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CHAPTER XV
MATHEMATICS
SUMMARY OF CONTENTS
| sections |
I. SOME GENERAL CONSIDERATIONS | 1-20 |
A. INTRODUCTORY | 1-2 |
B. ARITHMETIC AS A SUBJECT | 3-4 |
C. THREE ESSENTIAL STAGES IN THE TREATMENT OF ANY ARITHMETICAL TOPIC | 5-12 |
(i) Arousing interest through preliminary practical work | 6 |
(ii) Developing mechanical skill | 7-10 |
(iii) Applying skill to the working of problems | 11-12 |
D. MENTAL ARITHMETIC | 13 |
E. ORGANISATION OF THE SCHOOL COURSE | 14-19 |
F. INCLUSION IN THE COURSE OF OTHER FORMS OF MATHEMATICS | 20 |
II. THE INFANT SCHOOL STAGE | 21-28 |
III. MATHEMATICS AT THE JUNIOR SCHOOL STAGE | 29-37 |
IV. MATHEMATICS AT THE SENIOR SCHOOL STAGE | 38-51 |
A. INTRODUCTORY | 38 |
B. EARLY STAGES OF THE COURSE | 39-41 |
C. LATER STAGES OF THE COURSE | 42-48 |
D. BACKWARD PUPILS AND DULL PUPILS | 49-50 |
E. CONCLUSION | 51 |
I. SOME GENERAL CONSIDERATIONS
A. INTRODUCTORY
1. The threefold aim of mathematical instruction. The teaching of Mathematics in the Elementary School has three main purposes: first, to help the child to form clear ideas about certain relations of number, time and space; secondly, to make the more
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useful of these ideas firm and precise in his mind through practice in the appropriate calculations; and thirdly, to enable him to apply the resulting mechanical skill intelligently, speedily and accurately in the solution of everyday problems.
The expert teacher who realises the threefold nature of his task will not fall into the common danger of over-emphasising the second of these three aims at the expense of the other two. He will naturally be pleased if his pupils can work with speed and accuracy the ordinary mechanical sums in the "four rules" and in money, weights and measures, but he will always feel that this sort of achievement may be somewhat barren if it cannot be turned to real use. If, however, his pupils, as a result of their mathematical training, have learnt to apply their textbook knowledge to practical problems, the teacher will have succeeded in the main purposes of his instruction.
A good deal of the mathematical teaching in schools, indeed, which has as its aim nothing but the cultivation of speed and accuracy in working sums of a mechanical type, cannot be justified even as a form of mental training; for operations with numbers and quantities which cannot be applied to life-situations must be largely without meaning for many children who perform them. In the early stages, especially, teachers should restrict themselves to giving children facility only in such mathematical skills as they can use and see the point of using. Throughout the school course the speed and accuracy which will count for most in the long run will be that shown in work which has a direct application for the child who performs it.
2. The importance of making the work fit the capacity of the individual child. Syllabuses of instruction in mathematics are still very much under the
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influence of an older tradition. They need not only to be brought into closer relation with the requirements of today, but also to take more account of the natural proclivities of childhood. Mathematical conceptions are easier to understand and to apply when they arise out of the pupils' interests and experiences, and the duller the child the broader the concrete foundation should be. Present-day treatment is far too much influenced by the supposed needs of those who proceed to Secondary Schools. The wide differences in ability that occur within each age-group need to be more fully recognised. Where a two- or three-stream organisation exists, it is easier to arrange for alternative syllabuses. It is equally desirable, though perhaps not equally practicable, to do this in all schools.
There is probably no subject in which the requirement that the syllabus must be adapted to the interests and capacities of the children is so well recognised, though difficult to fulfil, as in Arithmetic. Not only do capacities and normal rates of progress differ widely, but the subject has a definite content which must be taught in a more or less definite order, and if progress is to be continuous, certain fundamental skills and certain essential kinds of knowledge must be acquired by each individual pupil at every stage of the course.
The usual Arithmetic course in most schools consists largely of short "sums" of which the pupil has to work out a large number one after another. If these "sums" are so difficult that he gets them wrong, his sense of failure rapidly grows, distaste for the subject inevitably follows, and in such circumstances he becomes unteachable. It is essential therefore that the exercises should be so graded in difficulty that every child can enjoy the stimulus of success and of steady progress. He should, moreover, at each stage of the course see their bearing upon the practical problems of life.
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B. ARITHMETIC AS A SUBJECT
3. The approach through children's games. Number relations are implicit in many children's games, for example, in counting-play, in scoring-games and games with numbered boards and dice, and in many of the occupations that they imitate from adult life: for example, shopping, weighing and measuring. Such games and occupations, if carefully selected and arranged, are valuable throughout the early stages, but most of all in the Infant School. They familiarise the child with numbers and with the common units of measure, and they import meaning and interest into Arithmetic by basing its conceptions on a wide range of experience. Some of them - for example, the scoring-games and shopping - involve much practice in the simpler arithmetical operations, and also have so obvious a meaning for the child as to provide him with a strong motive for success. Such meaningful practice shortens the labour of learning and leads to good habits of calculation.
Such familiar and interesting "make-believe" and "real-life" situations provide the best introduction both to pure Arithmetic and to problems. When John, for example, buys at the classroom shop, the class may be led to see that the problem is "What change should he get?" They may go on to describe in their own words concisely and accurately the transaction that takes place, and to say what particular arithmetical operation the transaction has called for. The skilful teacher, by varying the shopping situation, may lead up to a varied series of problems, graded in difficulty. The children will thus come to see how textbook problems arise and how to state them. By approaching problem-solving in this way thus early they learn to grasp the situation which the words of a problem represent.
4. The approach through the use of apparatus. But there comes a time when the very wealth and variety
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of these games may make abstraction difficult and may obscure the essential character of the operations involved. Arithmetic as such soon advances beyond their range, and the foundation for its more systematic procedure is needed. This should be provided for by formal practical exercises in calculating with the help of simple objects like balls and counters. Formal practical work of this kind with simplified material is a stage half way between the fully concrete situation of the game and the abstract sum. By using apparatus to help him in his calculation the pupil comes to see, in both senses of the word, what the operations mean: e.g. how the operation required for "27 plus 6" may be derived from the simpler calculation of "7 plus 6," or why six-eighths is the same as three-quarters.
Apparatus, however, may do harm if it is of the wrong kind. Apparatus, for example, that involves the moving about of objects one by one, tends to fix the low level habit of counting, if it is kept up too long, and to obscure the essential character of the higher level habits of addition, multiplication, etc. Apparatus may also do harm, if its use by individual children continues too long. If pupils are taught that it is more "grown-up" to work without apparatus, even when they have it, the teacher will soon discover when it may be withdrawn. Long after this, however, the teacher who can use skilfully one or two types of standard apparatus, e.g. the ball-frame, will find them useful in removing difficulties of understanding, especially those of children who may be relatively backward.
C. THREE ESSENTIAL STAGES IN THE TREATMENT OF ANY ARITHMETICAL TOPIC
5. Practical and oral work: mechanical work; problem work. The justification for ordered and systematic instruction in Mathematics, whether in Arithmetic, Algebra or Geometry, is that it enables the
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pupil to make faster progress than he otherwise might in acquiring the technique needed for the solution of the practical problems encountered in everyday life which call for the application of mathematical knowledge. The teaching, in particular, of any arithmetical topic or process in school should proceed by three clearly-marked stages. First, by way of introduction, should come practical and oral work designed to give meaning to, and create interest in, the new arithmetical conception - through deriving it from the child's own experience - and to give him confidence in dealing with it by first establishing in his mind correct notions of the numerical and quantitative relations involved in the operation. Next should follow "mechanical" work, the purpose of which is to help the child to form the mental habits in which skill in computation is rooted, so that he may be able to perform both speedily and accurately the particular arithmetical operations required. Finally, there should be problem-work: when the necessary skill has been acquired by each pupil, he will naturally apply it to solving the kind of problems which rendered it necessary for him to acquire it. Thus the treatment of each topic will end, as it began, by giving the pupil practical experience in dealing with situations which have meaning for him.
In applying the principles here indicated, however, the teacher should be on his guard against adopting any stereotyped procedure which is followed rigidly, either within the compass of a single lesson or over a series of lessons. In proceeding, too, from one stage to the next in his teaching of any particular topic or process, he should aim at preserving a proper balance as regards the time devoted to the various stages. He should not, for instance, give to mechanical work a disproportionate amount of the time at his disposal, nor should he make the fundamental mistake of deferring all practice in problem-solving until after the mechanical
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rules laid down in his class syllabus have been completely learnt. Above all, he should in this matter bear in mind the individual needs of his pupils, and should differentiate the syllabus so that he need not hurry the slower, or keep back the brighter workers among them.
(i) Arousing interest through preliminary practical work
6. The importance of introducing a new topic in arithmetic by means of practical and oral work. The importance of the introductory practical and oral work of the first of the three steps in teaching an arithmetical process lies in the fact that the child can learn to understand the meaning of numerical conceptions and operations by working exercises with small numbers and familiar quantities only. To increase the size and complexity of the numbers used is to demand greater skill in computation, but often, with young children, serves only to obscure the meaning of what is done. Facility in dealing with a new conception grows slowly, and is often reached only after long familiarity. Such familiarity is best acquired through oral work and the solution of a wide variety of simple problems involving quite small numbers and including as many as possible that are derived from the child's own experience.
(ii) Developing mechanical skill
7. The amount of mechanical work usually done should be reduced. To allow of a properly balanced course of instruction, the range and the amount of mechanical work usually attempted, especially in the Junior School, must be reduced - for many of the children, if not for most. This may be effected by postponing the teaching of the more difficult "rules" or by restricting the exercises set to give practice in them to examples involving small numbers. It is usually a waste of time to teach a mathematical process
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or technique to a child, unless he is likely to acquire a reasonable degree of skill in using it before he leaves the class in which it is taught. The duller child, in particular, must not only have his knowledge of forms of numerical calculation related more directly and more extensively to his own everyday experiences, but he needs more practice in applying to quite simple life-situations such mechanical skill as he acquires.
8. Memorisation of tables essential. At every stage of the school course, however, certain essential habits must be acquired as a foundation for further work. For example, the fundamental tables - addition-and-subtraction, multiplication-and-division - present the results of the preliminary oral and practical work in systematic arrangement and they derive their meaning from it. Their items must all be memorised, for they form the basis of the mechanical work which follows. The pupil must quite early be able to add any two numbers less than ten, for practically every lesson involves many of these operations. If he fails to learn the tables properly, i.e. to give the result automatically, without intermediate steps and without stopping to think, or if he fails to master them at the right time, he is hampered in all subsequent work and wastes, in the course of his school life, far more time than their accurate memorisation would have required.
9. Arithmetical "rules". The mechanical rules of written arithmetic are primarily devices for adding, subtracting, multiplying, or dividing quantities that are too large or too complex to be dealt with mentally. If the child were supplied with actual machines, to do the work of these rules automatically, his power of attacking problems would hardly suffer. Indeed it might be increased. The written rules, in fact, are best regarded as forms of mental technique or as complex habits to be formed. To teach them successfully means
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that the child will acquire, with the least expenditure of time and energy, such a degree of speed and accuracy that they can be readily applied.
It is often contended that the child should learn these rules intelligently. If this means that he should be able to recognise the kind of problem situation that leads to them, it is true. If it means that he should grasp the full logic of, say, the subtraction rule at the age when it is commonly learned, it is certainly untrue. Some insight into the logic may facilitate memory and computation and may therefore be desirable. This aspect, however, should not be over emphasised.
10. How to secure economy of time and effort in mechanical work. The importance of economising time and effort in mechanical work has already been stressed. The following suggestions have this end in view:
(a) Methods should be standardised. This should be done, if not throughout a district, then at least throughout the group of schools contributory to a given Senior School. The method of procedure should be quickly reduced to a final and simple form, (e.g. in dealing with "36 - 17", the child might say, almost from the start, "seven from sixteen, nine; two from three, one"), avoiding unnecessary statements. "Crutches" if used at all, should be discarded before they become fixed habits. The writing of "carried" figures, for example, is probably best not taught at all, at any rate with normal children.
(b) A definite standard of accuracy should be aimed at. For every rule the teacher should have in mind a standard of working accuracy which each child should reach before passing on to another. Any such standard, e.g. three right out of four is more quickly reached with short "sums". The fewer
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figures and operations involved the smaller the chance of going wrong. This implies that exercises of smaller number range should be used with the duller children.
(c) Excessive mechanical work should be avoided. The amount of mechanical work to be covered at any stage should not be so great as to upset the balance of the syllabus. A suggested minimum syllabus for normal children is given in Chapter II of the Board's pamphlet Senior School Mathematics. Further economy of time and effort may be secured by avoiding alternative methods, or by reducing the number range, as suggested above.
(d) Mechanical drill directed to a specific purpose is valuable. Much time may be saved if the mental habits that each new rule implies are considered separately, and special oral drills are devised when needed. Short division by six (for example, 351 ÷ 6) requires, (1) a familiarity with the division aspect of the table of sixes even when the table is known, (2) the ability to deal with exercises like 35 ÷ 6, and to "carry" the remainder, (3) the ability to take 48 from 51, (4) the ability to set down neatly. Any one of these may require special attention. Again, in fixing the rule for multiplication by decimals only one new habit is required, that of placing the point. This may be quickly established by specially devised oral work.
In short, mechanical drill work should be purposeful. It is sometimes claimed that the mere indiscriminate working of long "sums" may increase accuracy, but even if this were true the method would be wasteful of time.
(e) Brisk working promotes accuracy. Accuracy grows more quickly when reasonable speed is
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demanded. For exercises requiring thought procedure may be leisurely, but in mechanical exercises brisk working should be encouraged.
(f) Unnecessary written work to be avoided. There is no need for a child to write down every exercise that he attacks. Cards with half-a-dozen mechanical sums, so constructed that the card can be laid on the paper and the answers written beneath may save much time, especially if a standard answer card is used which can be laid on the child's book.
(iii) Applying skill to the working of problems
11. Mechanical practice valuable only in so far as it gives power to solve real problems. The teacher should see that the children understand that they are not doing mechanical arithmetic merely for the sake of getting sums right, stimulating and satisfying as that will always be. Mechanical practice should be taken for the same reason as practice with a new stitch in needlework, or fielding practice in cricket, i.e. in order that the skill acquired may be used, while it is still fresh, in coping with the difficulties of a real situation. In other words, the teacher will not take the point of view that instruction in arithmetic must inevitably take the form of: (a) teaching the bare skills and (b) looking round subsequently for sums in books that test the ability to apply the skills acquired. Rather, he should take the view that in approaching any new range of work, whether it be vulgar fractions, or proportion, or simple interest, his first task is to interest his pupils in a variety of easy problems, involving such small numbers and such simple quantities that written computation is not required.
Only when familiarity with new conceptions has been gained in this way will it be necessary to consider what mechanical rules and what formal written procedure
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need to be taught in order that problems involving bigger numbers and more complex quantities may be introduced. Though this formal instruction may cover a series of lessons, its ultimate purpose and justification should be borne clearly in mind all the time.
If the introductory work is well done, the primary notions and operations will be derived from the child's experience. Practice in solving a wide variety of problems that can be treated orally will help to strengthen the relation between experience and arithmetical operation. But when the child has learnt the mechanical rules, i.e. acquired the skill to handle larger numbers and quantities, the range of application of what he has learnt will be increased. He will not only be in a position to attack the traditional textbook problem, in which the data are selected and marshalled for him, but will be ready to meet a wider range of real life exercises. This kind of practical work is fully discussed in Senior School Mathematics. Simple forms of it are appropriate to all stages.
12. Problem work: some principles to be observed. A problem is an exercise that contains an element of novelty. In the time available only a few methods of calculation can be standardised, i.e. converted into mental habits, and these enable the pupil to attack only a small proportion of the exercises that he meets. For the remainder he must be able to modify or combine methods as circumstances demand, or to devise new ones.
. Problem-solving cannot be taught as rules are taught. The traditional method of dividing problems into types and teaching a standard method for each type has but a limited value in view of the large number of types. Training children to solve problems is in essence training them to meet and surmount difficulties for themselves, and the best training is the kind of teaching that grades
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difficulties so that the pupil can surmount them, and presents them in such a way that he has to do his own thinking.
But although problem solving cannot be directly taught certain suggestions for handling problems are worth bearing in mind:
(a) The pupil must face the problem unaided. To remove the new difficulties in advance, e.g. by suggesting the method to be used, is to destroy the essence of the problem, and reduce the exercises to mechanical work.
(b) Problems must be easy. This means that they must be such as the pupil can tackle unaided with some chance of success. The view sometimes expressed that backward children cannot do problems - which implies that they are learning what they cannot apply - really means that most textbook problems are too difficult for them. The experimental grading of problems in order of difficulty has hardly been begun.
(c) Approach through real-life situations. Problems are best approached through "real-life" situations, as already indicated. "Real-life" problems may be supplemented by varied oral exercises in which the answer only is written down. These may be introduced at an early stage, and up to the end of the Junior School backward children will hardly go beyond this type.
(d) Language used should be simple. Young children commonly fail with textbook problems, because they cannot read them, i.e. because they are unable to realise the exact situation that the words describe. In order to minimise this difficulty
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problems should deal with familiar and interesting situations. They should be expressed in short, crisp sentences and in simple language.
(e) Demonstration better than verbal explanation. Children will grasp a situation more easily, if the training begins with problems involving such simple numbers that practical apparatus can be used to illustrate the operations. For example the pupil, when he first meets such a question as "How many quarts in five gallons?", can easily be shown that five groups of four are involved and that multiplication is indicated. Demonstration where practicable is to be preferred to verbal explanation. If, at a later stage, pupils attempt to solve a problem by multiplying when they should divide, or fail to understand what the remainder stands for, they will often surmount the difficulty at once if the problem is restated with the simplest numbers and demonstrated with actual objects. But this difficulty should rarely arise, if care is taken from the start to link process with objective demonstration and with verbal statement.
(f) Full comprehension of the meaning of the problem the first essential. When written problems, involving several steps and the use of large numbers, are introduced, the teaching should not be directed primarily to suggesting the method by which they can be solved, but rather to training children to ask themselves what is given and what is required, to note what units are employed, to use diagrams when possible and, in general, to analyse the problem in such a way that the meaning will become clear. If children have been systematically trained, as they should be, to see how problems arise out of real-life situations, they will find the written problem much easier to understand.
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D. MENTAL ARITHMETIC
13. The uses of mental work in Arithmetic. "Mental" Arithmetic includes all exercises in which pen or pencil is not used, except perhaps to record the answer. Much of the Arithmetic of everyday life is "mental" in this sense. The value of such work, provided that the exercises are chosen to serve a definite purpose, has been stressed throughout this chapter. Its main purposes will be: (a) to give practice in solving a wide variety of problems, with or without the help of apparatus; (b) to give brisk drill in specific habits (e.g. in addition of fractions, changing the subject of a formula) or in the tables themselves; (c) to revise: e.g. a test might be given to discover, by means of a few very simple exercises, how much a class has remembered of the rules for operations with decimals or for finding the areas of plane figures.
Certain short cuts in computation may usefully be introduced through mental work. Only a few of these short cuts are important, e.g. the rules for finding the cost of a dozen articles, or for multiplying by twenty-five.
Few textbooks contain enough exercises in "Mental Arithmetic"; still fewer provide purposeful exercises of all types. For some purposes, e.g. brisk drill work, the exercises are best devised by the teacher who knows the needs of the class; but most teachers need a supplementary source-book for mental work. If ample exercises of this type are available, one section of a class can be set down to "mental work" by themselves. Even where classes are taught as a whole, there are many types of example which the child needs to read as well as to hear; some require the numbers only to be written on the blackboard.
Finally, it is well to remember that, although the use of pen and pencil in ordinary written work makes thought clearer by necessitating its clear exposition,
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the rule "Show your working" may easily lead to mental slackness. The complementary rule "Never do work on paper that can be done mentally" often needs to be emphasised; the bright child may often be set to do mentally as many as he can of an ordinary set of exercises.
E. ORGANISATION OF THE SCHOOL COURSE
14. Individual work and group work both essential. In some schools, provision is made for differences of ability by allowing each pupil to proceed at his own pace. In Arithmetic, where it is of special importance that the pupil shall surmount his own difficulties, there will always be much individual work. Teaching which relies solely upon it, however, misses the great stimulus and value of oral group work. A group which is put on to individual work, before sufficient group instruction has been given, will be in danger of making many mistakes. The teacher will be hard put to it to keep pace with the corrections and the pupil, through repetition of errors, will contract bad habits that are difficult to eradicate. Moreover, the method, if carried on throughout the Junior School results in a truncated course for all but the best children.
15. Adapting the course to the children's capacity. Where numbers admit of it, the course should be planned on the basis of the "two-stream" or "three-stream" classification which the Hadow reorganisation makes possible. There may be then a minimum syllabus, in which each major topic will be completed by all normal children at about the same age. There may also be a supplementary course with extra rules, larger numbers and more difficult problems, enough to provide ample work for the brightest children; on the other hand, there may be a few for whom the minimum will be too much. Many schools are not large enough
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to allow the classes to be organised in separate streams. Where it is possible at least for each age-group to have a teacher of its own, differences of ability can be met without much difficulty by sectional treatment, though to ensure adequate oral teaching the number of sections should not be too large. In such circumstances the supplementary course might be designed to enable the brighter children to work largely by themselves.
In a small school, where each teacher is responsible for more than one age-group, it is very difficult to allow for differences of ability as well as of age. If the age range is very wide, several sections may be necessary; but multiplication of the groups often leads to the neglect of one or other of them and to the cutting down of necessary oral work. For practical work such as measurement or out-of-door surveying it may be necessary to take the whole class together, but the tasks allotted to the children should be graded according to their ability. In very small schools it may be a useful plan to set a bright child occasionally to supervise the work of a small group which is less advanced or to arrange for children to work in pairs.
In any case, the syllabus must be considerably simplified in the school where children of a wide age range are taught by one teacher.
16. A course should not be on rigid lines: it should be modified to meet individual needs. But, although forethought and planning are necessary, it cannot be too strongly emphasised that the most suitable course for any given class cannot be laid down in advance, even by teachers who know the children, and still less by textbook writers who do not. Real success is only possible when the teacher is on the alert to notice where modification is needed and is resourceful in supplying it. He must distinguish between failures that are
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due to misunderstanding, to lack of skill, or to carelessness, and treat them differently; note children who habitually get their sums wrong and modify their course before they become discouraged; see that the brightest ones are kept on the stretch and fully employed; devise exercises that arise out of the children's special interests and prescribe drills calculated to help them to remove bad habits and to master essential forms of skill.
17. How a textbook should be used. Textbooks should not be followed too closely, nor should they be changed too often. The teacher will do well to note in an interleaved copy the modifications that experience suggests to him. He may note, for example, which exercises may be omitted; record additions that he has to make; note which exercises prove difficult and which easy, where difficulties of language occur and where alternative methods may be employed, where rough estimates or checks may be profitably employed, and where short cuts may be looked for from the brighter children. If experience is gathered and recorded in this way, especially when children's books are being corrected, the second passage through the course will be far more profitable than the first.
18. Methods of correcting written work. Correcting the pupils' written work - i.e. indicating their mistakes by means of signs or marks - has two primary uses. It enables the teacher to profit by experience, as suggested above, and it brings home to the pupil where he stands. The correction must be thorough, if interest is to be sustained, and if the teacher is to discover the sources of error in the pupil's work. Correction, however, in the sense of putting right what is wrong, is mainly the pupil's business. Different types of error need different treatment which will depend upon the diagnosis, and this should be indicated in the marking: an error in reasoning may necessitate further teaching
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before correction by the pupil; inaccuracy may be due to ignorance of tables, or to bad setting down, or merely to general slackness or to a casual slip. It may be unwise to insist on the reworking of an exercise just because of a single slip.
19. Value of short methods: Estimating and proving. The brighter children should be kept on the alert for short cuts and alternative methods. All pupils should often be required to make preliminary estimates of results. Many teachers show the children how to "prove" their sums and frequently require them to do so. The use of a book in which some exercises are starred to indicate that estimates should be made or the best method sought, is a stimulating device. Very few short cuts are of such general application that they should be specifically taught. Moreover specific teaching rarely ensures that they will be used at the appropriate occasion. Discussion of short methods has little effect on subsequent work unless, by the "starring" or some other device, the habit of looking for opportunities is continually encouraged. The more backward children should be taught standardised methods, and will rarely be able to depart from them.
F. INCLUSION IN THE COURSE OF OTHER FORMS OF MATHEMATICS
20. Possible lines of development. The Board's pamphlet Senior School Mathematics describes in some detail the forms of mathematical training other than Arithmetic that may be included in the Elementary School course.
The practical activities of the Infant and Junior Schools will extend the child's familiarity with size, shape, and direction. The Junior School teacher may
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do much to facilitate the growth of geometrical notions, which is necessarily slow, and to link them to correct description. In the Senior School such activities as Practical Surveying, Practical Drawing in connection with Bookcraft, Woodwork, and other forms of craftwork will develop these notions and make them more explicit. It should be the aim of the teacher of Mathematics to bring home to the pupils the geometrical facts and principles involved in these practical activities through discussion and by means of supplementary exercises, and so enable them to acquire a working knowledge of some of the more important generalisations of geometry.
Other forms of mathematical work that Senior Schools can profitably introduce are the reading and drawing of graphs (for which also a foundation may be laid in the Junior School stage); generalised Arithmetic, especially the construction and use of formulæ; and, where conditions are suitable, the use of logarithm tables and an introductory course of Mechanics.
II. THE INFANT SCHOOL STAGE
21. The aims of number teaching in Infant Schools. The aim of the Infant School should be to organise an environment in which the orderly development of children's early ideas of number and of their experiences of measurable quantities etc. can take place most easily. It is particularly important that the training which is concerned with the enumeration of objects and with the understanding of the simple numerical relationships arising out of it, should be accompanied by the introduction of situations in which the child meets with the commoner measures of money, weight, length, and capacity. A zeal for number analysis on the part of the teacher may lead to a rapid development in the Infant School child's ideas of number and
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numerical relations; but the ability to deal with numbers will not in itself help a child to understand what is meant by 2 lbs. of butter, a yard of dress material, half-a-pint of milk, or a 10s. 0d. note.
Instruction in number should, therefore, have, as its objective, a development of ability, suited to the capacity of the child, not only in counting and number analysis, but also in dealing intelligently with the simplest common quantities. Even before they begin to attend school, children have acquired some experience of number, distance, shape, size, weight and money. The extent and clearness of this knowledge will vary considerably with home conditions, but the children will probably have in their vocabulary some general terms associated with quantity such as "little", "high", "heavy", etc. and some of the number names.
22. The approach to number teaching through various activities. When children enter school they should not at first be expected to make any change in the ways by which they normally acquire number knowledge. Their environment should, however, be more stimulating and the teacher should from time to time draw attention to the quantitative side of their experiences and take steps to systematise what they are learning. She will, for example, make use of the traditional counting rhymes, games, and songs for teaching the number names and she will introduce occupations such as bead threading, which gives opportunities for counting, and the handling of objects designed to bring out differences of size and shape. Children enjoy counting and no special setting is required for a great part of the early teaching in enumeration. The child may count the buttons on his coat, he may play the make-believe games of laying plates for six and he will hear stories such as that of the three bears with their bowls of differing size. His rhythmic activities and his handwork
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will also add to his experiences of numbers and quantities. He will throughout be increasing his power to describe what he does in appropriate language.
23. Learning to count. Use of number patterns before figures are taught. In learning to count, the child first meets a number, say four, as one of an invariable sequence of words (one, two, three, four etc.). He also learns to associate the words, one by one, with a series of objects, touching or pointing out these as he does so. The abandonment of this habit later on marks a definite stage of progress. He also learns to associate the words three, four with the numbers of objects in a group rather than with those counted third and fourth.
It is at this stage that some teachers introduce orderly arrangements of dots to represent the numbers in easily recognisable patterns, such as
for five. Practice in recognising such patterns and counting the dots adds to the range of number activities and may be made a useful preliminary to the teaching of figures. Figures themselves add nothing to the understanding of numbers and they should, therefore, not be taught until the children can make confident use of the sounded names corresponding to them.
24. The introductory step in the teaching of number operations. At a later stage, when introducing simple operations, the teacher will be well-advised to display the same caution in teaching the symbols for adding, subtracting, and so on. The first essential is that the operation itself should be understood in a sufficient number of situations to give it generality. Adding must be associated not only with counting the total of two or more groups of balls, counters
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or dots, but with groups of all sorts of things in a great variety of circumstances. Again, subtracting may be associated either with the comparison of groups of objects or with removing a smaller group from a larger one.
It will be wise for the teacher to see that the children are able to perform such operations with a variety of actual objects, and to describe what they do, before she gives them exercises, for example, with counters, in which the only variety is the changing of the numbers involved. Thus, while he is still at the stage of learning what subtraction means, a child will say "I threw 5 balls into the basket and Jack threw in 2, so I threw in 3 more balls than Jack." Or, "I had 5 biscuits and gave Jack 2, so I have 3 left for myself." Later he will become interested in trying out the operation of subtraction with a great variety of numbers and should then proceed to memorise the addition-and-subtraction table. But the preliminary work done with the aid of objects and the practice in describing what he does will help to lay the foundation of an intelligent use of this table, and the child's actual experience will help him in constructing the table itself. A child is more likely to remember that his score of 5 was 3 more than Jack's 2 than he is to recall items of a series of routine exercise with counters. Moreover, in his eagerness to get at the answer to a little sum arising out of some game the child will tend to discard unnecessary aids to calculation.
25. Practice in simple operations with a variety of numbers. Skilful teachers have devised numerous ways of giving children plenty of practice in doing sums of graded difficulty. The use of apparatus sometimes adds the play element to the working of a sum; beads, cards or dominoes may also be used to represent the numbers concerned. Number patterns are particularly
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helpful when a number has to be broken into its component parts, e.g. the removal of 3 dots from the 7 exhibited thus
clearly leaves 4 dots
though this is less obvious if the pattern
be used for 7.
In using apparatus for this purpose it should be borne in mind that memorising the fundamental tables has to be completed early in the Junior School and that a habit of counting or of using other aids to finding out the answer should not be associated with calculations of which the results are already confidently known. The practice of visualising numbers and that of "building up tens" may retard memorisation, if they are adopted habitually instead of as helpful or explanatory stages. On the other hand, the importance of groupings in tens and in twelves justifies an early teaching of the composition of both these numbers in the informal stages of building up the complete addition table. The children will co-operate in the right use of apparatus if encouraged to approach little sums in an attitude of experimentation or with a desire to reach results quickly and accurately; sometimes the one and sometimes the other of these attitudes is appropriate to their stage of progress. The dangers of over-dependence on apparatus have been dealt with above in Section 4, page 502.
The range of operations to be dealt with in a practical manner will not necessarily be restricted to addition and subtraction. For a small range of numbers the children may learn to deal with all the four fundamental operations even if they do not get to the stage of making symbolic statements of their results.
26. Range of numbers for which automatic knowledge can be expected. There will probably be wide differences amongst children, who have reached the
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end of the Infant School stage, in the sureness of their knowledge of the addition-and-subtraction table; but normal children should be able to deal practically and orally with all sorts of operations involving numbers not greater than 12 with certainty and rapidity, i.e. without stopping to think. They may not, however, be so sure of the composition of numbers between 12 and 20. With children whose development is retarded, more time must be allowed for fundamental ideas to take root. If such children pass on too quickly to the symbolic statement of little sums, or even to formal practice with counters, they may for a time give the appearance of keeping up with their fellows, but they will pass from stage to stage without forming the mental habits which the graded exercises are designed to foster. Their attainments when they proceed to the Junior School stage will be superficial and their condition much inferior to that of a backward child whose experiences of number have been arranged to suit his stage of development.
27. Counting and notation. Counting should be extended beyond the range of numbers commonly used by children and they should be encouraged to count in groups as well as in ones, (e.g. they might count the children in a class as they sit in twos, the panes in the window, or the milk bottles in their crates). Children should also practise counting backwards in ones and naming in order the odd as well as the even numbers. Frequent use should be made of apparatus such as the ball frame, which exhibits clearly grouping by tens. This will help to give significance to the number names "ten", "twenty", "thirty", "fourteen", "twenty-four", etc.; and it will be an essential part of the teaching for the children to read and write numbers larger than ten. Before they complete the Infant School stage most children should have learnt to read and write
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numbers up to 100 and, in such a number as 14, to appreciate the significance of the 1 and the 4.
28. First ideas of magnitude, measurement, and money. The children's growing competence in dealing with numbers should be accompanied by increasing exactness in their ideas of magnitude and measurement. The earliest ideas of quantity should be a matter of comparison rather than measurement, words such as "longer" and "heaviest" sufficing at first. Some of the materials given to the children to handle should, however, be so graded in size as to suggest increase by a regular unit. Opportunity should then be given to the children to experiment with improvised measures of all kinds - strips, hand-breadths, paces for length, metal or cardboard discs for weight, cartons, bottles or toy pails for volume and so forth - without any attempt being made at first to force standard units on them. In this way rudimentary ideas of measurement will take root and meaning will be given to the use of numbers in such expressions as "a four pound weight", "six inches", "five years old". Moreover, among the children's experiences involving operations or counting should be included some which also involve measures such as those which occur in simple shopping or in counting the groups of five minutes round the clock.
Before children pass out of the Infant School stage they should have had some acquaintance with coins, with measures most commonly used, such as pints and quarts of milk, and with such practical matters as telling the time and using the calendar. They should have become accustomed to the use of scales, weights, measuring-tape and foot-rule, and should have grown familiar with the idea that coins have different values according to their size and the material of which they are made, and that giving change is a common feature of shopping;
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for it will not be difficult for the average child to give change in pence from 6d. and 1s. 0d.
In short, the more realistic the school room procedure, the more likely children are to gain clear ideas of measures of quantity and of number relations generally.
III. MATHEMATICS AT THE JUNIOR SCHOOL STAGE
29. Continuation in the Junior School of work begun in the Infant School. In the Infant School emphasis should fall not so much on the acquirement of a high degree of skill in computation as on bringing out and clarifying such notions of number and magnitude as are implicit in childish experience. The normal child on leaving the Infant School will probably have learnt rhythmic counting - e.g. in twos up to, say, forty - and will have memorised the addition and subtraction tables up to a total of twelve. He will have been familiarised with multiplication and division, as performed with the aid of objects, though he will not have memorised the multiplication table as such. He will also have become acquainted with the commoner units of weight, length, and capacity, in use around him and he will have an intelligent idea of their use. The value of such training should be judged by the interest aroused and by the range of quantitative experience that has been explored, rather than by the systematic knowledge and skill obtained. It cannot be judged merely by the results of a formal test, especially of a written test. The teacher of the lowest Junior School class should be familiar with the course that each child has followed in the Infant School, so that she will be able to provide equally well for those who have done more or less than the average.
30. The range of work in the Junior School. The importance of strictly limiting the amount of mechanical work required of each child, so that the balance of the
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syllabus may be maintained, has already been emphasised. In the Junior School, however, a certain emphasis must fall on the acquirement of skill in calculation, for the children are at the age when they most readily form simple mental habits, and if the right habits are not formed they are difficult to acquire afterwards. Accuracy should be secured in such "rules" as are taught, but no child should spend so much time in learning rules as to leave no time for simple problems or for the introductory practical and oral work that serves to give the rules meaning.
There are obvious practical advantages in assigning a minimum of knowledge and skill which every normal child will be expected to acquire before the end of the Junior School stage; this might well be settled by local agreement for each group of re-organised schools. A suitable minimum, to be thoroughly and permanently known, will be found suggested in Senior School Mathematics. It will be noted that this does not include long multiplication and division of money, weights and measures, or any work with decimals, and further, that restricted range of number is suggested. For example, exercises in weights and measures may well be confined to three-unit quantities, and fractions to simple denominators that will not involve teaching the rules for H.C.F. and L.C.M. It is, of course, assumed that the brighter children will attempt more, and there will be a small minority who cannot attempt so much.
Many teachers have been deterred from simplifying their syllabus of Arithmetic to the extent that is desirable by the requirements of the Special Place Examination. It is important that the scope of this examination should not be too wide, especially when the papers are set to a complete age group. A suitable range is that suggested in Chapter II of Senior School Mathematics. Further, the questions set should not be too
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complex or too long. Papers consisting of a large number of short problems ranging from very easy to very difficult have been found to do the work of discrimination at least as well as those consisting of a few difficult ones. A paper in mental arithmetic may have high selective value. When a paper of mechanical exercises is set it should be long enough to test speed as well as accuracy. Moreover, due emphasis should be given to the more important rules taught at the beginning of the Junior School course. The long rules in weights and measures, taught usually in the year preceding the examination, are relatively unimportant.
31. The importance of providing alternative courses. Throughout this chapter it has been emphasised that Arithmetic is taught in order to be applied, not merely in order that the child may pass tests in formal rules; and that young children are only able to apply what they are taught, when it has been sufficiently related at the outset to their experience by means of practical work. The younger or duller the child, the wider the basis of experience that is needed for the grasp of each abstract conception. For the "C" child in the Junior School, for example, it may be necessary to base most of the oral and written exercises on real-life problems. The classroom shop for example, may provide most of the exercises in money and weighing, and full use will be made of such real-life experiences as arise naturally in the course of the school life, e.g. planning an excursion or laying out a garden. The "C" child will also profit from much brisk oral work designed to give facility in handling small numbers. Such exercises, with the more formal practical work, which will also have to be emphasised, take a great deal of time. His formal written work on the other hand may have to be limited by omitting all or nearly all the long rules, by using only small numbers or two-unit quantities, and the simplest
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fractions. And it will have to be graded much less steeply than is customary, so that difficulties may be taken up one at a time and overcome before passing on. With such children in particular it is not only a waste of time but a source of great discouragement to teach formal rules that are not carried to the stage of working accurately and that will probably never be applied.
Even where brighter children are concerned interest and variety may be given to the work by the judicious use of catalogues and railway and other timetables and of games, puzzles and problems that arouse their interest.
32. The tables and table-learning. The practical work of the Infant School will be continued, and apparatus will be used to bring out the meaning of essential operations: e.g. that multiplication is an operation with groups and not with units - a fact which the child who builds up tables by moving objects one by one is apt to miss; how "17 + 8" and "27 + 8" follow at once from "7 + 8"; how the different multiplication tables are related to one another, e.g. the tables of fours and eights; how they are related to the number series, as shown for example by the patterns made on the number chart by the tables of nines and twelves. In short, the purpose of the practical work is not merely to discover the values of the items (e.g. that "6 X 7" = 42), but also to bring out other relations. Unless the teacher is alert to its wider purpose, table building may be as mechanical as table repetition.
The four fundamental tables have to be memorised. Every child, if he is to avoid wasting time in subsequent work, must reach a high standard of accuracy, not far from 100 per cent, in the items. For example, by the end of the first year of the Junior School the addition-and-subtraction table up to a total of 20 should have become automatic. Points worth noting are that the
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addition table is at least as important as the multiplication table; that such a combination as "31- 27" follows at once from "11- 7"; that memorising a table means being able to recall each item as required, not merely in table order; that attention to pattern, e.g. that of the table of nines, greatly facilitates learning; that the traditional order and range of the tables learnt is not necessarily the best. There is much to be said, for example, for learning the table of twelves before the tables of sevens and elevens; also for memorising the tables of fourteen and sixteen when they are required in connection with weights.
Table learning should be supplemented by oral exercises designed to give general facility in calculation (e.g. exercises in rapid successive addition), in finding as many pairs of factors as possible for a given number, in recognising the prime numbers (e.g. between 60 and 70) etc.
33. The simple rules. The weight of evidence suggests that accurate subtraction is best attained by the method of equal addition. Many teachers also favour approaching short division by the long division arrangement, thus showing the essential unity of the two methods. It is now generally accepted that the best method in long multiplication is to begin with the left-hand digit of the multiplier. In all the rules numbers involving one or more zero figures give much trouble and call for special attention.
34. The compound rules: money, weights and measures. It should not be assumed that all children have been sufficiently familiarised in the Infant School even with the commoner coins and units. Some, for example, will need teaching how to tell the time or to use a pair of scales. All will need practical work to familiarise them with less common units and their relations. If
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the children are to have the varied individual experiences which will ensure a real understanding of weights and measures, it is obvious that there must be adequate equipment of the right type. At the Junior School stage there should be rulers exactly twelve inches long marked in fourths, eighths and tenths, without angle measurements, yard sticks, plywood cut to one foot square size and also to one inch square, weights of ¼ oz. to 1 Ib., and separate measures of ¼, ½, 1 and 2 pint capacity. Children should also know certain useful and familiar measures that will serve as standards where estimates are made. For example, a child may know his own height or the height of the classroom door, his own weight or how much he can easily lift, how long it takes him to walk a mile, or how far it is from home to school. His attention may be drawn to familiar measures in common use, e.g. that a cricket pitch is one chain, that three pennies weigh an ounce, and that the school milk bottle contains a third of a pint. The teacher who is familiar with the history of our weights and measures will be able to draw much interesting material from it.
The compound rules involve a new operation, unit-changing - changing one step up to a larger unit and one step down to a smaller. This process, though no more than an extension of the notation principle, is of fundamental importance, and needs careful treatment. It not only forms the basis of all the rules, but its intelligent application saves much time in solving problems. The meaning of unit-changing should be carefully demonstrated, by using objects and diagrams where possible, in relation to all the tables. And before learning, for example, the formal rule for reduction of money, oral practice should be given in solving small number exercises involving changing shillings to sixpences, sixpences to half-crowns, pounds to florins etc. The first rule work should consist of two-unit exercises arising out of oral work.
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(a) The money rules. Shopping exercises, which may be graded in difficulty so that a wide range of calculation is involved, are invaluable for providing an interesting approach to the money rules.
The pence-table need not be separately learnt, if the tables of twelves and its pattern on the number chart are well known. Counting backwards and forwards by steps of 3d., 1½d., 2s. 6d. etc. is a useful exercise.
In long multiplication and long division of money and other compound quantities the superiority of the "column" method is now established.
(b) Weights and measures. Most of the applications to everyday life situations involve quantities of no more than two units, and the teaching of the rules should reflect this fact. Few problems involve quantities of more than three units. The "reduction" rules should not be taught until unit-changing is thoroughly grasped. Children who are taught them too soon and too thoroughly often handle the units met with in problems very unintelligently.
The factors and properties of certain numbers that occur often in relation to our English weights and measures are worthy of some special study, e.g. 112 and 1760. Further, certain tables lead naturally to the more important groups of fractions. The statement "one gallon = 8 pints" easily leads to its correlative "one pint = 1/8 gallon." Similarly, the 1/12 notation may be introduced in relation to shillings and pence or feet and inches.
35. Fractions. Fractions should be introduced gradually through practical exercises. The work might begin with the halves, quarters, eighths family, and go on to the thirds, sixths, twelfths family. So long as the work is confined to such simple families, problems and
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operations involving fractions should be based on commonsense methods rather than on formal rules. The extension of notation to sixteenths, twenty-fourths, and of fifths, tenths, and hundredths as an introduction to decimals and percentages at a later stage will probably include as much as most normal children can thoroughly master by the age of 11. The relation of unit fractional parts to simple division should be understood, e.g. that -¼ of 23 = 23/4. Though the course may be extended for brighter children to include fractions with larger denominators that may be treated by the factor method of finding H.C.F. and L.C.M., it is more important at first to work a wide variety of exercises with a small range of fractions than to learn the formal rules of manipulation.
If the emphasis is placed on the intelligent use of fractions rather than on learning the rules, the importance of adequate practical work will be realised. Some standard apparatus will probably be needed. One useful type of individual apparatus can be made from stout strawboard or plywood by taking strips 12" by 1", marking one whole strip into 24 equal parts, cutting the others into 2, 3, 4, 6, 8, and 12 equal pieces respectively and labelling the pieces on one side, some in figures and some in words, e.g. "1/3" or "one third". By matching and super-imposing various pieces the child can discover for himself the equivalence of fractions and can easily state orally (though not at first in writing) that 1 = 1/3 + ¼ + 1/6 + 3/12; that 1/3 is greater than ¼ by 1/12; that ¼ may be divided into 3 equal parts, each of which is 1/12; that four times 1/6 is 2/3; that 1/3 + 1/8 = 11/24; and so on. Measurement, especially measurement involving parts of an inch, is especially valuable. Enough of this should be included in the course for both boys and girls to form a basis for the appreciation of both fractional and decimal notation.
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Decimals, if taught, will arise naturally as alternative notations for tenths and hundredths, and will be illustrated by ruler work. The treatment of hundredths, however, involves estimation on the ruler, and other apparatus may be preferred. Children should know the decimal equivalents of halves, quarters and fifths. The decimal notation can be illustrated through addition and subtraction but the formal rules for multiplication and division of decimals are best left to the Senior School.
36. The connection of Mathematics with other subjects. The other subjects of the curriculum, e.g. Nature Study, Geography, Needlecraft and Handwork, will provide occasions and material that can be used as a basis for Geometry, Mensuration and Graph-work, such as measurements of all kinds, and practice in estimating; drawing simple shapes and patterns with the aid of ruler, set square, compass, squared paper, etc.; drawing rough diagrams and plans, not to scale; reading and drawing simple maps and plans to scale; compass-bearings and direction, involving the right angle and very simple fractions of it, but not the use of the protractor; reading and constructing simple graphs, e.g. a temperature chart or weather record.
It is for the teacher of Mathematics to discover what material is available, and to see how far it can be made to serve his own purposes. For example, he may teach the correct use of such geometrical terms as right angle, perpendicular, vertical, etc. in describing it. He may use it to illustrate mensuration e.g. the finding of areas. It is important, however, that in treating area problems (e.g. the formula for the area of a rectangle), the material used should be such as brings out the notion of surface area, i.e. it is better to cut out pieces of paper for the rectangle and the units of area, than to draw rectangular outlines.
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Some of the diagram work may be introductory to the graph. The lengths or areas of diagrams may represent quantities that are not necessarily spatial, e.g. the different speeds of ships, engines, etc. may be represented by pictures of different lengths. This diagrammatic work may be used to help children to grasp the relative size of large numbers.
37. The importance of tests. It is difficult in Arithmetic to assess the value of teaching without some form of test. It is probable that standardised tests, enabling each teacher to compare the results that he gets in any part of the subject with the "norm" for children of given age, will become increasingly available in the future. Such tests, if judiciously used and with due regard to circumstances, may be very helpful in setting a standard or in maintaining it from year to year.
Tests are useful not only to determine whether proper standards, e.g. of accuracy or speed, are attained at each stage. If properly constructed, they may be used to diagnose the specific mental habits which each child has formed or failed to form.
IV. MATHEMATICS AT THE SENIOR SCHOOL STAGE
A. INTRODUCTORY
38. Alternative courses desirable. The recent publication of the Board's pamphlet Senior School Mathematics* makes it unnecessary to consider here the general problems relating to the teaching of Mathematics. Teachers are advised to make a preliminary study of that pamphlet before undertaking any thorough study
*Board of Education Educational Pamphlet No. 101, printed and published by H.M. Stationery Office, London, price 1s. 0d.
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of this section of the Handbook, the main purpose of which is to deal more particularly with the preparation of schemes and with actual teaching methods.
The practice of dividing the Senior School into two, three or four streams has become common. In the selection of pupils for the various streams, arithmetical attainments will have carried considerable weight. For the sake of convenience, it will be assumed that the "A", "B" and "C" streams commonly found in the Senior School contain respectively, pupils of above average, average, and below average mathematical ability.
These wide differences in ability to which detailed reference is made in §39 and §40 of this Chapter, must be met by syllabuses which differ not only in content, but also in outlook and treatment. Each school therefore will normally provide at least two alternative courses in Mathematics.
The problems which arise in connection with the teaching of "A" and "B" pupils are here dealt with together, but where topics and methods are more suited to the abilities of the "B" than of the "A" pupils, this is indicated. The suggestions regarding the teaching of "C" pupils are dealt with separately.
B. EARLY STAGES OF THE COURSE
39. The link between the Senior School and the Junior School. The arithmetical attainments of pupils on entering a Senior School will vary considerably. Some of the pupils may not have covered the minimum syllabus indicated in §16 of Senior School Mathematics. The various tables and processes may be imperfectly known, roundabout and crude habits may have been learned and reasonable facility may not have been acquired. There will also be differences in the extent and variety of their practical experience.
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Carefully designed tests should be applied to find out what each pupil knows, and can do, and where weaknesses lie. Individual records of the results will guide the teacher in prescribing the appropriate remedy. In multiplication, for example, a pupil who knows his tables may yet make continual mistakes in "carrying": exercises should be designed to repair the real weakness, and more time than is necessary should not be spent on working long multiplication sums. Remedial exercises should not be applied indiscriminately to all members of a class. Reasonable facility in simple calculation and a fair knowledge of everyday weights and measures are all that is required at this stage. Methods should be the same as those taught at the Junior School stage, and where more than one school contributes pupils to a Senior School, uniformity of method should be agreed upon, if confusion and waste of time are to be avoided at the Senior School stage.
In revising and consolidating the foundation work of the Junior School it is essential that interest should be maintained. If a rule is imperfectly known, the skilful teacher will devise a fresh method of attack and will make sure that the pupil sees the need for any remedial exercise before he attempts it. Varied oral work, the introduction of a wide variety of real problems, team contests and the keeping of records by the pupils themselves are some of the ways of arousing and maintaining interest.
The work of consolidation should be spread over the first year, as the teacher finds it necessary.
40. Extension of the course in Pure Arithmetic. Throughout the first year the pupils should spend a large part of the time in covering the remaining topics in "Pure Arithmetic" outlined in §47 of Senior School Mathematics, so that in the later years they may devote their attention in Arithmetic mainly to "Practical
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Topics" as explained in Chapter IV, §§29-39, of that pamphlet. This will be possible with "B" pupils if the numbers involved are kept small and the problems are easy.
(a) Multiplication and Division of money, lengths, times, weights and capacities by numbers greater than 12. As these rules are relatively unimportant they need only be learnt by the brighter pupils.
(b) Vulgar Fractions. The introduction of unusual vulgar fractions with large denominators, and the manipulation and simplification of long fractions are now not so common as formerly. Even for the "A" pupils the rule for finding the L.C.M. should not figure prominently. It may be avoided altogether for the "B" pupils, if the work is mainly confined to the units found on the foot-rule used in craft-work.
Fractions should be associated with money and the standard weights and measures, and with groups of objects as well as with abstract numbers.
(c) Decimal Fractions. The decimal notation is employed in daily life in measuring with a rule marked in tenths, a clinical thermometer, a cyclometer or a surveyor's chain. As the need, however, of decimals of more than two or three places seldom occurs in everyday life, the introduction of long and complex decimal fractions is useless and wasteful.
If the pupils are familiar with the idea that a vulgar fraction is a way of expressing a division, the conversion of a vulgar fraction into a decimal is an easy step. The pupils should know the decimal equivalents of ½, ¼, ¾, 1/8, 3/8, 5/8, and 7/8.
In multiplication and division the main objective is to teach the pupils where to place the decimal point
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in the answer. The easiest and safest rule in multiplication is to ignore the decimal points, to multiply as in ordinary multiplication, and then to insert the point in the answer after counting the total number of decimal places in the two numbers being multiplied. This brings the multiplication of decimals into line with ordinary multiplication. If each of the two numbers to be multiplied is small and has only one place of decimals, the pupils will place the point in the correct position in the answer in accordance with what they know about the size of the numbers; but the rule can also be made intelligible by means of squared paper marked in inches and tenths of an inch.
The easiest and safest rule in division of decimals is to set down the sum in fractional form, with numerator and denominator, make the denominator a whole number, adjust the position of the point in the numerator and then proceed as in ordinary long division.
The "standard form" method of division is preferred by some teachers on the ground that a rough approximation to the answer is easily obtainable. The sum is first set down as in the above method, and the decimal point is adjusted so that the denominator has only one digit before the point.
The selection of the method to be used in multiplication and division must be left to the teacher; provided the method adopted is made intelligible to the pupils, their proficiency in using it is then the final test of the teaching. Whatever method is chosen, it should be used throughout the school.
Some reference should be made in the teaching to the Metric System, but the extent to which the topic is developed will be largely determined by the use made of it in the Science course. There is no need to introduce the Metric System as applied to money or the decimalisation of English money and its converse process until
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the rules are required in the "Practical Topics" taken later in the course.
It is desirable to discuss what is meant by "degree of accuracy" in measurement, and, with "A" pupils at any rate, care should be taken to ensure that calculations based upon measurement are not carried to a number of significant figures clearly unjustified by the data. No elaborate estimate of the reliability of results is necessary. For example, in the calculation of the area of a rectangle from measurements which are correct to the nearest tenth of an inch it is easy to identify the figures in the multiplication, which arise wholly or in part from approximate figures in the data; the conclusions can thence be drawn that it would be misleading to use the last figure of the answer, i.e. in this case, the hundredths of a square inch, and that some of the other figures are more or less doubtful. Similarly it can be shown that, if an approximate decimal form of the value of π is used (for example, in calculating the circumference of a circle from measurement) the number of figures which it is useful to retain in the value of π depends on the number of significant figures in the measure of the diameter.
(d) Ratio, Percentage and Rate. One way of comparing two quantities of the same kind, e.g. two heights, is to use the vulgar fraction or ratio.
The method of percentage enables two or more fractions to be compared at a glance. For example, where classes are of different sizes, the ratios of absentees to the number on roll may be more readily compared by expressing each ratio as a percentage or decimal than in fractional or ratio form. Percentage should, therefore, be taught in close association with fractions and decimals. The method should be applied to class attendances, lengths, areas, weights and other measurable quantities. Its application to money transactions
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should be left until the pupils are studying the practical topic of "Savings" later in the course.
The essential difference between a rate and a ratio is not always clearly understood. The former is also a comparison between two quantities, but the quantities are of different kinds; and whereas a ratio is an abstract fraction, a rate always involves a unit, e.g. feet per second, gallons per minute, pounds per annum.
(e) Proportion. The unitary method will have been used at the Junior School stage. Later, however, the method becomes cumbersome, often involves absurd statements, and tends to obscure the fact that proportion is a comparison of two ratios. It should, therefore, be superseded in the Senior School by the "fractional method."
The fundamental idea underlying proportion, viz. that of one quantity varying either directly or inversely with another, is best taught through a wide variety of real and, wherever possible, practical examples which should not be confined to Arithmetic. The subject should be linked up with "height problems" in Surveying and "area problems" in Map-reading, and with the relation between the dimensions and volumes of similar solids. Other illustrations may be drawn from Science, e.g. from experiments on the extension of a loaded spring, and on the variation in volume of a given mass of air under different pressures. The "A" pupils should, later in the course, learn to draw the graphs of related quantities, to read such graphs, to apply numerical and graphical tests for proportionality and to express a proportion by means of a formula. Further reference to this is made in §41(b) below.
(f) Averages. There is no need to define the term "average" for the pupils. The underlying idea can be understood by them after working a
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few examples in connection with such familiar matters as attendances, temperatures, rainfall and ages, weights and heights of pupils in the class. The idea of "average reading" in connection with linear and angular measurements is also important. The length of a classroom, for example, should be measured by several pupils and, after ignoring those readings which are obviously inaccurate, the average reading taken as being a more reliable estimate of the length than any of the individual readings. Frequent use should be made of the idea in outdoor Surveying.
(g) Factors, Square root. It is often necessary in calculations arising in mensuration to determine the square root of a number. The process should be taught as the need arises. The notion of squares and square roots may be given at an early age: when pupils have learnt that "7 X 7 = 49", it is easy to teach them the idea and notation of "7² = 49" and "√49 = 7". Practice should be given in finding the square roots of numbers which are readily resolvable into factors, and in making rough approximations to the square root of numbers by comparison with known squares and by checking through multiplication.
The pupils should also be made familiar with the use of a "table of squares" and a "curve of squares" for estimating both squares and square roots. It will be desirable, with "A" boys at all events, to teach the rule for the extraction of square roots. The rule is easy and may subsequently be illustrated by a diagram.
The notion of cube root and the recognition of the cube roots of the smaller numbers may be dealt with in a way similar to that adopted in the case of square root, but the rule for extraction of cube roots should not be taught. Pupils who have arrived at the stage when
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cube roots of more difficult numbers are required should be trained to use Tables of Logarithms.
41. Further new work in the First Year. Some form of new work apart from that in Arithmetic should be attempted from the outset of the course if the pupils' expectations on entering the Senior School are to be realised. Concurrently with work aiming at consolidating the foundations and with the teaching of fresh rules in Pure Arithmetic should go exercises in Geometry and Mensuration, and in Graphic Representation of Statistics.
(a) Geometry and Mensuration through practical activities: earlier work. The pupils will already have acquired a foundation of geometrical notions and a simple knowledge of shape. This preparatory knowledge is best extended through purposeful activities which give meaning to the work and which stimulate the pupils' interest and arouse in them a feeling of need for further knowledge and skill. Outdoor exercises in Surveying are eminently suitable for this purpose. Practical Drawing in connection with Bookcraft, Woodwork and other forms of Craftwork also provides opportunities for giving the pupils a knowledge of Geometrical facts. By itself, however, Practical Work is not sufficient to develop definite geometrical ideas. The experiences need to be analysed and adequately discussed, and the new facts and principles made explicit. Supplementary concrete problems, bearing on the work in hand, are desirable, and may even be found necessary to assure mastery of the facts.
The following are some types of exercise in Surveying, which will be found suitable in the early stages of Geometry and Mensuration:
(1) Measuring distances in the playground and in the playing-field by means of a tape-measure;
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finding the length of a pupil's pace; (2) estimating lengths of roads, fields, etc.; verification of estimates; (3) drawing plans of classrooms and finding their areas; (4) drawing a plan of the playing-field, using chain and cross-staff, and finding the area in square chains and acres; (5) drawing a simple plan of a winding road using sighting compass and tape-measure; (6) levelling of rising and undulating roads, using an improvised levelling-sight and levelling staff: plotting level sections.
The rules for the mensuration of solids may also be taught through Practical Drawing in connection with Woodwork. The actual handling and measuring of objects by the pupils themselves should form the foundation of the instruction.
There is for girls no one practical subject which takes the place of Woodwork in giving interest and purpose to their course of Mensuration and Geometry. The girls' course will be generally less extensive than that of the boys and will be drawn from a greater variety of fields.
Further reference to the teaching of Geometry is made in the latter part of this section, and details regarding the approach to Practical Drawing appear in §97 and §98 of Senior School Mathematics, under the heading "Mechanical Drawing".
(b) Graphic Representation of Statistics. The graphic representation of Statistics involves such simple ideas that it may profitably be taught to "B" as well as "A" pupils. The method should be introduced at an early stage both because of its intrinsic interest and because of the use to which graphs may be put in the teaching of Geography, Gardening, etc. The treatment suggested here is more exhaustive than is possible in the first year
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or with "B" pupils but is given somewhat fully for the sake of continuity.
The reading of pictorial and bar graphs of school attendances, rainfall, wages, etc., forms an easy approach to the subject, and it is an easy step to the more abstract form of graphic representation, namely, the smooth curve graph. A brief explanation is all that is needed to enable the pupils to supply information contained in bar graphs.
A smooth curve graph may similarly be introduced and the pupils asked to supply information contained in it. Elaborate explanations at this stage tend to cloud the issue and waste time. Statistical graphs relating to temperature readings, barometer readings, and ages and heights or ages and weights of pupils are appropriate for this purpose, but the Geography teacher will be ready to supply others.
The use of interpolation in finding probable values, where one of the observations is missing, can readily be shown by means of such graphs as age-and-weight and age-and-height graphs. Whether or no a comparison should be made at this stage or later between two such graphs as a barometric height graph and a graph relating to the area of circles and their radii, to bring out the validity of interpolation, will depend upon the mathematical ability of the pupil.
The idea of maximum and minimum values will readily follow from such questions as "What is the greatest length?", "What is the shortest length?", when applied to a graph representing the varying lengths of the shadow cast by a vertical stick at different times throughout the year.
The pupils' attention may be directed quite early to the idea of "rate of change". It is convenient to introduce this by examining actual changes in graphs of
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discontinuous quantities, in which the points do not lie on a smooth curve but are joined by straight lines in order to show the changes more clearly. Graphs relating to imports, exports, prices and wages are suitable for this purpose. "In which year or years were the exports greater than those of the preceding year? In which less?" "In which year was the increase or decrease greatest or least?" are examples of questions which will lead the pupils to associate the greater or smaller changes with the corresponding greater or less steepness of the lines joining the points. The general idea of associating rate of change with steepness of a continuous graph will follow easily, if the earliest example chosen shows strongly contrasting slopes at different points and refers to practical matters thoroughly well understood. For example, if a curve of growth shows any marked irregularity of rate of growth, this will generally be associated with the steepness of the curve. The drawing of freehand graphs from dictation will help still further to clarify and enforce the idea. At a later stage it will be possible to extend the idea of "rate of change" to smooth-curve graphs in which the rate of change at any point is measured by the slope of the tangent to the curve at that point.
It is essential that the pupils should come to realise the advantages that graphic representation of statistics possesses over the tabular form. When statistical graphs have been taught, use should be made of them by teachers of other subjects, and the pupils themselves should be encouraged to use the method on all suitable occasions. There is need for a greater variety of graphs than is usually found in schools: the teacher should gradually collect a portfolio of interesting graphs. Practice in reading graphs is even more important than drawing them. The pupils should, therefore, be frequently asked to give a description of the "story"
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contained in the graph, whether it is one which they have constructed for themselves or one derived from some other source.
The drawing of graphic ready-reckoners is also appropriate at this stage, and should deal with such matters as numerical equivalents, rates of exchange, gas and electricity costs, simple interest, Centigrade and Fahrenheit readings and squares and cubes of numbers. With very bright pupils, the construction of graphic ready-reckoners in connection with formulæ might profitably be taken, but the distinction between graphs of statistics and those graphs obtained from formulæ should come at the appropriate stage in the treatment of the formula.
C. LATER STAGES OF THE COURSE
42. Practice in the fundamental operations and problems illustrating them. The groundwork of Pure Arithmetic will normally be covered by the end of the first year even by the "B" pupils, provided that only small numbers and simple problems are involved. During that period efforts will have been concentrated on consolidating the work of the Junior School stage, on extending the fundamental processes to fractions, decimals, etc., and on giving the pupils practice in solving problems dealing with familiar matters and designed to illustrate the use of those processes. No scheme for the "B" pupils can, however, be considered satisfactory if it reduces the needful amount and variety of practical experiences and of easy problems, or sacrifices reasonable facility in computation, in order to cover the whole of the groundwork in one year.
So necessary is this facility and accuracy in computation that during the remainder of the course the pupils should have regular practice in reviewing and mastering 'the Arithmetic of the first year. "B" pupils may need
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more than one period a week for practice of this kind. These lessons should be kept fresh and interesting. A typical period might devote, say, the first ten minutes to brisk oral problem work or to mechanical exercises designed to develop speed and accuracy. The practice should be varied either by keeping the exercises to one topic or, at other times, by introducing exercises from different topics. The remainder of the lesson should be devoted to written work. In selecting examples, the aim in view should not be forgotten. Elaborate, artificial and complex sums are unprofitable; they test perseverance and patience rather than mastery and accuracy, and they often induce discouragement through failure. Interest will be added to the work, if pupils are encouraged to keep graphic records of their performances. In revision work of this kind there is no need to differentiate between the needs of boys and girls.
43. Applied Arithmetic. Selection of Practical Topics. The new material for study in the later years will be grouped round such practical topics as seem likely to be of most value in the domestic, social and economic life of the pupils when they leave school.
Some of the topics will be of interest and of use to both boys and girls, but others to boys or to girls only. Many of the topics may fittingly be studied by the "B" as well as the "A" pupils, but the extent to which a topic is developed will depend upon the capabilities of the pupils. Topics will also be selected that are appropriate to the conditions and environment of the school. For example, where Poultry-keeping is one of the practical activities in a rural school, the mathematics of this subject could well form one of the topics. It will probably be neither possible nor desirable to treat a topic exhaustively in one year: the better plan is to deal with the topics concentrically.
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The following are examples of the kind of "Money Transaction" topics which might be made the basis of exercises:
(1) The Family Income and Expenditure; (2) Purchasing and Planning a House; (3) Family Savings and Investments; (4) A motorcycle - buying, insuring; running costs and records; (5) The Money of the Local Authority; (6) The Money of the Nation.
The method of treatment accorded each topic is just as important as the right selection of topics. If they are used mainly as a means of obtaining further practice in arithmetical operations, they will have little educational value in training the pupils in that type of thinking and quantitative judgment which the topics are largely designed to develop. The aim will be further obscured, if the material is taken ready-made from a textbook. It will, therefore, be important to maintain a proper balance between preparatory work with oral or dictated problems and written exercises.
The preparatory work should usually consist of a general survey of the topic or of the particular situation arising out of it, discussion of the method to be adopted, and the collection of any necessary information or data. Use should sometimes be made of catalogues, daily papers, year books or timetables.
The treatment of the topic "Family Savings and Investments" might take the form of discussion of, and problems on:
(a) The Post Office Savings Bank. Information on how to deposit and withdraw money. Rate of interest and how calculated; (b) Local Banks, Deposit and Current accounts, Interest and Commission; (c) National Savings Certificates; (d) Insurance for children, Endowments at specified
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ages, Old Age Pensions, Insurance against accidents, National Health and Unemployment Insurances; (e) Methods of Hire Purchase, Moneylenders and their charges; (f) Renting or buying a house, Mortgages, Building Societies; (g) Investments suitably treated for young pupils, e.g. a continuous study of the shares of a well-known firm.
The mathematics of Poultry-keeping could be treated under such headings as:
(a) Housing; (b) Foodstuffs; (c) Produce; (d) Accounts.
The treatment of the topic "Housing" might include:
(a) Study of commercial price lists of houses and equipment. Estimating amount and cost of materials for constructing and equipping a homemade poultry house. Comparison of catalogue and own-construction costs; (b) Measuring houses and runs, with simple calculations on areas and capacity, allowing 3 square feet floor space per bird, and 200 birds to the acre; (c) Drawing a plan and elevation of a poultry house. Plan of farm layout: (d) Calculations on capacity of houses and runs relating to intensive, semi-intensive and free range systems of poultry-keeping; (e) Comparison of results from different systems tried in the locality.
44. Geometry and Mensuration in connection with outdoor activities. The general aims of the teaching of Geometry are dealt with at length in Senior School Mathematics.
Outdoor exercises in Surveying and Practical Drawing in connection with Bookcraft, Woodwork and Metalwork afford an excellent means of teaching and applying the principles of Geometry through practical activities.
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Useful supplementary exercises can be drawn from concrete problems in Map-reading, "Buried Treasures", Navigation and Compass Bearings, Latitude and Longitude. Gardening and Housecraft can be used to illustrate the application of geometric principles.
Already in the first year the teacher will have used Surveying as one of the best means of teaching the principles of Geometry in an interesting and a systematic form. It is suitable for boys and girls, for urban as well as rural schools, and many of the very easy exercises are within the powers of the less able pupils.
Unity and coherence are added to such a course if the problems are grouped under practical topics. Chain-surveying, traverse surveying, the measurement of inaccessible heights, surveying by triangulation, levelling and contouring, and the measurement of inaccessible distances have in many schools proved suitable topics. Practically all the necessary instruments can be constructed in the Handicraft room, and some can be improvised from quite simple material.
Within each of the foregoing topics it is possible to devise many practical problems which can be arranged systematically and in order of difficulty. It is not advisable to treat one or more topics exhaustively in any one year of the course. They should be dealt with on the concentric plan, easy problems from each topic appearing in the first year and more difficult ones in each of the succeeding years.
The following is an illustration of the development of a practical topic, but no attempt is made to allocate the problems to particular years. The suggested treatment is somewhat exhaustive, and only in a school where Geometry is a special feature would all the problems be covered in the three years.
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The Measurement of Heights
(i) Accessible objects including flagstaffs, buildings, and telegraph poles.
(1) Finding heights by means of: (a) a shadow stick; (b) a geometric square; (c) an Astrolabe; (d) a 45° set square (various methods); a 60° set square; (e) a plumb-bob clinometer; (f) an optical spirit-level clinometer; (g) a theodolite; (h) a sextant.
(2) Finding heights by means of various forms of direct height-finders.
(ii) Inaccessible objects including chimneys, steeples, etc., where the base-line points towards the object but does not extend up to it.
Finding heights by means of (a) a clinometer; (b) a theodolite; (c) various forms of direct height-finders.
(iii) Inaccessible objects: using any base line that does not point towards the object.
Finding heights by means of (a) angle-meter and clinometer; (b) a theodolite.
Nearly all the problems included in the above list involve exercises in scale drawing, while some lend themselves to numerical and trigonometrical calculations. The latter should be reserved until the pupils have made some advance in the construction and use of algebraic formulæ.
Every problem should be designed either to introduce a new principle or to provide further practice in applying one already known. Before introducing a problem, the teacher should have clearly in mind the principle he intends to teach and should see that the new principle is made explicit before leaving the problem. For example, finding the height of a flagstaff in the playground might be used to teach the fact that similar
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triangles have the ratios of corresponding sides equal. The pupils should come to realise the nature of the problem by having their attention drawn to the shadow cast by the flagstaff. Experiments with three or four shadow sticks of different lengths should follow. They should be first stood vertically in the sun, so that the pupils may observe the varying lengths of the shadows. The sticks should next be arranged in a straight line so that the ends of their shadows coincide. By laying a long stick from the end of the shadow and touching the tops of the sticks the pupils will observe that a number of right-angled triangles are formed of the same shape.
Discussion on similar triangles should follow. By measuring the shadow sticks and their respective shadows, the ratio property of similar triangles can be verified by the pupils. Since the flagstaff and its shadow form the two sides of a right-angled triangle similar to those cast by the shadow sticks, the ratio of the lengths of the flagstaff and its shadow will equal the ratio previously found. The pupils will thus be able to calculate the height of the flagstaff. The new principle will then be expressed in words and committed to memory. Further practice in the application of the principle will then be provided.
45. Geometry and mensuration in connection with indoor crafts. In the construction and decoration of books and in the making of working drawings in connection with Woodwork and Metalwork real problems arise, which may be used to teach many of the elementary principles of Geometry. Most, if not all, of the fundamental constructions, such as bisecting an angle, dividing a straight line into a number of equal parts will frequently occur. The properties of parallel lines, of plane figures, of symmetrical and similar figures, and of circles can well be taught through such work. The making of
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Geometric designs and rhythmic patterns can also be used for the purpose.
In making hand-sketches and ruled drawings the pupils should obtain their own measurements, and should frequently be set to work calculations on the size of the objects and on the amount and cost of the material. Further detailed suggestions appear in §98 and §99 of Senior School Mathematics under the heading of "Mechanical Drawing".
Material for additional exercises in Mensuration are provided in rural areas by such things as stacks, potato clamps, and tree-trunks. Urban schools, especially those in industrial areas, should be furnished with many suitable objects such as metal washers and nuts, small pieces of piping, metal plates and ball-bearings which may be used to provide a wide range of purposeful practical exercises.
When the pupils have made some advance in Practical Drawing, they should be allowed to construct cardboard models of regular solids, and the "A" pupils might attempt more difficult exercises involving sections of regular solids towards the end of the course.
46. The use of algebraic methods and graphic representation in numerical calculations in Arithmetic, Geometry and Science. (a) The construction and use of formulæ. Algebraic methods are best introduced through the construction of formulæ in connection with simple rules in Arithmetic and Mensuration. In this way meaning and purpose are at once given to the methods. They should not be introduced too early or abruptly, as the step of making a generalised verbal statement in Arithmetic is difficult for all but the more able pupils; but once made this statement can readily be changed into the algebraic form. From time to time during the early stages of the Arithmetic courses opportunity will thus be taken in oral lessons to arrive
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at generalised statements and to translate them into algebraic form, until the pupils are thoroughly familiar with the procedure.
The formula should be regarded merely as a piece of shorthand, providing a handy way of writing a rule. It should be first expressed in equational form and the important words should then be replaced by their initial letters, thus:
Area of a rectangle equals length multiplied by breadth:
A = L multiplied by B. = LB.
As the course in Mathematics proceeds, the pupils should always be allowed to translate rules of this kind into the shorthand of simple formulæ. The multiplication, division and index notation will be introduced as the need arises.
Exercises in substitution should follow the construction of formulæ in order to develop speed and accuracy in their use, and from the results of formulæ involving two quantities the pupils will occasionally construct graphic ready reckoners. Practice will follow in reading and interpreting simple formulæ.
Next the pupils should be shown how it is possible by transposing a formula to obtain new formulæ. Thus, by memorising one rule in shorthand form, there is no need to memorise others that may be obtained from it. For example, the formula for the area of a triangle is A = ½ bh. The new formulas to be derived from it are b = 2A/h and h = 2A/b. If these formulæ are stated verbally, two new rules will thus have been derived from the original rule. The step of deriving new formulæ is a difficult one and may only be fully understood by the brighter pupils.
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Most pupils delight in solving "missing number" problems and these can be used to introduce simple equations. They also afford practice in converting verbal statements into algebraic form and in transforming equations.
Factorisation or rearrangement of various simple types of formulæ - chiefly in connection with numerical calculations in Mensuration, such as "A = Ka ± Kb"; "A = Ka² - Kb²"; and "A = (a ± b)²" - can be readily appreciated as a time and labour-saving device of wide application. In each case the enquiry should arise out of a practical problem and be illustrated geometrically.
(b) The formula and its graphic representation. In some Elementary Schools, those, for example, which send on an appreciable proportion of their pupils to Evening Technical Schools of certain types, or retain their pupils for a fourth year, it may be profitable to proceed further with the brighter pupils. There is no need to teach any algebraic processes except in so far as they are actually needed in the solution of applied problems. The introduction of any but the simplest exercises in addition, subtraction, brackets etc. should be avoided. The subsequent work should deal with the formula and its graphic representations.
The distinction between graphs of statistics and graphs obtained from formulæ should be made clear to the pupils and lead them to the idea of related quantities. In order to give the pupils a basis of comparison between different types of functionality, a direct proportion graph and an inverse proportion graph should be compared with a curve of squares and a curve of cubes. Only the two simple types of functionality namely, direct and inverse proportion, however, need be dealt with in detail at this stage, and it is essential that these should be taught through concrete problems. The
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connection between a direct proportion formula and its corresponding graph should follow. This means that pupils should be able not only to draw a graph from its formula but translate a graph into its formula. It will then be possible for them to obtain working formulæ from experimental data obtained in the Science room. Inverse proportion graphs and formulæ should be treated and used in the same way.
The pupils will come to associate a straight line graph through the origin with direct proportion and with formulæ of the type y = kx and a rectangular hyperbola with inverse proportion and with formulæ of the type y = k/x.
The difficulty of recognising an inverse proportion curve is a convincing reason for considering the relation of the inverse proportion formula to the direct proportion formula. In an experiment leading to the law of levers, instead of plotting W (weight) and D (distance) and obtaining a curve which cannot be recognised at a glance, the pupils will plot 1/W and D and obtain a straight line, and from it the formula connecting D and 1/W will be obtained.
Similarly, simple experiments can be introduced to show the relation of the formula for the curve of squares or the curve of cubes to the direct proportion formula.
Another type of formula, e.g. the Simple Interest formula A = IT + P (where "I" is the interest on the Principal for one year, "T" the time in years and "P" the Principal) may be introduced at this stage. Formulae for calculating the cost, including the meter rent, of electricity and gas are familiar instances. The pupils should be able to draw the graph from its formula,
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and also to translate a graph into its formula. Where the study of simple machines forms part of the Mathematics course, practical use may be made of this latter method in obtaining formulæ from experimental data.
The solution of arithmetical puzzles and problems involving two unknowns forms an interesting introduction to the solution of simultaneous equations, leading on to the verification of given laws in connection with simple machines.
Quadratic equations might be approached through the solution of arithmetical puzzles and problems. The subject may be further developed through a number of interesting experiments, such as tracing the path of an oiled ball-bearing projected obliquely up a slightly inclined blackboard or tracing the path of a jet of water starting horizontally from a small hole in the side of a vessel. The graphical test for a parabola should be explained to the pupils and then they will proceed to the discovery and verification of formulæ connecting vertical heights and horizontal displacements obtained from the above experiments.
Where conditions are particularly favourable, the construction and use of a slide rule can be taught. The subject may be approached through the study of the unique property of all Compound Interest graphs, which enables the "Abscissa" axis to be so graduated that numbers may be multiplied and divided, and roots and powers of numbers found from it. It is then an easy step to the construction of two such scales which can be used as a slide rule.
47. Logarithms. Where logarithms are taught they should be introduced sufficiently early, say before the beginning of the third year, so that reasonable facility may be acquired before the end of the course.
Some teachers will probably decide to teach the mechanical use of logarithms. Others may take the
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view that the pupils ought to be convinced of the reasonableness of the methods, even if they are not completely understood. The first step then is to make the pupils realise that the figures in the table of logarithms are merely powers of 10. One way of doing this is to plot such powers of 10 as are easily calculable, and to show that the figures in the table agree with those obtained from the resulting graph. For example, assuming that the pupils are familiar with the methods of multiplying and dividing numbers using the index notation, and that the meaning of an index has been extended to fractional values, then the logarithms of 2, 3 and 7 can be obtained from the statements:
210 = 1024 = 10³ approximately.
34 = 81 = 8 x 10 approximately.
74 = 2401 = 3 x 8 x 10&178;2 approx.
From the logarithms of 2 and 3, those of the remaining integers up to 9 can be found. If the results are suitably plotted, from the resulting smooth curve, the logarithm of any number may be obtained correct to two decimal places and compared with that given in the table. The graph may be used to multiply and divide numbers, and to obtain any root or power, providing that in the case of numbers less than 1 or greater than 10, they are first expressed in "Standard Form". The use of the table should then follow without delay.
48. Mechanics. Where conditions are suitable Mechanics can usefully be taught in the Senior School as the subject provides a good field for experiment, observation and the discovery, interpretation and verification of algebraic laws. Much of the work will be done in the Science room, but the readings obtained from experiments can best be dealt with by the Mathematics teacher.
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Originally Mechanics was the science of mechanical contrivances such as the screw and the block and tackle, which men devised to enable them to handle weights too heavy for their unaided muscles. The study of such contrivances makes a good beginning in a Senior School. It is best to use real machines and to handle fairly heavy loads. Where this is not possible, the pupils should be acquainted with the working of the real machine, and the model used for experiments should resemble it closely in design and be strong enough to withstand fairly large forces. It may well be constructed by the pupil as an exercise in handicraft.
A qualitative study of "how it works" may be followed by plotting efforts and loads on a graph and perhaps by translating the graphs into formulæ. Where Algebra is carried far enough, it will be possible to study the connections between time and distance for a ball rolling down a groove or for a wheel with its pivot on metal runners. The notions of speed at a point and average speed should be dealt with. Simple apparatus can be devised to illustrate the rule that doubling the speed of a car quadruples the stopping distance. The motion of projectiles and the law of the pendulum may be studied by graphical methods.
In rural schools Mechanics is equally valuable, but the subject belongs to rural Science more than to Mathematics. Farm and garden implements, from the spade to the mechanical harvester, afford endless examples of the Simple Machines, either singly or in combination, but the calculations that arise mostly consist of ratios either of lengths or numbers of cogs.
In some schools the notions of thrust and tension in beams and girders may be studied practically and the notion of couples, e.g. in the suspension of a farm gate.
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D. BACKWARD PUPILS AND DULL PUPILS
49. The treatment of pupils who are backward but not dull. Among the "C" pupils will be those whose backwardness has been caused not by any natural dullness of mind but by circumstances beyond their control, such as long absences from school, frequent migrations from one school to another, home conditions and differences of efficiency in the teaching given at the Junior School stage. In their case, the remedy lies in the use of a simplified revision course aiming at securing reasonable facility in simple calculations with small numbers and a fair knowledge of the weights and measures actually used in daily life. They should then be ready to undertake much of the easy work arranged for "B" pupils, but should not be expected to deal with large numbers or difficult problems.
Amongst the backward pupils will often be found those whose failure to learn Arithmetic arises from their disinclination to try. It is for such pupils that freshness of attack and a new appeal to their interests are particularly valuable in dealing with processes already familiar but imperfectly mastered.
50. The treatment of dull pupils. (a) Their limitations. Some pupils are definitely much below the average in arithmetical ability. As a rule the proportion of these dull pupils in any year group is appreciable and the difficulty of educating them is so great compared with the more normal pupils that the adoption of separate schemes and special methods of treatment is necessary. A full realisation of their many limitations is essential before the teacher is able to adopt the right measures.
Dull pupils will be distinguished from their fellows in the "A" and "B" classes by their lack of sustained attention and their inability to concentrate for any length of time. The lessons for the dull pupils must,
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therefore, be shorter than those arranged for the more normal pupils. A lesson should seldom exceed 30 minutes, except for practical work, and even then the period should be broken and may include say a few minutes mental practice or written work in addition to time for individual or group activities. The practical work suggested for Juniors in §23, §25 and §26 of the Board's pamphlet on Senior School Mathematics should be continued. Again, their interest is aroused only through purposeful experiences. Dull children understand processes by observing and using them in concrete situations but have difficulty in realising situations described in words. Furthermore they take longer to grasp a rule, need a longer time to memorise it and must have practice in revision at regular and short intervals in order to retain their knowledge and skill. In oral and written work they can only deal with small numbers.
(b) Attainments on entry. A large proportion of the "C" class will have only reached the mental age of 12 by the time they have arrived at the statutory school-leaving age, while some may scarcely have reached 11. It is most unlikely, therefore, that dull pupils will have covered the suggested minimum syllabus outlined in §16 of Senior School Mathematics. The teacher's problem will be to find out, at the outset, the extent and limitations of the pupil's knowledge through diagnostic tests and close scrutiny of his work. Suitable individual records may be very helpful in meeting the varying needs. Particular defects of knowledge of number facts or "tables" should be met by specific remedies.
Revision work should never consist merely of a dull grind at mechanical exercises. The pupil's interest must be caught and held. A process that needs to be retaught may be approached through purposeful activities and games. Where one method of attack fails,
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the teacher should be ready with another. Nor should all the time be spent in revision. Fresh ground should be broken, for example, through practical exercises in Geometry, where lack of proficiency in the groundwork of Arithmetic is of little account.
(c) The arithmetical needs of dull pupils when they leave school. The most important need of dull pupils is close familiarity with a wide variety of situations and experiences that they will normally meet in their home life. These will often involve only "mental" calculations. The knowledge of rules required in everyday life is not so extensive as is commonly imagined; it is certainly much less than is covered by the ordinary school course. The need for working sums on paper seldom arises, and the range of rules is even more restricted. The minimum syllabus outlined in §16 of Senior School Mathematics includes more than is necessary and could only be covered by the dull pupils by encroaching on time that should be spent on other subjects. By the time the pupils leave school, however, confidence and proficiency should have been acquired in the limited range of calculations likely to be useful and some skill in the types of weighing, measuring and estimating needed in adult life.
(d) General lines of treatment. Dull pupils learn best by "doing" and by dealing with real problems in situations which are familiar and interesting. It follows that the most effective way of teaching is to deal as realistically as possible with the pupils' everyday activities. Daily routine duties which involve calculations may be given to individual pupils, e.g. recording and calculating the cost of the school's supply of milk.
A growing practice is to arrange the scheme as a series of projects, such as the construction of a garden-frame or the planning of a holiday trip, due regard being paid to sequence. Interest is increased if the projects arise spontaneously and the pupils obtain their own data.
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Similarly the work may be arranged under practical topics in connection with the home and various school activities like gardening or poultry-keeping. These will be not unlike the practical topics that have already been suggested for normal pupils, but their interest will be more personal, the treatment simpler and the arithmetical processes involved much more elementary.
In addition to practical topics in Arithmetic, some of the very easy outdoor exercises in Surveying have been found suitable for dull boys. Many opportunities for measuring, sketching, drawing and calculating with small numbers will also arise in connection with Woodwork.
In a girls' school, the general planning of a scheme based on Practical Topics might include:
(1) Keeping the household accounts; (2) The provision of meals; (3) The provision of clothes; (4) Soft furnishings and floor coverings for the home; (5) Wall coverings for the home; (6) Saving money.
Some of the above topics are equally suitable for boys, but in their case more emphasis might be placed on money topics or projects, e.g. keeping the accounts of the school camp, journey or sports.
The method of treatment is highly important; no topic should be used merely as a means of obtaining further practice in arithmetical operations.
The following is given as an illustration of the development of one of the above topics:
Soft furnishing and floor covering for the home:
(a) Planning of tray cloths, simple short curtains, cushion covers.
(b) Curtains with pelmets or frills. Cost of providing them in a four or six-roomed house. Curtains for modern windows and fittings for them.
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(c) Floor coverings. Stair carpets.
(d) Bedding: mattress covers, sheets, pillow cases, quilts, etc.
The material for such topics cannot be drawn from any textbook. The situations will arise either in connection with a housewifery course or with a project for fitting up a classroom or staff room as a sitting or bedroom. The whole purpose of such treatment is to train the children to extract for themselves the arithmetical data from the real situation. Essential facts, such as kinds of material, widths and prices, should be collected from reliable and if possible local sources. Technical questions will affect the difficulty of a problem, e.g. questions of cutting with warp, woof or on the cross, and with allowances for hems, joins or fullness. The teacher can to some extent grade her problems in difficulty and she can analyse them into easy stages for the pupils to tackle one by one. Some of the exercises under (a) in the example given above, for example, will be from one point of view harder than questions on stair carpets, as the latter involve length measures but not choice of width.
E. CONCLUSION
51. Selective Central Schools. No reference has been made in this Handbook to the teaching of Mathematics in Selective Central schools. Much of what has been suggested for "A" pupils is applicable to the pupils of these schools; and any special bias will be reflected in the choice and treatment of material. For pupils in Selective Central Schools as well as for pupils who remain in the non-selective Senior School beyond the age of 14 the topics suggested in the Board's pamphlet Senior School Mathematics will provide ample material for an extended course.
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APPENDIX
THE LEAGUE OF NATIONS
I. HISTORICAL BACKGROUND
To ensure the proper handling of this subject it is desirable, as indeed it is in the treatment of history in general, that the teacher should know a great deal more than he will ever have to teach. But it may be useful here to trace in the history of Europe those forces which have tended to strengthen the idea of international co-operation and to add a few notes on the more important articles of the Covenant and on the action that has been taken under them.
The establishment of the League of Nations was the direct result of the feeling which arose in every country during the Great War, especially in Great Britain and the United States, that, on the restoration of peace, some system or organisation must be set up to prevent the outbreak of a similar war in the future. The British Government appointed a Commission in 1917 to consider this problem, and on January 8, 1918, President Wilson declared publicly that at the peace:
"A general association of nations must be formed under specific covenants for the purpose of affording mutual guarantees of political independence and territorial integrity to great and small States alike."
In the early stages of the Peace Conference a "Covenant" was drawn up, largely based on proposals put forward by the British representatives, and was incorporated as a first chapter in all the Treaties of Peace. The object of the Scheme was, first, to establish a permanent Conference of the nations which should regulate matters of common interest and settle differences
[page 566]
by a regular procedure of consultation, mediation, and arbitration; and, secondly, in the last resort, to ensure joint action between the members of the League against any covenant-breaking State.
But these recent events do not really explain the League. The sense of unity and mutual obligation in the Western World and the idea of common organs for dealing with affairs of general interest are at least as old as the Roman Empire; and Christianity gave new meaning and a wider scope to these conceptions. Hence the attempt during the Middle Ages to express the unity of Western Christendom in the persons of the Emperor as the representative of the Roman tradition of a common secular government and the Pope as the head of the religious community to which all belonged. But the Emperor was a head without a body; his nominal Empire could not provide the peoples of Europe with a political organisation; political unity had to be realised in narrower spheres before it could be expressed in the larger. The growth of national monarchies and national patriotism, with the increasing use of modern languages in place of Latin, while it weakened the traditional sense of European unity, was thus an essential stage in the history of European civilisation.
The development of nationalities tended to substitute war between nations for the conflicts of feudalism, and this tendency was accentuated when the ecclesiastical unity of Europe, already weakened by the struggles between the Papacy and the Empire and almost dissolved by the dissensions of the 15th Century, was shattered by the Reformation. But the nations of Europe, while struggling to organise themselves as independent units and to define their frontiers, were also always seeking to establish a settled European system. The conception towards which they worked was that of balance; no nation was to be powerful enough to dominate Europe.
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The ideas of a "system" or "concert" of Europe and of a "balance of power" were not clearly defined until the 18th and 19th Centuries, but they can be discerned as motives of policy as early as the days of Wolsey's diplomacy. And while statesmen were thus working out methods of common international action by diplomatic missions and conferences, the thinkers of Europe, from Grotius (1583-1645) onwards, were endeavouring to lay down the principles of an international law. These, therefore, are the main tendencies which have to be traced through the confused history of four centuries, viz.:
(i) the development of stable national units;
(ii) international diplomacy and agreements;
(iii) a system of international law.
Progress was slow. In the great treaties of Westphalia (1648), by which the Thirty Years' War was concluded, and of Utrecht (1713), at the end of the wars of Louis XIV, a division of the soil of Europe between governments was established and guaranteed. By this time the organisation and frontiers of the States of Western Europe had become so far defined as to make it possible for Great Britain, France, Holland, Spain and Austria to join in an alliance to maintain peace (1717 to 1720), but Central and Eastern Europe was weak and divided. The wars of the 18th Century, apart from their effects in America and India, were largely concerned with the emergence of Prussia as a new centre of national organisation in Germany and of Russia as a new force in the East. The idea of the balance of power had to be adjusted and extended to deal with these new factors; and the partition of Poland (1772, 1793, 1795), together with the advance of Russia and Austria against the decaying Ottoman Empire showed how gravely the approximate balance hitherto secured in the West might be endangered by developments in the East.
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Another new factor was introduced by the revolutionary and Napoleonic wars. Napoleon's renewed attempt to establish the dominion of a single Power was defeated, as the previous attempts of Charles V, Philip II and Louis XIV had been defeated, by the general revolt of European opinion against it; but the struggles of this period led to a revival of national self-consciousness not only in Germany and in Italy, but also among the smaller nationalities of Europe. When, therefore, after the settlement of 1814-15 statesmen set themselves again to construct the "Concert of Europe", the European family of nations was still in a state of flux. Five wars had to be fought before Germany and Italy emerged as united nations; four great ones before the Balkan nations secured their independence; and it was not until after the Great War that the principle of nationality received something like full recognition in the new map of Europe.
Nevertheless, the "Concert of Europe" made some progress during this period. In its original form, that of the Holy Alliance (1815), it broke down because it developed the tendency to intervene in the internal affairs of other States and became identified with a campaign against liberalism. More practicable was Castlereagh's idea of an alliance between the Great Powers, but from this also, after a series of conferences and congresses, the British Government felt obliged to withdraw, as it tended to become dominated by the policy of the Holy Alliance group. But the practice of summoning conferences to deal with disputes which threatened the general peace survived these failures, e.g., the coalition of England, France and Russia which determined the independence of Greece (1829); the Conference at London which settled the separation of Belgium and Holland (1839); the Congresses of Paris and Berlin and the London Conference which dealt with the problems of Eastern Europe in 1856, 1878 and
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1913, and the Algeciras Conference (1906) which made the Moroccan settlement. These, however, were conferences specially summoned to meet emergencies or to conclude peace; there was no established organisation which could automatically bring a conference into being. Nonetheless the Concert of Europe even in this rudimentary form succeeded in realising some great reforms, such as the suppression of the slave trade; it laid down the lines of a regular diplomatic and consular system for the conduct of ordinary international relations, and it developed for the joint administration of affairs of common interest such important organs as the Danube Commission (1865), the Postal Union (1874), and the Sanitary Union (especially the conventions of 1892 and 1897 dealing with cholera and plague). Here we touch a subject of the greatest importance: the influence of Science in facilitating communication between nations and compelling them to consult and act together to meet common problems. An important development took place in 1884-5 during the Berlin Congress, summoned to deal with African problems, when the States directly interested in the colonisation and partition of Africa combined in adopting agreed measures for the control of the liquor traffic, the slave trade, and the traffic in arms in certain portions of the continent - the so-called "conventional basin" of the Congo - thus recognising the common duty of civilised states in dealing with uncivilised peoples.
Meanwhile, the need for a more effective method of ensuring peace was impressed on men's minds by the growth of naval and military armaments. Suggestions for a mutual agreement for the reduction of armaments had been made by the Czar as early as 1815, and the idea continued to be discussed throughout the century. At the same time the work begun by Grotius had reached a point where the conception of a law of nations had become familiar to all the peoples of Europe. It was
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inevitable, in view of the character of European society, that this law should in the first instance be mainly concerned with the rules of war and that even these rules should be worked out and applied by national courts, e.g., the prize courts of the Napoleonic and American Civil Wars. Nonetheless, by the end of the 19th Century the science of international law had grown greatly in authority and scope, and in many matters its principles seemed to have become definite enough to permit of the establishment of an international tribunal in whose decisions all nations could have confidence. The two ideas of "limitation of armaments" and "international arbitration" led to the Hague Conferences of 1897 and 1907.
These Conferences form an important landmark. Attended as they were by nations from every continent, they constituted a recognition of the fact that, not only Europe, but the civilised world as a whole, was bound together by a community of interests. The actual achievement of these Conferences was, however, small. A permanent Court of Arbitration was established at the Hague; an alternative procedure for settling disputes by means of Courts of enquiry was worked out; and treaties were drawn up laying down more definitely the rules of war. In the then existing condition of Europe nothing could be done for the limitation of armaments. In the language of the 18th Century, the balance of power in Europe was too unstable; in the language of Mazzinian nationalism "the social idea could not be realised before the reorganisation of Europe was effected". The League of Nations is the attempt made by the European family of nations, as it has emerged from the Great War, to realise this balance or this social idea which for centuries different schools of thought have sought to express in different language, and statesmen, diplomatists, lawyers and idealists have worked towards in different ways.
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This historical sketch has inevitably concerned itself mainly with European history, but during the last hundred years Europe has been profoundly affected by the political development of the New World and by the political resurgence of the Asiatic peoples. It would be impossible to deal at length with these factors, but before passing to a study of the League as it exists, reference must be made to what we in this country must regard as the most potent factor of all. Side by side with the growth in Europe of the principle of nationality another principle leading towards looser political organisation was being developed and tested elsewhere. At the end of the 18th Century American statesmen revived the idea of a federal union between sovereign states. The constitution of the United States was an attempt to work out on a grand scale the experiments in federalism made by Switzerland and the United Provinces of the Netherlands. Various forms of federal union had existed from early days in Europe, for instance in Germany and Northern Spain, but for the most part they had not been able to survive the centralising tendencies of European monarchies, and the history of the French Revolution showed that European Republics were no more friendly to the idea. German unity was later established on a federal basis and examples of the same idea are found in the constitutions of the great South American Republics, but the United States remains the best example of this type of political organisation. Under the stress of modern conditions, however, the distinction between these federal unions and what are sometimes called "unitary" states is in many ways tending to become less marked. The British Empire has found itself obliged to attempt an experiment in political organisation on an even grander scale, but on still looser lines. This experiment has developed into the voluntary co-operative union which we know as the British "Commonwealth of Nations".
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The nature of that union has been described in the introductory statement prefixed to the Summary of Proceedings of the Imperial Conference of 1926 (Cmd. 2768), entitled "Status of Great Britain and the Dominions". This is by far the most novel and striking effort of political invention which has been attempted since the Reformation; it is the peculiar contribution which our country has made to the solution of the problem of international co-operation; and a careful study of it is essential to any proper understanding of that problem.
II. THE COVENANT
1. How the League works. The League of Nations, in many of its aspects, developed out of the movement for co-operation between States which had grown with the growth of modern Europe, but comparison of the older machinery for international action with the modern methods of Geneva brings out contrasts as well as affinities. Two innovations in particular are changing and modifying the practice of diplomacy under the League Covenant. One is continuity, the other publicity. The fact that Conferences now take place at regular intervals, and not, as often heretofore, only under the shadow of impending crisis, and that a permanent organisation exists to carry out the decisions so taken, has entirely altered the atmosphere in which co-operation between States is fostered. It has also greatly increased the practical possibilities of producing results. The publicity given to the proceedings of the Council and Assembly profoundly influences decisions, by bringing public opinion to bear upon the points at issue. From another point of view, too, it is important, for it now becomes incumbent on every citizen of a State Member of the League to appreciate intelligently the work done at Geneva.
The value of public discussion depends largely on the educated good sense of the public. League Membership
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carries perhaps some peculiar responsibilities for the British nation, which must be considered by everyone of us who goes to Geneva, literally or figuratively. The League has now more than fifty Member States, and each government represented must explain to the others its own point of view and its own national position. League agreements are made by harmonising national interests, never by suppressing them. Strong national consciousness, strong national governments, are, therefore, necessities for strong international co-operation in the sense envisaged by the Covenant of the League. We in Britain have these essentials. As a people, however, we have a special difficulty of our own, namely, a lack of experience of many problems which pre-occupy continental nations. This part of our island heritage is apt to make us too unconscious of what other countries are feeling and doing. We may take the problem of general limitation and reduction of armaments as a case in point. This question must naturally present itself differently to States with land frontiers and compact territories, and to those with long sea-boards and responsibilities overseas.
The League of Nations works under a written Constitution, the Covenant. This consists of twenty-six Articles embodying the principles on which the League acts, and setting up the main organs through which it acts, namely, the Council, the Assembly, the Permanent Secretariat, certain of its great standing Committees, and the Court of International Justice. Almost all the developments of its activity and its machinery which have been made since its establishment in 1920 derive from these. Thus some familiarity with the Covenant is an absolute essential to any study of the League. Whatever changes it may now seem desirable to make in the working of the League in the light of sixteen years' experience, much that has happened in the intervening years has made the statesmanship displayed
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in drawing up this document even more apparent than it was in 1920. Framing a constitution is a difficult task, even when the conditions to be met are more or less known. The League of Nations was launched in 1920, itself an experiment, on a world of which nothing could be predicted except that it would differ greatly from the world before the Great War. Too much definition of the League's functions might have wrecked the hopes of a League at all at the beginning.
The idea of an undefined Constitution, adapting itself to circumstances as they arise, does not present so much difficulty to us, who have seen our own Commonwealth of British Nations shape itself successfully under those very conditions, as it does to the logical mind of the French and other Latin races. All points of view had to be considered, however, while the final draft of the Covenant was being hammered out during the Paris Peace Conference. Fortunately much thought had been given to it even before the Armistice, in America and Britain especially. This bore fruit, and the Covenant as we have it, is, to quote President Wilson's words, "not a strait-jacket, but a vehicle of life".
2. The Covenant. Only a detailed study of the Covenant in the light of the position at the close of the Great War will reveal the skill and care expended in its preparation. The aims of the League are stated in the Covenant in abstract form in the Preamble.* The first
*(THE PREAMBLE)
THE HIGH CONTRACTING PARTIES
In order to promote international co-operation and to achieve international peace and security
by the acceptance of obligations not to resort to war,
by the prescription of open, just and honourable relations between nations,
by the firm establishment of the understandings of international law as the actual rule of conduct among Governments, and by the maintenance of justice and a scrupulous respect for all .treaty obligations in the dealings of organised peoples with one another,
Agree to this Covenant of the League of Nations.
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seven Articles,* concerned with the machinery through which the League acts, are precise and detailed. But to
*Article 1 [Membership]
* * *
Article 2 [Executive Machinery]
The action of the League under this Covenant shall be effected through the instrumentality of an Assembly and of a Council, with a permanent Secretariat.
Article 3 [Assembly]
The Assembly shall consist of Representatives of the Members of the League.
The Assembly shall meet at stated intervals and from time to time as occasion may require at the Seat of the League or at such other place as may be decided upon.
* * *
Article 4 [Council]
The Council shall consist of Representatives of the Principal Allied and Associated Powers, together with Representatives of four other Members of the League. These four Members of the League shall be selected by the Assembly from time to time in its discretion.
* * *
The Council shall meet from time to time as occasion may require, and at least once a year, at the Seat of the League, or at such other place as may be decided upon.
* * *
[The number of Members of the Council selected by the Assembly was increased to six instead of four, by virtue of a resolution adopted by the Third Assembly, Sept. 25, 1922; and to nine instead of six, by virtue of a resolution adopted by the Seventh Assembly, Sept. 8, 1926.]
* * *
Article 5 [Voting and Procedure]
Except where otherwise expressly provided in this Covenant or by the terms of the present Treaty, decisions at any meeting of the Assembly or of the Council shall require the agreement of all the Members of the League represented at the meeting.
* * *
Article 6 [Secretariat]
The permanent Secretariat shall be established at the Seat of the League. The Secretariat shall comprise a Secretary-General and such secretaries and staff as may be required.
* * *
Article 7 [Seat. Qualifications for Officials. Immunities]
* * *
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a British mind, accustomed to two-Chamber and Cabinet Government and to decisions reached by the votes of a majority, they present difficulties. The precise relations of the Council and the Assembly, the constitution of the Council and the reasons for the allocation of seats to certain Powers deserve further study. Similarly the implications of the article which provides that decisions shall require the agreement of all the members need careful consideration. The League of Nations is not a "Super-State".
Article Eight,* however, which broaches the subject of reduction of armaments, suddenly breaks difficult ground and there is, so to speak, a change of pace in the very wording. The true relationship between the League and its Members is brought out very clearly in this Article. The League will impose nothing without the consent of the individual Governments. It provides
*Article 8 [Reduction of Armaments]
The Members of the League recognise that the maintenance of peace requires the reduction of national armaments to the lowest point consistent with national safety and the enforcement by common action of international obligations.
The Council, taking account of the geographical situation and circumstances of each State, shall formulate plans for such reduction for the consideration and action of the several Governments.
Such plans shall be subject to reconsideration and revision at least every ten years.
After these plans shall have been adopted by the several Governments, the limits of armaments therein fixed shall not be exceeded without the concurrence of the Council.
The Members of the League agree that the manufacture by private enterprise of munitions and implements of war is open to grave objections. The Council shall advise how the evil effects attendant upon such manufacture can be prevented, due regard being had to the necessities of those Members of the League which are not able to manufacture the munitions and implements of war necessary for their safety.
The Members of the League undertake to interchange full and frank information as to the scale of their armaments, their military, naval and air programmes and the condition of such of their industries as are adaptable to war-like purposes.
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for them, however, through the action of the Council, the power of considering their own position as part of the whole.
3. Preservation of Peace. Article Ten,* which pledges League Members to protect each other against external attack, has perhaps caused more misgivings in some States, which have pledged themselves to be loyal League Members, than any other Article, for fear it should drag them into foreign wars against the interest and judgment of their people. It is best considered in connection with the group of Articles† which succeed it which provide peaceful means of settling disputes before war breaks out. The sooner the use of these means becomes universal and habitual, the sooner will Article Ten which provides against cases of aggression become a dead letter.
4. Locarno Treaties. There is one school of thought which believes that the world-wide obligations of the Covenant would not in practice be honoured and that it is better to strengthen the Covenant by means of regional agreements between Powers specially interested in a certain area. We ourselves, under the Locarno Treaties of 1925, undertook a special obligation to lend immediate aid and assistance to France or Belgium if attacked by Germany and vice-versa.
The desirability of regional agreements is still warmly debated. On the one hand, the League's inability to solve satisfactorily the Sino-Japanese dispute of 1931 is used as an argument in favour of regional reinforcement of the Covenant; on the other hand, Germany's denunciation of Locarno in 1936, on the ground that it had been
*†See the "Extracts from the Covenant" given below on page 586.
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invalidated by the Franco-Soviet Pact, shows that the whole question is more complicated than had been imagined.
5. The Collective System in operation. The League has had to deal with a large number of political disputes, some trivial, others very serious. In general, it may be said that the collective system has worked well wherever the circumstances of a dispute have made its full application possible. An example of successful application is the settlement of the Greco-Bulgarian dispute in 1925. On October 19th, 1925, firing broke out between the sentries in a desolate region of the Greco-Bulgarian frontier. Greece assumed that the responsibility was Bulgaria's and invaded her territory in force. Bulgaria appealed to the League by telegram, the appeal reaching Geneva at 6 a.m. on October 23rd. The League Council met in Paris on October 26th and called on Greece and Bulgaria to withdraw their troops. The withdrawal was completed by October 30th. The League sent a Commission to the spot to investigate the origins of the quarrel. The Commission's report was presented to the Council in December. In accordance with its findings, Greece paid an indemnity to Bulgaria, and a Commission with a League chairman was set up on the frontier to prevent the recurrence of such incidents.
On many other occasions, the collective system has proved its value. In 1921, Albania was saved from invasion by Yugoslavia. In 1924 fighting on the frontier between Turkey and Iraq was stopped. In 1932 the League successfully obtained a "complete and final agreement" between Colombia and Peru over what threatened to be a dangerous dispute. As recently as 1934-5 the League scored two important successes. The plebiscite in the Saar, which was to settle the future of that area, threatened to give rise to a very grave dispute between France and Germany, as neither party
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trusted the other to stand aside and allow fair play. An international police force was constituted in pursuance of a resolution of the Council. Detachments of British, Italian, Dutch and Swedish troops, under a British Commander-in-Chief, were moved into the Saar territory, where they were placed at the disposal of the Governing Commission of the territory for the maintenance of order. The plebiscite passed off quietly. About the same time the King of Yugoslavia was assassinated in France and Hungary was accused of having sheltered the terrorists. A very similar situation in 1914 had led to the world war. This time the question was referred to the League. Responsibilities were duly assessed, offenders punished and precautions for the future taken.
On the other hand, experience to date has shown the great difficulty of getting the League States to act resolutely when they do not see their interests immediately affected, or when the offender is a large Power. In the case of Bolivia and Paraguay, fighting dragged on for years, without any resolute action being taken. When Japan invaded Manchuria in 1931 the League sent a Commission to the spot and proposed an equitable settlement, but did not attempt to enforce its recommendations, which thus remained on paper. When Italy invaded Abyssinia in 1935 fifty Members of the League found Italy guilty of violating the Covenant, and some economic sanctions were applied by 50 States, but the effect of the economic measures which the Members of the League were able to agree to apply collectively was not sufficiently rapid to prevent the conquest of Abyssinia, and it proved impossible to arrange for the drastic and effectual sanctions which alone might have saved Abyssinia.
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6. Article 18* provides that all treaties entered into by Members of the League must be registered with the League. Treaties not so registered are invalid. Article 19,† which provides for consideration of possible change in the existing order is designed to supply the necessary elasticity without which the collective system would be dangerously rigid. It is a frequent criticism of the League that this Article has never become operative, and that States desiring change have resorted to force rather than to constitutional methods. But no State has yet taken the opportunity afforded by this Article to bring its case to Geneva, and in consequence the procedure to be followed in applying the Article has never been worked out. One of the problems of the future is how to apply the principle of this Article in order to bring about peaceful change in a given international situation.
7. Mandates. Another outstanding Article is Twenty-two, that interesting experiment in practical idealism, on which the Mandate system is founded. The idea of a Mandate is not entirely new in its moral foundation in the sense of responsibility on the part of the administering government for the interests and welfare of the natives. British Colonial administration has been increasingly animated by the spirit of trusteeship for many years past. Attempts to formulate the idea are also found in some pre-war international Treaties.
*Article 18 [Registration and Publication of all Treaties]
Every treaty or international engagement entered into hereafter by any Member of the League shall be forthwith registered with the Secretariat and shall as soon as possible be published by it. No such treaty or international engagement shall be binding until so registered.
†Article 19 [Review of Treaties]
The Assembly may from time to time advise the reconsideration by Members of the League of treaties which have become inapplicable and the consideration of international conditions whose continuance might endanger the peace of the world.
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The complete novelty in the Covenant plan is the system by which the State holding a Mandate gives an account of its stewardship to the League through yearly reports.
The history of the Mandates themselves, however, is naturally developing somewhat apart from that of the League, since the League supervises, without itself governing, the territories. The record is very variegated, as it must be since it describes the work of several civilised Powers, to one or other of whom is allotted the charge of peoples who differ as much in their stage of culture as do the Arabs and Jews of Palestine from the savages of New Guinea. No more interesting episodes than those connected with Mandates are to be read in the literature that is growing up round League work. One Mandate came to an end when Iraq received her independence and entered the League in 1932.
8. Human Welfare. Article Twenty-three* follows
*Article 23 [Social Activities]
Subject to and in accordance with the provisions of international conventions existing or hereafter to be agreed upon, the Members of the League:
(a) will endeavour to secure and maintain fair and humane conditions of labour for men, women and children, both in their own countries and in all countries to which their commercial and industrial relations extend, and for that purpose will establish and maintain the necessary international organisations;
(b) undertake to secure just treatment of the native inhabitants of territories under their control;
(e) will entrust the League with the general supervision over the execution of agreements with regard to the traffic in women and children, and the traffic in opium and other dangerous drugs;
(d) will entrust the League with the general supervision of the trade in arms and ammunition with the countries in which the control of this traffic is necessary in the common interest;
(e) will make provision to secure and maintain freedom of communications and of transit and equitable treatment for the commerce of all Members of the League. In this connection, special necessities of the regions devastated during the war of 1914-1918 shall be borne in mind;
(f) will endeavour to take steps in matters of international concern for the prevention and control of disease.
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very naturally after Article Twenty-two. To many minds its constructive paragraphs for improving human health and happiness are the most interesting in the Covenant. Certainly the remarkable growth of activity that has sprung from them already seems to show that the times were ripe for them.
Thus the League has made determined efforts to cope with the drug problem. Several inquiries and conferences have been held and two important conventions adopted. One (1925) aims at restricting the manufacture of drugs and controlling and supervising the international trade. The second (1931) is a much more drastic instrument for limiting production. There are also special agreements affecting the Far Eastern countries.
Conventions of 1921 and 1933 aim at suppressing the traffic in women. Much subsequent work has been done in the same field, while a Child Welfare Committee has also attacked many problems. The League has also taken up problems relating to slavery and the slave trade. Temporary Committees on this subject were set up in 1924 and 1931; and an international convention was brought into force in 1926. In 1932 a Permanent Committee of Experts was constituted by the Assembly, on the initiative of the British Government, to advise the League on matters relating to slavery.
Any student of the League curious to understand the technique of its work on the social side should study the details of discussion of these questions. They present a lively picture of the process of harmonising various points of view that go to the make-up of sound international agreements. The zeal of experts, the discretion of governments, the cold criticism of the
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supervisors of League finance all play their necessary parts.
Article Twenty-three has been in a way the starting point from which the League has launched out to meet various kinds of emergency. The repatriation of great numbers of war prisoners from Central Europe, the relief and re-settlement of numbers of war refugees in the Levant, are examples of problems which private charity and even grants from single governments could not cope with alone. These episodes abound in picturesque stories.
The health work of the League has its own appeal to the imagination and is as varied as it is beneficent, ranging from general assistance in the organisation of health services in e.g. Bolivia, China and Greece to patient analysis and pooling of experience about malaria, tuberculosis, cancer, nutrition, infant welfare, etc., and the devising of methods of standardising potent biological remedies such as insulin, salvarsan and the various antitoxins. In these and in other ways the League has made valuable contributions to medical science, which knows no national boundaries.
Health and Welfare work is contributing more than its direct share to the process of binding League Members together by mutual interests, and producing a habit of mind in international relationships* which will in time go far in reducing the strain that arises in times of political crisis.
*Article 24 [International Bureaux]
There shall be placed under the direction of the League all international bureaux already established by general treaties of the parties to such treaties consent.
* * *
*Article 25 [Promotion of Red Cross]
[page 584]
9. European Reconstruction. This is no attempt at a complete account of the League's work. The flexibility of its Constitution has enabled it to undertake some tasks which were quite unforeseen when its Covenant was framed. The most important of these are, perhaps, in the sphere of finance and economics. After the War Austria was on the verge of bankruptcy, and it seemed as if her whole administration must crash, and her people famish, and that a new threat of disorder in the heart of Europe might put the whole peace settlement in the melting pot again. The Austrian Government appealed in despair to the League. From that appeal was evolved the idea of a loan, with its expenditure controlled by a League Commissioner, which has been applied since in Hungary and Greece and Bulgaria under different circumstances.
10. Protection of Minorities. The economist finds so much to interest him in the details of this experiment that it seems absurd to dismiss it with a word. This is even truer of the League's Protection of Minorities, of which likewise no mention is made in the Covenant, which was framed before the series of Treaties that brought Poland, Czecho-Slovakia and the Baltic States into existence and gave new frontiers and new populations to the Old States of Central Europe. These Treaties confer upon every citizen in these countries the right to equal treatment in matters of religion, language and civic rights.
In accepting this responsibility the League put itself in a position where it may be forced to express an opinion on the domestic legislation of States, which in every other sphere it scrupulously avoids doing. The League procedure under which are examined the complaints of minorities in those countries which have accepted minority obligations is, nevertheless in harmony
[page 585]
with the general tendency of modern thought to recognise the value in national life of the traditions and culture of racial minorities.
11. The Permanent Court. In accordance with Article 14 of the Covenant, the League set up the first World Court of Justice at the Hague. By February 1936 the Court has handed down 23 judgments and 27 advisory opinions, many of them of great political importance.
12. The International Labour Organisation was set up in 1919 by Pt. XIII of the Treaty of Versailles on the ground that "the failure of any nation to adopt humane conditions of labour is an obstacle in the way of other nations which desire to improve the conditions in their own countries." The Organisation consists of a general conference of representatives of the members and an International Labour Office controlled by a Governing Body. The conference meets at least once a year and is composed of delegates representing governments, employers and workers. It can determine whether its proposals on any subject shall take the form either of a recommendation for submission to the member states with a view to effect being given to it by legislation or otherwise, or of an international convention for ratification, i.e. for incorporation in the law of any state which decides to ratify. The Governing Body consists of 16 representatives of governments, 8 of employers and 8 of workpeople. It controls the work of the International Labour Office and is responsible for passing the budget and for fixing the agenda of the conference. The International Labour Office prepares the agenda and performs the secretarial work, and also carries out the important function of collecting and distributing information on industrial subjects.
All the members of the League of Nations, with the United States of America, Japan, Brazil and Egypt
[page 586]
in addition, are members of the Organisation. Up to July, 1936, 52 conventions and 47 recommendations had been adopted dealing with such questions as hours of work, the minimum age of admission to employment, the protection of women and young persons, social insurance, workmen's compensation, health, unemployment and the special problems of seamen, dockers, migrants, native labourers, etc.
EXTRACTS FROM THE COVENANT
Article 10 [Guarantees against Aggression]
The Members of the League undertake to respect and preserve as against external aggression the territorial integrity and existing political independence of all Members of the League. In case of any such aggression or in case of any threat or danger of such aggression the Council shall advise upon the means by which this obligation shall be fulfilled.
Article 11 [Action in Case of War or Danger of War]
Any war or threat of war, whether immediately affecting any of the Members of the League or not, is hereby declared a matter of concern to the whole League, and the League shall take any action that may be deemed wise and effectual to safeguard the peace of nations. In case any such emergency should arise the Secretary-General shall on the request of any Member of the League forthwith summon a meeting of the Council.
It is also declared to be the friendly right of each Member of the League to bring to the attention of the Assembly or of the Council any circumstance whatever affecting international relations which threatens to disturb international peace or the good understanding between nations upon which peace depends.
Article 12 [Disputes to be Submitted to Arbitration or Inquiry]
The Members of the League agree that, if there should arise between them any dispute likely to lead to a rupture, they will submit the matter either to arbitration or judicial settlement or to inquiry by the Council and they agree in no case to resort to war until three months after the award by the arbitrators or the judicial decision, or the report by the Council.
In any case under this Article, the award of the arbitrators or the judicial decision shall be made within a reasonable time,
[page 587]
and the report of the Council shall be made within six months after the submission of the dispute.
Article 13 [Arbitration of Disputes]
The Members of the League agree that, whenever any dispute shall arise between them which they recognise to be suitable for submission to arbitration or judicial settlement, and which cannot be satisfactorily settled by diplomacy, they will submit the whole subject-matter to arbitration or judicial settlement.
Disputes as to the interpretation of a treaty, as to any question of international law, as to the existence of any fact which, if established, would constitute a breach of any international obligation, or as to the extent and nature of the reparation to be made for any such breach, are declared to be among those which are generally suitable for submission to arbitration or judicial settlement.
For the consideration of any such dispute, the court to which the case is referred shall be the Permanent Court of International Justice, established in accordance with Article 14, or any tribunal agreed on by the parties to the dispute or stipulated in any convention existing between them.
The Members of the League agree that they will carry out in full good faith any award or decision that may be rendered, and that they will not resort to war against any Member of the League that complies therewith. In the event of any failure to carry out such an award or decision, the Council shall propose what steps should be taken to give effect thereto.
Article 14 [Permanent Court of International Justice]
The Council shall formulate and submit to the Members of the League for adoption plans for the establishment of a Permanent Court of International Justice. The Court shall be competent to hear and determine any dispute of an international character which the parties thereto submit to it. The Court may also give an advisory opinion upon any dispute or question referred to it by the Council or by the Assembly.
Article 15 [Disputes not submitted to Arbitration]
If there should arise between Members of the League any dispute likely to lead to a rupture, which is not submitted to arbitration or judicial settlement in accordance with Article 13, the Members of the League agree that they will submit the matter to the Council. Any party to the dispute may effect such submission by giving notice of the existence of the dispute to the Secretary-General, who will make all necessary arrangements for a full investigation and consideration thereof.
[page 588]
For this purpose the parties to the dispute will communicate to the Secretary-General, as promptly as possible, statements of their case, with all the relevant facts and papers, and the Council may forthwith direct the publication thereof.
The Council shall endeavour to effect a settlement of the dispute and if such efforts are successful, a statement shall be made public giving such facts and explanations regarding the dispute and the terms of settlement thereof as the Council may deem appropriate.
If the dispute is not thus settled, the Council either unanimously or by a majority vote shall make and publish a report containing a statement of the facts of the dispute and the recommendations which are deemed just and proper in regard thereto.
Any Member of the League represented on the Council may make public a statement of the facts of the dispute and of its conclusions regarding the same.
If the report by the Council is unanimously agreed to by the members thereof other than the Representatives of one or more of the parties to the dispute, the Members of the League agree that they will not go to war with any party to the dispute which complies with the recommendations of the report.
If the Council fails to reach a report which is unanimously agreed to by the members thereof, other than the Representatives of one or more of the parties to the dispute, the Members of the League reserve to themselves the right to take such action as they shall consider necessary for the maintenance of right and justice.
If the dispute between the parties is claimed by one of them, and is found by the Council, to arise out of a matter which by international law is solely within the domestic jurisdiction of that party, the Council shall so report, and shall make no recommendation as to its settlement.
The Council may in any case under this Article refer the dispute to the Assembly. The dispute shall be so referred at the request of either. party to the dispute, provided that such request be made within fourteen days after the submission of the dispute to the Council.
In any case referred to the Assembly, all the provisions of this Article and of Article 12 relating to the action and powers of the Council shall apply to the action and powers of the Assembly, provided that a report made by the Assembly, if concurred in by the Representatives of those Members of the League represented on the Council and of a majority of the other Members of the League, exclusive in each case of the Representatives of the parties to the dispute, shall have the same force as a report
[page 589]
by the Council concurred in by all the members thereof other than the Representatives of one or more of the parties to the dispute.
Article 16 ["Sanctions" of the League]
Should any Member of the League resort to war in disregard of its covenants under Article 12, 13, or 15, it shall ipso facto be deemed to have committed an act of war against all other Members of the League, which hereby undertake immediately to subject it to the severance of all trade or financial relations, the prohibition of all intercourse between their nationals and the nationals of the covenant-breaking State, and the prevention of all financial, commercial or personal intercourse between the nationals of the covenant-breaking State and the nationals of any other State, whether a Member of the League or not.
It shall be the duty of the Council in such case to recommend to the several Governments concerned what effective military, naval or air force the Members of the League shall severally contribute to the armed forces to be used to protect the covenants of the League.
The Members of the League agree, further, that they will mutually support one another in the financial and economic measures which are taken under this Article, in order to minimise the loss and inconvenience resulting from the above measures, and that they will mutually support one another in resisting any special measures aimed at one of their number by the covenant-breaking State, and that they will take the necessary steps to afford passage through their territory to the forces of any of the Members of the League which are co-operating to protect the covenants of the League.
Any Member of the League which has violated any covenant of the League may be declared to be no longer a Member of the League by a vote of the Council concurred in by the Representatives of all the other Members of the League represented thereon.
Article 17 [Disputes with Non-Members)
* * *
Article 22 [Mandatories, Control of Colonies and Territories]
To those colonies and territories which as a consequence of the late war have ceased to be under the sovereignty of the States which formerly governed them and which are inhabited by peoples not yet able to stand by themselves under the strenuous conditions of the modern world, there should be applied the principle that the well-being and development of such peoples
[page 590]
form a sacred trust of civilisation and that securities for the performance of this trust should be embodied in this Covenant.
The best method of giving practical effect to this principle is that the tutelage of such peoples should be entrusted to advanced nations who by reason of their resources, their experience or their geographical position can best undertake this responsibility, and who are willing to accept it, and that this tutelage should be exercised by them as Mandatories on behalf of the League.
The character of the mandate must differ according to the stage of the development of the people, the geographical situation of the territory, its economic conditions and other similar circumstances.
Certain communities formerly belonging to the Turkish Empire have reached a stage of development where their existence as independent nations can be provisionally recognised subject to the rendering of administrative advice and assistance by a Mandatory until such time as they are able to stand alone. The wishes of these communities must be a principal consideration in the selection of the Mandatory.
Other peoples, especially those of Central Africa, are at such a stage that the Mandatory must be responsible for the administration of the territory under conditions which will guarantee freedom of conscience and religion, subject only to the maintenance of public order and morals, the prohibition of abuses such as the slave trade, the arms traffic and the liquor traffic, and the prevention of the establishment of fortifications or military and naval bases and of military training of the natives for other than police purposes and the defence of territory, and will also secure equal opportunities for the trade and commerce of other Members of the League.
There are territories, such as South-West Africa and certain of the South Pacific Islands, which, owing to the sparseness of their population, or their small size, or their remoteness from the centres of civilisation, or their geographical contiguity to the territory of the Mandatory, and other circumstances, can be best administered under the laws of the Mandatory as integral portions of its territory, subject to the safeguards above-mentioned in the interests of the indigenous population.
In every case of mandate, the Mandatory shall render to the Council an annual report in reference to the territory committed to its charge.
The degree of authority, control, or administration to be exercised by the Mandatory shall, if not previously agreed upon by the Members of the League, be explicitly defined in each case by the council.
[page 591]
A permanent Commission shall be constituted to receive and examine the annual reports of the Mandatories and to advise the Council on all matters relating to the observance of the mandates.
LIST OF BOOKS ON THE LEAGUE OF NATIONS AND ITS WORK FOR FURTHER STUDY
A. GENERAL
I. (1) Webster, C. K. and Herbert, S. The League of Nations in Theory and Practice. 1933. (Allen and Unwin). 10s. 0d.
(2) Jackson, J. and King-Hall., S. The League Year Book, annual since 1932. (Nicholson and Watson). 10s. 6d. each.
(3) Geneva Institute of International Relations. Problems of Peace, annual since 1927. (Allen and Unwin). 7s. 6d. each.
(4) Stawell, F. M. The Growth of International Thought. 1929. (Thornton Butterworth). 2s. 6d.
(5) York, E. Leagues of Nations, Ancient, Mediaeval and Modern. 1919. (Allen and Unwin). 8s. 6d.
(6) Jones, R. and Shennan, S. S. The League of Nations from Idea to Reality. 1929. (Pitman). 3s. 6d.
(7) Smith, N. C. and Garnett, J. C. Maxwell. The Dawn of World Order, 1932. (Oxford University Press). 3s. 6d.
(8) Rappard, W. E. The Geneva Experiment. 1931. (Oxford University Press). 5s. 0d.
(9) Gibberd, K. The League of Nations, its Successes and Failures. 1936. (Dent). 2s. 6d.
(10) Zimmern, Sir A. The League of Nations and the Rule of Law. 1936. (Macmillan). 12s. 6d.
(11) Stone, J. International Guarantees of Minority Rights, Procedure of the Council of the League of Nations. 1932. (Oxford University Press). 14s. 0d.
II PUBLICATIONS OF THE SECRETARIAT OF THE LEAGUE, GENEVA
(12) Essential Facts about the League of Nations. 6th ed. 1936. (Allen and Unwin). 1s. 0d.
(13) The League from Year to Year. Annual 1926-1934 (Allen and Unwin). 1s. 0d. each.
[page 592]
(14) Ten Years of World Co-operation. 1930. (Allen and Unwin). 10s. 0d.
(15) The Aims, Methods and Activity of the League of Nations. 1936. (Allen and Unwin). 2s. 0d.
III. PUBLICATIONS OF THE LEAGUE OF NATIONS UNION, 15, GROSVENOR CRESCENT, LONDON, S.W.1
(16) The Covenant of the League of Nations. 1935. 1d. and 3d.
(17) The Covenant Explained, by Frederick Whelen, 7th ed. 1935. 1s. 0d.
(18) Organising Peace, by Maxwell Garnett. 9th ed. 1934. 3d.
(19) What the League has done, by M. Fanshawe and C. A. Macartney. 9th ed.,1936. 1s. 0d.
(20) Some Recent General Treaties. Texts of treaties. 1934. 1s. 0d.
(21) Maps. League of Nations wall Map of the World. 40s. 0d. and 45s. 0d.
A new map of Europe. 3s. 0d. and 65s. 0d.
B. PARTICULAR ACTIVITIES
I. (22) Bentwich, N. The Mandates System. 1930. (Longmans, Green). 15s. 0d.
(23) White, F. Mandates. 1936. (Cape). 3s. 6d.
(24) Macartney, C. A. National States and National Minorities. 1934. (Oxford University Press). 18s. 0d.
(25) Alexander, F. From Paris to Locarno and After. 1928. (Dent). 5s. 0d.
(26) Bennett, J. W. Wheeler. The Disarmament Deadlock. 1934. (Routledge). 15s. 0d.
(27) de Madariaga, S. Disarmament. 1929. (Oxford University Press). 15s. 0d.
(28) Shotwell, J. T. War as an Instrument of National Policy. 1929. (Constable). 15s. 0d.
II. PUBLICATIONS OF THE INTERNATIONAL LABOUR OFFICE, GENEVA
(29) The I.L.O.: The First Decade. 1931. (Allen and Unwin). 12s. 6d.
III. PUBLICATIONS OF THE LEAGUE OF NATIONS UNION
(30) Geneva 1935, and annually by F. White. An account of the League Assembly. 1s. 0d.
[page 593]
(31) Peace through Industry, by Oliver Bell. 1934. 3d.
(32) World Labour Problems, an annual account of the International Labour Conference since 1927. 3d. each.
(33) The League and Human Welfare. 1934. 3d.
(34) Minorities. 1930. 4d.
(35) World Disarmament by M. Fanshawe. 1931. 1s. 6d.
(36) What the League has Done. (1920-1936). M. Fanshawe and C. A. Macartney. 1s. 0d.
C. TEACHING METHODS
I. (37) Board of Education. The League of Nations and the Schools. Educational Pamphlet No. 90. (H.M. Stationery Office). 1934. 6d.
(38) Parnell, N. S. Education for Peace. 1934. (National Union of Women Teachers). 6d.
(39) Evans, F. (ed.). The Teaching of Geography in Relation to the World Community. 1934. (Cambridge University Press). 1s. 0d.
II. PUBLICATIONS OF THE SECRETARIAT OF THE LEAGUE, GENEVA
(40) Bulletin of League of Nations Teaching. Annually. (Allen and Unwin). 2s. 0d.
III. PUBLICATIONS OF THE LEAGUE OF NATIONS UNION
(41) Teachers and World Peace. 4th ed. 1935. 6d.
(42) *The Schools of Great Britain and the Peace of the World. 1927. 2d.
(43) *Education and the League of Nations. 1929. 3d.
(44) *Geography Teaching in Relation to World Citizenship. 1934. 4d.
(45) *Modern Language Teaching in Relation to World Citizenship. 1935. 6d.
(46) *Report of the Guildhall Conference on Teaching World Citizenship. 1935. 6d.
(47) *Report of National Conference on Junior Branches. 1936. 9d.
*Prepared by the Education Committee of the League of Nations Union in co-operation with the Associations of Teachers and of Local Education Authorities.
[page 594]
INDEX
This Index is not intended to be exhaustive. Reference should also be made to the Summary of Contents at the head of each Chapter.
The numbers of the pages on which a Summary of Contents is to be found are printed in the Index in heavier type.
In many cases where there are several references to a topic the principal reference is placed first.
A
Accuracy in computation 14, 500, 508
"Activities" and "subjects" 36
Adolescence 128, 129, 140
"Advanced instruction" 122
Aims of education 8 and 15, 23, 41, 42
(Infant School) 85 and 66, 67
(Junior School) 102 and 66, 67
(Senior School) 129 and 66, 67
Algebra 553 and 518
Amenities of school life 28
Apparatus and appliances:
(Gardening) 328 and 270, 271, 272
(Geography) 463, 467, 468
(Mathematics) 502, 550
(Nursery and Infant School) 75, 80, 83, 89
(Physical Training) 164, 169
(Rural activities) 271, 334, 335, 337
(Science) 489, 490, 493, 494
(Woodwork and Metalwork) 269, 270,
Appreciation of:
Art 222, 276, 280
Literature 352, 333
Music 212
Aquaria 473, 477, 494
Arithmetic 499 and 94, 116 144
Art and Craft 219 and 14, 91, 114, 140
Astronomy 434, 495
Athletic sports 167
[page 595]
B
Backward children:
classification and treatment 35, 70, 120, 150
curriculum (English) 378, 395
(Geography) 456, 478
(Mathematics) 527, 560
(Science) 487
Biology 482
Books for school use 47, 267, 293, 367, 369, 386, 405, 408, 464, 493
the uses of 49, 112, 146, 148, 358, 365, 366, 370, 389, 411, 464
Brighter children, needs of 119 and 10, 34, 54, 136
British Commonwealth of Nations 413, 416, 418, 433, 571, 574
Dominions 416, 426, 572
Empire 162, 413, 423, 445, 469, 571
Broadcast lessons (See Wireless)
C
Character training 9, 10, 23, 24, 87, 106, 111, 127, 128, 130
Chemistry 484
Child Guidance Clinics 36, 104
Citizenship 428 and 11, 130
Class teaching, its function 30 and 17, 84, 109, 152
Classification, principles of 31, 84, 109, 149
Clothing for Physical Training 163, 165
Clubs and school societies 29
Commercial subjects 145
Concerts for schools 215
Conferences in groups of reorganised schools 56
staff 55
Cookery 301, 307, 310
Corporal punishment 28
Country dances 206
Countryside, protection of the beauty of 42
Craft (See Art and Craft)
Crafts, traditional 159, 235, 255
Curriculum of the Elementary School 36, 85, 111, 135, 157
its underlying principles 36
main branches 39, 158
needs of boys and girls 43, 128
[page 596]
D
Dancing as part of Physical Training 346
training in Music 205
Design in art and in craft 219, 222, 225, 252, 265, 269
Development, natural order of children's 16, 18, 32, 65, 70
Dialect 377, 393
Dictation, its uses 384
Musical 179, 194
Differences in children's ability 7, 17, 81, 99
Discipline:
at different stages of school life 26, 27, 67, 87, 110, 130
changed conception of 27
its basis 24
its maintenance 25
Domestic subjects (See Housecraft)
Dramatic activity 29, 91, 108, 139, 357, 361, 375, 392
Drawing and painting 219 and 159, 220, 226, 233, 238
E
English Language and Literature 350 and 159, 494
Environment of the home 19
school 23
Examinations, external 21, 31, 32, 118, 123, 154
internal 57, 97, 117, 153
F
Fabric printing 261
Festivals, musical 214
Films, use of in school 51, 431, 468, 480, 492
Folk-dancing 201, 205
Foods and food values 307
Foreign languages 145
G
Games and athletic sports 164, 167, 168
Gardening 314 and 159, 271, 272, 491, 495, 550
"General education" 13, 15
Geography 434 and 14, 115, 118, 138, 142, 160,495
Geology 449, 485
[page 597]
Geometry 518, 542, 549, 552, 553, 562
Girls' special needs in Senior Schools 128
Good manners 132, 133
Grammar 14, 384, 398
Gymnasium 164
Gymnastics 168
H
Habits and skills 13, 15
Hadow principle of reorganisation 6, 33
Handwork 219 and 89, 91, 108, 114, 117, 141, 335, 494, 550
Handwriting 41, 94, 361, 384, 399
Head Teacher's duties 54
Health, training in 161 and 14, 15,39, 73, 77, 86, 104, 139, 158
History 401 and 107, 115, 118, 138, 142, 160, 494
Hobbies 19, 20, 132, 133
Home and school, contact between 19, 20, 78, 133
Home management 302, 310
Home leisure-time occupations 20
Housecraft 295 and 141, 159, 496, 564
"House system" 134
Hygiene (See Health)
I
Imperial Institute 468, 469
Individual differences among children 7, 17, 136
Individual work at various stages 30, 83, 109, 146
Infant School stage 68 and 26, 67
Interest and drudgery 25
Interests, instruction based on 13, 85, 101, 103, 127, 137
J
Junior School stage 100 and 26, 67
K
Knitting 285, 291
[page 598]
L
Land Utilisation Survey 451
Language training 10, 91, 112, 359, 364, 369, 373, 380, 382, 384, 391, 398
Laundry work 301, 309
League of Nations 565 and 418
Learning by heart of poetry 372
Literature 350 and 9, 14, 77, 90, 115, 144, 160, 352
Livestock, the keeping of 314, 332
Logarithms 557
M
Maps 458
Marks, competition for 37, 105
Mathematics 499 and 83, 94, 116, 114, 160, 495, 563
Mechanical aids to teaching 51
(See also Films and Wireless broadcasts)
Mechanical drawing 543
Mechanics 495, 518, 558
Mensuration 542, 549
Metalwork 263, 268
Metric System 538
Modelling 232
Modulator 190, 192
Music 174 and 83, 89, 92, 113, 117, 139
N
National Savings Movement 42
Nature Study and Science 470 and 90, 107, 115, 118, 143, 160
Needlework 281 and 256
Nursery School stage 68
O
Old Scholars' Clubs 135
Open-air schools 163
Orchestras in schools 217
Ordnance Survey maps 460, 476
Organisation of schools internally 29, 84, 108, 149
[page 599]
P
Parents, co-operation with school 11, 21, 133
Part-singing 187
Percussion bands 208
Physical training 161 and 89, 110, 139, 158
Physics 483
Pipe bands and pipe making 210
Play, individual and team games 16
its place in Infant Schools 75, 76, 82
Poetry 114, 370, 390
Posture 166
Pottery 261
Premises 28, 79, 108, 133
Probationary year 57
"Projects" 39, 139,271
Promotion by attainment, by age and by ability 31, 32, 33
from Infant to Junior School 69
Punishment 28
R
Reading 93, 357, 365, 385
"Reception Class" 71
Record cards for individual pupils 63, 97
Religious instruction 8, 36, 39, 95, 157
Rhythmic training 89, 113, 140, 188, 200
Rural England, Council for the Preservation of 42
Rural schools 44, 271
S
Safety First Council 42
School Journeys 150, 469
Science and Nature Study 470
Senior School stage 122 and 27
Singing 174, 180
Skill, development of 13, 15
Solfa, Tonic 190
Special Place Examination 119
Specialisation by teachers 40, 110, 152
Speech training 91, 113, 359, 373, 391, 393
Spelling 383
[page 600]
Staff conferences 55
Staff notation 191
Standardised tests 117
Statistics, graphic representation of 543
Subjects of instruction (See Curriculum)
Surveying 542, 550
Syllabuses, some general principles 53
T
Teams 134
Temperance 57
Tests 98, 117, 153, 438
"Three Rs" 158 and 22, 93, 98, 101, 116, 117, 438
Thrift 42
Time charts in History 432
Timetables, some general principles 52, 95, 96
V
Vivaria 473, 477
Vocabulary, development of children's 356, 368, 391
Vocational employment 135
Voice training 181
W
Weaving 257
Wireless broadcasts 29, 51, 179, 180, 213, 400, 431, 480, 492
Woodwork 262, 264, 270, 494, 563
LONDON
PRINTED AND PUBLISHED BY HIS MAJESTY'S STATIONERY OFFICE
To be purchased directly from H.M. STATIONERY OFFICE at the following addresses:
Adastral House, Kingsway, London, W.C.2; 120 George Street, Edinburgh 2;
26 York Street, Manchester 1; 1 St. Andrew's Crescent, Cardiff;
80 Chichester Street, Belfast:
or through any bookseller
1937
Price 2s. 0d. net
The Government does not accept responsibility for statements from non-Official sources made in the advertisement pages of this publication, and the inclusion of any particular advertisement is no guarantee that the goods advertised therein have received official approval. |
A Change from Textbooks
Textbooks of today, unlike those of the past, are often admirable, but there is still room for sources of information, not devised for school use, which will help the teacher to present his teaching in a lively form related to the environment and the period in which his pupils live.
¶ The physical environment, as illustrated in the geology of the surrounding area has, for instance, been treated in a popular series of Regional Surveys prepared by the Geological Survey. There are separate volumes on London and the Thames Valley, the Wealden District, Bristol and Gloucester District, South-West England, the Welsh borderland, Central England and North Wales. The price is 1s. 6d. each, and a full descriptive list of these handbooks will be supplied on request.
¶ Weather and climate form another aspect of physical environment dealt with in two books simple enough in style to facilitate an easy presentation of the subject by the teacher. These are a Short Course in Elementary Meteorology, price 2s. 6d. and The Weather Map, price 3s.
¶ The local history of districts containing ancient monuments under national care is illustrated by small guides sold at 2d. and 6d. While some are of value for local history, others, for example, the Corbridge Roman Station on the Roman Wall in Northumberland, Edinburgh Castle, Holyroodhouse Palace and Abbey, Richborough Castle in Kent (6d. each) can be utilised in illustrating general history. Each guide contains two or three photographs of the buildings.
¶ Of even greater service to an understanding and appreciation of the past are the three volumes of Regional Guides to Ancient Monuments, neat books bound in
green cloth, with about 20 illustrations and a sketch map of the area dealt with in each. They each contain an historical survey of the area, and short descriptive notes on the principal historical remains. Apart from their use as teaching aids they are ideal books to take on a touring holiday. Volume I deals with North England, Volume II, England South of the Thames, Volume III, East Anglia and Central England; price 1s. each volume. A short list of all guides and other volumes relating to Ancient Monuments is also available free.
¶ The publications prepared by the Ministry of Agriculture include some which will profit the rural school with a school garden, for example, Allotments, price 1s., Collected Leaflets on Birds, price 1s. 6d., which contains in a handy loose-leaf cover some 20 illustrated leaflets describing the habits of common birds of agricultural importance. For the domestic science class, the Bulletin on the Domestic Preservation of Fruits and Vegetables is invaluable. For the geography class there is a Map of the World prepared by the Admiralty, printed in two colours, size 60" x 33" (5s.) which shows diagrammatically the position of British Empire shipping at a selected date in March, 1936, and thus brings out pictorially the concentration of British shipping along the principal trade routes of the world.
¶ Lastly there are the publications prepared by the Board of Education for the direct guidance of teachers. Some of them have been mentioned already in the text of this book. A full list will be supplied on application.
¶ Overleaf will be found the post free prices of the publications to which reference has been made above. Those listed represent only a few titles from the many which will interest the teacher either in his professional capacity, or as a citizen interested in this country's affairs. These and any other official publications can be obtained by using the order form included as a loose inset in this volume.
LIST OF BOOKS
mentioned in the two preceding pages
PHYSICAL ENVIRONMENT
Regional Geological Surveys -
London and Thames Valley 1s 8d
The Wealden District 1s 8d
Bristol and Gloucester District 1s 9d
South-West England 1s 8d
The Welsh Borderland 1s 8d
The Central England District 1s 9d
North Wales 1s 9d
WEATHER AND CLIMATE
A short Course in Elementary Meteorology 2s 9d
The Weather Map 3s 2d
GENERAL HISTORY
Guides (H.M. Office of Works) -
Corbridge Roman Station 7d
Edinburgh Castle 7d
Holyroodhouse Palace and Abbey 8d
Richborough Castle 7d
Illustrated Regional Guides to Ancient Monuments -
Vol. I. Northern England 1s 2d
Vol. II. England, South of the Thames 1s 3d
Vol. III. East Anglia and Midlands 1s 2d
OF INTEREST TO RURAL SCHOOLS
Allotments (Ministry of Agriculture Bulletin No. 90) 1s 2d
Birds (Ministry of Agriculture Collected Leaflets No. 5) 1s 8d
Domestic Preservation of Fruit and Vegetables (Ministry of Agriculture Bulletin No. 21). Art Paper Cover 1s 2d. Quarter Bound 1s 9d
GEOGRAPHY
A Map - Geographical Distribution of British Empire Shipping, 1936 (Admiralty) B.R. 84 5s 3d
Obtainable from
HIS MAJESTY'S STATIONERY OFFICE
LONDON, W.C.2: Adastral House, Kingsway;
EDINBURGH 2: 120, George Street; MANCHESTER 1: 26, York Street;
CARDIFF: 1 St. Andrew's Crescent; BELFAST: 80, Chichester Street;
or through any bookseller.
Inspect the display of Government publications when you next visit one of these towns.
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