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INTRODUCTION

This introduction is intended (1) to take the place of a preface, and (2) to offer some helpful suggestions to the teachers who may use this book.

(a) Number facts and number applications are the things of which arithmetic treats.

Number facts are not only such facts as (1) 4 + 3 7, (2) 6 x 5 = = 30, and (3) + { = , but they are also the steps and laws by which the results of combining numbers are determined.

Number applications are the applications of number facts to concrete quantities or the concerns of life. The following are applications, respectively, of the above facts:

(1) If there are 4 birds in one cage and 3 birds in another, there are 7 birds in both cages.

(2) If the price of oranges is 5 cents each, 6 oranges will cost 30 cents. (3) of a dozen eggs and } of a dozen eggs are of a dozen eggs.

(6) Practically, the valuable part of arithmetic is to be sought in the applications. Without these the number facts are practically worthless. But without the number facts there can be no number applications. Figuratively, the facts are the tools with which the applications are made. A man may understand the theory of constructing houses, but his ability to do the work himself depends on his skill in the use of the hammer, the saw, and the plane. Similarly, proficiency in the applications of numbers requires a knowledge of number facts and the ability to combine numbers accurately and rapidly.

(c) Educationally, both the facts and applications of numbers are valuable in so far as they develop in the pupils the powers of observation, comparison, and generalization, as well as independent, persistent, and consecutive thinking. This consummation depends largely on the manner in which the facts and applications are presented. In the present book, and also in the “Grammar School Arithmetic ” of this two-book

series, an effort has been made to introduce suitable matter in such a manner as to secure the best results, educationally and practically. The main features of the book are based on, and are the outgrowth of, the following considerations.

(d) Induction and drilling. The primary number facts should be presented and taught in connection with concrete quantities. The objects selected should be those with which the pupils are most familiar, especially those which appeal to them most strongly. After the mean. ings of the terms, the signs, the operations, and the fundamental concepts have been thoroughly grasped by the pupils they should be drilled in the number facts pertaining thereto until they can name or perform the combinations readily and accurately. Teachers should not become so engrossed with the so-called “informational features” of the work as to lose sight of the “mechanical processes,” nor vice versa. It is a sad reflection on any school that the pupils thereof who have finished arithmetic are unable to add, subtract, multiply, and divide with facility, and to make a practical application of any abstract work they have performed. It is advisable to begin each recitation with a short, spirited review of number facts, especially those in which the pupils are least proficient; and it is well to require the pupil to make applications of some, if not all, the facts referred to.

(e) Unity of the whole. Pupils should be led, as soon as possible, to see the relation of the four fundamental operations. Because (1) they will then have a smaller number of independent facts to learn, (2) they will learn the facts more readily and remember them longer, and (3) they will get a broader and more fruitful knowledge of the subject.

The child who knows that 3 + 2 = 5 has no independent fact to learn in finding 5 – 3, when it is presented thus, 3 + ? = 5. Therefore addition and subtraction, and multiplication and division should be taught together, at least until the relations of the operations are learned. Furthermore, the pupil should be thoroughly taught how to multiply by addition and how to divide by subtraction. Thus the child learns the oneness of operations. But, in addition to this, he should come to see that arithmetic, as a whole, is pervaded by common principles and connected by cominon laws. The next paragraph will explain to some extent how this book attempts to compass this.

(f) Building normally and logically. The teacher and pupil should get the idea that learning arithmetic is a process of building, - begin.

ning with the simplest facts and applications as a base, and gradually building on these by the expansion of numbers and the introduction of new ideas. This book is divided into parts which form and emphasize important steps in the process of this construction. A glance at the Table of Contents will show the progressive and logical order of these steps. Part I treats of ones, from 1 to 10, introducing the ideas of addi. tion, subtraction, multiplication, division, fractions, and measurements; and each succeeding part is largely a review and extension of what has gone before. Thus it will be seen that the treatment is continuous, being both spiral and topical, avoiding the extremes of each method and securing the advantages of both.

It may be that the pupils have already had the substance of Part I in oral instruction; but it is advisable for them to go over it again, at least hurriedly. Thus they will crystallize what they know by reviewing it in writing, and also bring themselves into touch with the system which pervades the book.

(9) The higher applications, both in this and the “Grammar School Arithmetic,” relate mostly to what concerns the business man and the farmer. In the higher book special attention is given to cotton, corn, rice, hay, sugar cane, dairying, etc.; while in the present book an effort is made to interest and instruct pupils in farm and garden concerns, and in the kinds of birds and insects that are beneficial and hurtful. While this material affords useful and interesting practice in the manipulation and application of numbers, it meets, at the same time, the modern demand for the teaching of agriculture and the industries in the public schools. Evidently arithmetic may treat of things pertaining to the industries as well as of things relating to commercial institutions. Therefore, information about birds, insects, poultry, hogs, cotton, corn, dairying, cultivating and harvesting, farm drainage, the reckoning of farm crops, etc., is as appropriate ini arithmetic as information about banks, bonds, stocks, etc. The introduction of “farm arithmetic” into this book instead of into the higher one is designed to meet the wants of that vast army of children who leave school before they reach the grammar grades. As many of them spend their lives on farms, they have no occasion to apply the rules of stocks and bonds and kindred subjects. The problems relating to farming, etc., have been carefully graded for fourth or fifth grade pupils.

(h) Oral and written work in this book are distinguished by the kinds

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