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256 The Teaching of Mathematics in Preparatory Schools,

but if it is borne in mind that every proposition successfully grasped is a step on the ladder, and that no steps are of any use at the top if intermediate steps are unsound, real progress will be made, even though slow and steady.

The enunciations and corollaries should always be thoroughly learned by heart, and clearly understood, for it is these that constitute the directions for the way, so to speak, and they are besides the only parts of the bookwork that are quoted in subsequent propositions.

The amount of Euclid that can be learned up to the limit of preparatory age depends entirely on the individual. With the majority three books form an amount that can generally be managed, while the more mathematically disposed will add the fourth and sixth books without much trouble in the same time. It is a good practice to work the greater part of the fourth book as problems, and this is certainly true of the first half of the book. A knowledge of the opening propositions of the sixth book gives a geometrical interpretation of ordinary proportion to which, in analytical form, by this time the boy is well accustomed in both arithmetic and algebra. Two lessons weekly of 50 minutes or one hour will be ample for ensuring a good knowledge. In this will be included preparation by beginners, but in the case of more advanced boys, an allowance of fifteen minutes twice a week in preparation will be of great value. Propositions should be written out neatly, all references put in the right hand margin, and the wording of the text insisted upon.

It is not advisable to attempt the solution of complicated problems; easy work alone is suited to minds of this age. In this way it is manifest that nothing is to be gained by unduly limiting, as some would limit, the number of books to be read, for even though the time that would ensure a good elementary knowledge of six books be devoted to three books only, there remains the incontestable fact that problems are limited to those suitable to the age. Too difficult problems defeat their object; and in a recent scholarship examination, where the candidates, who expected, as usual, six books, were confined to three without notice, the differentiation of the better mathematicians was almost defeated by the fact that the increased difficulty of the problems tended to reduce all to the same level.

CONCLUSION.

To sum up, quality rather than quantity is the essential of good teaching. The aim should be to develop thinking power, and this is best attained by careful explanation being followed by plenty of practical examples, varied as much as the ingenuity of the teacher will permit. Allow no hard and fast rules; let method depend entirely on the interpretation that is to be placed on the definitions; cultivate style, and the result will be the development of a really mathematical mind, as opposed to a memory that is likely to be treacherous in the hour of need.

C. G. ALLUM,

NATURAL SCIENCE IN PREPARATORY SCHOOLS.

Of late years much has been written and said in favour of the more extensive teaching of scientific subjects. And to such an extent has this been the case that some of the advocates of science teaching appear to regard a boy, educated wholly on these lines, and illiterate in every other way, as a desirable product.

But the reaction from this early specialisation is sufficiently strong, in the majority of cases, to counteract the over-zealous advocacy; and in connection with Preparatory Schools there is probably no danger of its occurrence. In their case a more general education is the object, and there is little prospect of a small boy being induced to give so much of his time to science as to interfere with his general education.

In Public Schools the teaching of science has only recently begun to take reasonable shape, and ceased to be a series of fireworks, or isolated physical phenomena, presented in a casual and indigestible manner to the pupil; while there has been so little of it in Preparatory Schools that its past and present state in these institutions does not require any long exposition.

Nevertheless, now that the large number of subjects included under the head of Science are more reasonably taught to elder boys and others, there has arisen a fairly widespread feeling, amongst both parents and schoolmasters, that some elementary information on scientific subjects should be given to boys whilst still at Preparatory Schools, and that these subjects afford valuable material for educating the minds of such boys. To their credit be it said, Board Schools and Girls' Schools have for sometime realised this fact, and in many of them scientific subjects find a place in the curriculum.

In Preparatory Schools the result of this inclination has been that several tentative efforts in scientific instruction have been made, and are still in progress at many of them, though nothing approaching the systematic "nature-study systematic "nature-study" of the young American has as yet been achieved.

The following short account seems to represent the various schemes at present in force, and, as will be seen, they appear to afford possibilities of much success with a slight amount of direction and co-ordination.

The practice which has found most favour is probably the occasional lecture. Either one of the staff or a stranger gives a lecture, with or without lantern slides, on some more or less scientific subject.

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The next place is occupied by Botany of some sort-but, unfortunately, mere Systematic Botany, consisting of the finding and naming of various flowers and weeds, is the rule.

After these two efforts the instruction is of an even more vicarious nature, consisting of scraps tacked on to geography or some other work, ranging from cyclones and thermometers to the distribution of animals. Lastly, in one or two places systematic attempts are made at teaching some given part of Chemical or Physical Science, such as the properties of Air or the Laws of Heat

Now, it will be seen that such attempts as the above, in most cases, are singularly lacking in those essentials which are supposed to constitute good teaching. There is no uniformity, no continuity-in fact, in their nature they too much resemble the "General Information" column of the modern cheap newspaper. And yet some good results have been produced, for these courses have tended to stimulate the mind and improve the reasoning powers of those boys who have had sufficient intellect to select the good from the chaos offered to them. So that for this reason alone one is tempted to consider whether there are not claims for, and advantages in, the teaching of scientific subjects such as to justify their inclusion in the curriculum of Preparatory Schools.

If properly managed, there seems to be little doubt that scientific work tends to truly educate the minds of even quite young boys. Certainly, their powers of manipulation and dexterity are visibly improved by a small amount of practical work entailing the use of their fingers and eyes.

As regards the use of the latter, the difference between a small boy's powers of seeing the features of some given natural object, when he has been taught to use his eyes, and his inability with an untrained eye to see the same things, until they are pointed out to him, is worthy of more than passing notice.

In the same connection this ability to see more leads to a wider range of thought and a greater knowledge of the powers of language for descriptive purposes. Moreover, the powers of reasoning are given fuller play in this manner than in the majority of taught subjects, if it be so arranged that the pupil has to suggest explanations and to arrive at conclusions for himself, subject to the correction of the master.

So that the advocates of this teaching of science would maintain that in the sum the advantages of increased powers of observation and manual dexterity gained from it justify it as a convenient and teachable subject for those ends. This leads to perhaps the most debatable part of the question, viz., the subjects to be taught and the methods to be employed.

Considering the various possibilities in turn, Chemistry, in virtue of its long-standing position, as the subject most taught in Public Schools, naturally suggests itself.

But the teaching of Chemistry involves a considerable amount of apparatus and, a room more or less adapted to the purpose, and it

is far too difficult, in any form in which it has been so far suggested to teach it, for preparatory school divisions.

These objections seem insuperable, and, in addition, in the one or two cases where it has been systematically tried, to the writer's knowledge, it resolved itself into qualitative analysis which, though pretty and instructive to a fair chemist, is an unjustifiable waste of time for a young boy.

Of course, it is possible to draw up a series of chemical experiments, of a more or less "fireworks" nature, which will entertain a small boy, and possibly to a slight extent add to his stock of knowledge-for instance, a series illustrative of breathing, burning, and decay.

But at the same time the probability is that such a course will not, to any true extent, educate a boy without any preliminary knowledge of the subject. It is far more likely to be to him a series of isolated facts, to be learnt like so many grammar rules, than a means of improving and training his powers of reasoning and deduction in the manner which can be effected with other subjects.

Moreover, it is not safe to set small boys to perform chemical experiments for themselves, and it seems, as will be insisted on below, that scientific teaching unaccompanied by individual practical work is not of much value.

Finally, it becomes more evident every year that the study of Chemistry is far more profitable after some elementary knowledge of the physical properties of matter has been gained; and it is certainly far easier to teach Chemistry to boys having some such preliminary knowledge than to those who lack it. On all these counts, then-viz., expense, impossibility of practical work, and advantage of postponing its study, Chemistry seems to be an unsuitable subject.

Physics naturally follow; and if by this term one means the normal courses of Electricity, Light, Sound, etc., then Physics are as useless as Chemistry for the present purpose. But if, on the other hand, one includes under this term instruction in common sense and manipulation, by means of experiments dealing with physical apparatus and phenomena, the case is entirely altered.

Since the development of science teaching for small boys is of comparatively recent origin, it is not amiss to indicate what is intended by the above.

Take such a subject as Heat. As commonly taught from text-books with a view to examinations, it consists of formule of expansibilities, radiation, etc., and the suitability of the subject is not obvious. But limit the instruction to proof of the effects of heat and their application, eg, making of cannon, tyring of cart-wheels, laying of railways, bracing of buildings, bursting of frozen waterpipes, etc., and a thoroughly suitable course, well illustrated by experiments, can with a little trouble be evolved. Or for the higher forms, the uses and making of thermometers, of barometers, proofs of atmospheric pressure, the working of Ꭱ ?

pumps, and so forth, will afford a groundwork which can be built into a course of instruction thoroughly within the grasp of Preparatory School boys.

At the same time a series on these lines can be easily arranged so as not to consist of isolated scraps of information, but of a continuous course in which the pupil depends as much on his own brain as on the information supplied to him. But as in the case of Chemistry these courses may appear only to provide lecture material, though of a suitable kind. So now to come to the most vital point of the whole subject. Whether the matter to be taught be designated Physics or no is of little importance, but the one certain thing is that the work must be chiefly of a practical nature. This result is slowly being achieved in those Public Schools where the science work is under intelligent supervision; and it steadily becomes more evident that even older boys derive but little benefit from a weekly or bi-weekly science lecture unaccompanied by practical work of their own.

However well the lecture may be given, and however well illustrated by experiments, it in no way compares in value to the time spent by boys in doing similar work themselves. One cannot overcome the ingrained habit, acquired from long hours spent in classical work, of regarding a lecture as providing an isolated selection of facts and theories to be remembered.

The above, of course, must not be understood to detract from the value of lectures when accompanied by the pupils' own practical work. In this case lectures afford, if not the only, at all events the most satisfactory method of instilling the theories and amplifying the information bearing on the work in hand; but, comparatively, the lecture alone is far out-distanced in educational value by the combination of lecture and practical work. For the abstract conception of the subject, gained from lecture work alone, is lost when the individual is performing his own experiment, and finds the hundred-and-one small difficulties to overcome in bringing it to a successful issue. And although the value of practical work for elder boys in this branch of education cannot be over-estimated, it is even of more importance when dealing with the younger mind.

The advantage derived from the information being conveyed in a concrete rather than an abstract form, the gain in manual dexterity and in accuracy of observation are so self-evident at the end of a term's trial as to fully reward the extra expenditure of time and trouble in arriving at the result.

Now this plea for practical work may conjure up such alarming ideas of laboratories, apparatus, and so forth, as to seem to forbid it out of hand.

It is,

But this alarm, if it exists, is wholly unwarranted. like most of the objections raised to starting scientific work, the result of approaching the consideration of the subject from the standpoint of a wholly classical education and without

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