MYERS ARITHMETIC FOR GRAMMAR SCHOOLS BY GEORGE W. MYERS, Ph.D. PROFESSOR OF THE TEACHING OF MATHEMATICS, AND MATHEMATICAL SUPER- SCOTT, FORESMAN AND COMPANY CHICAGO NEW YORK PREFACE This book continues the work of the Myers Elementary Arithmetic and covers comprehensively the work of the last four years of the elementary school. It is a topical treatment of the subject and is thus in itself a complete arithmetic. The educational reason for arithmetic as a school subject is that it is a type of the organization of experience that the world deems useful in daily life. The experiences which arithmetic should organize are sensations of measuring and numbering. In order that this measuring and numbering experience may be organized into the most efficient practical knowledge, the work must be done in harmony with the mind's way of construing these sensations into relationships. Nothing is more fundamentally important than that the pupil actually have the experiences that are to be organized. The ready-made experiences of others cannot be substituted for those of the learner himself. Furthermore, the pupil should organize as well as experience these sensations. The requirement that the sensations to be organized shall be the pupil's own can be met only by the plentiful use of concrete material. The objects, pictures, real problems, and other concrete materials must be not merely talked about by the teacher, the pupil passively listening and looking on. The pupil must handle the objects, draw the sketches, examine them and even make verbal problems. He must be allowed to make discoveries for himself. The concrete material used may be very simple indeed, as sticks, blocks, dots, marks, squares, oblongs, verbal problems, if the pupil himself do the handling, 3 counting, grouping, measuring, to the end that he experience the relations to be organized. A variety of simple objects will in fact serve the purposes of number sensations better than will a few elaborate and artificial objects, made for the pure purpose of exemplifying number relations. For example, so simple a group of number truths, as 2 dots and 3 dots are 5 dots, 2 marks and 3 marks are 5 marks, 2 inches and 3 inches are 5 inches, 2 children and 3 children are 5 children, 2 sticks and 3 sticks are 5 sticks, 2 things and 3 things are 5 things, given in quick succession, leading the child to say, or to feel, "Why, it's always that way; 2 and 3 are always 5," is for number purposes as good an exercise in child-like generalizing as can be devised. Then the correlative subtraction truths 3 from 5 and 2 from 5, and the supplementary addition truth 3 and 2 are 5 should be presented at the same time. All the simple addition combinations should first be arrived at by the pupil in this fashion. Such truths as 3+2=5, 4+5=9, would then stand for organized experience. But important as is the concrete material as an aid in early work, this material is only an aid. The chief business is to organize the number truths exemplified by this material into definite arithmetical knowledge. It is important not only at the beginning but all along the way that the arithmetical truths be based on the concrete, but the weightier matter is to bring out clearly and to teach thoroughly the number truths. . The use of concrete material is to a large extent a function of the teaching. And yet a text-book, by its plan, by the nature of its problems and problem material, and by the mode and order of presentation, can be of real assistance to the busy teacher. Accordingly, the problems and modes of treatment of this book may be regarded as in the nature of types and suggestions. It may even be that teachers may here and there prefer problem material of a more distinctly local interest than are the problems of the text. The treatments here given will suggest how such substituted material may best be handled to secure the desired arithmetical result. Experimental pedagogy has been active in recent years in testing school work by objective standards. No subject has come in for a larger share of this testing than arithmetic. Some of the findings for arithmetic are:— 1. That what we speak of as arithmetical ability is in reality a complex of many separate abilities. 2. That the several abilities, such as those used in the process of addition and subtraction, and that for analysis, may or may not represent a single unified arithmetical ability. When the separate abilities do not combine to produce a unified ability, high power and skill in any one line may co-exist with very low power and skill in others. When they do combine, power and skill of one kind serve to heighten power and skill of related kinds. The more completely the several processes and kinds of work are kept together in teaching, and the more persistently they are returned to from time to time, and from year to year, the more completely are the several particular abilities fused into a composite single ability. 3. That methods of teaching can be weighed and tested with precise objective certainty. 4. That the most effective method is that in which arithmetical truths are grouped into small unities in the early grades, into ever larger unities as the grades are passed, and into topical units in the later grades. 5. That in the grammar school a forecast of the main theme of each year's work should be given at the close of the preceding year, while its basal notions are most vivid. and that at the beginning of each year a summarizing review and synopsis of the work of the preceding year should be given. This brings each main topic repeatedly into the learner's consciousness in a fuller way as he proceeds. |