In our first article we arrived at certain conclusions: It is a mistake to require children in the first year's work to take up the formal study of numbers. It is a mistake in the study of any number to present to the child's mind a great variety of combinations. It is a mistake to deal with these combinations graphically. It is a mistake to allow children to solve many problems at their desks.

The rational plan is obvious. Have children begin the formal study of numbers when they are ready for it. For most children, the third year of school life would be soon enough. For two years make the work purely mental, having all the operations performed under the immediate supervision of the teacher. These problems should involve all of the fundamental processes, addition, subtraction, multiplication, division, together with common and decimal fractions and denominate numbers.

In the preceding article we have indicated clearly that numbers should be taught incidentally during the first two years of the child's school life, taught in connection with science and language work. In the third and fourth years this teaching should be continued. Beyond a doubt, considerable written work could be presented during this time, but we do not consider this desirable, because there is no positive gain in mental power by dealing with written work until the habits of performing the operations are mentally well established.

The reader will no doubt feel that something more specific should be said concerning just what should be taught in the consideration of any particular number. In illustrating we will consider the numbers 8 and 9; 4 plus 4 equals 8, 5 plus 3 equals 8, 6 plus 2 equals 8, 7 plus 1 equals 8; 8 minus 7 equals 1, 8 minus 6 equals 2, 8 minus 5 equals 3, 8 minus 4 equals 4, 8 minus 3 equals 5, 8 minus 2 equals 6, 8 minus 1 equals 7. Two fours are eight. In eight there are two fours. One half of eight is four. 5 plus 4 equals 9, 6 plus 3 equals 9, 7 plus 2 equals 9, 8 plus 1 equals 9, 9 minus 8 equals 1, 9 minus 7 equals 2, 9 minus 6 equals 3, 9 minus 5 equals 4, 9 minus 5 equals 4, 9 minus 3 equals 6, 9 minus 2 equals 7, 9 minus 1 equals 8. One third of nine equals three. The reader will observe that in the consideration of these numbers we have broken them, with a single exception, into two parts. No hard and fast rule can be used in determining just how much we should teach in relation to these numbers. To the mind of the writer, one this is evident enough and that is, that the schools of today are demanding altogether too much.

It is acknowledged that the ability of the child to read depends in part upon his power to call words at sight. Just as soon as there is a conscious effort on his part to determine what the word is the child is shut off from the possibility of expressing thought with any degree of ease or attractiveness.

In teaching numbers the same principle should be recognized. Have the child thoroughly taught a few combinations, and only a few. If he has thoroughly mastered these few combinations he will be able to use them just as he uses the knowledge of words. Some critic will was, why not present the written along with the oral? We answer, that for the simple reason that nearly all of the elementary schools have within the last fifteen or twenty years drifted towards the exclusive use of pencil and slate, or pencil and paper. This usage violates present social conditions. This may best be illustrated by considering a common error in teaching language. The majority of mankind use chiefly spoken language. All of the important relations of mankind are conducted through the use of speech. This does not imply that the written is valueless. It simply implies that the spoken should receive the attention it merits. In numbers no one will contend that in emphasizing the mental we sacrifice anything in the way of discipline. In fact it is all a matter of gain. If one were to acquire a knowledge of arithmetic solely for business purposes everything would be gained by emphasizing the mental rather than the written.

So long as all the smaller numbers are broken into as many parts as the ingenuity of the teacher can suggest, just so long will the majority of pupils in all the grades fail utterly in accuracy and rapidity. The child's mind cannot store up an infinite number of combinations. A few combinations mastered, a few combinations always at instantaneous command forms the only sure basis for higher arithmetical achievements.

Lest some readers should misapprehend, I wish to emphasize further the importance of outlining written work. If for example the numbers 8 and 9 as outlined on this paper are put before the child's eye in graphic form he's obliged to consider the written language of numbers. At this time he has no special use for it, and unless the teacher is more than ordinarily successful the child that answers every one of the questions promptly and correctly, if asked independent of the black-board, will not infrequently fail when asked in relation to written questions.

For some undiscovered reason the child often proceeds in adding by analysis, and analysis by ones, a form of analysis that is thoroughly destructive of discipline.

The thoughtful reader has already asked how fractions are to be taught orally. While we have said very little about how we should teach the numbers even to ten, we have assumed that the larger part of the work, at first, would be taught in relation to things, and that the things, as such would be discarded just as soon as the pupil has power to picture them in the mind. His home experience and his school experience has done much to acquaint him with objects that are bought and sold. These are not presented if the memory has stored away the pictures. Problems in relation to things, without so much as hinting that it is difficult, will have no more trouble in teaching orally the subject of fractions than he has had in teaching integers. Nine tenths of all the difficultly in teaching fractions grows out of the use of the blackboard and slate at the wrong time. The average boy understands all about half an apple before he enters school. If the teacher continues to use the half just as the child has used it he will find no difficulty whatever. The same thing may be said of teaching one third, one fourth, one eighth and one tenth. In relation to things, fractions can be added and subtracted just as readily as integers can be added and subtracted. If the work in multiplication and division is properly presented there will be no multiplying a fraction or dividing a fraction by a fraction. It matters not whether an apple be used, an orange, a potato, a stick or some other object. Possibly if there is a choice, pieces of paper six or eight inches square are most serviceable in teacher fractions. The paper can be folded so as to show two halves. The remaining crease will show the division. Folded again quarters are clearly shown. The work in addition, subtraction, multiplication and division can be performed in relation to this one sheet of paper. Many problems can be given in relation to objects with which they are familiar.

In considering denominate numbers the children will especially enjoy the work in so much as they will have occasion to deal with matter by means of some form of measure. If the work is done in the usual routine manner, only one thing will be accomplished. The child will become disgusted with the so called application of arithmetic to the material world about him. If he learns to handle the foot rule and yard measure, if he learns to manipulate a pair of steelyards in weighing scores of objects to be found in his own school room, there will be an enjoyment in the application of arithmetic never to be forgotten.