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PREFACE.

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DEVELOP in the pupil's mind the idea of number and its uses, to give him facility in numerical operations, and

to lead him to grasp firmly the principles involved requires the serious effort of the most skilful teacher. The most any author can do to aid him is to present, in more or less detail, an outline of a practical scheme of instruction based upon the latest and best thought in number teaching; and even this is a task that none but the superficial would undertake without fear and trembling.” Neither theory nor practice alone is

sufficient endowment for the work. Every theory, every method, every device must be weighed in the class-room balance. That which is to be taught is the child—not arithmetic.

Holding these views, the writers of this book have spared no effort in its preparation. With a wholesome respect for the injunction, “ Prove all things; hold fast that which is good," they have endeavored to steer carefully between the Scylla of modern fad and the Charybdis of mechanical drudgery and stupefying monotony. Their aim has been to furnish for the teacher a sound and practical work presenting the matter in an approved order and suggesting rational methods of obtaining the most desirable results. As to how well they have succeeded the teacher must decide.

ARRANGEMENT.-Before entering school the child learns a little of a great many things, not a great deal about one thing. There is no topical arrangement in nature. “And Nature, the dear old nurse," permits the order of the child's mental development to determine the arrangement of the material, which is always a progressive one. The child learns a little about many things to-day, a little more to-morrow, adds an increment to his knowledge and widens his range of subjects day by day. This arrangement keeps interest alive, without which there is no substantial progress. This book aims to follow the natural

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order as closely as may be, considering the conditions under which teachers are required to work.

" FROM THE CONCRETE TO THE ABSTRACT.”—The child's mind must be approached through those ever open doors, the senses. Primary number teaching must be begun by the pupil's observing and handling objects, yet the mere presence of concrete things does not guarantee the presence of definite numerical ideas in the mind of the pupil. They must be so presented as to stimulate and aid the mental movement of discriminating and relating which leads to definite ideas of number. The mind must (a) recognize the like objects as distinct individuals, and (b) group (put together) the objects into a whole. The child's own activity must conceive a whole of parts, and relate the parts in a definite uhole. It is not enough that he knows our country as 45 states; he must know the 45 states as one country. He should recognize five, for example, not only as five ones, but also as one fire, considering the component ones not chiefly for their own sake, but as giving definite value to the whole, the group. By repeated acts of such measurement or valuation the mind advances naturally and inevitably from things to relations, from facts to principles, from the concrete to the abstract. Believing that the concept of ratio-an abstract idea—must be evolved in the young mind by its own activity, and that therefore it should not be thrown at the child in the earlier stages of his progress, the authors have not explicitly stated the ratio idea until it has had time to grow into a recognizable and useful product of the pupil's mind.

MATERIAL.—The objects called for in this book, such as cubes, splints, foot rules, quart measures, etc., can be provided by the teacher with little trouble, and their use should precede that of pictures and words. How long their use should be continued with any given class must be determined by the teacher. They should certainly not be laid aside until the child can bring their images into consciousness without the presentation of the actual objects to his senses; but when these images are within easy reach, their use should supersede that of objects. The number pictures given in this book are an important aid to the child's mental movement of abstracting and relating, and must be used for economy of energy. Five dots analytically arranged

are more easily grasped than five dots placed

as

promiscuously, or even in a row. Moreover, they are perceived

one whole, rather than as separate, unrelated parts, and they should symbolize any five units whatever-five $2 or five 10-ft. as well as five $1 or five 1-ft. In the perceptive form

twelve is seen as three fours and as four threes—as four three times and as three four times—and the child's idea of the number is more definite than when the whole was measured by but one unit, the individual. Besides, this use of pictures suggests to the pupil a method by which he can discover for himself the measuring units of quantity or determine the definite value of a whole. In notation, fractions, measurement of surfaces, etc., the illustrations will be found highly beneficial; their purpose is to aid, not to ornament.

COUNTING—MEASURING.—Dr. Harris says, “ The first lessons in arithmetic should be based on the practice of measuring in its varied applications.” In its earlier stages measurement is a vague estimate, a crude guess. Later an undefined unit of measure is used to make a whole more definite, as when a class is measured by the unit boy-an unmeasured unit. But it is only when the unit itself is accurately defined that precise numerical value is reached. The immediate purpose of measuring is to make some whole more definite-to find how much there is of it. This can be done by learning how many units are in it, and the “how many can be found only by counting; hence measuring involves rational counting--the finding of the sum of the units that compose a whole. It seems clear, therefore, that the child learning to count should be led to number the parts that make up some whole, as the desks in a row-to count with a definite end in view; that the starting point should be a group of things—a whole to measure, parts to count and relate. As soon as possible there should also be exact measurements. The foot may be taken as the whole and measured by 4-inch units, 3-inch units, etc. Not only the idea of number as the repetition of a unit of measure to equal a magnitude, but also the law of commutation will be slowly, perhaps, but surely developed through the use of the facts supplied by senseperception in the rational use of objects and number pictures.

FIGURE PROCESSES ONLY THE MEANS TO AN END.—The child must learn figure processes, not for their own sake, but as an aid in discerning relation. Figures are not at all necessary to

numerical perception, but when this is to be supplanted by numerical facility they should be employed as a convenient tool. The real subjects of thought are magnitudes (of value, of space, of weight, etc.) and their relations; figures are mere symbols of these. The essential thing is the discernment of relation by comparing magnitudes; figure processes may or may not be an aid in securing this end-they can never be anything more. It is evident, therefore, that the mere manipulation of figures—the mechanical work—is of small value in comparison with the thought processes—the mental work; yet as the means to an end it is of great importance, and to secure absolute accuracy therein should be the constant aim of every teacher.

SUGGESTIVE QUESTIONS AND DIRECTIONS.—The exercises of this book are more abundant and varied than is usual in books of this kind, and much additional work will be called forth by the suggestive nature of many of the questions and directions. There is an especial need of an abundance of constructive exercises. The pupil must be active, must do as well as see. Knowing and doing are correlative. Things are grasped most readily when there is a necessity, real or apparent, for knowing them. Boys who sell papers or fruit on the streets, being under the necessity of computing and making change, enter school with a surprising stock of number facts. Excellent number work may be had in connection with the occupations of the school-room-passing pencils, making boxes, modeling, playing number games, measuring things used in nature study and manual training, etc. The questions and directions given indicate the kind of work that should be done, but the quantity of it needed must be determined by the teacher, and by him supplied when necessary. Figures and signs are not used until the child has acquired some familiarity with the names of numbers to twelve and with a few of the simpler processes and facts. This should at least suggest that the figure symbol should not be given until the child can understand that it is a mere mechanical tool by which he can obtain rapid and at the same time equally true results.

It is neither necessary nor desirable to keep this book out of the hands of young pupils until they are able to read it with ease.

They should at least get the benefit of the number pictures and the picture exercises. Anything too difficult for the

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