From the standpoint of utility, there is no occasion for teaching greatest common divisor. From the standpoint of factoring and the further application of the principles of division, problems in greatest common divisor may have a place in a good all-around arithmetic. All of these problems should involve numbers that can be factored readily without the use of pencil and paper. In order to determine the greatest common divisor, the factoring method should be employed exclusively.

Least common multiple is of special value from its application in changing fractions to equivalent fractions having a least common denominator. Again, we ask teachers to use problems involving all numbers such as can be factored mentally. Here we advise that the factoring method be employed.

In cancellation a strong effort should be made to emphasize factoring. It is an excellent opportunity to extend the pupil's practice in this valuable line of work. From this standpoint much can be gained by separating the terms of the dividend and divisor into prime factors. Then the application of the principle that dividing both dividend and divisor by the same number does not alter the value of the quotient, can be brought into use by the pupil. Teachers who have required their pupils to solve all problems in cancellation after this method have acknowledged that there was very little loss of time, and that what was gained in factoring was ample compensation for the extra time used. The second aim in teaching should be to impress upon the mind of the pupil the practical value of cancellation in computing interest, paper hanging, carpeting, stone and brick laying, etc.

In preceding articles we have made an effort to impress upon the mind of the teacher the importance of simplicity in teaching arithmetic. Greatest common divisor, least common multiple and cancellation are only special applications in division. They are not to be considered by the teacher or the pupil as something strangely new.

It would be exceedingly difficult to name a subject that taxes the ingenuity of the average teacher to a greater extent than does fractions. In the judgment of a writer, this arises largely from the excessive use of the pencil and tablet. In other words, mental arithmetic has received so little attention that very little arithmetical power has been developed. Other hindrances arise from requiring very young pupils to take up the subject. True, it is absolutely impossible to think at all without giving consideration to idea of number.

This does not argue that arithmetical computations are necessary to the development and training of the children of the lower grades. In fact, modern methods of teaching arithmetic in the lower grades have a tendency to hinder and paralyze mental effort. As a consequence, when the class reaches the subject of fractions in so-called written arithmetic, the pupils are unable to think; they are totally unprepared for the work that confronts them. It is our business to consider things as they are. The successful teacher must always do this. Let every teacher who reads this article give the suggestions we offer upon the subject of fractions serious attention. It is unwise to attempt to do written work in the ordinary class that is just taking up the subject of fractions. The work should lie along the line of developing in the minds of the class clear notions of fractions. The teacher will be obliged to use objects. One of the simplest aids lies in the use of paper cut in squares of uniform size. These squares can be folded under the direction of the teacher in many ways, and many interesting problems can be made from these foldings. When the square is opened the creases show clearly the divisions. The pupil should be required to solve every problem connected with the paper through the use of the paper. A two weeks' or even a four weeks' development course can be pursued with profit. The teacher having developed in the minds of the children clear notions of fractions, easy problems relating to the things with which the children are familiar can be given by a successful teacher, and the teacher thus secure a foundation even for the little written work he feels compelled to give. If pupils can perform mentally operations required, they cannot fail when they come to use pencil and paper.

Two things in teaching arithmetic are ever to be kept in mind. First, developing of the reasoning faculty. Second, securing of skill through training and drill. Assuming that the class has had the necessary mental training in fractions; assuming that they are thoroughly familiar with the different cases, the teacher may wisely attempt to do more or less written work. Under each case of written work a large number of mental problems should be given. Another point to be kept in mind is that even in written work the pupils should be trained to avoid the unnecessary use of the pencil. The pupil should be taught that the unnecessary use of the pencil is a waste of time and hinders mental activity. Another important fact is with reference to picturing numbers mentally.

Where paper and pencil are used habitually pupils utterly fail in picturing numbers in their mind. We have had occasion to examine several arithmetical experts - experts in adding, subtracting, multiplying and dividing. In every instance we found that they had the power of holding figures in their mind; the power of performing the operations with marvelous rapidity. The modern pupil not infrequently obtains the correct answer but can't recall the problem. Others forget the problem before they have succeeded in completing the solution. So far as the writer is able to learn, no special effort has been made on the part of the teacher to cultivate this picturing power with the view of making arithmetic work more thorough and more satisfactory. We have wandered a little from our line of work, but in this wandering recommendations have been made that cannot be repeated too often.

We assume a third point in the teaching of fractions in written arithmetic. I refer to the student's having a thorough knowledge of the definitions and laws used in fractions. With this preparation the class may be given for their first case, as it is commonly termed, the reduction of fractions to equivalent fractions in their simplest form. Right at this point we would caution teachers in regard to using fractions having for their terms large numbers. An immense amount of time is lost and there is no compensation for this loss. Fractions with large terms do not appear anywhere in business. They are not necessary for mental discipline, in fact, they are a hindrance to mental discipline. There can be no good reason for their use unless it is to teach patience, and this important virtue can be taught in a better way. Suppose the pupils are asked to reduce to equivalent fractions in their simplest form the following: ^{48}⁄_{96}, ^{24}⁄_{72}, ^{15}⁄_{60}, ^{75}⁄_{100}, etc. If the pupils had had the necessary training, they would give the results without the use of pencil and paper. Require them to separate the terms of each fraction into their prime factors and thus make it a problem in cancellation. The pupil will not get too much practice in factoring. He will be surprised to discover that he has in reality nothing to learn. He is again studying division in a form identical with the form he has already learned. Before leaving this case, I would advise teachers to have problems solved without the work of factoring. Require pupils to become adepts in rejecting a large common factor from both terms of the fraction. In this paper we have gone into preliminaries extensively in order to secure the mastery of fractions. Any class of boys and girls that can solve readily the miscellaneous fractions in our best written arithmetics and explain the same, can master percentage and most of the remaining subjects of an arithmetic with a degree of ease that is little less than marvelous. In fact, in percentage the pupil has little more than the language of business to acquire.