[page 499]
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
[page 500]
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
[page 501]
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.
[page 502]
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
[page 503]
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
[page 504]
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
[page 505]
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
[page 506]
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
[page 507]
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
[page 508]
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
[page 509]
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
[page 510]
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
[page 511]
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
[page 512]
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.
[page 513]
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,
[page 514]
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
[page 515]
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
[page 516]
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
[page 517]
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
[page 518]
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
[page 519]
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
[page 520]
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
[page 521]
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
[page 522]
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
[page 523]
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
[page 524]
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;
[page 525]
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
[page 526]
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
[page 527]
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
[page 528]
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
[page 529]
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
[page 530]
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.
[page 531]
(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
[page 532]
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.
[page 533]
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.
[page 534]
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.
[page 535]
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.
[page 536]
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
[page 537]
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
[page 538]
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
[page 539]
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
[page 540]
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
[page 541]
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
[page 542]
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;
[page 543]
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
[page 544]
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
[page 545]
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"
[page 546]
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