and soul.) But these external forms are preliminary conditions in which the realities thrive best. Automatic obedience is better than disobedience, and furnishes a suitable garment in which to clothe the obedience that flows from a sense of right, sense of duty, or from love of the one obeyed. The earnest, consecrated teacher will use the machinery of the school-room in such a way as to develop all that is highest and best in the human soul. To cultivate the higher realms of the soul much more than mere machinery must be used-so much more that many pages would be required to set forth a method of arousing in a child the highest and noblest ideals, impulses, and aspirations.

The following books on "School Management" present a more extended discussion of some of the topics mentioned in the foregoing pages:

School Management-E. E. White, American Book Co.

The Philosophy of School Management, Arnold Tompkins, Ginn & Co.

School Management-Samuel T. Dutton, Charles Scribner's Sons, N. Y.

Art of Class Management-Joseph S. Taylor, E. L. Kellogg & Co.

A New School Management-Levi Seeley, Hinds & Noble, N. Y.

Department of Methods

Easy Steps in Subtraction

Much of the labor attending the mastery of subtraction is eliminated by the use of the Austrian or "additive" method of finding differences or remainders*. By this method, the facts of addition are used in subtraction. Thus it is that the mastery of a single set of number facts suffices for both of these operations. There remains, however, the marking out of progressive steps by which the difficulties involved shall be taken up singly, and in the desired order.

The relation which obtains between the facts of addition and the corresponding facts of subtraction suggests that the time relation between the mastery of these two operations should be a close one. To determine exactly how close that relation should be is a difficult task. The mistake is often made of crowding the work in subtraction too closely upon the work in addition. This is the extreme to which the revolt against the topical arrangement has been carried. The golden mean is to be found between these two extremes. The teacher must find this mean for her own class. In doing so, she must guard against introducing the subject of subtraction before the class has been given a good start in addition. Again, she must not delay it until it becomes something wholly apart from the corresponding work in addition. Whatever the method followed in teaching addition is, much the same plan must eventually be pursued in teaching subtraction. But little headway can be made in column addition until several facts of addition have been perfectly memorized, and the work in subtraction should not be undertaken until such is the case.

As an illustration, let us presume that the class has mastered the following addition combinations (together with the inverse form of each) and has been drilled in the addition of columns involving these combinations:

As a step preliminary to the teaching of subtraction, let the teacher review these combinations orally, using the following form:

* For a more extended explanation of this method, see A Course in Elementary Arith. metic, McClymonds and Jones, pp. 37-39 (American Book Co.).

How many are 2 and 3? The pupils should be required to answer in the following form: 2 and 3 are 5. Next, let the teacher call for the same number facts, using the following form: 2 and how many are 5? In replying, the pupils should be required to use the following form: 2 and 3 are 5. From the previous work in addi3

tion, the pupils should know that the written form +2 asks, How

many are 2 and 3? They are now ready for the written form

5

which asks, 2 and how many are 5. It is -2. The following sub

traction forms that correspond to the several facts in addition should now be placed upon the board, as follows:

The class should be given a thorough drill in reading these expressions, until every pupil can read each without any hesitation. Beginning at the right, they should be read as follows: 6 and how many are 10? 3 and how many are 5? etc. Next, have the class read the expressions and give the required answers, thus: 6 and how many are 10? 6 and 4 are 10, etc. Follow this with exercises 59 95 in subtraction in which there is no "carrying," such as -24, -53, -124

-32. Do not fail to have the children begin at the right of each

means

exercise to subtract. After the pupils have become perfectly familiar with what is required in each of the above expressions, require them to give the answers without reading the expressions, as follows: 6 and 4 are 10, etc. Finally, require them to give the differences without naming the subtrahend or minuend. Give much drill in doing this. Teach the class that the sign less, that the lower number is taken away from the upper number. Dictate to the class thus: From 59 take away 24. Call their attention to the fact that the lower number is never greater than the upper number, and that when they are the same the answer is zero. Later, use the terms minuend and subtrahend. The pupils should become familiar with these terms through hearing them used.

In preparation for the more difficult task of subtracting when carrying" is involved, the pupils must be trained to tell at a

glance which expressions can be solved and which can not. Such expressions as the following should be read, and pronounced as ones 2 3 2 0 9

which can or cannot be solved: -5, -0, -2,

-6, -4. The pupils

should be required to state why some of the expressions can be solved, and why the others can not be solved. The teacher must here accept reasons such as the following: "Because the little number is on top," etc. To teach "carrying," begin with such an

2

expression as -9. The class will pronounce it one that can not be solved. Place the figure 1 before the minuend, thus changing it to

12

- 9, and ask if it can now be solved. Treat in a similar manner:

2 0 0

-3,

-4, -6. Ask the pupils to tell how one that can not be solved

12 52 may be changed into one that can be solved. Pass from 9 to -29.

The class will see that the 2 in the minuend must be changed to 12. Require them to make this change mentally, and to retain it without indicating the change in the exercise. After they have solved that part of the exercise, tell them that since they changed the 2 above they must now add 1 to the next lower number. So the 2 in the subtrahend is changed to 3. The class should learn that this change must always be made in the next lower number, following a change in the upper number. The reason for such a change as told to them is, "Because the upper number was changed." Making only a single change in the character of each succeeding exercise, solve such exercises as the following:

52 52 52 52 50 50 50 50 4022 -29, 19, -13, -23, -24, -26, -14, -24, -1329

The class has now had exercises in which "no carrying" and others in which "carrying" was involved. As the next step, exercises in which both occur should be given. Here it is that much confusion and many errors will be avoided by the fact that the pupils have, in the meantime, been required to distinguish carefully between the expressions that cannot be solved and those that can be solved. Notwithstanding this, the teacher must be prepared to meet the confusion that will result from lack of experience in dealing with exercises in which both occur. Exercises such as these should

924

5003

follow: -492, -1363, etc.

The number of such exercises that

can be made from the group of subtraction facts given is very great. The teacher must determine the amount of drill needed by her class. She must recognize, however, that among the many exercises which she may provide there must be the conscious introduction of exercises which contain the several types of difficul524

ties that must be met in this work. In the exercise, 92, after

subtracting 9 from 12, the pupils must be led to see that since there is nothing under the 5, the 1 is added to nothing, then sub902

tracted from the 5. In the exercise, -493, the 9 in the subtra

hend becomes 10. The question then is, 10 and how many are 10? As the 0 in the minuend was changed to 10, 1 must be added to the 4 in the subtrahend. The teacher should explain such difficulties to the class before assigning exercises in which they occur.

The following exercises illustrate the several difficulties that must be foreseen in the assignment of work to the class:

595 550 526 320 502 502 594 520 1022 -245 -230-290 -284 -193 -103 -42 - 86

83

5624
92

carrying "

Although the explanation of the changes made in " seems meager, it will serve the purpose better than a more elaborate one. What the child needs is something that will help him to do his task later, understanding will have its place.

:

D. R. JONES

State Normal School, San Francisco.

Some Useful Pictures in Composition

BY HENRY MEADE BLAND.

Repeat a story to your children till you have led up to its most interesting part. Request the children to finish the story by drawing from their own imagination for the material. Use only stories entirely new to the children. Suit the story to the age of the child.

It is suggested that, for example, in the "Story of Vulcan," you leave the children to tell how Juno got out of the wonderful