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It is perhaps natural that a growing impatience with the meagre results of the time given to arithmetic in the traditional course of the schools should result in attacks upon that study. While not all educational experts would agree that it is the most useless of all subjects” taught, there is an increasing tendency to think of it and speak of it as a necessary evil, and therefore to be kept within the smallest possible bounds. However natural this reaction, it is none the less unwise when turned against arithmetic itself, and not against stupid and stupefying ways of teaching it. ceived, the movement stands only for an aimless swing of the scholastic pendulum, sure to be followed by an equally unreasonable swing to the other extreme. If methods which cut across the natural grain of the mental structure and resist the straightforward workings of the mental machinery, waste time, create apathy and disgust, dull the power of quick perception, and cultivate habits of inaccurate and disconnected attention, what occasion for surprise ? Because wrong methods breed bad results, it hardly follows that education can be made symmetrical by omitting a subject which stands par excellence for clear and clean cut methods

of thought, which forms the introduction to all effective interpretation of Nature, and is a powerful instrument in the regulation of social intercourse.

It is customary now to divide studies into “form” studies and “content" studies, and to depreciate arithmetic on the ground that it is merely formal. But how are we to separate form and content, and regard one as good in itself and the other as, at best, a necessary evil ? If we may paraphrase a celebrated saying of Kant's, while form without content is barren, content without form is mushy. An education which neglects the formal relationships constituting the framework of the subject-matter taught is inert and supine. The pedagogical problem is not solved by railing at “form,” but in discovering what kind of form we are dealing with, how it is related to its own content, and in working out the educational methods which answer to this relationship. Because, in the case of number, "form" represents the measured adjustment of means to an end, the rhythmical balancing of parts in a whole, the mastery of form represents directness, accuracy, and economy of perception, the power to discriminate the relevant from the irrelevant, and ability to mass and converge relevant material upon a destined end—represents, in short, precisely what we understand by good sense, by good judgment, the power to put two and two together. When taught as this sort of form, arithmetic affords in its own place an unrivalled means of mental discipline. It is, perhaps, more than a coincidence that the particular school of educational thought which is most active in urging the merely “ formal” quality of arithmetic is also the one which stands most sys

tematically for what is condemned in the following pages as the “fixed unit” method of teaching.

As for the counterpart objection that number work is lacking in ethical substance and stimulus, much may be learned from a study of Greek civilization, from the recognition of the part which Greek theory and practice assigned to the ideas of rhythm, of balance, of measure, in moral and æsthetic culture. That the Greeks also kept their arithmetical training in closest connection with the study of spatial forms, with measurement, may again be more than a coincidence. Even upon its merely formal side, a study which requires exactitude, continuity, patience, which automatically rejects all falsification of data, all slovenly manipulation, which sets up a controlling standard of balance at every point, can hardly be condemned as lacking in the ethical element. But this idea of balance, of compensation, is more than formal. Number represents, as is shown in the following pages, valuation ; number is the tool whereby modern society in its vast and intricate processes of exchange introduces system, balance and economy into those relationships upon which our daily life depends. Properly conceived and presented, neither geography nor history is a more effective mode of bringing home to the pupil the realities of the social environment in which he lives than is arithmetic. Society has its form also, and it is found in the processes of fixing standards of value and methods of valuation, the processes of weighing and counting, whether distance, size, or quality; of measuring and fixing bounds, whether in space or time, and of balancing the various resulting values against one another. Arithmetic can

not be properly taught without being an introduction to this form.

Thanks are due to Mr. William Scott, of the Toronto Normal School, for some assistance with the proofs; and to Mr. Alfred T. De Lury, of University College, Lecturer on Methods in Mathematics in Ontario School of Pedagogy, for valued practical assistance.

August 13, 1895.

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