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Primary Education (1959) (page numbers in brackets) Notes on the text
Part 1 Historical
Part 2 The Primary Schools
Part 3 The Fields of Learning
Part 4 The Special Problems of Wales
Index (331-334) |
Primary Education (1959)
Suggestions for the consideration of teachers and others concerned with the work of primary schools London: Her Majesty's Stationery Office 1959
[page 179] [See my note at the end of this chapter if you need explanations of the imperial measures and pre-decimal coinage mentioned.] There is no subject in the primary school curriculum which gives rise to more thought at the present time than mathematics. It is significant not only that it is the theme of discussions and conferences in all parts of the country among primary teachers themselves, but that mathematicians are increasingly giving thought and attention to the development of their subject in the minds of young children.* This interest springs partly from the contemporary need to have as many good mathematicians as the country can produce and as widespread an understanding of mathematical communication as is possible, and partly - and here teachers have been especially active - from the feeling that the time and energy spent on teaching mathematics (or 'number' or 'arithmetic') in primary schools produces too seldom in children lasting interest, general competence or the confident attitude towards mathematical problems that might have been expected. What follows in this chapter makes no pretensions to giving complete answers to the many questions that arise. It offers ideas and suggestions which are based on what has been observed in schools, in the hope that teachers will find something to stimulate and guide their own thinking and so lead them to more effective practice. In all primary schools some mathematics is taught, whether it is called 'number', or 'arithmetic' or 'practical mathematics', and whether it is in set lessons or as an incidental part of children's general work. Are the schools merely following tradition or are there compelling reasons why a proportion of children's primary school life should be spent in the learning of mathematics? *See for example the Mathematical Association's Mathematics in the Primary School. [page 180] Tradition certainly plays a part, perhaps as a legacy from the days of payment by results. Mathematics was, and still is, a subject in which performance can be assessed with a fair amount of objectivity. Also, in the past, much time was spent on the learning of mathematics because there was a strong belief that the training given was of general value, that the virtues of accuracy, concentration and logical thinking which are encouraged in mathematical training were automatically transferred to other activities. In its crude form this idea is now discredited but it is true that mathematics, well taught, may have an influence on children's general attitude to learning, and that ways of tackling problems in other situations are influenced for good by sound mathematical training. Mathematics has retained its traditional importance as an examination subject; it is widely used in examinations for allocation to secondary education and for the measurement of children's progress; also parents and teachers often quote a child's success in arithmetic as evidence of his progress at school. The question which may reasonably be asked by teachers today is: 'Obviously we must teach mathematics in our schools if our children are to pass external examinations, but what other reasons are there for teaching it?' If the learning of mathematics is conceived of as nothing more than the daily grinding out of pages of mechanical sums, then probably the answer is that there is little justification for spending a great deal of time on it. If, however, it is conceived of as something which will help children (and adults) to solve some of the problems of living, to order and extend their everyday experiences so that they may better deal with their affairs, then it becomes of great importance. The attempt to achieve this aim makes more demands on the teacher, but there will be compensation in the added interest and stimulation which both teacher and children will enjoy. It may be useful to consider some further reasons for learning mathematics in the primary school. (a) Historical Mathematical thought is part, and a great part, of the heritage of the race. Children should not remain unaware of this activity [page 181] of the human mind, by which, to some extent, our own individual minds have been moulded. By its aid man has measured the distances to the stars, forecast eclipses, navigated the seas and the air, made maps of the earth, built cathedrals and bridges, split atoms and designed machines from the simple lever to the most complicated space satellite or electronic computer; all the elaborate business transactions between men, between groups of men, and between nations are founded on a knowledge of mathematics. And the subject is growing; the need to know more about the structure of the atom led to the development of new algebras and geometries. To quote a great mathematician, it is 'a study which did not begin with Pythagorus and will not end with Einstein, but it is the oldest and the youngest of them all'.* It is a continuing and unique way of thought and children should become acquainted with it, and experience it, at however humble a level. (b) Utilitarian As babies, children are surrounded by shapes, and their lives are subject to the rhythm of time. Later they are concerned with houses, and buses, and pages of books, all of which have numbers, with clocks, the hands of which turn through angles, thus measuring the passage of time, with calendars which register dates, with the ages of various people and the differences in their ages. Our environment is daily becoming more mathematical in its implications; the vocabularies of even young children contain words such as wavelength, supersonic speed, acceleration, gradient, interplanetary and atomic energy - words, which until comparatively recently were part of the vocabulary of the scientist and the mathematician only. Children and adults are confronted by these mathematical ideas at different levels; if they are to talk intelligently about, and deal efficiently with, their environment it is essential that words should have some precision of meaning and only through mathematics can some of these words be defined. Every adult needs a certain amount of mathematics in order to live in this complex society and feel some sense of mastery over his environment. Everyone must be able to deal with numbers, *GH Hardy, A Mathematician's Apology Cambridge University Press. [page 182] with money, and with the quantities such as length, weight, time and capacity. In addition, increasing numbers of adults are needed to follow occupations which require more technical mathematical knowledge. Society must have its clerks, engineers of all types, and atomic physicists. It is true that with the advent of calculating machines there is no longer the need for man to undertake long computations; but the design, programming and maintenance of these machines require more advanced mathematical knowledge than mere computation. It may be argued that the mathematics of the engineer and the atomic physicist is remote from the work done in the primary school. Nevertheless, it is in the primary school that children's attitudes towards mathematics are created; if children become confused and unable to master the work at this age, the feeling of fear and frustration may remain with them throughout their lives. (c) Aesthetic Aesthetic satisfaction may seem a somewhat unusual value to expect in the mathematical work of a primary school. As the subject is often taught, it may seldom emerge, but if it does not, the fault lies, not in the subject, but in the teaching. The elegance, the order, the pattern and the generality which are inherent in even the most elementary work can be appreciated by all children in some measure if the work is presented with this aim in mind. Clearly the teacher must be aware of these qualities himself, and appreciate them. It may be that some primary teachers need to know more about mathematics. They are now fortunate in that they are all within reach, either by post or by a personal visit, of an educational library where there is a librarian who will help in suggesting suitable reading. If more primary teachers developed an interest in mathematics there might be a different emphasis in the teaching, and children, most of whom begin by enjoying their work with numbers and shapes, would continue to enjoy mathematics to the end of their lives; unfortunately, their enthusiasm is often allowed to die. Children find pure arithmetic fascinating; the often heard statement that children love doing pages of little sums has probably much truth in it; they find enjoyment in manipulating numbers and in gaining power over them. But discussion of the sometimes strange properties possessed by numbers plays [page 183] far too little part in much arithmetic teaching. They might discuss, for example, the nine times table, the casting out of nines, triangular numbers and their relations with square numbers. There are many even simpler results. For instance, because 9 + 7 = 16, then 19 + 7 = 26, and 29 + 7 = 36. Facts like these many children never discover for themselves, but they should become obvious if the decimal notation is fully understood in all its simplicity and economy. It is often said that if children are to enjoy mathematics, and if they are to apply what they learn in school to the affairs of life, then they must understand what it is about. What is meant by understanding? It is a difficult question which exercises the minds of mathematical teachers at all levels. Obviously children cannot wait to use the idea of 'four' until they are capable of understanding it fully. This is a postgraduate exercise, and if children never use the idea of 'four', the stage of postgraduate mathematics will never be reached. In any case whether teachers wish them to use it or not they will use it; they live lives apart from that in the classroom. When learning a new mathematical process children should at least know for what purpose they will use it, and be convinced that it gives the 'right answer' to the problem. This implies that they can find the 'right answer' by a more rudimentary method or by using concrete material. For example, if they are being taught the multiplication process they should realise that this is an economical way of doing repeated addition and they should satisfy themselves that the new process gives the same answer. They should also be capable of using this process in significant situations. This does not mean that teachers should continually return to the early stages and thus retard progress. Once children feel that they have reached at least the degree of understanding mentioned above, and that their 'sentiment of rationality' has been satisfied, then the new ideas become a part of them and they use them, as they use language. When they speak they do not stop to think out the definition of every word they use, or they would never talk; but they must have some understanding of the words and sentences they use, and the teacher's aim should be to [page 184] increase and develop this understanding, by widening and deepening the meaning. Responsibility for being aware of what depth of understanding can be demanded from particular children rests with the teacher, as he alone watches the learning process from day to day. Very little is known of the mental processes by which children learn mathematics but research which has been done suggests that children are too often expected to deal with mathematical ideas which they do not sufficiently understand and cannot assimilate in their early school days. There may, however, be some processes and ideas which the teacher thinks it important to teach at a particular stage, even though he knows that it is difficult to give adequate justification for them. For example, he may think that by the time children are, say, seven or eight they should be able to use the subtraction process. There are many ways of performing this process, some of which are more easily explained than others. If he believes that the 'equal additions' method is the best he may find it difficult to show why the method is valid. He can however show that it works in a variety of examples which can be tested practically and so lead the children to have belief in the method, as opposed merely to belief in his statement that it will work. This is a method which permeates all science - the observation of many particular cases leads to a belief in a general result, which can be tested by further experiment. Later the general result may be seen as a deduction from simpler results. The teacher's desire that children should believe in their own reasoning powers will be a guide for him in this notoriously difficult problem of children's understanding. How are children to be taught mathematics in the primary school so that their interest in the subject will grow all the time and they will leave the primary school feeling that they are looking forward to the next stage and are competent to deal with it? Mere accuracy of computation is not enough. Certainly children must learn addition and multiplication facts and they must learn the addition, subtraction, multiplication and division processes as applied to numbers and quantities, and to use these processes accurately - otherwise no satisfaction can be obtained from the attempts to solve problems. There is always a sense of frustration if a child or an adult wishes to do a piece of work and finds either that he has not the requisite tools, or, if he has them, [page 185] that they are not sharp enough for his purpose. Nevertheless he must also know what work he wishes to do with his tools and what tools are required for a given purpose. It would not be thought sufficient if a workman had sharpened a tool to a fine edge and then practised for weeks the correct method of using it; he would be expected to make something with the tool. Similarly it would seem wasteful if a pianist practised scales for years and never attempted to use his technique to interpret a piece of music. Both the workman and the pianist, though probably deriving pleasure from the mastery of their technique, would have stopped short of the important end of their work. Similarly children derive pleasure from their mastery of mathematical techniques, but these are learned so that they may be used, and so that they may be seen as part of a wider mathematical structure. It is not generally realised how, in mathematics, a class can develop its own mathematical knowledge. When history, for example, is being taught children will need to be given some facts; they cannot deduce the story of the Armada from their previous knowledge of Queen Elizabeth I's reign. With skilful teaching, however, they can be led to discover the process of addition as a more economical method than counting in ones, and multiplication as an economical method of performing repeated additions. If they know how to multiply numbers they can then deduce how to multiply, say, shillings and pence by a number; they are merely changing the base from ten to twelve. In the same way, at a later stage, in the secondary school, a knowledge of the geometry of the triangle enables them to deduce the geometry of many-sided figures, of the circle and of the conic sections. This 'awakening of the learner's belief in reason and his confidence in the truth of what is being demonstrated'* is one of the fundamental aims of the teaching of mathematics and can permeate any syllabus, however stereotyped. If children are to play this important part in their own learning of mathematics then there will need to be interplay between the minds of the teacher and of the children, and between the minds of the children themselves. This will entail discussion; therefore it is important that there should be talking in mathe- *Bertrand Russell: 'The Study of Mathematics' from Mysticism and Logic. George Allen and Unwin Ltd, London. [page 186] matics lessons, the discussion of new situations for which the children's present knowledge is inadequate, the development of that knowledge further, perhaps to the stage of evolving a new process, and the seeing of the implications of the new discovery. Another important part of mathematical teaching which can be dealt with by discussion is the development of an interest in general number relations and in interesting facts about particular numbers. For example, that if any number be multiplied by an even number the result is an even number, that the square numbers 0, 1, 4, 9, 16, 25, 36 ... have the odd numbers in succession as differences, that 60 is not only 6 x 10, but also 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12 etc. Time spent in encouraging children to notice these things and to become familiar with them gives a two-fold benefit - it gives the children great pleasure and increases their confidence in dealing with numbers, and it also simplifies the learning process in that they have far fewer unrelated facts to deal with. The emphasis on discussion does not mean that all children within a class or group should be expected to work at the same speed. If they do, then something is wrong with the teaching; either the slower ones are being dragged along in a confused state of mind or the quicker are bored. When a discussion is finished there should be opportunity for the children to work individually and for the teacher to deal with the difficulties of a child or of a group who have not grasped the ideas. Where a class works in groups it may sometimes be of value to have all the groups dealing with the same topic and to allow the ablest group to range more widely over the field. The brightest children in the primary school are capable of doing far more than is usually asked of them; their powers are developed in only a very limited way by giving them more and harder mechanical sums of the same type. Of more value is work, perhaps still on the same topic, which stimulates ingenuity and originality. The general principles of teaching which emerge from the preceding paragraphs may be applied to any syllabus and at any stage of teaching. They may be summarised as follows: 1. The teacher should know, well in advance, what he means to teach and how he means to teach it. He should ensure progression at all times, keeping in mind the abilities of the children. It is essential that this progression should apply to the brighter children as well as to the duller. Too [page 187] often the bright child, who learns something very quickly, is not allowed to progress until the child of average ability has caught up with him. 2. The teacher should be preparing well in advance for the introduction of a new idea or process, using situations that arise in the classroom and in other subjects and in practical work. 3. Having prepared the ground he will need to draw together all the various incidental information and facts in a discussion with the children, and will probably then find that some children, with judicious guidance, can work out on their own initiative the steps necessary to deal with the new problem. 4. There will be a period of consolidation when the children practise the new work until the teacher is satisfied that they have mastered the new process or the new skill sufficiently for the time being. Even though errors are occasionally made it may be better to let the topic rest after a time and to return to it later. This calls for experienced judgement on the part of the teacher; the important facts are that children must practise calculating techniques, but that calculation should not occupy too much of any child's time in mathematics. 5. The techniques have been introduced because they are seen as superior methods of solving introductory problems. They will continue to be used to solve problems, perhaps chosen from a textbook, or made up by the teacher. The use of problems occurring in children's everyday lives, while admirable if the problems exist or can be contrived, needs careful forethought, especially with a large class. Too often such problems provide very little arithmetic, and that too difficult for children to deal with. It is more important that problems should be significant, that is, that they should refer to situations which children understand. To take an example - here are four problems of greatly differing significance to primary children and yet all involve the simple arithmetical sum 7½ ÷ 3: (a) Three ounces of sweets cost 7½d. How much does one ounce cost? [page 188] (c) What is the volume of a cone whose base and height are the same as those of a cylinder whose volume is 7.5 litres?6. If work previously learned is constantly used in the subsequent teaching there should be less need for formal revision; but this will always have a place in teaching. 7. Along with the continued extension of ideas and techniques, their practice and application, time should also be given to the learning of number facts, and whether this becomes wearisome depends mostly on the teacher. He will try to keep a balance between the two opposing ideas, that children must sometimes spend time learning facts that are not intrinsically interesting, and that, on the other hand, children learn more quickly and lastingly if they are interested. Children living in a normal environment will learn their mother tongue without direct teaching; so inevitably will they learn to deal with mathematical ideas. As in their daily lives they learn language through contact with the external world, a world which includes other children and adults, so do they consciously and unconsciously absorb mathematical ideas while manipulating objects and while watching and listening to others. Too frequently in dealing with children on their entry to the infant school teachers tend to ignore the very considerable background of mathematical knowledge which the children already possess. This knowledge will vary according to their experiences during the first five years of their lives. Children who have gone shopping with their mothers, or perhaps alone, who have helped with the household chores, who have watched men at work, who have played in the garden with other children, putting water or earth into cans or jars, may have a far better mathematical background than those who have not had these advantages. A valuable experience for the teacher of the youngest class would be to [page 189] make a list of the words and ideas concerned with mathematics which children use by the time they first come to school. Most children will know the number names (although perhaps only as a meaningless jingle with no reference to counting); many will know words such as heavy, light, ounce, year, month, week, day, just right, big, little, pound, shilling, penny, quick, slow, half, both, pair (of shoes), lots, thin, thick, wide, long, high, low, tall, short; they will use words involving comparison such as bigger, biggest, less, more, twice, double; they will recognise certain shapes such as circles (sun, plate, clock face), spheres (ball) and rectangles (table, room, carpet); they will know that they find halves by dividing into two parts (not necessarily equal!); they may be able to find quarters; they may be able to appreciate the idea of ratio (e.g. in the story of The Three Bears the sizes of the individual bowls, chairs and beds are proportional to the sizes of the bears). In all these cases there may be little precise knowledge but the words they use certainly mean something to the children. They will also have used numerals in many different ways: 'I am five years old', 'It is five o'clock (TV time)', 'I have five bricks', 'I catch a No. 5 bus', 'My house is the fifth down the street'. They will almost certainly see no connection between all these different uses of the word 'five', but if the teaching which they receive in the infant school is to have significance for them this rudimentary knowledge should be used skilfully. To teach number as though it were a foreign language with which children have had no previous acquaintance is wasteful of time and children's experience; even more important is the loss of an opportunity to illustrate a fundamental principle - the continuity of mathematical learning. There are no categories such as infant number, junior arithmetic, grammar school or university mathematics; those ideas which children discover and learn at the beginning of their lives are those which (after the ideas have gradually become more and more precise) they will use at the university. As Sir Percy Nunn said, 'The teacher of infant number is a teacher of mathematics'. For example, when children learn to count they are using the idea of a one-one correspondence, that is, to one object corresponds one number-name. This same idea is used by the pure mathematician in his work on transfinite numbers and projective geometry. Again, when children learn to measure, say, length, [page 190] they may decide that two lengths are equal if by putting a ruler alongside each length they find that each length corresponds to the same marking on the ruler. This is a particular case of the abstract law that if a = b and c = b then a = c. (a) The infant school It would not be desirable to lay down how much work should be covered during this stage of children's education; so much depends on personal factors, on the ability of the children and on their early experience, and on the knowledge and experience of the teacher. After children have entered the infant school they will continue to meet situations in their home lives which will increase their store of mathematical knowledge; but this knowledge will still be unordered and vague. The first function of the teacher in the infant school is to make this knowledge more orderly and more precise and to help children to build up a vocabulary in which the words have a greater precision of meaning. Some children will need to have experiences which will give to them that background knowledge which others have gained from their home environment, and it is very important to realise that this stage cannot be hurried. All children need an environment which will stimulate discussion about shape, size, number and quantity. It is only by talking with children that the teacher can find out what they know and how vague or how clear their conceptions are. She must create situations out of which mathematical ideas will arise, though some such situations may arise spontaneously in the day-to-day life in the classroom and the teacher will use them to increase the children's vocabularies and their awareness of mathematical content. Although to an outsider this may appear a haphazard method it is valuable if the teacher knows what experience she wishes children to have at any particular time. For example the distribution of milk can be an experience at several different levels. It can be a counting exercise, it can later create a situation in which the meaning of 1 can be discussed, the bottles can be counted in groups of 3, the number of girls wanting milk can be added to the number of boys wanting milk and the result checked, and eventually the process of division can be illustrated in finding how many pints of milk have been drunk. To the teacher this apparently incidental teaching is following a [page 191] definite pattern which she has thought out in advance, though this does not preclude side-stepping. An intelligent child may evolve a method of working out a problem which leads along a path different from that which the teacher intended. Such discoveries on the part of children are certainly to be welcomed; they are moments of insight and a teacher will have to decide whether she will discuss the matter individually with the child, leading him on to further insight, or whether she will make it a teaching point for all the children who can appreciate it. From this stage of clarifying the children's ideas and giving them a vocabulary the teacher will proceed in the same way to more definite teaching of number and quantity. By putting together 3 objects and 2 objects, by adding 3 inches to 2 inches, by mixing 3 ounces and 2 ounces and finding that 3 and 2 always give 5, children will be gaining experience which will enable them to deal with abstract addition; they will realise the generality of the statement that 3 + 2 = 5. They will know it because they have discovered it, not because a teacher says that it is true. (i) Apparatus and games What specially designed apparatus should be used in the infant school? How far should children's experience of numbers and quantities come by way of artificially contrived situations? These again are very personal questions and depend on the attitude of the teacher. All apparatus used should have a purpose which the teacher understands clearly even if the children do not. If all their mathematical learning arises from the use of artificial apparatus children tend to think of this learning as something divorced from everyday life, and the way is paved for 'learning to do sums' which have no connection with the world in which children live. Sticks, counters, beads, and shells are all useful aids for counting; apparatus which encourages counting in groups is needed because counting in ones, if carried on too long, becomes an obstruction to economical thought. Bead frames are valuable for many purposes and especially for the teaching of notation; indeed it is probable that our present highly efficient decimal notation grew out of the use of the bead frame, known as the abacus, which in turn probably originated in man's ten fingers. Apparatus can be used, among other things, for developing [page 192] the basic ideas of relationship (for example, by matching and grading); clarifying ideas of quantity (clocks, weight scales, tape measures); the making automatic of recognition and of skills (associating a picture of three things with the word three and the symbol 3, flash cards on which are written the number bonds); and the learning of processes (bead frames, shops, cardboard or real money, apparatus which associates number with length and perhaps colour. Self-correcting apparatus is often useful for learning but when used for testing defeats its end). There is a place in the infant school for number games, whether communal or individual, but it should not be assumed that all games are of equal value; some might even develop false ideas from the beginning. Young children will sing rhymes involving the numerals and these may involve counting their fingers or counting each other; situations which give some purpose to counting, together with the enjoyment of a song and game, have value. Later the children may play games with skittles and dominoes, and perhaps counting-on games such as Ludo and Snakes and Ladders. From all these games the teacher should try to extract as much mathematical experience as possible if she is considering them as aids to mathematical education; otherwise she may assume that children have gained experience which they have not. She will need to watch the games in action and make comments which will bring out the numerical ideas. For example, if children are given dominoes to play with and are merely told the rules of the game, they might as well be playing with small blocks with matching pictures on the ends - the five-spot may have no connection with 'five-ness', it may just be a pattern that matches a pattern which is on the table. Similarly counting-on games may perpetuate counting in ones to the stage where it hinders progress. At the beginning of their school life children will obviously count on in this type of game, but later they can be encouraged to add (23 + 3 = 26, therefore the counter is moved on to square 26). Playing at shops can be of great value, but much thought is needed on the part of the teacher if time is not to be wasted; it may be that children are gaining valuable experience in other directions, but it should not be assumed too easily that they are gaining mathematical experience. The teacher should see that playing at shops makes progressive demands on children's [page 193] intelligence. At the earliest stages a shop will probably provide an occasion for making things, including price tickets with 2d. or 3d. written on them; it will also involve simple counting, whether counting the number of pennies required or the cost of two articles bought (leading up to statements of the type 2 + 3 = 5). Later it will provide an occasion for clarifying vague ideas on money and coins which children already possess. It may also provide an opportunity for introducing the decimal notation if something (say buttons) is sold in packets of ten and a child who wants 13 buttons buys (and records) that he had bought one packet and 3 buttons. Still later, children may use measuring rules for the making of cartons or boxes or other equipment for the shop, and may also gain experience with weight scales. They can learn that weight is not directly proportional to size if the materials are different, e.g. that a container filled with sand will weigh more than the same container filled with paper cuttings and that a cardboard penny is lighter than a real one. When children can read, cards can be made on which there are instructions for several pieces of work connected with a shop, e.g. 'Take 2 half-crowns from the money box. How much have you? Record this. Buy ..... at the shop and pay with one half crown. How much money have you left? Record this. Buy 3d. bars of chocolate with the money left. How many bars can you buy? Have you any money left? Write down on your paper the story of buying the bars of chocolate'. The teacher who is working with the children will invent a variety of games which can be played with the shop and she will always have in mind what she wants the children to learn and how they are preparing themselves to understand a new and more difficult idea. (ii) Experience of quantity By considering discrete objects children gain a store of factual knowledge of numbers and of their relationships; by dealing with measures of various types not only are they gaining more specific knowledge of the inch, foot, ounce, pint, etc, and how to measure with them, they are also gaining a deeper insight into number. Numbers are used to measure quantities and, in measuring, relations between numbers can be seen. Hence it is important to have apparatus such as rulers and scales as part of the classroom equipment. Whatever measuring is done should have an aim; it is possible for children to use rulers in an [page 194] apparently skilful fashion and yet have little idea of the properties of length. It is, for example, common to find that children who have been in the habit of measuring their heights against a wall ruler during all their time in the infant school, will estimate the height of one of themselves as 3 inches! That is, they have looked at the marks opposite a head which may be the 3 of 4 feet 3 inches, and in recording the height they have written the 4 feet as something which is merely part of the ritual. By using measuring apparatus of many kinds children will comprehend the meaning of words which they already know. They will see that an inch is small, a foot longer, and a yard still longer, and they will gain some idea of the relationship between inch and foot. Towards the end of their infant course children may well have learned that 12 inches are equivalent to one foot, that 12 pennies are equivalent to one shilling, and 16 ounces to one pound weight. They may also be able to estimate lengths roughly, even if only to decide whether a length would be more satisfactorily measured in inches, feet or yards. In general the teacher will select those properties of length, weight, time, etc, which are important and likely to be understood by children and will build firm ideas from incidental knowledge through experiment and discussion. (iii) Written work Written 'sums' should not be begun too early since premature written work often has the effect of producing work which is done completely by rote - 'this is the way to do this sum' - and children suffer throughout their lives from lack of understanding. They are incapable of applying the processes of addition, subtraction, multiplication and division in any situation, because the 'sum' may be a set of meaningless symbols which children, being intelligent, have learned to manipulate, but which have no connection with the money they spend or with the lengths they measure. The first introduction to written work should be as recording. Children use the number four as a word, they know some of its meanings, they may have seen the symbol 4 on a peg in the cloakroom, they will have used it in the jingle 1, 2, 3, 4, 5. If they have counted four objects then they might well record the number as 4 (and also perhaps as the word 'four'). If they have threaded beads in the order 2, 4, 7, 6 then they might record this [page 195] pattern of threading in symbols. Later when they have solved, with counters, or other objects, some problem involving the addition of 2 and 3 they can record their findings in the shorthand form 2 + 3 = 5. But this should come only after they have dealt with this sort of situation sufficiently often to realise that 2 things and 3 things are always 5 things, and this is a situation that children need to meet repeatedly. Nothing is gained by labelling the sums 2 cats + 3 cats = 5 cats, or 2 pennies + 3 pennies = 5 pennies. What is recorded is that 2 + 3 = 5 irrespective of what the numbers being added represent. After much experience of dealing with actual situations involving facts of this type children can practise the addition bonds in abstract form and begin to deal with the structure of mathematics. Besides being able to recognise and write the numerals and later the numbers of two or more digits children must learn to deal with the signs +, -, x, ÷, and =. Teachers read these symbols in various ways. Some will say that '+' means 'add', others 'and', and others 'count on', others 'plus'; some will say '=' means 'are', others 'make', others 'come to', others 'equals'. It is important that the statement 2 + 3 = 5 should be read to make sense of at least one of the meanings to the children. Later, perhaps towards the end of the primary course, the statement 3 + 2 = 5 (read as 'three plus two equals five') can be shown to cover a whole range of meanings such as those to be found in ¾ + ½ = 1¼, where the plus sign could not be sensibly read as 'count on' since by count we mean progression by ones. The same applies to the signs for subtraction, multiplication and division. In the infant school many children are not ready to deal with written statements involving division, such as 12 ÷ 3 = 4; the mathematical structure involved will be discussed in the section on the junior school. Some teachers find difficulty in teaching children to deal with zero. If when numeration is discussed zero is thought of as the starting point to the number scale there should be less difficulty. A number is the answer to 'How many?' It is perfectly reasonable to give the answer 'none', which is written '0'. 'If I have two pennies and spend two pennies how many have I left?' can lead without difficulty to the writing of 2 - 2 = 0. A scale marked in equal divisions 0, 1, 2, 3, 4, 5 is helpful to children. They can see that 0 is one less than 1. Twenty (written 20) means 2 tens and 0 units, just as 25 means 2 tens and 5 units. [page 196] Probably, some of the difficulty arises because children do not often give the word 'nought' as an answer to the question 'how many?' They usually reply 'none' or 'no pennies'. If, when they record this fact, they say the word 'nought' as they write, the symbol '0' will become familiar to them as a number. (iv) Cooperation with the junior school Besides being aware of what knowledge is possessed by children on entering school the infant teacher needs to be concerned with the learning process in the junior school. More cooperation between teachers in infant schools and those who receive the children in junior schools would help to preserve continuity of mathematical teaching. If the teacher of a first-year junior class were to discuss with the infant teacher how the children have learned mathematics, what they have achieved in ideas as well as in processes and what they have not learned, there would be a great economy of effort in the junior school, and far less sense of frustration among children. Many teachers find that although such meetings take time and trouble, more accurate and useful knowledge is gained from them than from the time-consuming testing which otherwise takes place during the first weeks of children's life in a junior school. The results of such testing after a long holiday, in strange conditions and sometimes in unfamiliar language, are less reliable than the knowledge and the judgement of the infant teacher. While written records are better than no information it would be still better to see the methods employed and to talk informally with the infant teachers. In a combined junior and infant school there is generally no difficulty in finding opportunities for such discussions; it is more difficult when the junior school is in a different building and when children are admitted from several infant schools, but the effort made to establish such contacts is amply repaid. (b) The junior school In the junior school children have much to learn and much experience on which to build; fortunately most of them, at this age, want to learn and will bring enthusiasm to any learning which satisfies their curiosity and gives them a feeling of power over their environment. Mathematics, well taught, will do both. [page 197] It is often said that the allocation examinations* restrict the scope of junior school mathematics. It is probable that a good performance could best be achieved by a broader content and treatment than one which keeps strictly to examination requirements. The teaching of mathematics, or any other subject, implies far more than preparation for an examination. In considering the work of the junior school the far-reaching interests and curiosity of children must be fully taken into account. Many mechanical devices are now part of their everyday lives; they are interested in speeds, in time, in navigation and in maps; they see graphical representation on posters, in magazines and on television. All these can be used to give purpose to the teaching of mathematics and to broaden the whole conception of the work. It is not suggested that these topics should be made items of a new junior school syllabus; but they should be used to give point to the teaching. Geometrical ideas and graphical representation, besides being of value in themselves, often illuminate the work being done in arithmetic. There may be opportunities for the introduction of some of the simple ideas of mechanics such as the principle of the lever, a principle learned, in action, by every child who has played with a see-saw. Infants will have had experience of addition facts, such as 3 + 2 = 5, with the corresponding facts 5 - 3 = 2 and 5 - 2 = 3; they may have learned facts such as 2 x 8 = 16 with its corresponding facts 8 x 2 = 16, 16 ÷ 8 = 2. At some stage in the junior school they must be expected to learn both addition and multiplication facts systematically. Progress is slow until these facts are known. As long as children have to think out, or work out, each time they use it, what eight sevens are, or as long as they have to count on their fingers when they wish to know 8 + 5, they will find it difficult to deal with the processes which they are learning. Once these facts become automatic, energies are left free for more advanced thought. In the same way, it is essential that processes should eventually become automatic in order that children may use them in significant situations, without having to give too much thought to the manipulation of the numbers. The number and variety of mechanical processes cannot be laid down categorically, but it is probable that, in the past, the *See Chapter VI, Section on effects of allocation to secondary education. [page 198] mathematical education of most children has been too exclusively restricted to computation. A reasonable standard of accuracy in computation is only one criterion of successful teaching. If children are to see the need for learning new work and how it links on to their previous knowledge, and if they are to be able to apply it to significant situations, the teacher will find that the time spent on mere calculation must, of necessity, be reduced. (i) Number relationships The multiplication tables form, for the child, an early acquaintance with ordered number relationships. Any teaching of tables which does not make the pattern and order clear to the children will be missing a great opportunity of helping them to see some of the fascination of mathematics. There are many opinions as to how tables should be learned; some teachers would build them up in the sequence of the natural numbers 1, 2, 3, 4, 5 ..., others would not make up the tables until nearly all the number facts are learned; it would seem more reasonable to learn in sequence, as ordered tables, those which have some relation to each other; for example 2, 4, 8 are closely related, so are 3, 6, 9 and so are 5 and 10; the table of sevens must be learned independently, but probably by the time the children are ready to learn this table all the individual terms except 7 x 7 will have been met in other tables. Rhythmic counting in 3s, 5s, 7s, etc, helps to bring out the patterns. Later, a number square will emphasise the patterns of, and the interrelation between, the various tables.* This recognition of the grouping and patterns, interesting though it may be, is not sufficient. Ultimately the facts must be learned, and it is possible for the routine necessary for their learning to give satisfaction and enjoyment. Besides the tables there are other relations between numbers which should be emphasised; these will have been introduced in the infant school but should continue to be an important part of junior school teaching. Some children will notice that there are numbers which have no factors, some will catch a glimpse of the economy of thought which our Arabic notation has produced (especially if their attention is drawn to the Roman notation as seen on clock faces); many children will need to have these *See the Mathematical Association report on The Teaching of Mathematics in Primary Schools, Chapter 3. [page 199] relations pointed out to them when opportunity arises, and a few will see them only with difficulty. ii) Apparatus and practical work Teachers are becoming increasingly aware that children may still need to use apparatus and concrete material after they leave the infant school. Some children will need counters and bead frames if they have not yet mastered the processes of counting and simple addition and subtraction, and many will need further experience in handling money and measures. In general, however, the purpose of apparatus will change as children grow older. During their years at school they are building up a body of knowledge which is part of their way of thought. They can think out what will happen if a certain situation is presented to them - they do not need the concrete material in their hands. Practical work may still be necessary when a new process of calculation or a new topic is being introduced, for example fractions or area; it may help children to see familiar ideas in new contexts. Often the so-called practical work is relegated to a special period called 'Practical Arithmetic' on the timetable, and the work done in that period, while it may be interesting in itself, may have little relevance to the work done in other arithmetic lessons. The need for practical work may arise in any lesson. If the idea of area is to be developed at some future date, reference to it might be made when patterns are being drawn or there may be discussion of what is meant by the area of the netball pitch, or by the area of a country as shown on a map in an atlas; a teacher's ingenuity or a child's curiosity will provide an opportunity for wishing to find the area of a circle. This is all practical work and leads naturally to discussion of economical methods of finding areas and also to a clearer conception of area, so that children will not automatically say that it is length times breadth, only to be confounded when the shape is not rectangular. This is not to suggest that the junior school is the place where methods of finding the area of circles should be taught to all children, though some might be ready to learn them; the importance of introducing non-rectangular shapes at an early stage is that children shall not assume, as they often do, that the word area applies exclusively to rectangles. In some schools boys have lessons concerned with [page 200] geometrical drawing and model-making while girls are being taught needlework. This arrangement tends to divorce the work done in these periods from that done in other mathematics lessons, chiefly because only half the arithmetic class is acquainted with the ideas met in the geometrical drawing lesson. Where enterprising work of this kind is done with the whole class, as an integral part of the mathematics syllabus, both boys and girls benefit. The teacher is able to link all the work more closely and to use measurement to amplify and illustrate the rest of the work; the girls gain valuable experience in measuring purposefully and develop geometrical ideas. It is sometimes said that girls in secondary schools appear to have less aptitude than boys for mathematics because they have missed this 'exploratory' stage in their primary schools. (iii) Problems The solution of problems has already been discussed in preceding paragraphs. What is a problem for one child is not a problem for another, or for the same child when he is older. The dividing line between non-problems and problems separates those questions which children can tackle without reflection from those questions to which they do not immediately see how to find an answer. The mechanical sum 12 x 3 may be a problem to a child who has not yet learned multiplication tables. Thus, training children to solve problems is training them to overcome difficulties for themselves, and they do not receive this training if the teacher removes all the difficulties in advance. If problems are taught as types, only the first one or two remain as problems; the rest are merely mechanical sums, and even less valuable than the usual mechanical sums, because children waste much time reading the words in order to abstract the numbers which they know in advance will be, say, added together when found. The problems become what Ballard called 'An ounce of arithmetic to a pound of padding'. Children may solve a problem practically at the class shop or post office, they may solve it by using counters or they may solve it mentally. In all these cases they may use rudimentary methods, such as repeated addition or subtraction; the teacher's task is then to show them, at the point where he thinks it useful, how much more elegantly and economically they could solve the problem by using, say, multiplication or division. Further, the [page 201] children should be able to deal with the arithmetic, in some fashion, when they have eventually decided what must be done with the numbers; and lastly the problems should be of reasonable difficulty, so that confidence is not damaged by too much failure to solve them or boredom engendered by too easy success. For these reasons some teachers use work cards so that children attempt problems individually matched to their ability, experience and interest. The solution of problems, whether arising from actual situations or referring to imaginary ones, whether invented specifically by the teacher or by the compiler of a textbook, is one of the most important parts of mathematical learning. Much of the work done in the primary school is meant to be used in the world outside the school, both at the time and in later life, in situations which can be roughly reproduced or imagined in the classroom. The only way in which we can learn to apply knowledge is by actually trying to apply it. (a) The structure of mathematics Whatever views are held about the possibility of a logically developing course of mathematics for children there can be no doubt that teachers themselves should have carefully considered elementary numerical and quantitative ideas and relations. Without such consideration they cannot see the structure of the body of mathematical knowledge which they wish the children to have. The meaning of one idea presupposes the meaning of others and there are steps from one idea to another. A teacher who is unaware of these relationships finds it impossible to diagnose a child's difficulties and to further his understanding. It may be that some of the most intelligent children would, towards the end of their junior school course, gain much satisfaction from looking back and analysing the structure in the same way that the teacher should have done before starting to teach. (i) Notation Although young children will not appreciate the power and the economy of the Arabic notation it is essential that the teacher should. This notation is often taken for granted and yet it is one of man's greatest inventions. With ten symbols [page 202] (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) we can write any number however large and, if we use a decimal point, however small. Not only is it a perfect recording instrument (the Greeks and Romans could record numbers, but only clumsily, in their notations) but it is also a calculating instrument of great power and elegance, as is readily seen by comparing attempts to multiply one hundred and forty-six by seventy-three in Roman and Arabic notations. If decimal fractions are taught in the junior school the teacher may, at the same time, discuss the knowledge of place value, already vaguely possessed by the children, and demonstrate to them the wonder of the Arabic notation. (ii) Addition Addition saves recounting all the individuals in two or more groups when the number in each group is known, but until the addition facts are known automatically there will still be counting on from the number in one group through the individuals of the other. This is a crutch which should be discarded as early as possible. (iii) Subtraction The ideas of subtraction are much more difficult than those of addition. Subtraction arises from such questions as: (i) What must be added to 7 to make 10?and further difficulty arises from the addition question (v) What must 7 be taken from to leave 3?The language here puzzles many children; in questions (i) and (ii) the word 'add' comes into the question and yet the process required is subtraction. In question (v) we have the words 'taken from' and the process required is addition. These examples are given in order that the teacher may see the difficulties inherent in the idea of subtraction and the variety of problems that give rise to the statement 10 - 7 = 3. If children are to deal competently with questions of this type they must know the relationship between 7, 3 and 10 in all its forms: 7 + 3 = 10, 3 + 7 = 10, 10 - 3 = 7 and 10 - 7 = 3. With small numbers children will probably answer a question by intuition or quick trial and error; good teaching will enable them to see that the process [page 203] used is subtraction, and so help them to answer questions involving larger numbers. In general, subtraction arises as one way of comparing numbers or quantities, and determines how much larger or smaller is one thing than another. (iv) Multiplication Multiplication arises in adding equal collections. Just as the answer to an addition sum can be found by counting, so the answer to a multiplication sum can be found in counting in twos, threes or whatever is the number in each of the equal collections. If children are asked how many dots there are in : : : : : they may say 2, 4, 6, 8, 10, or they may write 2 + 2 + 2 + 2 + 2 = 10 or 2 + 2 + 2 + 2 + 2 = five twos = 10. Here the conventional sign for multiply could be introduced and the sum written as 5 x 2, read at first as 'five twos' or 'five times two'. The rectangular pattern of dots shows that they could also have been counted as 5 + 5, hence the statement 2 x 5 = 10 can be made. This shows that 5 x 2 = 2 x 5 (the commutative law). Children at this stage can use harder words than 'multiply' and where appropriate they should be encouraged to use this word in preference to the ugly word 'timesing'. Some difficulty arises with the words applied to multiplication of quantities. 5d. x 2 may be read as five-pence multiplied by two, but 2 x 5d., if read as 'two multiplied by five-pence', is nonsense; it would have to be read 'two five-pences'. Since, however, it can be seen that the commutative law still holds and that (5 x 2) pennies = (2 x 5) pennies, there is no lasting difficulty. (v) Division This is probably one of the most difficult ideas which children have to grasp, especially when dealing with remainders. Just as subtraction and addition are different aspects of the same relation, so are multiplication and division. As one recent textbook writer states: 'Addition is counting from zero in ones.*Bass and Dowty Counting and Arithmetic in the Infant School. Harrap, 1956. [page 204] Children meet division at an early stage as sharing - 'I share 12 apples between 3 children. How many apples each?' The working out of this problem with concrete material involves giving each child one apple, then each child another apple, then a third and a fourth and then finding that no apples remain. Each child counts his apples and the answer is '4 apples each'. On the other hand a problem such as 'How many 3d. bars of chocolate can I buy for 1/- [one shilling]?' involves an entirely different method of approach practically. The shilling is replaced by 12 pennies, a group of 3 pennies is taken for one bar of chocolate, and another and so on, until all the pennies are used. The groups of 3 pennies are counted, there are 4 groups, so 4 bars of chocolate can be bought. In the first problem the children are making three equal groups from 12 but they do not know, until they have finished the sharing, how many will be in each group. In the second problem they have to take out groups of three and then see how many groups they have. Both problems (and still other different types) give rise to the statement 12 ÷ 3 = 4. In the first type of problem the idea of a remainder is simpler to understand than in the second. If 14 apples and 3 children are taken as the data it is obvious when only 2 apples are left that another round of sharing is impossible, therefore there are 2 apples left, i.e. 14 = (4 x 3) + 2. In the second type if 1s. 2d. and 3d. bars of chocolate are the data, after making piles of 3 pennies there are 2 pennies left. The answer to the question 'How many bars of chocolate?' cannot be given as 4 and remainder 2, it should be '4 bars of chocolate and 2d. left over'. From this type of question the idea of a remainder as a fraction is more easily seen. The answer could be 4 2/3 bars of chocolate (if the shopkeeper were willing to divide up the bars!). It is not suggested that children should have all these ideas put before them, but the teacher must be aware of them if he is to help children solve their difficulties. (vi) Fractions Fractions were invented as an extension of the number system, initially the sequence of positive integers. The need for fractions arises because the unit (whether of length, or weight, or time or whatever it is) is too large for a particular purpose and must be broken up into parts. The integers, which until this [page 205] new invention have been 'the numbers', now become 'the whole numbers', which have wider application than the fractional numbers. For example, we can talk meaningfully of 7 dogs but not of 1/7 of a dog; on the other hand there is meaning in half a cake. Before children can be expected to apply the four rules to fractions they must first understand the notation thoroughly, that is they must become familiar with fractions of the form 1/n and they must know what is meant by 1/7 or 1/11. They must also learn that 1/n gets smaller as n gets larger although they may only know it in the particular cases 1/3 > 1/4 and 1/7 > 1/10. 'If I divide something up into 4 equal parts, will each part be bigger or less than if I divide it into 3?' Eventually the child arrives at the general law (unformulated) that 1/n > 1/m where m > n. Meaning has then to be given to the form a/n. This may arise in several different ways. If ¾ is considered, there may have been a collection of quarters and three of them taken, or there may have been three wholes and 4 equal partitions made. When the meaning of 1/n and a/n is understood children can then begin to discuss equivalence, e.g. 3/4 = 6/8 = 30/40, from which methods for adding and subtracting follow readily. The learning of what is meant by multiplying or dividing by a fraction may be considered as a generalisation of methods of multiplying and dividing by integers. A set of problems which gives meaning to the idea of multiplication by a fraction may be as follows: 2 lb. at 4d. a lb. cost 8d.Similarly with division, although here children may find difficulty in understanding that the division process may result in a larger answer. For example, (1) How many 2d. stamps can be bought for 1/-?Children can see that the answer to (1) is obtained by writing 12 ÷ 2, and may follow on, with skilful questioning, to see that the answer to (2) is similarly obtained by writing 12 ÷ ½, which [page 206] gives the answer 24, but this does not imply that they can understand that 12 ÷ ½ = 12/1 x 2/1 = 24. From the last and some foregoing paragraphs it will be seen that the general idea of division is difficult and some teachers may think that, if children are to understand how to apply the process, the teaching of it should be left until later in their lives. They will be able to deal with simple problems which can be solved intuitively or by rudimentary methods, and some intelligent children may be led to an understanding of the process. It is unlikely that to the slower children of 10 years it will be more than a 'ritual'. (vii) The general laws of mathematics Underlying all work on the four rules are the notational rule, and the commutative, associative and distributive laws of combination of numbers. Commutative a + b = b + a and a x b = b x a Associative (a + b) + c = a + (b + c) and a x (bc) = (ab) x c Distributive a (b + c) = ab + ac For example 23 x 2 = (20 x 2) + (3 x 2) (the distributive law together with the notational rule, 23 = 20 + 3) = 40 + 6 = 46 (notational rule) Similarly 26 x 2 = 20 x 2 + 6 x 2 = 40 + 12 = 40 + (10 + 2) = (40 + 10) + 2 (associative law) = 50 + 2 = 52 This process is almost identical with the following: 2s. 8d. x 2 = (2s. x 2) + (8d. x 2) = 4s. + 16d. = 4s. + (12d. + 4d.) = (4s. + 1s.) + 4d. = 5s. 4d. There is one important point of difference in notation here; 4s. means 4 shillings, 40 (in the previous example) means, not 4 noughts, but 4 tens. [page 207] Similarly 144/3 = (100 + 40 + 4)/3 (notational rule) = ((100 + 20) + (20 + 4))/3 (associative rule) = (120 + 24)/3 (notational rule) = 120/3 + 24/3 (associative law) = 40 + 8 = 48 (notational rule) The process of multiplication by units and larger numbers, and of short and long division, are merely extensions of these applications of general laws. (b) Extensions of work to fields other than arithmetic While any systematic course in algebra would be out of place in the normal junior school, the use of letters to represent numbers is an idea which many juniors do not find difficult. It is valuable that children should see that general statements about numbers can be economically made. For example a + b = b + a; 99x = 100x - x; a/d + b/d = (a + b)/d; x things costing y pence cost xy/12 shillings. Difficulties of notation such as the dropping of the multiplication sign, the writing of x/y for x ÷ y, and the use of brackets can be introduced gradually. Not only should letters be used to represent numbers in general in this way and in suitable formulae, but they should also be used to represent unknown particular numbers, and arguments made leading to simple equations and their solution. The setting down of a problem in symbolic form will often be enough to secure its solution. Thus a child using A = LB and wishing to find the breadth of a rectangle whose area is 100 sq. ft. and whose length is 12 feet will gain much from seeing 100 = 12 x B written down and being left to his own devices to find B. A sequence such as 4, 7,10,13 ... can be developed in an algebraical way. What is the rule for finding the nth number in the sequence? How many numbers in the sequence must be written down before reaching 100? [page 208] All junior school children have some experience of spatial relations through their work in infant classes and their everyday lives; they know the shapes of various figures, such as the triangle, sphere, circle. In the junior school this experience can be extended; they may become acquainted with angles, especially the right angle, they may learn something about parallel lines and the properties of triangles. They may meet the mathematical ideas of locus and envelope. They enjoy using ruler and compasses and making constructions; children of 9 years old have been seen thinking out a construction for a quatrefoil window in the neighbouring church. Representational and graphical work of all kinds, at the children's own level, can provide much material for stimulating thought and much practice in dealing with numbers, especially if it is connected with the work being done in some other subject, say geography or nature study. For example children in a top junior class, after looking in their atlases, asked what the lines of longitude and latitude on the maps were. A simple explanation was given with great clarity by the teacher; then followed what the children considered to be a most exciting 'game' - they found the approximate latitude and longitude of various towns, and, given latitude and longitude, found the towns. (Incidentally this was a good exercise in interpolation; they guessed what the latitude would be when they found a town lying between the 40th and the 50th parallels). At the end of this lesson the teacher said (not to the children) 'In my next arithmetic lesson I shall put two rectangular axes on the board for my brightest group and see if they can plot points. I shall leave them to do it alone, telling them to remember latitude and longitude.' This is a good example of awareness of mathematical implications, though the teacher himself was not a mathematician and was probably unaware that he was about to set the children discovering coordinate geometry! The idea of time is a fruitful topic for junior schools; the pendulum, the hour glass and home-made water clocks can all be used to record the passing of time, with extensions to the idea of the hands moving through angles on the face of a clock. The use of a pendulum gives rise to much questioning and discovery on the part of the children, and the whole topic can be connected with history. The principle of the pendulum is a part of mechanics and [page 209] there may be other ideas from mechanics which could be introduced with profit; as yet little has been done in this direction in the junior school. Children are interested in speed and many of them have a vague idea of the meaning of acceleration and so on. The equinoctial sundial, in which the shadow of a stick moves at 15 degrees an hour, is well within the comprehension of some 9 or 10 year old children, and from it they will learn much about the sun's apparent motion. But most of these children under 11 years are not yet ready to have the principles of mechanics, or formal geometry, or the motions of the heavenly bodies explained to them mathematically. What is suggested is that every opportunity should be taken to relate the mathematics which the children are learning in school to their own interests and to the external world. The topics discussed above should be taken when they arise and when they are likely to make a contribution to the more formal work in hand. (c) Class organisation and correction of work There is considerable variation in the methods used in the classroom. In some classes children work with textbooks or work cards at their own speed, coming to the teacher for explanation and correction of work. Here the advantage is that each child discusses his difficulties and errors personally with the teacher. But there are also many disadvantages, among them lack of communal interest and discussion, the fact that the teacher repeats the same explanation to different children in turn, and the fact that no textbook can take advantage of a mathematical situation arising in the classroom, or provide illustrations as spontaneous and significant as those of the teacher. On the other hand if the class is working as a whole children may have the same test in mental arithmetic, listen to the same lesson and be required to work out the same set of exercises from the board or textbook. Answers may then be marked right or wrong by the children or by the teacher, often after the exercises have been worked out on the blackboard. This method, while providing opportunity for discussion, makes no provision for differing capabilities, and it wastes the time of those children who need no help on the points discussed, and does not ensure that those in difficulty can locate the cause. [page 210] Most teachers find a mean between these extremes and vary their methods so as to reap the advantages of individual work and correction, while not losing the stimulus of communal discussion. There are times for classwork, for group work, for individual work; there are times for exploring new ideas together, for enabling children to make their own discoveries, for practice and consolidation in groups or individually, and, at intervals, for testing. Ideally every child should make progress at his own pace, but this does not mean that he should always work in isolation. His pace may, in fact, be determined to a certain extent by the group with which he is working; he may be stimulated by opportunities for discussion at his own level. Whatever the method of teaching the teacher should be aware that mathematics is a subject in which success in new techniques depends on confident earlier knowledge. For example, children cannot successfully deal with long division until they can add, subtract and multiply. Without this previous knowledge not only will they fail to learn the process, however long the teacher may try to teach it to them, but they may eventually set up a resistance to all mathematical teaching and be unable to use what little knowledge they possess. No teacher should say of a child, 'He ought to have learned this' and then proceed on the assumption that he has. If he has not, future teaching will create confusion and dislike for the subject. Emotional difficulties and backwardness are closely related; for some children success comes very slowly and only with careful and sympathetic matching of work to their capabilities. If the teacher is to be fully aware of children's difficulties then correction of work is important. It is not enough that a child should know that a mistake has been made; he should know as precisely as possible what the mistake is - an error in copying, an error in tables, a mistake in method, a failure in the understanding of words or a 'conventional' error, such as writing 107/- for £5 7s. 0d. The teacher need not necessarily point out the errors, but either teacher or child should find them. Many teachers allow children access to answers; this encourages them to seek for their own mistakes and also helps to dispense with any waiting for the teacher to correct exercises, a procedure which is wasteful of learning time. If children are to be allowed to mark their own answers then class relationships must be good, so that no child is driven to cheating by insecurity or undue [page 211] competitiveness. Fundamental mistakes should be discussed with the child and, if they are general, with the group or the class. From time to time, very careful scrutiny of each child's work is necessary, for nothing can replace the constructive help and guidance of the teacher. A powerful weapon against inaccuracy is the teacher's own example in the careful setting out of work on the blackboard, and the expectation (or perhaps insistence) that children will set theirs out likewise. He can also draw attention to ways in which work can be checked; for example, by adding successively up and down columns of figures, by using addition to check subtraction, by considering the size and sense of answers to problems. But checks of calculation should not normally be harder than the original calculation; it is, for example, unsuitable to use long division to check long multiplication. Inaccuracy in calculation may be due to many causes, among them insecurity of previous knowledge, faulty knowledge of tables, working at too quick or too slow a rate, anxiety, or weariness of the same sort of work. Diagnostic testing can reveal persistent errors and should be followed up by remedial work. Rapid calculation, where only the answer is to be written down, may be stimulating for many children, but may damage the confidence of a slow-thinking child. It is essential that none of the children should lose interest, the weak ones by continual failure, the quick ones by too easily gained success. It is probable that the most efficient method is to fix speed and complication at a level where expectation of success for each child is high. It follows then that on occasions when the purpose of the teaching is to increase accuracy the whole class may not be the best teaching unit. The task envisaged for the successful teacher of primary school mathematics is seen to be a heavy one and all the more so because he is normally teaching several other subjects, in some of which he may have a more personal interest. But if the subject is worth teaching, and it is to be hoped that every teacher will feel that it is, then it is worth teaching well. It is well known that many young people are leaving our schools at the age of 15 or older with a positive distaste for mathematics; teachers in primary [page 212] schools can do much to prevent the development of this attitude, which so often arises because children have learned their mathematics by rote and have seldom seen enough meaning in what they are being required to do. Some primary teachers maintain that they themselves have this same attitude and that they have little mathematical ability. Many of them are too modest, and could, by reading and discussion, gain considerable insight into the fundamental ideas and the applications of mathematics. Their growing awareness of the fascination and importance of mathematics would not only be of interest and delight to themselves, but would communicate itself to the children, who would then have a far better chance of realising their potential mathematical capacities, whether limited to the simple calculations of everyday life or reaching forward to the discoveries of a Newton or an Einstein.
[Note Imperial measures Pounds and ounces (written as, for example, 2lb. 4oz.) are still in use, alongside metric measures. The pound is divided into 16 ounces. (1lb. = 454g). Pre-decimal coinage Before 1971, the pound (£) was divided into 20 shillings (written as, for example, 2s. or 2/-). A shilling was divided into 12 pennies or pence (written as, for example, 6d.) There were also half-pence (a ha'penny) and quarter pence (a farthing).] 
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