this defect to give the fullest and clearest meaning to the various mathematical operations carried out. The rationalisation of the arithmetic so as to prevent it from ever degenerating into unintelligent mechanical work is the very soul and secret of the practical arithmetic movement. And if this end is to be achieved, the practical arithmetic must come first and the mechanical drill come afterwards. With these principles in mind, we will proceed to estimate the value of the practical arithmetic instruction given in the schools. In every school some practical arithmetic is done; in some schools nearly all the arithmetic is practical. In a few schools it is well done; in a few others it is badly done; in the bulk of the schools it is indifferently done, but is improving. The gravest defect is the relegation of the practical arithmetic to the end of the school year. The mechanical and theoretical part is done first, and then-if there is time the practical work is attacked. This reversal of the true order of things is partly due to the fact that certain arithmetic books which are almost universally used in the schools provide a large number of the older style of examples, and append merely a few pages of practical work as a sort of grudging concession to newer views; and partly due to the opinion still tenaciously held by a few teachers that mechanical drill should take precedence over every other form of mathematical training. It should come first in order of time, and have the biggest share of practice. They call it dividing the difficulties and presenting them one at a time.

The second fault I have to find is that the practical work is often done by the wrong person. Let us consider an extreme case. The teacher is about to give a class of 60 boys in an ordinary class-room a lesson in weighing. He has on the table in front of him a balance and a set of weights. It is obvious that only one at a time can use the balance; and he elects that he shall be that one, or he lets about half-a-dozen boys do the weighing one after the other, while the rest of the class look on. A boy who watches another boy weighing finds no more pleasure or profit in the business than he does in watching another boy eat an apple. The difficulty of the case under consideration will readily be conceded, but it should not be met by so wasteful an expenditure of the time of the class. A better plan would be to provide each boy with a written problem or set of problems to be solved by weighing, and allow each boy in turn to get out his own results, while the rest of the class is engaged in drawing, silent reading, or ordinary arithmetic. A better plan still may be suggested. It would probably be found possible to furnish each desk with a cheap balance quite delicate enough for practical purposes. The upright could be permanently attached to the desk and rendered collapsible, the crossbeam and scales being detachable. An arrangement of this kind would enable all the class to do the practical work at the same time.

Another undesirable aspect of much of the practical work is its purposelessness. Weighing and measuring are essential features of practical arithmetic, but they are not the only features. Simple exercises in pure weighing or measuring are both allowable and desirable at the beginning of a course when it is necessary to make the children familiar with the instruments and the units; but these exercises soon become extremely tedious. Make the weighing or the measuring, however, subservient to some end in which the child is interested, such as the solution of a problem or the construction of an object, and the boredom immediately disappears. The simple processes should in fact form part of a larger purpose within the grasp of the pupil. This subservience of processes to purpose seems to me to be vitally essential to all sound education.

The part of the arithmetic course that seems at present to stand in greatest need of diagrammatic illustration is the theory of fractions. Children of 12 years of age frequently fail to show by the division of a line or rectangle the validity of such simple relations as 2/3 = 4/6 and 1/2 + 1/3 = 5/6. Successful attempts to deal diagrammatically with the division of fractions are extremely rare.

The desirability of reference to ultimate sensible data is still more apparent in dealing with decimal fractions. Whether it is because decimals are introduced rather late in the course, or whether it is that the subject presents inherent difficulties to the learner, the fact remains that the pupils seem to feel much more at home with vulgar fractions than with decimal fractions. They cannot think in decimals: they have to convert them into vulgar fractions before they can grasp their full significance. I have frequently tested the children in the top class orally with the following question: Which is the larger, 7 or 72? The answer I almost invariably get is 7; and the reason I get is that the first is tenths, and the second is hundredths, and tenths are larger than hundredths. The notion that 72 is seven tenths, plus a little bit more, does not seem to occur to them. There are, however, exceptions. In a few schools I have had questions of this kind answered promptly and correctly; and in all these cases it turns out that the pupils have had much practice in measuring with decimal scales. In the written exercises children frequently convert decimal fractions into vulgar, even when it would be much more convenient to deal with them in their original form; but they never seem to reverse the process. Even if the school is well provided with rulers divided decimally, the children should themselves construct such rulers of paper or cardboard and use them frequently.

A movement which has for its aim a closer correlation between mathematics and manual training is rapidly spreading. Conferences take place between the instructor at the woodwork centre and the head teachers who send boys to that centre.

They draw up a scheme which will benefit both parties. The woodwork instructor prepares models for use in the practical arithmetic lesson in the school, and the mathematical course is so devised as to facilitate and render more intelligible the instruction given at the woodwork centre.

The tendency towards practical measurement has promoted the development of the mathematical aspect of art, science, and geography. The official science syllabus issued by the London County Council as a suggested scheme of practical work consists mainly of exercises in measuring and weighing. It is in fact a course of practical and applied mathematics.

In a few schools the map studied during the geography lesson is made the basis of calculations respecting distances, areas, and the time and cost of travelling from one town to another.

In a large number of schools graphs representing the daily changes in temperature and barometric pressure are kept up to date by monitors and hung upon the walls of nearly every class-room. Everywhere, in fact, we find traces of the invasion of practical arithmetic.

Theoretical side of the Instruction in Arithmetic.

Some weaknesses in the more purely theoretical side of the instruction call for comment. My experience leads me to the conviction that the theory of proportion is very inefficiently taught. The symbols between the terms remain cryptic, the abstract nature of a ratio is not grasped, and the notion of equality is not utilised. A large number of children tested could not build up a complete proportion. There would, I think, be a considerable gain in clearness if the older symbols were discarded, and the proportion set forth in the form

a

с

a ÷ b = c ÷ d; or, better still, = d. To find a missing term would then be to solve a simple equation. Fractions and the simple equation known, the principles of proportion become an easy inference. Attempts should be made to connect the notion of ratio and proportion with similar geometrical figures.

It has been customary to postpone the study of averages till the last year of the school course (see Appendix I.). The nature of the modern mathematical programme renders this postponement undesirable. There is nothing intrinsically difficult in the notion of average; children in the lower part of the school can understand it; and it is extremely useful in lending greater variety and accuracy to the problems involving measurement. Where exact measurement is desirable, it is always better to take several measurements and strike an average. In finding the length of one's stride or the thickness of a piece of paper, the advantages of measuring several at once and finding the average size of one will be apparent even to pupils in the lower classes.

There are two distinct ideals which make a claim upon the teacher of Arithmetic. The one is speed and accuracy in

calculation, the other is intelligence in the solution of problems. The ideals are not by any means incompatible. It is possible to secure both, but they compete for possession of the limited time at the disposal of the teacher. To adjust the claims of these ideals has always been a difficult problem in the school; and I think we have come nearer than ever before to striking a reasonable balance between them. There is, and I presume always will be, a considerable difference between school and school in the relative importance attached to these two aims. The general tendency during the last two years has been to increase the intelligent work at the expense, if need be, of the mechanical accuracy. The children must understand what they are doing, even if they fumble at it. There are a few teachers who still think that "rules" must be learnt first and applied afterwards; that it is no use, for example, to set problems involving long division until the children can work quite difficult examples with facility; that, in fact, the greater part of the year should be spent in working abstract examples and the tail end devoted to problems. The one thing to be said about excessive drill is that it deadens interest; and to say this, is to condemn it root and branch. Even when the purport of the exercises is understood, they are to most children very tedious. When this purport is not understood they become the veriest drudgery. Facility should first be fostered by practice in the solution of simple problems. When the meaning and application of the various operations are thoroughly understood, there is no harm done by giving practice in these operations to secure rapidity and accuracy. But there are limits. It is possible to make too great a sacrifice in securing this result.

There is a prevailing opinion that the London elementary school children of to-day are slower and less accurate in computation than they were ten years ago. I have searched for evidence in support of this contention, but have failed to find it. I am, therefore, inclined to relegate the belief to that group of opinions which have reference to the annual deterioration of Academy pictures, the increasing degeneracy of each new generation of men, and other palpable fictions. But even if there has been a slight loss of accuracy, there has been a great gain in intelligence; and intelligence is an equipment incomparably more valuable than facility in calculation.

Progress has taken place, too, in the ability to explain the various operations and rules in the understanding of the theory of pure number. It has been contended that mechanical processes should not be used until the reason for the processes has been thoroughly understood. This is a counsel of perfection. Although such a standard has probably never been completely attained in any school, it will readily be conceded that no process should be used until some meaning is attached

*There is, however, abundant evidence that the arithmetic is not so accurate as it was 20 years ago, when much more time was devoted to the subject.

to it. A child should, for instance, not be set to work a subtraction sum on paper unless he can state it in the form of a concrete problem and give a reasonable interpretation to the result. He must have some idea of what he is aiming at; but if the teacher waits until every child in the class understands the rationale of what used to be called "borrowing" in subtraction before the device is practised, progress will be indefinitely blocked. As a matter of fact some attempt is made in the schools to render the process intelligible; and the reasoning seems to be understood by the brighter children. The question of method in teaching subtraction is sometimes a source of contention between the head teachers of the Senior Departments and the head teacher of the Infants' Department, for it is manifestly desirable that the same method should be adopted throughout the school. The head teacher of the Infants' Department finds the method of decomposition the easier to explain; and the head teachers of the Senior Departments find the method of equal addition the easier for practical calculation.

While there is much excuse for neglecting to make clear the reason for all mechanical operations in the lowest classes, there is no excuse for it when dealing with the older children. As I have already pointed out, fractions and proportions are not, as a rule, well understood; neither are L.C.M. and H.C.F. rationally taught. The difficulty with L.C.M. and H.C.F. is that the more convenient methods are the most difficult to understand. The same is true of square and cube root. In many cases the teachers get over the difficulty by using factors as an intelligible process, and teaching the ordinary methods as frankly rule-ofthumb. Square root is usually required in the application of Euclid I. 47 to mensuration, and here an approximate result may be obtained by drawing to scale and measuring. This method was found largely used in one school. If lines or bits of string were used to illustrate the method of repeated division in finding the H.C.F. the reasoning would not, I think, be too difficult for a bright lad of 12. Much may be said, however, for entirely omitting this method from the elementary syllabus. Its usefulness is virtually limited to operations with vulgar fractions of large numnerators and denominators. But these fractions have in actual life been superseded by decimals. To find the H.C.F. of big numbers is, in fact, waste of time.

It is lamentable to find, as one occasionally does, the top class in the school spending a whole year on a syllabus confined almost exclusively to percentages, averages, discount, and stocks. From the point of view neither of general nor of vocational training can such a programme be defended. The pupils waste their time (and very precious time, too) in applying the principle of proportion, which they do not understand, to commercial transactions which they will never carry out. Fortunately cases of this kind are rapidly disappearing, and there is

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