The authors found their method upon the following passage from Mill’s Logic:—“The fundamental truths of the science of Number all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child’s mind along with them in learning arithmetic, all who wish to teach numbers and not mere ciphers, now teach it through the evidence of the senses in the manner we have described.”
Here we may, I think, trace the solitary source of weakness in a surpassingly excellent manual. It is quite true that the fundamental truths of the science of number all rest on the evidence of sense; but, having used eyes and fingers upon ten balls or twenty balls, upon ten nuts, or leaves, or sheep, or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. In fact, he begins to think in numbers and not in objects, that is, he begins mathematics. Therefore I incline to think that an elaborate system of staves, cubes, etc., instead of tens, hundreds, thousands, errs by embarrassing the child’s mind with too much teaching, and by making the illustration occupy a more prominent place than the thing illustrated.
Dominoes, beans, graphic figures drawn on the blackboard, and the like, are, on the other hand, aids to the child when it is necessary for him to conceive of a great number with the material of a small one; but to see a symbol of the great numbers and to work with such a symbol are quite different matters.
With the above trifling exception, which does not interfere at all with the use of the books, nothing can be more delightful than the careful analysis of numbers and the beautiful graduation of the work, “only one difficulty at a time being presented to the mind.” The examples and the little problems could only have been invented by writers in sympathy with children. I advise the reader who is interested in the teaching of arithmetic to turn to Mr Sonnenschein’s paper on ‘The Teaching of Arithmetic in Elementary Schools,’ in one of the volumes published by the Board of Education.[25]
Preparation for Mathematics.—In the ‘forties’ and ‘fifties’ it was currently held that the continual sight of the outward and visible signs (geometrical forms and figures) should beget the inward and spiritual grace of mathematical genius, or, at any rate, of an inclination to mathematics. But the educationalists of those days forgot, when they gave children boxes of ‘form’ and stuck up cubes, hexagons, pentagons, and what not, in every available schoolroom space, the immense capacity for being bored which is common to us all, and is far more strongly developed in children than in grown-up people. The objects which bore us, or the persons who bore us, appear to wear a bald place in the mind, and thought turns from them with sick aversion. Dickens showed us the pathos of it in the schoolroom of the little Gradgrinds, which was bountifully supplied with objects of uncompromising outline. Ruskin, more genially, exposes the fallacy. No doubt geometric forms abound,—the skeletons of which living beauty, in contour and gesture, in hill and plant, is the covering; and the skeleton is beautiful and wonderful to the mind which has already entered within the portals of geometry. But children should not be presented with the skeleton, but with the living forms which clothe it. Besides, is it not an inverse method to familiarise the child’s eye with patterns made by his compasses, or stitched upon his card, in the hope that the form will beget the idea? For the novice, it is probably the rule that the idea must beget the form, and any suggestion of an idea from a form comes only to the initiated. I do not think that any direct preparation for mathematics is desirable. The child, who has been allowed to think and not compelled to cram, hails the new study with delight when the due time for it arrives. The reason why mathematics are a great study is because there exists in the normal mind an affinity and capacity for this study; and too great an elaboration, whether of teaching or of preparation, has, I think, a tendency to take the edge off this manner of intellectual interest.
XVI.—NATURAL PHILOSOPHY
A Basis of Facts.—Of the teaching of Natural Philosophy, I will only remind the reader of what was said in an earlier chapter—that there is no part of a child’s education more important than that he should lay, by his own observation, a wide basis of facts towards scientific knowledge in the future. He must live hours daily in the open air, and, as far as possible, in the country; must look and touch and listen; must be quick to note, consciously, every peculiarity of habit or structure, in beast, bird, or insect; the manner of growth and fructification of every plant. He must be accustomed to ask why—Why does the wind blow? Why does the river flow? Why is a leaf-bud sticky? And do not hurry to answer his questions for him; let him think his difficulties out so far as his small experience will carry him. Above all, when you come to the rescue, let it not be in the ‘cut and dried’ formula of some miserable little text-book; let him have all the insight available, and you will find that on many scientific questions the child may be brought at once to the level of modern thought. Do not embarrass him with too much scientific nomenclature. If he discover for himself (helped, perhaps, by a leading question or two), by comparing an oyster and his cat, that some animals have backbones and some have not, it is less important that he should learn the terms vertebrate and invertebrate than that he should class the animals he meets with according to this difference.
Eyes and No-eyes.—The method of this sort of instruction is shown in Evenings at Home, where ‘Eyes and No-eyes’ go for a walk. No-eyes comes home bored; he has seen nothing, been interested in nothing: while Eyes is all agog to discuss a hundred things that have interested him. As I have already tried to point out, to get this sort of instruction for himself is simply the nature of a child: the business of the parent is to afford him abundant and varied opportunities, and to direct his observations, so that, knowing little of the principles of scientific classification, he is, unconsciously, furnishing himself with the materials for such classification. It is needless to repeat what has already been said on this subject; but, indeed, the future of the man or woman depends very largely on the store of real knowledge gathered, and the habits of intelligent observation acquired, by the child. “Think you,” says Mr Herbert Spencer, “that the rounded rock marked with parallel scratches calls up as much poetry in an ignorant mind as in the mind of the geologist, who knows that over this rock a glacier slid a million of years ago? The truth is, that those who have never entered on scientific pursuits are blind to most of the poetry by which they are surrounded. Whoever has not in youth collected plants and insects, knows not half the halo of interest which lanes and hedgerows can assume.”
Principles.—In this connection I should like to recommend The Sciences, by Mr Holden. America comes to the fore with a schoolbook after my own heart. The Sciences is a forbidding title, but since the era of Joyce’s Scientific Dialogues I have met with nothing on the same lines which makes so fit an approach to the sensible and intelligent mind of a child. This is what we may call a ‘first-hand’ book. The knowledge has of course all been acquired; but then it has been assimilated, and Mr Holden writes freely out of his own knowledge both of his subject-matter and of his readers. The book has been thrown into the form of conversations between children—simple conversations without padding. About three hundred topics are treated of: Sand-dunes, Back-ice, Herculaneum, Dredging, Hurricanes, Echoes, the Prism, the Diving-bell, the Milky Way, and, shall I say, everything else? But the amazing skill of the author is shown in the fact that there is nothing scrappy and nothing hurried in the treatment of any topic, but each falls naturally and easily under the head of some principle which it elucidates. Many simple experiments are included, which the author insists shall be performed by the children themselves. I venture to quote from the singularly wise preface, a vade mecum for teachers:—
“The object of the present volume is to present chapters to be read in school or at home that shall materially widen the outlook of American schoolchildren in the domain of science, and of the applications of science to the arts and to daily life. It is in no sense a text-book, although the fundamental principles underlying the sciences treated are here laid down. Its main object is to help the child to understand the material world about him.”