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off three apples. Keep the six apples so they can not be seen; but show the three apples, say Mary has eaten them and they are gone, and throw them away. Ask, How many apples here on the paper behind me?" They will all know at once. They can illustrate this problem by drawing the nine apples, and then crossing out or erasing the three that are eaten. Counters of some kind should be used in illustrating subtraction very often. Do not think that time spent in this way is thrown away. You are teaching the science of number, and the time spent now will be saved over and over again when we come to use figures. I have seen a pupil try to illustrate a problem like the one given above by first taking nine crayons for the nine apples Mary had, then three more for the three she ate, and finally six more for the six she had left. He was one of the kind who, when asked why they subtract, point triumphantly to the word "left."

This amount of work is about as much as should be expected from a young class the first year. If it has been done so that the teacher is convinced that they thoroughly understand it all, and there is still time left, they may be taught to count as far as 100. Wooden toothpicks and the smallest rubber bands are the handiest things I have found for this counting. Let them count ten and bunch them, or illustrate:

Have a number of toothpicks fastened into bunches of ten, to remain so, in order to save time. Tell them to show you eleven, and have them take a bunch and one loose toothpick, or illustrate:

Notation.-Except with a very bright class, leave figures and notation till the second year. To teach them during the first year is very apt to efface the idea of number we have so carefully tried to build up, and put figures in the place of the mental conceptions we have been trying to give our class. Unless you are very sure that these conceptions have become a part of the thought of every pupil, do not, even in the second year, begin with figures until after a thorough review of all the number work already done; and then work with your illustrations and counters fully as much as you do with figures. Teach the first nine numbers by writing and making them write:

When you get this far, tell them you have no more figures, and ask what is to be done. Call attention to the bunches and to the illustration:

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Ask if they think it would do to say one bunch" instead of "ten." Ask if they do not think it might be a good plan to write it :

They will probably be very willing to accept that; but tell them that, while it may be a good plan, we long ago found out a better and easier way, by moving the figure to the left, and that we always write:

Go on and explain that eleven is one ten and one over, as they know. That the figure 1 for the ten is written just as before, and means ten, combined, tied into a bunch, put in a pen; and the one over is written just as any other one:

Explain that the first figure is one, but that it means one ten, and stands for a bundle or a

and the second figure also means one, but stands for the toothpick not in the bundle, or the mark not in the pen.

Explain twelve, thirteen, etc., in the same way. When you get to twenty ask how you are to express that in figures. Call attention to the two bundles or the

If none of them catch the idea, ask if you shall write it

2

Tell them we do not write figures that way, and ask how we write one ten, one bundle, etc. They will probably then tell you to put the 2 where you put the 1 before. If they do not, go back to ten and illustrate it again, and follow it by twenty at once.

Tell them again that your figures
Suggest that you put ten bundles

Continue in the same way to one hundred. are exhausted, and ask what you shall do. into one big one, and make them do it. Give the illustration:

Give a few examples, such as 103, 107, 110, 120, 156, 204, 320, etc. You may after this use counting, either with toothpicks or by the illustrations, from time to time, when your pupils need a change of work or to be kept busy.

As a last exercise write, “Bring me fifty-three toothpicks." Ask what the figure 5 means, bundles or toothpicks. If they can not tell you, illustrate:

Point to the five, and have them bring you five bundles; then to the 3, and have the three single toothpicks brought. Do the same with many other numbers, not forgetting some that contain hundreds. Continue the exercise until you are sure that they have the idea of numbers so firmly in their minds that they will never think of 53 as a figure 5 and a figure 3, but as five tens and three units; possibly as five bundles of toothpicks and three loose ones, as five

and three units; but it is number they have and not figures. This idea of number behind the figures is essential for any clear understanding of arithmetic. Hearing children usually get it for themselves outside of school, but in deaf children it must be carefully built up by steady, persistent drill in the schoolroom.

You have now started your class in the habit of reasoning about numbers and not of trying to find the answers to puzzles with figures. Give them in the future the same kind of instruction and you need never fear that they will ever take eight from four and have six left. They have also a definite and accurate idea of how many a hundred is, and from this you can easily build the ideas of higher numbers when the time comes. Should you still have more time to teach numbers, you will do no harm by extending your work in addition and subtraction to larger numbers in the same way. Do not use figures. Do not teach the signs―+, -, X, ÷, =. Do not set them at work on examples like:

1+4-2+7=?
5-2+3=?

Confine all their work with figures to what has been shown here in counting, and to writing the names of figures, and to expressing numbers in figures. Wait for more extensive work with figures until after you have laid the foundations for multiplication and division.

Gradually increase the size of the numbers you work with until you get to 20, working in exactly the same way that you did with the larger numbers. After reaching 20 as a limit for the sum of additions and the largest number used in subtraction, you may begin mental arithmetic.

Have your class sit near your slate, and you write out the problems. Be sure they understand the language. Tell them to think the answers. At first they can not do this, partly because they do not understand exactly what you require them to do and partly because they can not, as yet, perform the operations without some aid from visible illustrations. Help them at first, using toothpicks, etc., or by drawing the illustrations of a number of problems yourself or letting one of them do so. Give them the same problems next day, and others very much like them, and gradually lead them to think out the problems for themselves.

After this you must spell problems to them and have them solved mentally. You will at first be compelled to spell very slowly and repeat often. You may even have to act out the problem as you spell it, thus:

Give a boy three counters. Have him stand before the class, and you spell: "A boy has three cents." (Let him show them.) "His brother gives him two cents." (Giving him two counters.) How many has he?"

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After some practice in this way, get them to answer very easy questions without the illustration. Then gradually increase the difficulty of your problems. Some teachers give too much attention to the idea of a separate ten, in numbers between ten and twenty. Of course, logically, we can say that fifteen is one ten and five, just as we say that seventy-five is seven tens and five; but we can teach our children to realize clearly how many units there are in fifteen without separating the ten, which we can hardly do with seventy-five. We must, for rapid work, teach them to add and subtract between ten and twenty just as they did under ten. I should, therefore, illustrate these examples in exactly the same way, and not as we shall illustrate “carrying" after a while.

After the pupils begin to show considerable ability in solving mental problems, it will be time for us to begin to substitute the memory of different combinations in addition and subtraction for the reasoning we have been doing. The reasoning is necessary at first; but afterwards the operations are performed by memory alone, and the elementary combinations should suggest their results as quickly as combinations of letters suggest their sounds to those who read. We see the letters i t, and think "it" at once. In a similar way our pupils should see the figures 3 and 4, and think " seven." Rapid adders go further and think at once the sums of combinations of three, four, or even more figures; but, in addition, certainty, with moderate speed, is better than the greatest speed with the least bit of uncertainty. In these days a boy frequently leaves school without the ability to add a column of figures of any length with either rapidity or certainty. We wish all of our pupils to have this ability, and must begin now to lay the foundation on which it is to be built.

Questions in psychology may seem out of place in teaching a class of deaf beginners, and yet with them more than with any other class of children these questions constantly occur. Other children have their "concepts" and modes of thought in elementary knowledge fully formed before they go to school. It is not so with the deaf child. He is dependent on the teaching he receives, not only for what he thinks, but for how he thinks. We now have before us the very important task of giving to our children a mode of thinking in addition. This question does not seem to have received from teachers of the deaf the consideration which its importance demands.

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Probably different minds act in different ways. For my own part the fundamental operation of my mind in adding is the memory of the sound of the old addition tables drilled into me by a stately old lady in a lace cap, and were I to think three and three are five" I could almost feel the thump of her ruler, which surely followed such a mistake. What are our pupils to have in place of this memory of sound, which we can never give them? Shall we try to give them a printed, written, or spelled expression? Shall it be "three and four are seven or 3 and 4 are 7" ? All the combinations in addition and subtraction, thus written or spelled out, form a great bulk to be exactly and quickly remembered, and the drill must be constant and long-continued that will so impress it that almost without any effort, certainly without an effort taking any appreciable time, the desired result will come exactly. The same objections apply to 3+4=7 and

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and

Shall we drill our pupils to think of two hands, one making the sign "3 the other "4" and the result, "7," on another imaginary hand? These questions are important, and on a proper answer to them the progress of this class will depend to a great extent. Think the matter over carefully, decide it for yourself, and then make your drill in the method you decide upon very thorough and persistent. Drill by this one method till all the combinations in addition and subtraction are remembered almost instantly. Do this if it takes all the time you can devote to number work for the next three years.

A plan that I have found successful, but which I am not prepared to say is perfectly satisfactory, is to prepare a table of all the numbers from three to eighteen, giving all the combinations of two figures which will make each number.

When you begin to develop this table, have each square drawn on a card 10 or 12 inches square. Show them. Devote considerable time, at least two days, to each of these cards, practicing on all the combinations on it. Explain that the opposite small figures added produce the large number in the middle, and the small numbers on either side subtracted from the large one in the middle leave the opposite small numbers.

After you have explained each card and practiced on it long enough to be sure that the children understand how to use it hang it on the wall; and when you have finished with all of them replace them with a table large enough to be easily read from any part of the room. Have your pupils copy this occasionally, and refer to it constantly, especially when you have to correct a mistake. I believe that in time each pupil will have a mental picture of this table impressed upon his mind so firmly and clearly that all operations in addition and subtraction will be done instantly by seeing it mentally.

To review this work take one of your cards which you used in building up the table to the opposite end of the room, so that the pupils can not see the table. Cover part of the card and let them tell what the covered part is.

If you do not care to adopt this plan-which, by having the whole subject constantly in sight and often referred to, teaches a great deal without laborbe sure that you do have some plan which will give your children a clear, short, mental conception. Do not, for some time yet, attempt to form combinations above 20.

Gradually introduce your pupils to examples in multiplication and division. Begin by very simple examples, and at first allow them to be done by addition