How many ones in 4? How many twos? Threes? One-half of 4 is what? Two is of what number?

Problems containing the foregoing combinations are then given in great variety by the teacher until all of the facts about the number 4 in its relation with the smaller numbers are fully mastered.

In teaching any number, no larger number must appear in any way whatever. During the study of 4, it is not permissible to ask 4 twos, or that 4 is 1 less than what, etc., etc.

The work proceeds slowly and thoroughly, at least a year being devoted to the numbers below 10. The second year is given to the numbers from 10 to 20, and the third year to those from 20 to 100. This is probably as far as the method is carried in this country.

In the greater number of the schools using this method, systematic instruction in the fundamental processes is commenced by the beginning of the third year; while in some, the Grube method is used for oral work, and the teaching of slate addition is carried on at the same time, even during the first year.

Slate Problems. When, instead of receiving oral instruction for some time, children are taught processes from the outset, it frequently happens that many of them show little ability in solving problems. While some attention should be given in the early years to this side of arithmetic, it should not be permitted to retard too much the advancement of pupils. Many of them have to leave school soon, and they should be taught as rapidly as is consistent with real progress to perform accurately the ordinary operations in whole numbers, simple fractions, and decimals. Being familiar with these tools, greater maturity will, of itself, show which is to be used in such questions as are likely to come up in ordinary avocations.

The teacher should exercise much care to give only such problems as can readily be understood by the pupil, and which do not contain too many conditions or numbers that bewilder

the learner. While a beginner will have no difficulty in determining whether to add or subtract in a mental problem suited to his capacity, the same kind of problem with larger figures will give him much difficulty. For this reason, the earlier slate problems should be the merest trifle beyond his ability to solve mentally. In his attempt to work them out in his head, he will determine whether addition or subtraction is needed, etc.

Problems in all grades should be "miscellaneous," and pupils should be allowed as far as possible to determine for themselves what operation is necessary to solve any given one.

IV

NOTES ON CHAPTER ONE

THE hints given as to the work of this chapter are intended chiefly for the guidance of teachers of young children that are beginning slate work in the fundamental processes without much preliminary oral instruction. Pupils that have been taught for two years by the Grube method should not be required to spend unnecessary time on the simpler portions of the work.

Art. 4. In teaching notation of numbers of two figures to young children that have not been previously taught by the Grube method, it is not advisable to lay stress on the local value of the tens' figure. Show them how to read and write 10, 11, 12, etc., to 20; then 30, 40, 50, etc., to 90. After this, there is but little difficulty.

7. By working an example for the pupils, teach them to place under each column its sum. As their tendency is to begin working at the left, be careful to see that they always commence to add at the right.

9. The problems will present no difficulty, as they involve only addition.

11. These sight exercises may first be employed as drills to teach children to use in blackboard addition as few words as possible. The first figure should not be named, only the sum of the first and the second, then this total added to the third. In subsequent drills upon these combinations, each pupil should, in turn, give the sum of any set indicated by the teacher. work should be done rapidly to be of value.

The

13. The making of original problems by the pupils should be a feature of every grade.

15 and 16. Subtraction is here introduced by the "buildingup" method. Pupils find it easier to ascertain the difference between two numbers by going forward from the smaller to the larger, than by "taking away" one from the other.

17 consists of sight exercises in the form of addition, leading to the subtraction exercises in Art. 19.

21. While in adding, the use of the word and is considered unnecessary; in subtracting, it is used just before the figure that is to be written.

For some advantages obtained by employing the "buildingup" method, see Art. 384, where it is used to obtain in one operation the difference between 1000 and 643 + 287 + 25. In Art. 385, it is used to find a remainder in long division without writing the product of the divisor by the quotient.

23. Here begins the real problem work, as the pupil has now to determine for the first time in slate examples whether the result is to be reached by addition or subtraction. When the pupils are able to solve one of these problems without using the pencil, it should be repeated, but with such a change in one of the numbers as will render necessary the use of the slate. For the 10 cents in the first example, for instance, 14 cents or 24 cents may be substituted.

As many pupils attend rather to the numbers in a problem than to its terms, some may subtract when they should add, especially as this seems the natural operation when only two numbers are involved. It is important that they should be led to see that the size of the numbers does not change the nature of the example, and that they can easily determine whether addition or subtraction is required, by considering what operation

they would employ in a similar example containing very small figures.

It is not advisable as a regular thing to follow an oral problem by a written one of exactly the same nature, as this tends to make children inattentive to the terms of the latter when they already know from the oral problem what operation is required.

28. It is inadvisable to waste time in endeavoring to make clear to very young children the reason for "carrying."

37. Teachers should require pupils to write the proper sign. before working an example, as this tends to make them listen more carefully in order to determine whether addition or subtraction is involved. In some problems that are too simple to need the use of the pencil, changes may be made in the numbers employed; great care, however, should be taken not to use numbers so large as to confuse the pupils.

38. Have children understand that when a number contains the word "hundred," it should consist of three figures. Do not explain.

54. These exercises are intended to lead up to the subtraction with "borrowing" in the next article. Perhaps the following would be a better arrangement:

As children are generally taught to begin with the bottom figure in addition, they will naturally say in the first example, 9 and 2 are 11, writing the 2 in its place, etc.

55. Subtraction with "borrowing" is generally taught in one of three ways. The "building-up" method given in the text is the most readily taken hold of by young pupils.