DEPARTMENT OF EDUCATION LELAND STANFORD JUNIOR UNIVERSITY THE NORMAL RUDIMENTS OF ARITHMETIC, ORAL AND WRITTEN, DESIGNED FOR PRIMARY AND INTERMEDIATE CLASSES IN BY EDWARD BROOKS, A. M., Pн. D., SUPERINTENDENT OF PUBLIC SCHOOLS OF PHILADELPHIA; LATE PRINCIPAL OF STATE NORMAL SCHOOL, PENNSYLVANIA, AND AUTHOR OF SPHERICAL TRIGONOMETRY, PHILOSOPHY OF ARITHMETIC, "The highest science is the greatest simplicity." PHILADELPHIA: CHRISTOPHER SOWER COMPANY, 614 ARCH STREET. 1899. PREFACE. THIS little work is intended for the child's first lessons in Arithmetic. It is based on a careful study of the natural development of the numerical idea in children, and aims to aid the teacher in giving this development. Some of its leading peculiarities will be briefly stated. SPECIAL FEATURES.-Attention is first called to the following special features which distinguish this work: 1. The first object in the course of instruction is to develop the idea of number with children. This idea arises in the mind in connection with objects. A child sees the objects and thinks the number. All primary instruction is therefore to be given in the concrete. This is made a prominent feature of the work. 2. Objects and numbers, however, are not identical. One is a thing of sense, the other is an idea of the mind. This distinction, often lost sight of by modern educators, should be clearly seen by the teacher. The attempt to hold the mind of the pupil down to objects and explain all numerical processes by means of blocks and sticks would be to enfeeble its powers and give it a wrong idea of numbers. The aim has been, therefore, to gradually lift the mind up to the conception of number in the abstract. 3. The basis of Arithmetic consists of the fundamental processes of Addition, Subtraction, Multiplication, and Division. These operations are based on what may be called "elementary results," by which is meant the elementary sums and differences as far as 9 and 9 are 18, and the elementary products as far as "9 times 9" with the corresponding quotients. These elementary results constitute the alphabet of Arithmetic, and are to be committed to memory. Their extent is limited by our decimal system of numeration and notation to nine, though custom has fixed the limit at twelve. 4. These elementary processes are to be taught in their relation to one another. Addition and Subtraction are the primary processes, and are the converse of each other. They are therefore to be taught together in deriving the elementary results. Multiplication is a special process of Addition, and should therefore be so taught as to exhibit this relation. Division is the converse of Multiplication, and is presented in this relation rather than as an independent process. These four processes are to be combined in elementary instruction in as close a relation as the natural development of the mind of a child will permit. 5. The ideas of these elementary processes can be developed along with the ideas of numbers. In the development of these ideas it has been found convenient to begin with the first numbers and treat them in regular order as far as twenty. This will give all the elementary results of addition and subtraction, and several of the elementary products and quotients. The first section of the book therefore treats of numbers from one to twenty. In deriving the quotients up to this limit no inexact divisions are required, the object being merely to reverse the elementary products. 6. After reaching the number twenty, instead of continuing the numbers in order to 100 or 144, as in the Grube system, the order of the elementary products is followed, the object being to have the pupils derive and commit the elementary products and quotients. These exercises constitute the second section of the work. 7. In connection with these exercises the elements of fractions and denominate numbers are incidentally introduced, so that by the time pupils have reached the formal treatment of these subjects a solid foundation has been laid for them. 8. During all these exercises in becoming familiar with numerical ideas and the elementary results, pupils are gradually taught to write numbers according to the Arabic system. The elements of the fundamental operations with numbers exceeding the elementary results are also gradually introduced as preparatory to the formal treatment of them. 9. Having laid this foundation of the science and the art in the mind of the learner, the subject is taken up in regular order as indicated by the science under the heads of Numeration and Notation, Fundamental Operations, Fractions, Decimals, Denominate Numbers, etc. GENERAL FEATURES.-In addition to these special features, attention is called to the following general features of the work: 1. The aim has been to make the work thoroughly inductive in its character. Ideas are presented before words, processes before rules, analyses before inferences, and in every case the pupil is led to the ideas by easy, natural, and gradual steps. As a result of this method, all statements of principles, methods, definitions, etc. on the part of the pupil are merely the expression of what he already clearly understands. He is thus trained, not merely to follow old paths, but to be an independent truth-seeker. 2. The proper gradation of the work will be found a distinctive and valuable feature. Great pains have been taken to avoid those sudden transitions from the easy to the difficult which confuse and discourage pupils. The object has been to make each lesson an easy steppingstone to the one which follows, so that pupils may pass from the simple to the complex with ease and delight. 3. The effort has also been to give a practical character to the work. It deals so far as possible with the objects most familiar to pupils. Its problems are designed to represent the actual things and events of life. Its language and methods are suggestive of business life, and a number of problems on historical events, biographical dates, dates of celebrated inventions, etc. add interest to the study and present valuable information to the pupil. 4. Lastly, it is believed that the work will be found to embody the educational spirit of the times. The work is designed not merely to teach Arithmetic, but to train the intelligence of the child. Its method of reasoning is that of analysis, by which each truth flows in simple sequence from a preceding one. From these processes of analysis methods are derived by the simple inferences of induction. Analysis and induction thus run like a golden thread through the entire work, brightening its processes and binding its parts together into logical unity. It is thus adapted to teach pupils to think as well as to work problems—to develop mind as well as the power of computation. Appreciating the favor extended to my previous works, and cherishing the hope that this new book will aid teachers in their work and become a favorite of many of the boys and girls of the country, I intrust it, as the expression of my latest thought in teaching the elements of Arithmetic, to the decision of a kind and indulgent public. EDWARD BROOKS. PHILADELPHIA, PA., May 10, 1895. |