LONG DIVISION.

The necessity of the long division form grows out of the inability of the mind to grasp and hold all of the relations involved in large numbers: The memory does not try to retain the relations existing between all numbers, but relies on the judgment in the form of comparison to assist in discovering anew, each relation formed. The result of each comparison is then tested as to its truth. After pupils can divide accurately and quickly any problem using no divisor greater than 10, or better, 12, they are ready to have the form and process of long division. In beginning the form, have nothing to distract the attention from the process which leads to the form; therefore, use old problems in teaching the form. Give the class some simple, familiar problem, and require them to solve it by short division first; then lead them from the old form into the new by the same problem. Take the problem: Divide 724 by 4. Have it solved thus:

4) 724
181

and analyzed. One-fourth of 7 hundred is one hundred, with a remainder of 3 hundred; 3 hundred reduced to tens is 30 tens; 30 tens plus 2 tens make 32 tens. One-fourth of 32 tens is 8 tens. One-fourth of 4 units is 1 unit.

Show that the 3 hundreds remaining came by adding together the 4 different 1 hundreds and subtracting their sum from the 7 hundreds. That is 4 times the 1 hundred was taken from the dividend; or the product of the quotient by the divisior was taken from the dividend. Show that this is the case in each step. Pupils have been unconsciously doing this thing all the time, but they are now to be made conscious of it. Show that they

have been doing it mentally and have not written down. the result, but that they can write it out thus:

Now analyze the problem and the analysis will be found to be identical with the other analysis. After comparing many problems in this way, pupils will begin to decide on the advantage of this form. After skill has been secured in the use of this form, the pupils can be permitted to unite the partial quotients as they obtain them and omit filling out the orders in the several products. They can be shown to test the correctness of the work by adding together the several products and comparing the same with the dividend.

When the class is ready to use other divisors than the ones with which they are familiar, do not give them divisors indiscriminately, but help them to strengthen their faculty of comparison by allowing it time and opportunity to grow. The very act of comparison is difficult, so do not complicate difficulties by giving hard comparisons at first--those in which so much allowance has to be made in forming the products. After 12, 20 is much easier to divide by than 13. But be very careful here else the class will infer that the division of a number by a divisor of more than one order is analogous to the multiplication of a number by a multiplier of more than one order; and this, at this time, would plunge them into difficulties with which they are not able to cope, so have them use the divisor as a whole. Some short cuts can be taken after awhile. Therefore, after 12 use 20, 30, 40, etc., then 21, 31, 41, etc., 100, 200, 300, etc., 101, 201, 301, 102, 202, 302, etc. That is, use divisors first in

which no reductions will have to be made in forming the products, or no allowance made in the comparison; then the ones in which very little allowance has to be made, etc.

If pupils are led carefully through this work there will not be so much hap-hazard, guess work, and division will mean something more than a mere formal manipulation of figures. Of course, this means a great deal of work and preparation on the part of the teacher. I once knew a teacher whose class in division solved between 1,500 and 2,000 problems in division, and she had every one of them solved in her own note-book before she asked the class to solve them. But when those pupils were promoted to the next grade they did not have to make up "back work" in division, but they could solve neatly, quickly and accurately any problem in division with an integral divisor. This was the teacher's reward.

THE SCHOOL ROOM.

J. S. T.

[Conducted by GEORGE F. BASS, Surpervising Principal in Indianapolis Schools ]

QUESTIONING.

Much time and strength is wasted in the school room by poor questioning. In addition to this, bad mental habits are formed. It is an old adage "that any fool can ask questions." So he can, but they are not always good questions. Some people seem to take the view of school teaching that all there is to do is to appear on the scene about time to begin and ask the pupils questions to see if they have their lessons, and punish those who are neglectful, and help those who are weak. This help is in the shape of telling them the answers to questions in grammar, history and geography and "doing their sums" for them.

No teacher will admit that this is teaching, yet there are persons called teachers who act in their school rooms. as if they believe it. They ask questions seemingly, just to see whether the pupils can answer them. They ask, "What is a noun?,, The pupil says, "It is the name of an object." "What is an adjective?" "It is a word that describes or limits the meaning of a noun. "What is the size of North America?" "North America is more than 3,000 miles in width and about 4,000 miles in length. Its northern shores are covered with the snow and ice of the polar regions, while its countries in the south enjoy the green and the bloom of perpetual

summer.

Now if some one, the superintendent for example, should say that such questions are of little value and that there is no teaching in this sort of question and . answer business, there would be talk of an over-worked, over-supervised teacher who gets "beautiful" results in school when "let alone," but cannot do well in the presence of the superintendent, or visitors who have the reputation of knowing a good teacher when they see him at work in his school.

But let us turn our attention to these questions and the answers. The question, What is a noun? is a good question and the answer, It is a name of an object, is not a bad answer. But what does the pupil think when he gives this answer? What idea has he in mind? What is his idea of an object? Does he see that a noun expresses his idea of some object?

Does he see that the

Does he know that

noun is only the "sign of his idea?" he might express that idea by some other sign? Does he see that a noun is like everything else known to the senses in that it expresses an idea?

"What is an adjective?" "It is a word that describes

So it does, but does Does he see that the

or limits the meaning of a noun." the pupil know what he has said? noun and the adjective are alike in expressing ideas? How do the ideas expressed by each differ? Ask a pupil who has been asked such questions only, to parse the word red in the following sentence: I have a large red apple. He will say that the word red describes the noun apple by expressing the kind it is. Such an answer shows loose thinking and careless expression. He does not distinguish between the "sign and the thing signified." Does he know the difference between describing and limiting? Does he see that the idea expressed by the word red is related to the idea expressed by the noun? He may see all these and more and say just what he did in answer to the question, What is an adjective? But because he gave that answer it does not necessarily follow that he thinks or has ever thought these things. He has not properly thought the noun and the adjective until he has experienced the mental activities hinted at in the foregoing.

"What is the size of the earth?" In answering this question, the pupil gave the first paragraph on page 17 of the Complete Geography, Indiana Series. While only a line and a half has anything to do with the size of the earth, the rest is very interesting and expresses that which results from size and position of the continent. We are satisfied to hear the pupil say it all provided, he thinks the ideas expressed and thinks them in their proper relation. We should be sorry to have him think it absolutely necessary to tell about the everlasting snows of. the polar regions and the perpetual bloom of the summer regions in order to give the size of North America. But this would not be as bad as saying the paragraph without thinking, feeling and enjoying the thoughts expressed.