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NARVÁRD COLLEGE LIBRARY
FROM THE GIFT OF
MAR 5 1926
Copyright, 1893, 1895,
child an experience, instead of presupposing one for him. Pupils become practical, not by learning forms of reasoning, but by exercising the reason upon their own plane of comprehension.
In such a spirit this ELEMENTARY ARITHMETIC has been prepared. It presents three years' work, based upon carefully graded exercises which may be used as a means of training pupils to think, and of teaching at the same time the practical application of numbers to ordinary business transactions.
In order to enter satisfactorily upon the work provided for in the following pages, the pupils should be familiar with all the combinations of numbers through twenty. The training necessary to attain this familiarity is best secured by leading pupils to discover the relations of numbers by means of objects, and through this teaching to gradually free thought from dependence upon sense representation.
PART I. (Third Year in School.) The first and hardest step in solving an arithmetical question is to determine the processes required; the second, to state the different steps of the solution in proper arithmetical form.
It is very important that children should master the fundamental processes so thoroughly that they come to serve thought without loss of time or energy. The patient following of these graded exercises and drills should secure this result. The tables of “ Endings,” in addition (see pages 58, 61, etc.), have the same practical use as the multiplication table, and should be as thoroughly applied.
Each chapter presents, in general, division and multiplication as converse processes, followed by subtraction and addition on the same general plan. In the beginning each number is viewed as a whole, divisible into equal parts, and the parts are viewed in relation to the whole and to each other. A number is separated by division, united by multiplication; separated by subtraction, united by addition.
No formula should be taught with the thought that it will do the thinking for the pupil. Let the problem be pictured, and this followed by the expression in figures, before any formal expression in words is attempted. Give a very thorough drill, as on page 13, Article 2, before pupils are required to find these relations in concrete problems.
The object of picturing problems is not to teach children to make pictures (though all this work should be done with reasonable care), but to give a method of representation by which they can make their thoughts clear to themselves. It is a means, not an end, and should be so regarded. When problems can be stated clearly and solved correctly there is no further necessity for picture representation, except as a means of testing the pupil's comprehension of spoken or written forms. Let not objective work be underrated, however. It is a very necessary means, which, rightly used, will secure an accurate knowledge and use of terms, and save much time and confusion later on. Pupils should learn early to show objectively the difference between six and one-sixth of six, between one-sixth of six and one-sixth of one, etc.
The concrete problems of the book, involving numbers less than one hundred, may be first pictured and expressed in written form; second, used as drill in oral arithmetic, the pupil reading the problem and giving the solution orally; third, read by the teacher, the pupils solving silently, and either giving the answers orally or writing them on their slates.
Two-step and three-step problems, which, when worked out orally in the recitation, give an impetus to slow pupils, will often be found too difficult for a written test.
All measures introduced should be learned by actual use (see page 65). The standards in common use, such as the yard, foot, ounce, pound, quart, etc., can be obtained easily, and should form a part of the regular school supplies. Exercises in estimating volume and extension train the judgment while giving practical results in knowledge, and there is no time in the course when pupils can better afford to do this work.
The first two chapters contain the elements of everything that follows. Time will be gained by doing this part of the work very thoroughly, since its mastery will enable pupils to do the remaining chapters in reasonable time and without the constant aid of the teacher.
PART II. (Fourth and Fifth Years in School.) Review Part I. if pupils have not previously studied and mastered like work.
Definitions are now given in simple form. The rules should be made by the pupils after the process is learned from which the rule is derived. Be sure that all definitions are clearly understood.
Long Division is one of the difficult processes for children. At first they are unable to judge how many times the divisor is contained in the dividend or partial dividend. When about to commence Long Division much mental practice should be given with small numbers, 13, 14, 15, 16, etc., - such drill as 13 in 14, 13 in 15, 13 in 16, etc., to 13 in 117; same drill with 14, 15, 16, etc.
Some writers advocate having the children form a table of multiples of the divisor before they begin the example. Thus, in the example, 16530 ; 18 = ?
18 X 2 = 36 18 X 6 = 108 18)16530(918
18 X 7 = 126
Such plan may save time at first, but it should be abandoned early; if too long continued, pupils learn to depend upon it entirely, and are as much at a loss as at the outset. For additional suggestions, see “ Advanced Arithmetic,” page 58.
Review the first three chapters of Part II. to insure accuracy of results with reasonable rapidity.
The last chapters of Part II. present Common Fractions, Decimals, Compound Numbers, and Percentage. An extended treatment of these subjects is not attempted. Enough has been presented to provide an entrance to a more thorough study later in the course, and also to give to that large number of boys who are compelled to leave school early in life some knowledge of the business application of arithmetic.
This book has grown from experience, and is offered to fellowteachers as a thoroughly systematic work-book.
N. C. INDIANAPOLIS, January, 1893.