In the closing of our third paper, we made an earnest protest against wasting time in explaining long problems in addition. We now make a like protest against long explanations in subtraction, multiplication, and division. The work should be done mechanically and every result should be verified. Rapidity and accuracy are the ends sought.

In solving problems the first time over in multiplication, pupils should not be required to learn short processes. Beyond a doubt, much valuable time is wasted in attempting to teach short methods even to advanced pupils. In this brief life, a man is indeed fortunate if he knows one good way of doing a thing. Where he is familiar with more than one way, he frequently loses time in making a choice. In this, it is not implied that the student should go through tedious processes in order to multiply by ten, one hundred or one thousand. We simply hold that whatever process is taught, it should be one that is to be used throughout life. Even separating the multiplier into factors and multiplying by each is a doubtful value, for the simple reason that in the countingroom this method will not be used.

In division, much emphasis should be placed on short division for the simple reason that the work is chiefly mental. It is also the form of division that is used more frequently by business men. Pupils should be trained to avoid long division whenever it is possible to do so. In long division children frequently encounter serious difficulties, the chief of which is to determine how many times the divisor is contained in a given number of figures of the dividend. In long problems the difficulty can be obviated by training the student to place the products of the divisor by 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively, in a column. In order to determine each figure of the quotient a product is compared with the dividend every time a division is performed. For the beginner this is a real saving of time and labor. Advanced students should not be allowed this privilege under any circumstances. What has already been said of multiplying by ten, one hundred or one thousand, may be said of dividing by ten, one hundred or one thousand.

This wise teacher has been applying addition, subtraction, multiplication and division more or less to things of real life; in other words, the pupil has already been introduced to the calculations of the business world; nevertheless, at this point, what are commonly termed miscellaneous problems should be introduced and the child should be taught all of the applications to business involved in buying and selling. He will have occasion to use these forms of reasoning in fractions, denominate numbers, measurements, percentage, etc.

The mistake of mistakes in teaching arithmetic has arisen from the belief that in arithmetic there are many distinct cases, whereas there are comparatively few. The principles of analysis employed at this stage of the work are identical with those employed in nearly all the other parts of arithmetic. We will illustrate by the use of mental problems: (1) What will twelve oranges cost at five cents apiece? (2) How many oranges can be bought with forty five cents at five cents apiece? (3) If three oranges cost fifteen cents, what will seven oranges cost? (4) If three oranges cost fifteen cents, how many oranges can be bought for thirty-five cents? These four problems are typical, in reality they involve but two problems, the third and fourth being resolvable into the first and second. The analysis of each of these problems is so simple that we will not consume valuable space in Education Extension by giving it. These four typical problems reappear in fractions, in profit and loss, commission, in fact, in every application of percentage.

By the old method of teaching arithmetic the children were taught a rule for every case in every so-called division of arithmetic. This was positively pernicious because the grandest powers of the human mind were put to no use. Modern education demands that the laws of relation should be obeyed. To the mind of the writer, the revolution of revolutions in teaching involves the problem of relations. The new geography, the new zoology, the new sociology, etc., make relations fundamental. A man out of relation to his fellow-man, isolated, so to speak, has no use for the Decalogue. The case in arithmetic, isolated from other cases, has no significance, no value. Every case in arithmetic is related to some other case, in other words, arithmetic is a science of relations. When the pupils see clearly that there are but two forms of analysis, four when the two are amplified, and that these four forms will serve him in all the difference cases of arithmetic, he will be delighted with the simplicity of the study. Most pupils dislike miscellaneous problems. When they fully realize that the daily affairs of life involve only these four problems, they will find arithmetic a delightful study. Therefore, continue giving these miscellaneous problems until the pupils can analyze and solve them without hesitation. The teacher should bear in mind that addition, subtraction, multiplication and division may have been thoroughly mastered and yet the pupil remain powerless to solve miscellaneous problems. Realizing this, the teacher will give much time and attention to this work. If needs be, hundreds of problems will be selected from different textbooks. The teacher will direct the pupil's attention to a new topic only when the pupil shall have become thoroughly familiar with the principles of analysis. Should the teacher neglect this, the pupil will be absolutely helpless when he attempts to solve miscellaneous problems in fractions; miscellaneous problems in decimals; miscellaneous problems anywhere in the arithmetic. The teacher cannot excuse himself on the ground that it is difficult to teach pupils this analysis.

A little reflection will lead the reader to recognize the fact that even men and women who have not attended school, who know little or nothing of books, are constantly using these forms of analysis in daily life. The success with which men and women use these forms of analysis determines in a large measure, their success in life. The man who uses these forms of analysis readily is said to have good sense, practical sense, gumption. The man who cannot thus analyze is constantly making blunders; "making two holes in the door, one for the old cat and one for the kitten," is always doing the wrong thing at the wrong time. Things are out of relation with that class of people. We have devoted much time to emphasizing analysis, inasmuch as we believe that nearly all of the difficulties of arithmetic fall within the embrace of the word analysis. Granting that a pupil is rapid and accurate in addition, subtraction, multiplication and division, very little remains for him to learn except analysis and the application through analysis of his four fundamental processes to the requirements of business.

Inasmuch as it is not the purpose of the writer to present a text book of arithmetic through the columns of Education Extension, only a few words need be said concerning that portion of arithmetic which ordinarily allows miscellaneous problems after division. The next topic is labeled "Properties of Numbers." Most authors give some little attention to the terms prime and composite, odd and even, etc. Not enough attention is usually given to learning the prime numbers from one to one hundred. Very few high school graduates can give correctly the prime numbers from one to fifty inclusive. If successful work is done in least common multiple, in greatest common divisor, in factoring, the pupil must have mastered prime numbers as already indicated.