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PRACTICAL MATHEMATICS AT PUBLIC

SCHOOLS.

INTRODUCTION.

Changes in recent years. In recent years great changes have been introduced into mathematical teaching. It has been realised that the methods formerly followed, however suitable they might be for the training of "born mathematicians," were not suited to the needs of the average boy; and the teaching has been re-modelled in an earnest endeavour to meet those needs. Although it is too early to pronounce a final opinion upon the results, the success of the venture so far is highly encouraging; the boys enjoy their work and learn with alacrity instead of with reluctance. They may fail to surmount, or even to encounter, some of the more difficult lessons of the older regime, to which importance has been hitherto attached; but it is quite possible that this importance was overrated, and in any case the losses may not be sufficient to set against the undoubted gains in other directions. A brief survey of the situation is rendered possible by the appended reports from four different Public Schools.

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"Born" mathematicians.--Before proceeding to consider what has been done I may recur for a moment to remark made in the first paragraph about the "born mathematician.' A Congress of Mathematicians will naturally be anxious for his welfare, and alert against any short-sighted sacrificing of his interests and I hope I may claim to share this anxiety, as one who owes everything to the care lavished on boys who showed mathematical promise under what may be called the old regime. But I say with some confidence that the new regime is not likely to differ from the old in this respect. I have not seen the slightest tendency to relax vigilance in looking for mathematicians or care in training them when found. On the contrary, I firmly believe that there is on the modern plan an increased chance of finding the boys with special ability, since the net is spread much wider than before. Before he is recognised as such, a boy may possibly learn his Mathematics in a manner which the older school might think slipshod; he might, for instance, learn his geometry without the thoroughness of Euclid's demonstrations; but this stage is, from the nature of the case, not likely to last long, and when the boy of ability is once detected he can be, and is, treated specially. In my opinion the great gains to the average boy have not been accompanied by any loss to the specialist worth mentioning.

Leading idea of new Methods.-The leading idea which runs through the new methods is stated by Mr. Brewster of Oundle School thus:

"It is found that a boy will understand and remember a mathematical process with much greater ease if it is introduced to him in connection with a problem of practical value."

So long as our attention is fixed on this principle alone we should probably all accept it as reasonable enough; it is only when it comes into contact with other principles that objections arise. The main conflict is with the principle of logical order. Euclid presented his geometrical propositions, not as they were arrived at historically, nor as they might arise from practical problems, but in an arrangement which aimed at logical sequence. And he has been followed in this aim by the great majority of subsequent writers of text books on Mathematics. To adopt the principle above enunciated was therefore to throw over the existing methods in some degree. How far should they be abandoned? The answers have varied greatly, straggling the whole distance from complete adhesion to complete abandonment. Those who prefer to keep as close as possible to the old methods have made a minimum concession by introducing a few models and a little squared paper into their class rooms; the thorough-going reformers have boldly left the class-room altogether and installed themselves in the workshop. "An attempt is made to look upon the class as a staff of workmen," writes Mr. Sanderson of Oundle, "actually engaged in some 'live' work Mathematics come

in incidentally and are learnt as need arises. The various mathematical principles and methods are thus acquired indirectly by continually applying them, and the boy learns his elements of Mathematics in much the same way that he learns to walk. At a later stage he may put his knowledge into a logical framework." In other reports it will be found that less drastic measures have been adopted. Special mathematical laboratories have been organised, where Mathematics are learnt practically by weighing and measuring. Or without setting apart special rooms for this purpose, physical laboratories have been utilised for infusing practical knowledge into the theoretical; or, as above remarked, models and apparatus have been used in the class-room, with but little departure from the old traditions.

Keen interest of the boys.-But in whatever degree practical work has been introduced, it seems to have been successful in arousing the interest of the boys and stimulating the teaching. It would almost seem as though those who had gone furthest in this direction wrote with the greatest conviction; but it would be rash to draw the obvious inference, or at any rate to act upon it at present. Rather may we welcome the probability of a period of varied experiment and comparison of results. Such periods may involve strenuous exertion, but the exertion is likely to be of a stimulating and not necessarily of an exhausting

kind. That at any rate is the impression produced by my experience of the keenness of discussion in the Mathematical Association, and among the Science Masters whom I have met in connection with it.

Meanwhile we may note with satisfaction the awakening of interest which seems to characterise the new methods through"This work is the boy's first experience of experimental science," writes Mr. Bell of Winchester, "and he enjoys it

He makes a good deal of noise over it, and a master with nerves or a classical division in the room below-may be worried by the dropping of weights and water; but the boy is keen on the work, and there is no difficulty about discipline." It is surely significant that the discipline should be good even in the presence of noise, for nothing is more conducive to disturbance.

The interest shown crops out at various points. Mr. Siddons of Harrow refers to the increased interest in Arithmetic, i.e., in getting sums actually right instead of merely showing knowledge of method. As one who has lectured on Astronomy for nearly 20 years, I can well appreciate the point. Mathematicians, especially those of considerable ability, have often been curiously unable to get a correct numerical result, chiefly through want of attention. It has been my custom to draw the attention of my classes to the importance of correcting this habit by the example of Nautical Astronomy. "If you get your sum wrong, you may sink the ship." If interest is aroused in the result for its own sake, there is little difficulty in getting the sums correctly done.

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Mr. Brewster of Oundle mentions yet another point. The secondary aim is to foster an interest in the higher parts of Mathematics at the earliest stage possible." That this aim can even be stated complacently says much for the new methods. Older generations fixed the limit of their interest in Mathematics at a very short distance from zero.

Co-operation. It will be seen from the reports that the new system is distinguished from the old by an extensive use of the principle of co-operation. Boys like working together, and in former times this desire has been an inconvenience to be repressed for it led to "copying," or to one boy doing the work for others. But in the modern system co-operation has positive advantages. They appear most prominently in the Oundle report where the whole plan of work depends on co-operation. "În the workshop method of teaching, the boys are regarded as forming a body of workmen engaged in carrying out some of the regular work of the shop. Each boy is a member of the staff and has his definite part of the work to do-and he finds that whatever this work may be it is of importance to the whole

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In mathematical laboratories there is not of course so complete a co-operative scheme, but the boys work in pairs or

threes. "There is one obvious disadvantage in working in pairs," writes Mr. Siddons, "the possibility of a pair being very unequal in ability, so that the clever boy is kept back and the weak boy. scores marks for work mostly done by his partner; to avoid such cases occurring it is sometimes necessary to rearrange pairs. But this possible danger, which is easily obviated, is iminaterial compared with the advantages gained by the boys in discussing the experiments together and frequently succeeding in doing an experiment which neither could do alone. Of course all calculations are done by each partner and the results compared," another valuable lesson in the proper method of doing calculations, well known to astronomers.

Wholesale co-operation need not, however, be confined to the workshop. In the laboratory it has been found of great service in the construction of models, which would be far too serious a labour for an individual, but becomes easily manageable for a number working in concert. If, for instance, each boy constructs a plane section of some solid figure, when the whole is built up the excellence or defect of individual sections will teach the lesson (above mentioned in connection with the workshop) of the importance of each definite piece of work to the whole.

Imagination.--With practical work it is much easier to set a boy to think out a thing for himself. Teachers have long realised the difficulty of providing problems which call for spontaneity and not merely for the remembering of some rule. There were, for instance, Euclid "riders," but the good ones soon got tabulated and new ones were either too hard or too easy. In practical work it is only necessary to vary the dimensions of some of the apparatus in an experiment to call for new ingenuity from a boy who may have just done almost the same experiment.

Expenditure of Time.-In one point the new methods compare, at any rate in the first instance, unfavourably with the old. Progress is not so rapid, or at any rate not so obvious. It will be seen from the reports, however, that the advocates of practical methods claim compensating advantages in sureness and reality.

"It is true that the number of mathematical questions or examples [a boy] will get through in the hour or two hours 'lesson may be very few, but the questions arising in a concrete and visible form will make a more vivid and abiding impression on the mind, and both the methods and results will become an integral part of his being. On the other hand, although a boy will be able to work through many more examples and problems in a class-room, yet it may safely be asserted that the impression left by each one is comparatively feeble (Oundle report.)

This characteristic of practical work-that it takes more time is indeed well known. Against it are to be set, not only the considerations of thoroughness above mentioned, but the possibility of according actually more time for work. The

number of hours spent by a boy in the class-room at our public schools has hitherto been rather small, chiefly because the work was of an exhausting kind to both boys and masters. Practical work need not tire the boys in the same way, especially when they are keenly interested. It is in many respects little different from the carpentering and similar work with which some boys were and are accustomed to employ part of their leisure. Hence we may contemplate without anxiety a possible extension of school hours into those hitherto given up to mere amusement.

Examinations.-Allied to the characteristic just mentioned is the objection that practical work cannot readily be tested by examinations. "Results as tested by an examination may be disappointing," writes Mr. Bell. "It is hard to devise an examination which is a fair test, but this does not condemn the work. The boy understands what he is doing, as shown by the fact that he can make use of previous experiments, and make correct deductions from them; but it needs a much higher standard of training before he can show an examiner that he understands.'

This difficulty of examinations is met with elsewhere. To take a very different instance which happens to have come within my experience :-The working classes have recently appealed to Oxford and other Universities for Higher Education. The appeal was not made by large numbers, but on behalf of a very small percentage indeed, who are of special earnestness and (as the event proved) of special ability. Looking back on what has happened, we might perhaps have foreseen that when twenty people out of (say) 20,000 presented such an appeal, they would be men out of the common run: but the ability shown certainly took us by surprise. After only a few months' training (by a capable teacher) these men were able to produce essays of sterling merit--comparing favourably with essays written by first-class men at Oxford. But they were, and are, almost wholly unable to face an examination; it would be difficult to devise an examination which would bring out their merits in any adequate degree. To give another instance somewhat nearer the matter which concerns us :-I have been accustomed for many years to examine the students who have attended University Extension lectures in Astronomy, and am familiar with what may be expected from them. One of the lecturers has recently taken a rather new departure by encouraging his students to do practical observing for themselves at their homes instead of taking their facts from him or from books-to note the positions of the stars, and the variations in the height of the tides and trace them to their causes. I have been able to satisfy myself in a variety of ways that his students have thoroughly appreciated the new departure and have gained much from it: some of them have even gone on to do astronomical work of real value. But the examination results are disappointing. It seems difficult to make sufficient allowance for the time taken to work out a problem of

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