appreciation, and it should not be reduced to automatic facility. This is equally true of most of what is taught in literature, geography, nature study, geometry, and the like. But the greater part of arithmetic will be of genuine value to the individual only as it can be used without any conscious effort or direction. Again, one who must "think" how he should spell a familiar word has not been well trained in spelling. The same is true of writing; and it is just as true of most of what will be used in arithmetic. However, many teachers regard the latter subject as one appealing to reason, and they teach it with a view to training reason rather than to acquiring facile habits. But a pupil who is left to "reason out" the product of two numbers when they are to be multiplied, or the sum when they are to be added, or the remainder when they are to be subtracted, or the quotient when they are to be divided, is not properly prepared in arithmetic for every-day needs. In the same way, an adult who must "reason out" the application of most of the tables to the practical situations of life has not been well trained.

Take two children who are sent on a shopping errand. They must go to a grocery, purchase certain articles, and pay for them. One calculates automatically the amount to be paid. He does not have to go through a conscious and therefore more or less

laborious process of comparing the numbers involved. He has so often seen certain figures together with their sum or product or quotient or remainder that now the moment he beholds them in any given relation the inevitable result appears instantly in his mind. To illustrate the principle: an adult, when he sees two and two together in the relation of addition, immediately sees four. He does not have to go through the process of building up from two to the amount of two more. He automatically associates 4 with 2+2. In the same way, any properly trained adult who sees eight and four in the relation of multiplication instantly thinks of thirty-two. It is not necessary that he should count up four eights to see how many they make in total. But some teachers who endeavor to make these elementary processes automatic with their pupils, never attempt to do the same in dealing with more complicated relations. The second boy mentioned above suffers from this latter kind of training. He must take time to work consciously through all his processes; he must "reason out" everything he does. Which boy is better trained in the use of numbers? What has the second boy gained from his reasoning process that the first boy has missed? The first one jumps over a number of steps which the second must take slowly. There is no more value in taking these short

steps in arithmetical work than there would be in walking. One person might take a six-inch step while another could step three feet, covering all the intermediate points in one stride.

Of course when a pupil first performs any new process it is necessary that he should see the reason why it must be done in a certain way. This is important so

Relation of reasoning to automatic facility in arithmetic

that in the future if need be he can work his way through problems somewhat like this, but the solution of which he has not made automatic. However, in all arithmetical processes which will need to be repeated often in practical life, the plan should be to make them so facile that the pupil can execute them without hesitation, or even without consciousness of the detailed steps involved; and this can be accomplished only by causing him to repeat the application of a principle until he gets beyond the point of having to think it through, in the sense that he will not wonder what he should do, and his mind will not run here and there, because it has learned to proceed in a certain definite direction.

Much of what teachers call reasoning in arithmetic is nothing but the mind of the novice wandering into by-paths because he has not learned which path will lead to the goal he wishes to reach. In the affairs of daily life, the expert can go straight to

the mark, while the novice must try this route and that one, because he does not know which is the proper one. There is no advantage in the development of the mind for the novice to keep trying wrong routes, and having to return when he has discovered his errors. Many people are left in some such a condition in respect to many of the adjustments of daily life. They wander here and there because they have not learned how to decide at once which course to pursue, and so they proceed slowly, and are always making errors. The expert in any field, who has made many processes automatic, rarely makes errors, and he gets forward rapidly; while the beginner never knows when he is right, and he is apt to be halting and indecisive in all he does.

In the teaching of arithmetic, it is the common failing of teachers to leave principles hanging in the

Applying principles until their right application becomes "second nature"

air, so that every time a pupil has occasion to use them he must try various routes before he happens more or less by acBut the wise teacher,

cident upon the right one. whenever any principle is developed, will have the pupil apply it in so many ways that he will get a secure feeling for the way in which the thing is to be done. It will become "second nature" to him.

This means that he must be presented with a great variety of concrete situations in which any principle is to be applied; and the more skilful the teacher is in making interesting and practical problems involving the application of principles, the greater success she will have in making her teaching secure in the pupil's life. Take, for instance, the process of computing areas. When this principle is being considered, the good teacher will have her pupils determine the areas of so many surfaces in the schoolroom and outside that a habit of proceeding in a certain definite way when area is to be determined will become thoroughly fixed. But if only a few book problems are solved one week this year, say, and then a few more solved next year, there never can be developed automatic facility in handling these problems. The principle applies to the teaching of all processes in arithmetic.

analysis

Many teachers think there is some extraordinary value in having pupils "analyze" every problem they Danger of over- solve. Often one sees a pupil emphasizing who can grasp the relations presented at once, and reach the solution without delay; but when he comes to state what he has done according to the formula insisted upon by the teacher, he may be slow and incompetent. The writer knows a pupil who is marked very low in