for example, the ratio of two time periods = 2:3 or 2/3; the ratio of the length and breadth of a rectangle = 5:3 or 5/3; the map has a scale of 1:4 or 1/4; the mixture of flour to fat is 2:1.

Ratios of areas, volumes and capacities might be postponed for some time until the concepts of these measures have been well understood. The introduction to the misleading term 'similar' and even the well defined words 'mathematically similar', might well be delayed until the secondary school. Instead, children might be helped to understand what is meant by two plane objects having 'the same shape' - all squares have the same shape, regardless of size, but rectangles of the same shape will have the same length to breadth ratios.

Structural apparatus and aids

When structural apparatus* is used it should not necessarily be the initial introduction to the work, nor is it the final abstraction at any stage or in any branch. In arithmetic it may be a bridge between concrete environmental experience and final abstraction. It helps some pupils to gain understanding and insight in algebra. The most able quickly realise that it is only one way of expressing a more abstract idea.

This realisation often does not come to average pupils, but experience with structural apparatus can help them to have some understanding of algebraic ideas. Purists object to this, but average pupils may not be future mathematicians, scientists or technologists, and will use mathematics only where it is applicable to situations in life. The use of structural apparatus to aid logical thought has already been mentioned (page 15).

Restriction to one type of apparatus can limit a child's understanding of number although he may be able to manipulate the pieces of material with confidence. This is particularly true where a school relies wholly on one type of apparatus. It is necessary to have other apparatus as well as that of the structural type. Structural apparatus for arithmetic is one of two kinds - that which emphasises the number-bonds and that which emphasises place value. The second often involves work in addition to bases other than ten. This extension means that the patterns of number can be more fully investigated. It also promotes the mathematical as well as the computational aspects of arithmetic. When selecting

*See Appendix II for descriptions of the more commonly used types of structural apparatus.

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apparatus to aid children in learning their number-bonds, it is necessary to provide material which synthesises as well as analyses number. (Synthesis occurs where children see '3 and 5' being equivalent to 8 and analysis where children see that '3 separate ones and 5 separate ones' are equivalent to '8 separate ones').

The common error is for the 'play stage' to be curtailed or even to be omitted: children often need to revert to play after the work has been formalised. At the other extreme, some schools never go beyond the experimental stage and children are not helped to extract or crystallise mathematical - not only arithmetical - ideas.

Additionally, teachers need to recognise that children enter school having already solved problems; it is often the teacher who introduces 'pure' number. It is necessary to ascertain at each stage that the child can apply his knowledge to the real situation. This is especially important when structural apparatus is being used: it is one stage further removed from the real world than the more traditional and still essential apparatus such as conkers, stones, counters and fingers. The child should be able to interpret the problem, solve it and give an oral reply as an answer.

There is now a wide range of structural apparatus available from which to choose, but some styles are more popular than others. Work in different number bases is probably less appropriate at the infant stage, although it does have its adherents. It is more frequently used at the lower junior level, but it is with the oldest and most able juniors that work at a higher level of generalisation leading to algebra can be developed and investigated. Difficulties often arise since there is a tendency to work through 'ten'. For example, the point of an exercise is lost if, in changing from binary to octal, base ten is used. It is important that, if work in different bases is attempted, other apparatus as well as blocks of wood is used: certain shapes, abaci and coloured counters are all useful. As in many other activities, the teacher should become thoroughly familiar with the material, appreciating its strengths and weaknesses before introducing it to her children. What is important is that the children should acquire understanding.

Many of the ideas now being developed using structural apparatus are sound, but most ideas based on discrimination are more interesting to the children if they are engaged, for example, in sorting things that are familiar to them out of school, such as toy vehicles into car and lorry parks. In general, if teachers understand the basic ideas they can make the activities more applicable to young children. With some apparatus, only difference can be considered directly; the 'taking away' aspect of subtraction demands decomposition, even for small numbers. It often helps if teachers work through the printed material supplied for children (even though this is seldom suitable for the children themselves) and do the work suggested or outlined: it assists them in becoming knowledgeable about a particular type of apparatus. However, the printed paper does not always give enough guidance for full appreciation of the possibilities: structural apparatus is best studied in a working group. Many schools possess apparatus, but it is not unusual to find it unused.

It is important for the teacher to recognise that some apparatus can help with work in number bonds, emphasising the introduction of place value with numbers from 10 to 20, the four operations, work with

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fractions, and later on, algebraic ideas which are connected with the laws of arithmetic. Other apparatus can be used to reinforce the ideas of arithmetical generalisation. This is often within the competence of the more able older junior children. Symbolism is usually inappropriate at this stage but it is important that children have the opportunity for talking and perhaps writing about what they have learned: this is an important aspect of the experience.

Some of the apparatus devised for algebraic development with junior children gives rise to geometrical ideas, particularly those of mathematical similarity. These examples will indicate the flexibility commercial structural apparatus can have in the hands of imaginative teachers. Ideas introduced in the junior school may be developed further in the secondary school, especially in the early years.

A way to look at apparatus

1 It is essential that there is progression both in the ways in which environmental materials, eg stones, conkers, water and sand, are used by the children, and in the mathematical ideas learned as the child progresses through the primary school.

2 With commercially produced structural apparatus it is helpful to select material which can be used in a variety of ways.

3. Generally, it is important that more than one type of apparatus is provided; it is essential to ensure that there is appropriate material to cover the different aspects of the work. The apparatus should not be blamed for children's failure to learn their number bonds, if the material chosen is essentially used to emphasise place value.

4 Little structural apparatus has been developed specifically for geometrical ideas at the primary stage, but material used for algebra and logic helps to develop geometrical ideas.

5 Apparatus which synthesises number treats the numbers as a composite whole. For example, 'five' and 'three' added gives 'eight'. Apparatus which analyses number shows that 'five ones' added to three ones' yield 'eight ones'.

6 Apparatus which synthesises number is usually dependent upon length. Consequently, the apparatus should only be used for 'play' and not for number until the conservation of length has been established. Conservation of length for many children is achieved later than the conservation of number; this will inevitably delay the beginning of work with number unless apparatus which analyses number is also available.

Apparatus which synthesises number is a model of the rational numbers (ie whole numbers and fractions), not only of the natural numbers (1, 2, 3, 4, 5 .... ). Unless the teacher appreciates this difference, the apparatus which synthesises number should not be used.

7 The apparatus is in various sizes. Generally, the larger material is more suitable for younger children since it is easier to handle.

Conclusion

By the age of ten, nearly all children should have become aware, through use, of the laws of arithmetic. In the 10-13 age range, recognition of the commutative, the associative and the distributive laws should become explicit for the great majority, although the terminology need not be used.

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Many teachers give a great deal of attention to computation but unfortunately, in too many schools, this is treated in isolation from the rest of mathematics and from any applications, so that it loses the greater part of its value. Number knowledge should be secure. Here, the practice of computation is important, though over-practice can result in inaccuracy and boredom. Mental arithmetic, especially of the kind involved in good number conversation, should not be neglected by the teacher. Questions need to be pitched at a level simple enough to ensure a very high success rate. Children who are weak at mental arithmetic can all too easily be shamed in front of their fellows, but, with sensitivity, mental arithmetic can be used as a means of encouraging children to improve their facility in calculation, to acquire the skill of quick or immediate recall of number facts and to gain confidence in simple number processes and knowledge. Computational methods should be devised by the pupils themselves but a stage is often reached when it is advisable to decide on a reasonable method which pupils can understand and practise until it is retained. As much work as possible should be based on practical experience and it is important to encourage a flexible approach to problem solving.

In conclusion, it is perhaps worth re-considering briefly the nature and function of mathematics. When children measure the school hall, or survey the school field, or plan a traffic count, it might he said that they are engaged on a scientific rather than a mathematical enquiry. The mathematics begins when the situation is symbolised in numerals or algebraic terms, and progresses as these symbols are manipulated to yield new relationships. The mathematician abstracts from experience, manipulates almost 'unthinkingly' and translates the symbolic result back into experience. To yield real power, he must be able to manipulate symbols easily enough to maintain his grasp of the whole process - yet this essential facility is often denigrated as 'mere' manipulation or 'mechanical' arithmetic.

It is essential that these fundamental processes of arithmetic become automatic before the child leaves the primary school. Unless he can add, subtract, multiply and divide accurately, quickly and without hesitation, his future progress will be severely handicapped. This means that he must know his addition and multiplication tables through and through as certainly as his own name.

The Report of the Consultative Committee
on the Primary School 1931
Chairman: Sir W H Hadow

In recent years, and certainly since the Hadow Report was written, less emphasis has been given in the development of primary mathematics to arithmetical skills. Some people may ask if there is time for the breadth of practical activity and for the variety of applications of mathematical ideas advocated here, as well as for the practice which is needed for efficiency in the commonly occurring calculations. But since the change to decimal currency, it has not been necessary to teach the intricate calculations with money which were taught until 1971, and there has been a substantial reduction of calculations with weights and measures in Imperial units - though measurement with these units needs to continue; therefore the broader range of work now being undertaken in primary schools should help towards efficiency in the reduced amount of computational arithmetic which remains - as the

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National Primary Survey (Primary education in England, HMSO, 1978) indicated.

Establishing a proper balance between investigation, experiment, memorisation and routine practice is a problem for every school.

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6 Measurement

Measurement is mathematics at work in the interpretation of the environment.
Measurement for the child is the connecting link between mathematics and the real world.

IGW Sealey

There are two aspects of quantity: discrete quantities are counted whereas continuous quantities such as a length of ribbon, weight of sand, volume of water, capacity of a bottle or the surface area of a leaf are measured in a different way. Counted measures of discrete quantities such as school attendance, cricket scores, vehicles in a car park are usually exact: with large counts, a reasonable approximation is often used to express the answer eg the population of Greater London, attendance at a cricket match or the number of starlings in a flock. Once a unit has been defined, counting can be used to measure a continuous quantity in terms of this unit - but inherent in this process is the idea that we can only ever measure a continuous quantity to a certain degree of accuracy. The degree of accuracy depends on the measuring instruments which we have available and the purpose we have in mind. In the section below, the quantities measured are all continuous quantities and the stages involved in the teaching of measures are discussed in detail.

The main measures which are introduced to children in the primary school are weight, length, capacity and volume, area, time, angle and temperature.

Other measures for consideration within the age range of 8 to 11 include speed, force, pressure and density, but many of these would be introduced as mathematical concepts only to the older and more able children in the primary school.

Measurement provides one natural approach to mathematics. When the child measures any property he needs to acquire the language to describe his experience. Discussing his activity with his teacher helps to provide a more precise vocabulary. The child first compares weights and orders lengths, often without the need for measurement or counting. His first use of number in measuring is with improvised units, and he is eventually brought to see the necessity for standard units. In all his measuring activities, a child should be encouraged to make an estimate in appropriate units and then to compare this with the result of his measuring. When measurement is taught, 'finding the measure' is not the all important aspect of the enquiry. That a desk measures 66cm is often not what matters: the real question might be, 'Is it narrow enough to fit into this space? etc. It is the relationship of the measure to other measures which matters.

Although measurement is inexact, we can state exactly the degree of accuracy which is being claimed: 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', the answer is absolutely true. Children should not be allowed to think that an answer is

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necessarily more accurate because it has more numbers in it.

Practical measuring activities provide ample opportunities for recording narratively, pictorially or numerically. Written calculations, always linked with the appropriate stage in number, are necessary. For young children, arithmetic outside the context of counting or measurement can hardly make sense. Some form of concrete reference is necessary to give meaning to the numbers they are expected to manipulate. In other words, measurement can be the source from which the child may abstract his number concepts. For example, decimal notation can begin to make sense when the child can measure and appreciate that a way of recording 1 metre 6 decimetres is 1.6 metres. Similarly, pure computation can very easily become a meaningless activity unless at the early stage it is associated with the real world of measurement. (The use of units such as the decimetre is considered later.)

In the early years, it is important that children should be introduced to a wide variety of environmental objects and material of many different types: water, sand, ribbons, clay, plasticine, (all continuous materials) and bottles, boxes and containers of various shapes, from which measuring can arise.

Children can be brought to a gradual understanding of the concepts of the four operations of number (ie addition, subtraction, multiplication and division) through their own early experiences of manipulating continuous quantities.

The ideas of counting and formal measurement are not essential at this stage, although the child may be developing these concurrently. Only later is it necessary for the teaching scheme to relate these distinct ideas.

Stages in learning and experience

Certain experiences are similar for all aspects of measuring: weight, length, capacity and volume, area, time, angle and temperature. However, the ideas involved can be seen much more clearly in some embodiments than in others, eg we may add two lengths directly by placing them end to end but we cannot handle two temperatures in the same direct fashion. And again, for example, to order four children by height is within the competence of many infant children but to order three objects by weight may demand three balancings and two facts to be remembered, followed by logical reasoning to establish the order.

Ordering, inequality, equality, conservation, estimation and approximation are all involved in these early stages of comparison, before the introduction of any numerical measure.

Various stages may be distinguished

• Direct comparison using matching, with no actual measuring.

The development of appropriate language is crucial at this stage, and it cannot be hurried.

• Using non-standard units

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measurement:

length - using parts of their own bodies; hands, feet, fingers, spans
capacity - teaspoonfuls, cupfuls, jugfuls, tinfuls
weight - pebbles, conkers, beans, pieces of wood, marbles
time - leaking tin filled with water, pendulum, sand timer.

Until children appreciate how a physical quantity remains unchanged (conserved), measuring this quantity has no meaning. Piaget and others confirmed by detailed experimentation what perceptive teachers already knew from their classroom observations: that children require a great deal of experience before they grasp the idea of conservation of length, weight, volume capacity, and area. For example, when a piece of paper is cut into two or perhaps three pieces, does the total surface area of the pieces remain unchanged?

• Use of standard units

A knowledge of the appropriate weights, lengths, heights and, later on, the capacities of some common objects is helpful in learning to estimate.

• Using two units, or one unit and a decimal fraction of the unit

Measurement is essentially comparison. Such expressions as 'heavier than, ... lighter than, .... faster than, ... more than ...' all imply comparing either directly or indirectly through measurement. Generally we make our comparisons with accepted units of measurement.

The teacher needs, all the time, to bear in mind that there are two essentially distinct types of comparison - comparison by difference and comparison by ratio. Failure to appreciate this distinction can cause much confusion to the learner later on. The child may calculate successfully when asked to add or to multiply, but if he is given a problem where he has to decide which operation is involved, his underlying uncertainty may be quickly revealed. Thus, given the problem: A man walks at 5 km/h and a second man cycles five times as fast, at what speed does he cycle? a child may know that he has to do something with three and five, but may not be clear about which operation to use. A secure grasp of these concepts can be established only by continuing perceptive and sympathetic teaching - and it takes a long time.

Mass and weight

Whereas it is natural that the term weight should generally be used throughout the primary years, it is important to extend the vocabulary of the child who becomes interested in space travel and gravitational force or acceleration on other planets, Such terms as 'weightlessness' and 'the moon's gravitational field' may prompt children to ask about

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the difference in the weight of a body according to its position in space. Mass is the term used to define the quantity of matter in a body. It remains constant wherever the body is, ie mass is absolute; weight varies depending not only on the mass of a body but also on the gravitational force, which is different in different places in space.

In the International System of metric units (SI units), the unit of mass is the kilogram for which the abbreviation is kg. A kilogram contains 1000 grams for which the abbreviation is g.

Metric units

The choice of most units is arbitrary and we can (and often do) invent our own if we wish to do so, provided they are sensible and convenient.

For common usage, there is an obvious need for standard units of measurement but this need is not so obvious to young children. Older children should be helped to understand that the development of appropriate measures is an essential element in making progress in any new branch of science or technology.

Most standard systems of measurement are built up around one basic unit. Fractional parts and multiples of this unit make it applicable throughout a wider range. A good example of this is the metre, from which we derive the millimetre to measure small lengths and the kilometre to measure long distances. The centimetre and indeed the decimetre are important intermediate measures, and their use is appropriate throughout the primary school, even though in the long term both may be phased out. The decimetre provides a useful estimating unit of length or height which can be easily expressed in terms of centimetres or, later, as a decimal fraction of a metre. It facilitates the recording of lengths involving metres and centimetres to support the concept of place value, eg

The square decimetre (dm²) is an intermediate measure of surface area which can be used where the square centimetre is too small and the square metre too large. Through the use of the cubic decimetre, there are interconnections between the metric measures of length, area, volume, capacity and mass.

A cubic decimetre of water is a most important quantity. It has an edge of 1 decimetre (10cm), a face of surface area 1 square decimetre

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(100 cm²), a volume of 1 cubic decimetre (1000 cm³), fills a container of capacity 1 litre (1000 ml) and has a mass of 1 kilogram (1000g).

The teacher should know that there is an internationally agreed system of units with precisely defined abbreviations and conventions.

The proper usage of the International System should be developed in schools and by the age of 16, the pupil should be familiar with it. Early teaching should be compatible with this aim, but always appropriate to the stage of development of the child. Proper usage is described in How to write metric - a style guide for teaching and using SI units (HMSO, 1977).

Where Imperial units are still in use in the shops or at home for domestic use, they will be referred to spontaneously in genuine environmental work. Children should be familiar with the size of units which continue to be in common use but do not need to know the full range of theoretical possibilities. Written calculations with Imperial units are not normally necessary for primary children, and they should be encouraged to think and work in metric units. (See also DES Administrative Memorandum 9/74, Metrication).

There is a case for leaving a consideration of the history of measurement until the middle of the junior stage. By this time, the child's wide range of measuring experiences will give added meaning to this particular aspect of the work. A simple historical approach can summarise in a vivid way man's need to measure and the ways in which he developed this skill. It can represent a useful recapitulation of work the child has already done.

Measuring skills

Listed below are some important skills associated with measurement which children need to acquire within the primary school. This process is a gradual one dependent upon a wealth of experience sustained over a long period of time.

The ability:

a to choose the appropriate unit for different measuring situations. What units to choose, for example, to measure the weight of a house brick, the length of a field or the thickness of a page.
b to choose suitable methods of measuring which might not always be apparent, eg how to find the capacity of a container, the area of the surface of a stone, or the area of an irregularly shaped field.
c to use measuring instruments correctly.
d to estimate in the appropriate unit or units.
e to make their own measuring instruments (where appropriate) and to calibrate them in the appropriate units.
f to determine the degree of accuracy of a given measurement.
g to combine measurements.
h to invent methods of solving problems connected with measurement.
i to record findings in an appropriate way, both pictorially and numerically.

Stages of development in measurement

Every school will need to work out in detail the application of the principles discussed above. The following outline may help in the planning of the work, though care must alway, be taken to allow for

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children who progress more quickly or slowly than average.

Stage A. infant level

i. Experiences with different environmental objects: solid, hollow, of different sizes and different materials; floating/sinking, natural and man-made; feeling, handling, observing, classifying and sorting,
ii. Comparison of objects: comparing two things and later three; balancing, comparing length, height, shapes, surfaces, how much each holds, which sinks faster, which is warmer, - leading to an understanding of the notions of weight, length, capacity, volume, area, time and temperature,
(Note that ordering, inequality, equality, conservation, estimation and approximation are all involved in these early stages of comparison.)
iii. Use of improvised units: some of the stages of (ii) are repeated, using units which have been improvised by the child - comparing, ordering, estimating and approximating, using the improvised units,
iv. Use of agreed improvised units: stages (ii) and (iii) repeated, using agreed equal units.
v. Calibration of scales and dials, using home made instruments and agreed units.

The previous experiences have concentrated mainly on an introduction to the different concepts of measurement, comparison and approximate measure by counting the home-made units, leading to the need for equal and then for standard units.

i. Introduction to standard units: this stage of development will be arrived at by individual children at different times according to the unit of measure being introduced,
ii Calibration of own instruments, scales and dials using standard units, and later, combinations of units,
iii. Comparing, ordering, inequality and equality of measures, using standard units: how much difference; how many times lighter; longer; heavier,
iv. Measuring with standard units, involving estimation, approximation, the reading of scales and dials; choice of appropriate units,
v. Recording and notation, using the different measures and combinations of different units: notation and place value.
vi. Practical situations involving measurement and the use of the simple arithmetical operations,
vii. Small and large measures: introduction of smaller and larger units of measures.

i. Further practical work involving arithmetical operations using calculating aids, (tables, graphs, calculating machines) where appropriate.
ii. Use or common units of measure and relationships between

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them: quick access to appropriate units, conversion of units, notation and interpretation.
iii. Calculations with measures: use of four operations; scale, ratio: averages, fractions, model making, use of maps, scale drawings.
iv. The idea of a variable: distance and temperature varying with time, relationship graphs involving measures: conversion graphs, distance/time graphs, speed/time graphs, temperature/time graphs, area of a square, area of a circle, circumference/radius relationship; growth of squares, cubes, equilateral triangles, regular polygons etc ...
v. Historical aspects of measures, units and measuring instruments: length, weight, time, etc.

vi. Practical enquiry into relationships between measures of different shapes: volume of cone/volume of cylinder, area of rectangles and parallelograms on same base with equal heights,
vii. Practical experience involving further measuring apparatus: height finding using clinometer, simple theodolite work, simple pendulums of varying lengths,
viii. Experience with refined measuring instruments as appropriate, micrometer, feeler gauge, and fine balances,
ix. Practical situation involving experience of compound measures, as appropriate: speed, density, pressure, consumption (miles, gallon) etc.
x. Easy calculations with compound measures, as appropriate.

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7 Pictorial and graphical representation

Pictorial and graphical representation is a most important aspect of mathematics. The action of recording mathematical information in pictorial form is often the first introduction to graphical work and the ability to do so is a necessary and fundamental skill well within the compass of most children. Representation should permeate the whole of the teaching and not be dealt with in isolation. Graphs should be developed out of what children are learning rather than taught as an isolated topic. They are a means of communication and, when appropriate, children should be encouraged to record and represent information graphically as soon as they have the mental and physical abilities to do so; indeed, the act of recording and representing should contribute to the development of these skills.

The main functions of simple pictorial and graphical representation are:

i. to present information and data visually in a form that is more arresting, more significant and more easily interpreted than a collection of figures and words,
ii. to reveal relationships that might otherwise pass unnoticed, or, which is equally important, to show that a relationship does not exist.

Teachers need to ensure a careful development of pictorial and graphical representation. The illustration of statistical data must be progressive. It is important for children to learn how data can be organised, that is, collected, recorded, tabulated and presented. In an age when so much information is represented and misrepresented pictorially, it is of crucial importance that children should learn to use and interpret graphs, charts and diagrams of all kinds. Graphical representation is a further development of mathematical communication and there are many forms in which children can express their information visually. Other forms of communication include tabulations, networks, sketches, scale drawings, maps, charts and three-dimensional models.

In the early stages real objects are often used for graphs indicating counts, and this leads to the use of equal units and a base line, initially for nominal values, (names of children, pets, colours, days etc), and later for the numerical values of discrete variables (numbers, shoe sizes, scores etc).

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Throughout the primary school this work needs gradually to be extended, care being taken to ensure that, at every stage, children become increasingly aware of the need not only for reasonable presentation, but for the inclusion of such essentials as keys, scales and the labelling of axes, whenever these are appropriate. The composition of suitable titles is in itself a worthwhile challenge. But all this is only the beginning. The types of graph that children make are forms of communication - succinct and dramatic ways of presenting information. As such, they have an important role to play in fields of learning other than mathematics. There are times when a graphical representation is the most effective and natural way of recording what has been discovered. Children should be encouraged to develop this extension of their language, deciding for themselves the most suitable form of presentation to use.

Stages of development in representation

For many years, most people did not question that counting was the obvious beginning of mathematics. Recently, a number of teachers have attempted to go back further and to devote more attention to the activities of children which precede counting and without which counting can be no more than the meaningless recitation of number names. These activities include identifying things to count, collecting them together, matching objects with one another, and representing an object by a counter or a conventional sign. While sensitive teachers have always appreciated these things, controversy arises over the extent to which children can (or should) be encouraged to be aware of these pre-counting notions and to record this kind of thinking.

The earliest graphs

Initially some of the pictorial representations which arise from pre-counting activities are described. The teacher who wishes to begin with counting activities can join in at the appropriate place.

Stage A: no framework or base line.

i. Direct recording with real objects.
ii. Simple pictographs (pictograms): pictures drawn or cut out and placed in appropriate sets.
iii. Use of various discrete materials for concrete recording: use of counters, beads, pebbles, cotton-reels etc to represent each child (say) (one-to-one correspondence)
iv. Relations and mappings: children's names linked to favourite pets. (Later leading to the tabular form: Favourite fruit, Name.)

i. 'Children at our table'.

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ii. Use of conkers to represent this:

iii. Relations:

Stage B: count graphs with the base line emphasised.

i. Use of a natural baseboard: traffic or bird counts using cubes piled on top of one another or match-box graphs. (The beginnings of block graphs).
ii. Uniform representation: through discussion, the need is realised for identical units and for units to be edge-to-edge without gaps. Children can then be given three-dimensional uniform counters to mount in columns on a base line, or identical pieces of paper for drawings.
iii. Frameworks can be drawn on squared paper with a horizontal axis prepared by the children after discussion with the teacher. The spaces below each set (column) are labelled with the name of each set.
Further abstraction: columns built on squared paper, first using coloured squares and then shading squared paper with coloured crayons to represent counts. At this stage, these graphs might be 'preference graphs', eg my favourite pet, food, television programme. The order of the names is arbitrary along the horizontal base line.
iv. Nominal values ordered:
When the enquiry concerns sets which can be ordered, for example birthday months, days of the week, the spaces will be labelled in the appropriate order. It is always valuable to discuss whether there is an appropriate order, eg whether or not to put colours or children's names in alphabetical order.

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v. Introduction of numerical values:
By means of enquiries about, for instance, shoe sizes, a number scale on the horizontal axis can be introduced.

A 'vertical' axis is neither drawn nor labelled; squares can be counted as necessary where comparison of number is required.

Examples:

i. Base line emphasised: Use of a natural baseboard.

ii. Equal sized counts - equally spaced.
Our favourite pet: One-to-one correspondence - one pet for each child.

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iii. Our means of transport to school:

iv. Cars owned by parents:

v . Nominal values ordered: number of children absent on each day of a week

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vi. Numerical values ordered: number of families of different sizes.

Stage C: The introduction to block graphs.

As the size of the count increases, it is convenient and necessary to label the 'vertical' axis when the child is ready to use the numbers. At first, the spaces may be labelled 1, 2, 3 ... eventually, however, it is necessary to ensure that on the 'vertical' axis, the lines and not the spaces are labelled.

As before, spaces on the horizontal axis should be labelled using either nominal or numerical values, ordered where necessary. At this stage, a continuous strip or block can be used to represent the total count or frequency. Teachers need to ensure that this method of representation is appropriate for the child since it constitutes a further abstraction of the count. The following examples show how these graphs can be developed naturally over a period of some years.

i. Spaces numbered along the vertical axis.
Shoes sizes - numerical values ordered along the base line.

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ii. Scores resulting from 36 throws of a die:
Horizontal axis-numerical values, ordered
Vertical axis-lines marked, not spaces

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Block graph: Individual blocks used to represent the data. There are 8 children who come from families with 2 children.

Bar graph: A continuous bar is used to represent the count. A family size of 2 children has a frequency of 8.

Bar line graph: A 'bar-line' is used to represent the frequency. A clear scale is essential in both the bar and bar-line graphs.

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Later on, for older children, the following graph is appropriate:

Frequency graph for a continuous variable

This example shows an extension of frequency graphs using more complex data with a continuous variable, ie height measured in cm, for a group of junior children. In this case, it must be grouped into 'classes' having pre-determined 'class-intervals'. In this case the classes are 80-90-cm, 90-100-cm, 100-110-cm, etc. (Note that the interval 80-90- will contain all heights of children between 80 and 90, including 80 cm but not including 90 cm).

The term 'histogram' has a technical meaning which is explained in more advanced books. Unless the teacher understands this representation for a continuous variable, it is better to avoid the word entirely with primary school children.

Graphs which 'make themselves'

These are not count graphs but are obtained by using continuous quantities.

Examples:

i. Graphs of the shadow of a vertical post at half-hourly intervals during the day from early morning until evening: although the lengths are measured at intervals, intervening lengths can be included, if required.
ii. A set of similar elastic bands suspended from a set or equally spaced nails arranged in a horizontal line on which are fastened similar metal washers; one washer on the first band, two on the second and so on.
iii. Candle graphs: eight candles lit at the same time and blown out one by one, at intervals of 10 minutes.

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Strip graphs

i. These are not 'count' graphs but are obtained by direct matching using continuous material with no measurement: a ribbon or strip for each child representing an actual length (size of head, span, wrist, neck etc). The strips can be mounted in any order, either vertically or horizontally, from a base line. Ordering can take place and discussion can involve the terms 'the largest, smallest, longest, shortest'.
ii. Strip graphs to indicate relationships:
By direct matching, with thin ribbon or string the diameters and circumferences of a set of circular lids can be considered. The diameters can be marked directly along the horizontal axis and the corresponding 'string' circumferences can then be placed 'vertically' at appropriate points on the horizontal axis.
iii. Strip graphs involving scale:
By considering height and corresponding reach (arms outstretched), both variables directly recorded in string, a scale model of a child with arms outstretched could be represented. A relationship graph could then be plotted of the two variables.

i. Comparison of span sizes: using ribbon to match:

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ii. Strip graph to indicate relationship.
Circumference/diameter relationship. (Each ordinate is represented by a string).

*refer to example ii on page 63

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iii. Strip graph involving scale.

Heights of children

Line graphs or charts

Line graphs are often introduced through count graphs using pegboards, where the axes are drawn and the counts are represented by coloured pegs. Alternatively, graphs may be plotted using concrete material such as commercial structural apparatus to construct the two times tables or the table of threes etc.

Interesting points for discussion involve the question of when points are to be joined up (with continuous variables), and when left disconnected (with discrete variables). It is usual in science to plot the variable over which there is some control across the 'horizontal' axis, and the variable which depends upon it up the 'vertical' axis. Thus, travel graphs are most commonly drawn with the time axis 'across' and the distance axis 'up'. Where there are established conventions, teachers would be well advised to follow them. At the same time, children should not be discouraged if they volunteer unconventional ideas of their own.

At this stage, consolidation may be valuable with investigations which will give rise to the plotting of various types of graphs and to the discovery of relationships.

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There are many interesting examples to choose from, especially in science or drawn from newspaper information.

Examples:

i. Multiplication table using structural apparatus:

ii. Plots of multiplication tables:

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iii. Graph showing extension in length of a loaded spring.

iv. Temperature graph

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Coordinate graphs

The introduction to coordinates, ie the location of a point in a plane, by means of a grid, may be introduced gradually through several stages of development. These stages include the labelling of spaces and the labelling of the lines. Both systems are in everyday use and both are important.

Spaces labelled: street maps: eg references to labelled squares H2, E4; games of battleships, again using squares: position of a desk in a classroom.

Lines labelled: location of treasure on a map drawn on a rectangular grid - each point being located by its coordinates referred to the two intersecting axes. By custom, the distance from the 'vertical' axis, measured along the horizontal to the right, is stated first. The two distances are called the coordinates of the point - (x, y). It is Important always to state them in the proper order

There is much work with graphs that abler children in the upper junior classes can do with profit. Graphs which mean something are to be preferred. Graphs which embody information, such as the results of experiments the children have conducted, are much more valuable than graphs drawn from fictitious data. Graphs drawn only to illustrate algebraic equations are suitable for only a small minority of children at this stage.

Pie charts

These circular charts are sometimes used to represent information where there are not too many variables. They are for reading rather than making but children may enjoy making a few for themselves. The difficulties of sub-division of the circle or angle at the centre may be overcome by aids such as conversion graphs or even a simple slide rule. A pie chart can be drawn only when the complete data are to hand; whereas a bar graph, for example, can be extended as the data accumulate.

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A circle is divided into sectors which represent the different proportions of several pieces of data in the whole. A pie chart is used only when meaning can be given to the complete circle eg the circle could represent the number of hours of a complete day, the population of a certain group of people, the exports of a country etc. It would have no meaning, for example, if comparison was being made of the number of cars produced from year to year.

Example: How I spend my day

Initially, the pie chart can be drawn as if on the face of a 24-hour clock and later on, the proportion may be expressed as a fraction of 360°. For the more able child, these proportions can finally be calculated as percentages. The construction of these graphs cannot be given priority but we hope that they will be studied when they are a natural way of learning about other subjects eg geography or current affairs.

Scatter graphs

These are the graphs in which two pieces of information (two variables or statistics) about each member of a set are represented simultaneously. This type of graph, which would be introduced only to able children, involves the first notions of correlation. Scatter graphs reveal relationships that might easily pass unnoticed or, which is equally important, show that a relationship does not exist when the scatter is random. A simple example of a scatter graph would be the plotting of height against reach (with arms outstretched) for each child within the class. This might reveal a scatter of points which has an approximate linear relationship.

The three-dimensional scatter graph is a frequency graph of two variables. One example of such a graph would arise when considering the number of boys and the number of girls in the family of each child within the class. If these are plotted on the axes in the horizontal plane a three-dimensional scatter graph emerges where the number of children with, say, two brothers and one sister is plotted at right angles to the plane. These counts can be conveniently shown by using interlocking units, a number of kinds of which are produced commercially. In this case, the spaces and not the lines are numbered.

Examples of a scatter graph are:

height of child/total span;
weight of child/height;
estimates of length and breadth of a room.

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Examples of three-dimensional frequency graphs are:

number of brothers/number of sisters;
number of boys/number of girls in a family;
score of die A/score of die B

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8 Shape

One of the purposes of this document is to help teachers to identify those parts of mathematics that must be given priority. This is much harder to do with shape than it is with number. It could he said that no single detail of geometry is indispensable - but if the teacher neglects this area of work entirely, the children are certainly deprived. The problem is to select a range of suitable work, when we cannot declare that certain specific topics are essential.

In the very early years before the child comes to school, he is endlessly preoccupied with his position and mobility in space. He explores the geometry of his environment by reaching for and touching everything he can. Even the ability to focus on an object is an acquired skill. Is children's delight in pattern acquired from the highly regular man-made and natural environments into which they are born, or is it purely innate?

In the pre-school years, children have the opportunity to touch and handle many shapes, both natural and man-made, in the three-dimensional world in which they live. Through experiences they develop an awareness of shapes and their attributes in terms of texture, colour and form. These are later described and identified as the child acquires mathematical language.

In the infant classrooms focal points of interest, for example displays of shape, arouse curiosity and may be used as a stimulus for language. They may also create a feeling of wonder and curiosity in young children and increase their awareness of the aesthetic qualities in the subject. Collections of natural and man-made shapes provide opportunities for the child to become aware of many three-dimensional objects, their textures, surfaces, outline and form.

When shapes can be distinguished, the ideas of congruence and similarity (scale) begin to develop at the same time. Children come to recognise circles of all sizes - shape being the common property. The development of these abilities is enhanced by handling containers and shapes of many kinds and later on by the use of more structured apparatus. Classifying and discriminating are essential elements in a good mathematics programme. Each depends for its growth on the development of language and symbolism. The study of shape can promote a rich mathematical language and the vocabulary will become refined as the mathematical experiences become enriched.

Teachers need to help children to develop their natural appreciation of symmetry and to make them aware of times when their models, and later their art, have symmetry. So much of the work introduced to young children emphasises two-dimensional symmetry, to the almost total exclusion of three-dimensional symmetry. But often within the first term, children build models with waste materials and bricks which show three-dimensional symmetry. Children usually accept the word 'balance' to describe such models.

In the middle and upper infant school, patterns may be made with paint by cutting and folding paper, or on pegboards to give two-

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dimensional symmetry, but this is far more satisfying if it arises incidentally. Ideas of symmetry should he sought in the world around us - reflections in ponds and mirrors and the symmetry of natural things. Examples of rotational symmetry should also be sought. We should delight in a child's natural talent for appreciating shape and form and should encourage his growing awareness of symmetry.

Geometry with juniors will move on to three-dimensional solids and their properties, followed by the study of plane shapes. It is likely that for most children the idea of a plane will not be formed without explicit discussion. In the course of this, such ideas as angle, area, perimeter, symmetry and rigidity will be introduced. Through practical experience of model making, both solid and skeletal, plane-filling tessellations and paper folding, children should come to recognise the geometrical properties of common two- and three-dimensional shapes. They will have an appreciation of symmetry in two and three dimensions, both reflectional and rotational. An awareness of symmetry, not only in geometry, but in graphs and number might also be developed. Ideas of mathematical similarity and scale recur in other aspects of the curriculum and these might be developed for some children. Later, for the more able, the teacher may wish to systematize the learning. There is here, within the development of shape, the richest mine of opportunity for generalisation. There is also plenty of scope for logical argument and some older and abler children may be ready to appreciate stricter ideas of proof, for example, that tiles of some particular shape cannot possibly cover a plane without leaving gaps, or that there cannot be more than the five regular solids (described in a variety of books).

Stages of development in shape

Set out below is a detailed catalogue of experiences for children at different stages of development. Many teachers find time to provide some of these opportunities, and some teachers provide a great many. In all these stages, as the child's ability in language develops, he should be encouraged to speak and write about what he does under the various headings.

We would not want some less experienced teachers to be disconcerted by the length of the list and we feel that confidence and mastery over a more limited range is preferable to superficiality over a wider range. In view of the wide discretion which primary schools have in this matter, it is all the more important that they should have satisfactory arrangements for informing secondary schools of the work which children have done.

Stage A: infant level

1 Informal examination of a variety of shapes

Collections of natural and man-made shapes: shells, leaves, stones, driftwood, flowers, cartons and containers of all types, shapes and sizes. Touching, feeling and talking about them. (This stage of development is vital and needs considerable time in which to allow children extensive experience.)

2 Sorting

Extension of vocabulary by sorting, discussion and making collections

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of three-dimensional objects. Assorted shapes need to be chosen, including ones with holes, with both inside and outside surfaces, hollow, twisted, solid and non-uniform surfaces, providing opportunities for learning new words.

3 Patterns

Making patterns from objects or shapes, making mosaics with bricks or tiles. Printing with the faces of shapes, their outlines or sections (potato and card-cuts). Drawing round or tracing the faces of shapes - abstracting two-dimensional shapes from three-dimensional objects. Painting faces of three-dimensional shapes, such as cubes or boxes, and making further patterns.

4 Solids

Making and discussing shapes in clay, plasticine and sand. Shapes in pastry. Constructions in large and small bricks, Lego, Poleidoblocs, off-cuts.

5 Surfaces

Covering surfaces and faces, painting and colouring faces of different shapes and textures. Drawing round or tracing the faces of three-dimensional shapes, and covering the two-dimensional shapes with different materials.

6 Extended sorting

Using three-dimensional shapes and later, two-dimensional shapes. Making collections and extending vocabulary through discussion. Discriminating between attributes, classifying and identifying.

Description of shape: looking for differences and similarities.

7 Exploring space

Fitting three-dimensional shapes together; exploring space by packing these shapes. Exploring two dimensional space: covering a plane surface with different shapes - with overlap and without overlap eg tangrams. Covering a surface with a given set of shapes. Finding the number of ways in which a given shape will fit back into the hole from which it was cut. Exploring the relationships between different two-dimensional shapes, and between three-dimensional shapes, for instance the building of a tower with a cuboid and a square-based pyramid.

8 Line symmetry

Opportunities for children to explore symmetry - the balance of shape. Making ink/paint 'devils' or symmetric patterns by using folded paper or card. Folding paper and cutting patterns. The use of mirrors to produce symmetrical patterns.

Stage B: lower juniors

1 Solids

Making models by fitting together identical blocks, or assorted shapes and containers, exploring irregular as well as regular solids. Discussions of different 'solids' to develop vocabulary, and to identify the number and the different types of faces, the number of corners (or

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vertices) and edges. The plane sections of solids. Consideration of skeleton models of three-dimensional shapes, and of rigidity.

2 Sorting and classifying: plane shapes

Consideration of sets of shapes designed to exemplify the vocabulary of two-dimensional shapes: triangle, square, rectangle, circle, quadrilateral, pentagon, hexagon, kite. Regular and irregular shapes.

Once the word 'regular' is used, there must be careful discussion of what it means, applied to polygons, namely that a polygon is regular if and only if all the sides are equal in length and all the angles are equal.

Making plane shapes on geo-boards, with square and isometric (equilateral triangular) grids. Drawing shapes on square and isometric paper. Continuing discussion of rigidity. Identifying different interior angles, diagonals, and sections of plane shapes.

3 Symmetry: three-dimensional shapes

Consideration of plane symmetry: collecting of symmetrical shapes, containers, solids, and classifying these. Making symmetrical models using three-dimensional shapes. Cutting solids made of clay, plasticine or polystyrene and comparing elements to determine symmetry of original shape. Consideration of symmetry using skeletal models.

4 Patterns

Patterns based on simple shapes, without overlap. Looking at mosaic patterns. Circle patterns as practice in using a pair of compasses.

Varied practice in drawing straight line patterns, initially freely but moving towards more controlled situations. Mixed circle and straight line patterns. Thread patterns which introduce children to the ideas of curves formed from an envelope of lines, tangents and families of lines (curve stitching).

5 Solids (continued)

Construction of regular and irregular skeletal models using pipe cleaners, or straws. A study of sets of regular and irregular three-dimensional shapes, by analysis of packets, containers, boxes, cones. Consideration of ways of covering a solid body which children have handled.

6 Angle as a measure of turn

Consideration of angle through turning: working with clock faces or a compass with eight points: angle of rotation of wheels: of pendulum, metronome, cogs and pulleys, angle of turn not necessarily restricted to one complete turn.

Language to include: complete turn, half and quarter turn, right angle, half right angle, rotate, rotation, ray.

Stage C: (upper juniors)

1 Extension of sorting and classification:

Classification of three-dimensional shapes: cube, cuboid, prism, pyramid, sphere, cylinder, cone, tetrahedron, spiral.

Recognition and description of these shapes, classification by tabulation (or punched card). Discussion and descriptive writing about shapes.

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Language might include words such as cube, cuboid, prism, pyramid, cylinder, cone, sphere, tetrahedron, surface, diagonal, face, edge, vertex, regular, slant, spherical, spiral.

Classification of two-dimensional shapes: circle, triangle, quadrilateral, parallelogram, rhombus, trapezium, pentagon, hexagon, heptagon, octagon, decagon.

Language might include words such as parallel, regular, irregular, isosceles, equilateral, scalene, convex, concave, vertex, radius, circumference, diameter, centre, diagonal, segment, side, perimeter, chord, polygon, arc.

2 Patterns: tessellations (tile patterns)

Consideration of various tessellations: hexominoes on a squared grid, and similar shapes on isometric paper. Tessellations of given shapes including scalene triangle, and of more regular shapes.

Tessellations in architecture, mosaics; lino cuts and fabric printing.

3 Measuring angles

Consideration of right angles, 45° angles, by folding. Angles of 60°, 30°, and 120° by folding or, later, using compasses. Comparison of angles with standard angles, by tracing or using set squares. Discovery that the sum of the angles of a triangle is equal to half a turn (180°). Consideration of the sum of the angles of triangles from tessellation for equilateral, isosceles and scalene triangles. Similar work with tessellations of parallelograms, of hexagons and of regular octagons with squares.

4 Solids

Consideration of prisms, solids and polyhedra. Making simple regular and irregular three-dimensional shapes from a net designed by the pupil. (If a polyhedron is unfolded and opened out in such a way that all the faces lie in one plane, the shape, including the lines on it formed by the edges of the polyhedron is called a net of the polyhedron. Generally, a polyhedron can have several different nets.)

Extended experiences for the more able top junior children:

i. Symmetry of two- and three-dimensional shapes:

Making symmetric shapes on squared paper and isometric paper. Using mirrors, consideration of reflections of shapes. (The joining of the point and its image on such diagrams will establish that the mirror line bisects all these lines at right angles.) Constructing of images on squared, isometric and plain paper using this property.

Consideration and discussion of simple symmetry of three-dimensional objects.

ii. Angles: use of a protractor:

a Measures in degrees of fixed angles. The ideal protractor is circular and has a scale marked in degrees.

b Angle as a measure of turn: turning about a fixed point: whole turn; half turn, extending to the physical world: movement in PE, locomotive engines, levers, pulleys, swing bridges, cranes,

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rotation in mechanics.

c Rotation measurement in degrees:

Measurement of angles: hands of clock, work with a compass including three-figure bearings (036°, 168°) and compass bearings (N48°W). Angles between lines and the formal naming of an angle: angle ABE = 75°.

iii. Rotational symmetry:

Introduction to, and investigation of, shapes possessing rotational symmetry. Making patterns based on rotation using congruent shapes. Practical work on the determination of the order or rotational symmetry.

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9 Conclusion

An attempt has been made in this hooklet to distinguish between those parts of mathematics which are required by all - or virtually all - children in primary schools, and those that can usefully be tackled by children who show a flair for the work with the help of teachers confident and knowledgeable in the subject. The distinction between these two parts cannot be precise, and this booklet has been issued in the hope that it will provoke discussion, among other things, about whether this distinction is on the right lines or whether another would be more appropriate.

It is certainly true that the essential parts of the subject, as defined here, provide a great deal of work for teachers and children; more, perhaps, than it would be reasonable to expect if every teacher had to work in isolation and unsupported. But that ought not to be so.

Almost half of the two-form entry primary schools in the HMI national survey of primary education had teachers with a special responsibility for the teaching of mathematics within the school. Just under a third of smaller schools and nearly two-thirds or larger schools had such teachers. Teachers with this responsibility can give vitally important support to other members of staff, concerning themselves with far more than the organisation of the school's stock of mathematical books and apparatus. Some instigate in-service training within the school, work alongside other teachers in their classrooms to further the process and may, when necessary, take on the teaching of groups or children from another class for a shorter or longer period. They may draw up, in consultation with the head and other teachers in their own school and outside it, schemes of work that establish a framework for the teaching of mathematics in the school.

In addition, both the semi-specialist and other teachers may be helped through teachers' centres, the local advisory services and in-service training courses, and by means of published material such as that arising from Schools Council projects, consultations within LEAs and the deliberations of professional subject associations.

Given this support, even teachers who lack confidence in the subject can manage better. It remains true that the ground to he covered is substantial.

There are few safe short cuts. While it is desirable that all children should reach a satisfactory level of mathematical competence by the time they are 11, those who find the work difficult lapse into bewilderment if essential stages are rushed. On the other hand, time is too precious to be spent on the unnecessary repetition of work already grasped, especially if further practice can readily be undertaken while acquiring a new skill or idea.

The pacing of the work calls for careful professional judgment based on a knowledge of both children and the subject. The timing is not so critical, however, that each child must always work separately and individually. The grouping of children can encourage effort, and certainly makes it easier for the teacher to stimulate the children and to supervise what is being done.

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The last twenty years have seen many encouraging developments in primary school teaching, not least in the teaching of mathematics. We hope that this book will contribute to the continuing discussion of what ought to be taught and what might be taught, and that further clarification of these two matters will also help those concerned to decide on the methods and forms of organisation best suited to their purposes.

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Appendix I

Objectives

These should be considered to be the main objectives for the majority of children at the age of 11 years.

i. The development of appropriate language; qualitative description, the recognition of objects from description; discriminating, classifying and sorting of objects; identifying objects and describing them unambiguously;

ii. The recognition of common, simple mathematical relationships, both numerical and spatial; reasoning and logical deduction in connection with everyday things, geometrical shapes, number arrangements in order etc;

iii. The ability to describe quantitatively: the use of number in counting, describing, estimating and approximating;

iv. The understanding of whole numbers and their relationships with one another;

v. The appreciation of place value, the number system and number notation, including whole numbers, decimal fractions and vulgar fractions. The ability to recognise simple number patterns (odds and evens, multiples, divisors, squares etc);

vi. The appreciation of the measures in common use. Sensible estimation using the appropriate units. The ability to measure length, weight, volume and capacity, area, time, angle and temperature, to an everyday level of accuracy;

vii. The understanding of money, contributing to a sense or the value of money, and the ability to carry out simple purchases;

viii. The ability to carry out practical activities involving the ideas of addition, subtraction, multiplication and division;

ix. The ability to perform simple calculations involving the mathematical processes indicated by the signs +, -, X, ÷ with whole numbers (maintaining rapid recall of the sums, differences and products of pairs of numbers from 0 to 10);

x. The ability to carry out with confidence and accuracy simple examples in the four operations of number, including two places of decimals as for pounds and pence and the measures as used;

xi. The ability to approximate and to check whether the result of a calculation is reasonable;

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xii. A sound understanding of place value applied to the decimal notation for numbers. The ability to carry out the addition and subtraction of numbers with up to two decimal places and the multiplication and division or such numbers by whole numbers up to and including 9;

xiii. The multiplication and division of numbers with up to two decimal places by 10 and 100;

xiv. An appreciation of the connections between fractions, decimal fractions and the most common percentages;

xv. The ability to use fractions in the sequence 1/2, 1/4, 1.8, 1/16 or 1/3, 1/6, 1/12 or 1/5, 1/10 including the idea of equivalence in the discussion of everyday experiences;

xvi. An appreciation of the broader aspects of number, such as bases other than ten and easy tests of divisibility;

xvii. The ability to use and interpret the following: diagrams; drawings; tables; simple networks; maps; scale drawings; diagrammatic, pictorial and graphical representation, statistical charts and three-dimensional representation;

xviii. An appreciation of two- and three-dimensional shapes and their relationships with one another. The ability to recognise simple properties; to handle, create, discuss and describe them with confidence and appreciate spatial relationships, symmetry and similarity;

xix. An ability to read with understanding mathematics from books, and to use appropriate reference skills;

xx. An ability to write clearly, to record mathematics in statements, neatly and systematically.

Extension of concepts and skills:

Further skills and concepts which experienced teachers with the appropriate knowledge might consider appropriate for some children include:

i. calculations involving the four operations using numbers with up to two decimal places, and the addition and subtraction of vulgar fractions;

ii. further measures, including speed, force, pressure and density;

iii. simple ideas of probability;

iv. a simple approach to topological notions, (ideas or connectedness, as explained in a number of mathematical topic books for children);

v. an extended appreciation of broader aspects of number such as modular arithmetic, index notation and standard form, number sequences, number games and puzzles;

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vi. symbols of algebra for stating logical patterns, following abstraction and generalisation in arithmetic.

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Appendix II

Types of structural apparatus commonly used in 1979

Unifix interlocking plastic cubes are ¾-inch in section; the exclusive interlocking device forms an integral part of each cube by which the child is enabled to construct rods of any length, colour, or combination of colours from individual cubes. This facility is of importance in the practical presentation and operation of those early stages of number which involve the composition and decomposition of values.

The unit cube is the foundation of unifix rods and the dual aspect of values, first as a collection of units (analysis) and secondly, as composite entities (synthesis).

Unifix cubes are available in ten distinct colours but the colour has no significance as a representation of value; any significance attached is arbitrary and cannot be more important than the limitations inherent in any personal 'scheme'.

There is a very large range of apparatus available with ancillary equipment and a teacher's manual for reference.

Stern structural material is a generous size of ¾-inch square sections of wooden pieces. Each whole number between 1 and 10 is represented by a coloured rod comprising a number of clearly marked cube-units. This composite material has ten colours, each colour associated with the particular number it represents. The coloured rods are supported by an extensive range of apparatus including counting boards, pattern boards, unit box with blocks, number tracks and cases, twenty trays and box of 100 cubes etc.

A Cuisenaire set consists of wooden rods cut in lengths which vary by 1cm from length 1cm to length 10cm, all of cross section 1 square cm. Each of the 10 different lengths is coloured one of 10 different colours and this identifying association of colours and lengths is consistently applied throughout the set. The standard set contains 291 rods in 10 separate compartments. There is a large range of materials and publications to support the use of Cuisenaire equipment, including several text books for teachers and a pupil's textbook series.

The Colour-Factor set consists of 308 coloured wood sections, 1cm square cut in lengths from 1 to 12cm. Each length has a distinctive colour for quick identification and selection and all sections of the same length share the same colour. This composite material is similar to Cuisenaire but with the 11 and 12 rods.

The Colour-Factor set dictates no system or method: teachers are free to use this material to supplement any method of teaching number that they favour. For those teachers seeking guidance there are textbooks explaining different approaches which can be followed.

The Centicube is a precision made plastic centimetre cube which weighs 1g and is supplied in sets of 1,000 (200 each of five colours). The Centicube has 4 slots, 1 tongue and 1 blank face for marking or numbering, and interlocks to illustrate number. It is a versatile material, used not only as a structured apparatus for number experience but also for shape and measurement in metric units.

The Structa apparatus consists of cylindrical plastic pegs in twelve bright colours. The pegs fit into number peg boards for structuring

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groups of pegs into any pattern but also fit into each other to form vertically structured groups of pegs. There are also sets of pegs permanently structured into number rods varying in length from one to 10 pegs. These integrated lengths of pegs represent the numbers from 1 to 10. There is a large range of materials available to support the above apparatus, including 20 base-boards, peg-boards, number cards and notation cards etc.

Multibase Arithmetic blocks (including Dienes' and Tillich's blocks) are in natural wood, using the 1cm cube as the unit. The decimal number system is represented as follows: small cubes represent units; rods of 10 units ('longs') represent tens; slabs of 10 longs ('flats') represent hundreds, and large cubes of 10 flats ('blocks') represent thousands. A feature of this material is that it deals with the place-value concept and provides experience in number systems other than the decimal; this facilitates the abstraction of those mathematical and notational generalisations underlying different number systems, and this, in turn, enables the child to understand the operation of these generalisations in the decimal system.

Five other number systems of base numbers 2, 3, 4, 5 and 6 are represented similarly. In addition to the complete set, some supplementary materials are available including extended material, triangular/trapezoidal material, work cards and teacher's manuals.

The Algebraical Experience Material follows the same principles as the Multibase Arithmetic blocks but is intended for use at a later stage as a means of introducing algebraic ideas. The materials are in natural wood using a variety of flat pieces of different shapes and sizes: small and large squares and other rectangles, including strips; equilateral triangles, small trapezia and rhombi. Additional materials include a balance and weights and a pegboard; they are supported by children's work cards and teacher's instruction manual.

The use of this material is a skilful matter and primary teachers need to recognise the range of experience to which children can be introduced in the different aspects of the work. They should be familiar with some of the extensive literature which is available and engage in discussion of the material with other teachers within the school.

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Appendix III

Recommended reading list

The teaching of mathematics in primary schools. A report prepared for the Mathematical Association for consideration by all concerned with the development of young children. Bell (1956).

Primary education. Suggestions for the consideration of teachers and others concerned with the work of primary schools. HMSO (1959).

Mathematics in primary schools. Curriculum Bulletin No 1. HMSO (1965).

Nuffield mathematics project. Teachers' guides and materials. Chambers/Murray (1967).

Notes on mathematics in primary schools. Members of the Association of Teachers of Mathematics. Cambridge University Press (1967).

Primary mathematics today. EM Williams and H Shuard. Longmans (1970).

Primary mathematics - a further report for the Mathematical Association. Bell (1970).

The psychology of learning mathematics. RR Skemp. Pelican (1971).

Metric units in the primary school. The Royal Society (1969).

Nuffield/CEDO. Handbooks for Teachers:
Mathematics: the first 3 years. Chambers/Murray (1970).
Mathematics: the later primary years. Chambers/Murray/Wiley (1972).

Notes on mathematics for children. Members of the Association of Teachers of Mathematics. Cambridge University Press (1978).

The third R. Towards a numerate society. Edited by J A Glenn. Harper and Row (1978).

Nuffield/British Council. Handbook for Teachers. Mathematics: from primary to secondary. Chambers/Murray/Wiley (1978).