accumulate data regarding all these matters in his own city, and have them for his pupils year after year. The gathering of the problems at the outset would involve some labor, but once collected they would be of service without modification for a considerable period. The point is that what is needed in arithmetic is to apply it to the practical situations presented in the pupil's daily life. Most of the necessary drill can be secured through the solution of such problems. It is not intended to say that problems should all be of this character; but most of them should be. It is cause for rejoicing that the new text-books in arithmetic are eliminating the formal, remote problems, and bringing the study right to the door of the pupil, and making it interpret and illumine his environment. This kind of arithmetic will make a pupil more appreciative of what is going on around him than he would be without it. He can be led to think of the amount of rainfall in his region in precise terms, the amount of energy generated by a ton of coal, the average growth of plants per day, the relative amount of heat energy expended by the sun at different seasons, and so on at any length. If arithmetic could be generally treated in this way, it would become of far greater interest and greater dignity than it has been in the past, and than it now is in many communities where the traditional attitude toward it is maintained. We may look now for a moment at inaccurate thinking in this field. Many of the errors in arithmetic made by pupils after the fourth grade are due to their in The cure for inaccurate thinking in this field ability correctly to interpret the relations expressed in problems. When a teacher finds pupils inaccurate in this way it will do no good for her to say, "Now, be more careful next time," or, "If you do not pay closer attention, I will keep you after school", or, "I will put you back in a lower grade", and the like. Pupils who have not formed habits of accuracy can not correct their inaccuracy by simply saying to themselves, "Now, I must not make any errors." It is an easy thing for us to assume that a pupil can on his own initiative eliminate errors from his work, if he only wills so to do; but experience should teach us that no good comes from threatening or exhorting pupils who have developed inaccurate methods of work. The efficient teacher will analyze the situation before her, seeking to discover the cause of a pupil's errors; and then she will set about developing new habits. For after all, inaccuracy is a habit of mind which is the result of a relatively long process of doing a thing in a certain way. This habit can not be broken up in an instant; it can be corrected only by slowly building up a different sort of habit, which will in time replace the undesirable one. This last point will bear elaboration. Most teachers find that their chief difficulty in the teaching of The evil of in- arithmetic to young children is to get accuracy in them to be accurate in their work on school work their own initiative. Indeed, in some schools the only trouble teachers encounter is in respect to the errors which even the brightest children make. Any observing teacher knows that young pupils do not readily detect their own errors; and this is, of course, true of them in other work than in arithmetic. When the novice executes anything, whether it be a process in arithmetic, or spelling a word, or writing a sentence, he is practically unable to go back over the detailed steps, and detect the one that is wrong. It is a trait of the child mind to view things as wholes; and once executed they must be right. This is one reason why many teachers accomplish so little when they give the following direction to their pupils: "Now look over your work, and see that it is correct." It is difficult enough for even an adult to detect errors in what he has done. The very fact that he has solved a problem in a certain way, or constructed a paragraph after a given pattern, is evidence that he thinks it is correct; and when he goes over it he tends always to see it as correct and not as erroneous. If it is difficult for the adult to restrain the tendency to see as correct what has once been executed, how much more likely is this to be characteristic of young pupils. And how futile it must be to keep urging them to "look over your work to see that it is right." And yet we must, to the fullest extent possible, develop in our pupils the ability to review their work Self-correction of and detect errors. They can be inaccurate work made self-helpful in eliminating mistakes from all their work, but especially from their arithmetic, by requiring them always to check every process and "prove" every problem. No solution of a problem should be accepted from a pupil until it has been checked. When a pupil goes back over his own work and discovers his error, this furnishes the greatest precaution against his making the error again. In this way he learns what his tendencies are, and he will be on his guard against them. Then in this work of checking, pupils receive valuable drill in performing the fundamental operations, and in seeing the relations in a problem in every way they can be viewed. To know how to check a problem is just as valuable as to know how to solve it. Of course, as pupils go on into the fifth or sixth grade, there may be no need for checking when they |