or Central Schools at the present time. As a separate, formal, and academic subject it is slowly disappearing, being replaced by generalised arithmetic and the use of the equation in the solution of problems. Some examining bodies still demand it as a separate subject and hence its retention in the syllabus submitted, being required for "Trade" Entrance Examinations. The old stodgy type of sum is, however, now seldom met with, and it is increasingly felt that time spent on complicated highest common factors, least common multiples, algebraic fractions and the like may be more profitably employed by the boy who leaves school at 15 years of age, or earlier, to become an artisan. Thus although algebra appears somewhat prominent in the representative working syllabus attached, the writer suggests that the subject be reduced as may from time to time be found practicable until finally it embraces only such exercises as are merely extensions of Arithmetic, together with symbolical expression, the solution of equations and problems

π r2h,

solved by them, the application of a formula, e.g., V = 3 some elementary graphic work; the use of logarithmic tables and the little trigonometry necessary for the determination of heights and distances.

The graphic work may well be limited in the first and second years to graphic representation of the usual kind :the growth of a bean, class attendance, temperature, barometric pressure, rainfall, bright sunshine, market prices, and the results of work in the laboratories. In the third year, the straight line graph may be used for illustration and the solution of suitable problems, and in the fourth year the graph applied to the solution of simple quadratics. Abundant practice should be given in graphic work for the illustration of laboratory results so that the pupil may acquire facility both in the construction and reading of his graph.

Geometry.-Geometry in such a syllabus as that submitted must be regarded as a subject of primary importance, since it is related so closely to other mathematical work and to handicraft, mechanical drawing, and practical geography. No subject of the higher grade curriculum presents greater difficulty or needs more careful treatment. There are conflicting elements and points of view which make the necessity for compromise quickly apparent. The text-book, too, exactly suitable for our purpose is still wanting.

The general aim is determined by (a) the industrial character of the course, and (b) some considerations of that mental training value already discussed. It is therefore twofold. In a course including handicraft in wood and metal, mechanical drawing and the development of workshop methods, some constructional plane geometry of the old type is obviously necessary and to this must be added a really useful course of solid geometry. On the other hand it is desirable that the pupil should obtain some knowledge of reasoning processes as applied to geometrical figures.

Practical Work.-Plane Geometry.-The mechanical character of the old type of constructional work has already been indicated, and it is suggested that it be here limited to constructions in common use such as are required in concurrent subjects, e.g., the drawing of lines and angles, bisections, perpendiculars, parallels, proportionals, circles, tangents, the simple geometrical figures and some work on patterns.

This part of the course aims at giving (a) facility in handling pencil, ruler, compass and protractor; (b) accuracy and finish in the work; (e) a working stock-in-trade of the geometrical constructions required in science, machine drawing and handicraft; (d) opportunities for exercises in measurement and experimental geometry.

Solid Geometry must be introduced at the earliest stage. The work should not be abstract or ambitious, but thoroughly practical in scope and aim, one object being the quick and clean production of a working drawing. The fourth-year pupil should be able to make a plan, elevation and simple section of any of the common geometrical solids, or if working from the actual object, of any simple machine detail.

Geometry as Mental Training.-The question here is between the experimental and Euclidean methods of reasoning. To what extent shall the experimental method of illustration, investigation and so-called proof be carried? Shall the Euclidean method supersede it, and, if so, at what stage?

In an industrial course, although the maximum amount of mental training is sought, an extended course of abstract reasoning is scarcely possible. In the illustrative syllabus, therefore, the experimental method is employed throughout the first and second-year course, much of the constructional work lending itself to this treatment. Simple Euclidean reasoning is introduced towards the end of the second year, developing simply and naturally from the experimental work.

The extent to which the pure deductive method shall be carried will furnish further matter for controversy. It is here intended that the more important propositions of Book I. be proved by the modernised Euclidean method, that Book II. be illustrated algebraically, and that Book III., wherever possible, be worked experimentally by measurement, superposition, &c. The pupil will thus get an acquaintance with the geometry of the first three books, and some insight into the method of pure deductive reasoning, the time saved being given to more practical work. In the poorer type of Higher Elementary or Central School the study of pure geometrical reasoning after the manner of Euclid will probably be entirely omitted.

The mechanical drawing of the course embraces the production of working drawings from models or actual specimens of some 30 or 40 machine details, such as nuts, bolts, pedestals, fork end, angle irons, knuckle joint, eccentric, piston, &c.

In conclusion the following suggestions on the method of teaching arithmetic to get from it its full utility value, and

as much as possible of the mental training value, are additions to, or definite and concrete maxims based on, the principles of method advocated in this memorandum.

(i) Arithmetical meanings, relations, processes, &c., to be obtained as far as possible practically and as a result of individual experiment on the part of the child himself.

(ii) Wherever possible, especially when the rule or process comes first, get the answer by actually solving the problem by means of concrete material :-counters, cardboard coins, rulers, measures, squared paper, &c. Use the experimental method, see that the children clearly understand what is to be done, let them try for themselves and evolve the best way.

(iii) Let the children have realities behind their symbols:actual measurements from concrete objects, diagrams, standards of length, area, volume, paper folding, use of squared paper, graphic illustration and experimental demonstration.

(iv) Find out and use alternative methods of work, using one as a check upon the other.

(v) Let the lessons be concerned with things within the experience of the children, or possible to their experience, or within their imagination.

(vi) Let written work follow and be rather an appendage to oral work.

(vii) Introduce decimals early and tenths earlier and illustrate from the ruler.

(viii) Rely upon the few fundamental principles and processes rather than upon a number of fixed rules. Try to apply commonsense methods to all operations.

(ix) Gradually "generalise the ordinary arithmetic, make the fullest use of the symbol "x" and introduce the algebraic equation at an early stage.

(x) Deal with small numbers, and so cover much more. ground in the lesson.

(xi) Cover all the course, and every kind of problem, process and principle by oral work, using small numbers. To ensure that every child does his share of the work let all answers be written down. Let the blackboard be freely used as an aid. in oral work, which should be an integral part of most lessons and form part of each “term test."

(xii) Aim at real correlation with other school subjects where the gain to arithmetic is a real gain,

(xiii) Introduce symbols only when needed or their value appreciated. Make the children understand clearly the shorthand nature of symbols, and accustom them to pass at will from concretes to symbols and symbols to concretes.

(xiv) As soon as children of their own accord begin to drop the concrete, encourage them to work mostly with the abstract in each field of experience as it is met.

(xv) It does not necessarily follow that because children reproduce the language of reasoning, they have reasoned.

H. J. SPENCER.

THE TEACHING OF MATHEMATICS TO YOUNG

CHILDREN.

That skill in handling numbers is one of the fundamental bases on which to rear an educational structure is no new idea. Number has formed a part of the most scanty and elementary schemes of education through all historic time, and we may therefore assume that its value is undisputed even by those unable to realise it in exact terms of intellectual training and

power.

Those, however, who are deeply interested in the teaching of the science of number realise that, even though it may never be of practical use to the student, yet a true knowledge of this subject will give him such important knowledge as will stand him in good stead in his future dealings with men and affairs. To them no apology is necessary for the exceeding care which we consider must be bestowed on teaching children to really "think mathematically."

Taking as our working definition that "education is an atmosphere, a discipline, a life," it follows that we realise that education must surround and be a part of the child from his infancy; but until he is ready for school at the end of his sixth year it is to be an education by means of his senses, of his unstudied games, by means of his natural and not of an artificially prepared environment.

The conscious teaching then of number, as of other definite lines of thought, is to be begun in the schoolroom with a pupil whose age is not less than six years.

School Mathematics.-When children begin their regular school course, lessons last for two hours, or two-and-a-half hours, every morning, with a long interval; number for 20 minutes a day is one of the lessons. We generally find that the children, when they enter school, are able to count, but know nothing of the properties of numbers.

The number one is taken during the first lesson; the children point out to the teacher one window, one fireplace, one piano; in fact, everything in the room which exists singly; then the symbol for one is learnt. Whenever we see a stroke 1 we know that it stands for one of something. The children pick out the ones from groups of figures, and finally learn to write one; getting it as straight and perfect as possible.

The children have a small blackboard and piece of chalk each. and on these they first write the numbers; afterwards a book ruled in 4 inch or inch squares and a lead pencil are requisitioned. The next number to one is two, as the child probably knows; he learns then to write '2,' first on his board,

and then in his book; picks out 2 from a group of figures, and does little sums involving the number 2. Three is taken in the same way; and then four, which the pupil must realise is made up of two twos, or of 3 and 1, by very simple little problems such as will readily suggest themselves to any teacher. He learns to count up to 4, and backwards from 4; thus realising slowly the idea of a series of symbols denoting a series of quantities whose magnitudes continue to grow greater. The idea of an order of things, which is conveyed by a number, is perhaps grasped most easily by counting a series of things; and that of the relative magnitudes represented by numbers by the little sums in addition and subtraction.

In this way all the numbers from one to nine are learnt, the examples becoming more numerous as the numbers grow larger, and involving, besides simple subtraction, simple factors such as two threes make six, and three threes make nine. Each number is begun from a concrete set of things, beads, &c., and several questions are asked and answered with the help of the beads. Then these are put away, and for the next lesson work is done on the number without the aid of the concrete.

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When several of the numbers have been learnt the meanings of the signs,, and are explained to the child; + means "is added to," or "is put together with," means is taken away from," and = means "is the same thing as." Now we have the added joy of being able to write sums in our books. This is always considered a privilege, and is only indulged in on mornings when the children are working well, and during the final lesson on some particular number. Writing is still a laborious effort, and is apt to take attention away from the most important matter in hand. The sums are of course always worked orally first, and then written down, e.g., if your little sister is two years old now, how old will she be in two more years? When the answer 4 has been obtained the children write in their books 2+2 = 4; then they read it; two years added to two years make four years. This writing of sums, however, is very sparingly used, and all the work is oral.

During this stage too we give occasional examples dealing with pure number; there are mornings when the little ones are bright and eager, and more than ever anxious to do innumerable sums; this is an opportunity to be seized by the teacher; let us leave the boxes of beads and counters alone, let us even leave out sheep and motor cars, and have nothing but numbers. "How much left if you take 3 from 5?" "How much to be added to 4 to make 7?," and so on, quick question and quick answer, all easy and simple, so that the children may feel at home with the numbers, and feel that they have a real grasp of them, for, though the ability to work with pure number is undoubtedly a function of some minds only, yet it, like an ear for music, can, to a certain extent, be cultivated, to a very limited extent it may be, but even that is worth striving after with our pupils.