was the two, the fourth the four, etc. After preliminary work of this character I then applied the series to miscellaneous things in the room.

I early began the association of the number name and its symbol. I did this by writing the symbols, at first from one to ten, in regular succession on the board before the class. I then called on the class to repeat in unison the number series which they by this time could give without difficulty. As they repeated the series I pointed to the proper symbol. After running over the series in this fashion a few times I began skipping about from one symbol to another with my pointer, calling on individuals as I did so. After considerable work of this nature I erased the figures which I had on the board and gave some quick work in the following way: writing a "9" on the board I asked a child to name it. Erasing this quickly I put down a 3 with the same question. A few minutes spent each day in systematic drill of this kind made the children very accurate in recognizing the symbols. In a similar way I extended the range of the work beyond 10.

In the way of devices to secure variety and interest in these reviews I sometimes drew a house or ship on the board. In it I put all the numbers with which the children were familiar to represent furniture and people. The numbers (the hardest ones) which represented the people, I put in with red crayon. The easiest ones,

which I called furniture, were written in with white crayon. I then pictured the house as being on fire. Any child recognizing a red number upon erasing it rescued a

person. If he could not recognize one of the red ones perhaps he could a white one. By so doing he saved an article of furniture.

Another device used in the same work was what I called building a stone wall. Each pupil in turn wrote a number, which I dictated, on the board. If written correctly I allowed him to draw a rectan

[Receiving class.] My first work in this grade was to find out what the children knew about counting. I found that only one or two knew the number series to 10; the others not even to five. I had my class first learn the series to 25. I did this by having the children repeat the number names after me. I then began having the children write and read the figures from 1 to 25. In doing this I gave a number name such as six and

wrote the symbol on the board. The children imitated my motions and then repeated the name, thus establishing the association between the symbol and the name. (MISS KATHRYN MURRAY.)

[First grade.] In teaching my class to count serially I made no application until the series became a reflex. I found in establishing this reflex that counting in unison was of great value as by this means, the attention of the entire class is focused on the work, whereas in doing purely individual work frequently the attention of but the one reciting is gotten.

After I found the children could give the number series readily I began applying it to objects which came within the children's experiences.

After the children could count and apply the number series to ten objects I began teaching the symbols. I had the children pass to the board and draw one apple. I asked if anyone knew how to tell me with the crayon that there was just one apple there. One hand finally went up. She turned about and wrote the figure "1." Of course, if no one had known, I would have shown them by writing it myself. In a similar way I had the pictures of two apples drawn and associated with it the symbol 2. As we worked out the series in this way I placed the results along the top of the board in this manner:

1

2

3

4

5

6

7

9

19

By drilling a few moments each day for a few days the association between name and symbol was firmly established As soon as the children could point correctly to any of the figures given upon dictation I focused their attention on symbol writing. I wrote slowly on the board, say a figure 2. They watched the motion of my arm and hand and imitated the movement in the air a number of times. I then sent the children to the board, had them take up a piece of chalk and very quickly make the same movement. The first results were awkward and crude but they had the merit of free arm movement. Each day I gave a few minutes quick mechanical drill in writing numbers. The children were always eager for this drill. (MISS IDA DUNNE.)

FIG. 3.

[First grade.] I did much in having the children recognize such numbers as 21, 92, 142, 109, etc. This work came very readily to the children, since they had had work in 90 reading at sight and writing the result of such combinations as +8. The children had 98

difficulty with such numbers as 123. The tendency was to write it 1023. I broke this up by placing the number (123) in a 3-roomed house, one figure in each room as 1|2|3. I told them whenever a number was written so that it could be placed in a 3-roomed house it was to be read "hundreds."

I got excellent drill in writing numbers of 3 figures by dictating columns of figures 231 for addition as 323. I was careful in this never to give a combination which the children 445

had not had. Also I was sure that the sum of each was less than 10, for at this time they knew nothing of "carrying." (MISS NELLIE O'CONNOR.)

[Second grade.] I found that many of my children had incorrect ways of reading numbers. Some read 165 as one hundred and sixty-five, and others one sixty-five. The only correct form is one hundred sixty-five, for the "and" is used to stand for a decimal point. I insisted always on the correct form. It required much persistence, however, to establish this. (MISS CAROLYN HORTOP.)

In our third grades much of the formal work is done by dictation, hence the children get much practice in writing and reading numbers. Incidentally, however, to supplement this we give once a week a drill for a few minutes on the numbers which give children most trouble.

We make no attempt to teach the place value of figures in the Arabic system of notation. The primary grade child is interested in getting results, not in philosophizing upon the principles underlying our system of notation. He takes pleasure in learning things by rote and by rule of thumb, as Prof. Wm. James, the psychologist, says. He does not concern himself with the future usefulness of his acquisitions. If left to his own initiative he will as likely learn a string of nonsense syllables as the number series. It is the teacher's business to discriminate for him, to give him what he needs, and to give it, at times, regardless as to whether or not it appeals to his reason. As a matter of fact, the adult mind never has any occasion in performing calculations to think of place value. We presume that the fact that 10 units make 1 ten, 10 tens make 1 hundred, etc., never occurred to most of us until we began teaching young children. Our point is this: the knowledge of the place value of figures, though interesting enough in itself, is a bit of information that has no vital connection with the needs of either children or adults.

Combinations.

Success in calculation is dependent upon memorizing perfectly the 45 combinations of numbers between the units 2 and 18 inclusive. The sight of any two figures arranged in additive form or the hearing of their names similarly associated must immediately without conscious mental effort suggest their sum. This degree of accuracy and facility can be gained only through systematic and thorough drill, limited not to the first two or three grades as is frequently the case, but carried throughout the entire school

course.

The value of objective work in this connection has been entirely overrated. Holding children down to counting out 4 toothpicks and 3 more toothpicks, and then requiring them to count up the sum furnishes amusement for a short time, but the children will never thereby automatically 4 3.

4

think 7 when they see +3 or 12 when they see Recently we discovered a boy in our seventh grade who did not know his tables. He had had much objective work in the grades but had never really buckled down to learn them. The cry now going up all over our land that the children in our schools are exceedingly inaccurate and slow in calculation is proof enough that much concrete work in combinations and tables will not give working ability with figures.

Our use of objects in connection with combinations is limited to illustrating the parts of the first 10 units. Even here, however, no attempt is made to memorize the results directly from the objects. We attempt only

by this means to teach the child to habitually see things in groups. The drill in memorizing comes later and as we indicated in the introduction, is kept distinct from this preliminary objective work.

After completing this work we begin systematically breaking the units up into their complementary parts. At the same time we establish, associating both name and symbol, the 10 series through to 110 as, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110. The next step is to combine, orally and written, each of the digits with the ten series, a very easy matter. In doing this we have much sight reading, as

systematically so that nothing is omitted. By doing this it will be seen that we establish what may be termed new series, as

The purpose of establishing these series is to enable the child, when learning the combinations within the 1-18 number space, to apply them to the number space between 10 and 100. For example, when he learns that

Since subtraction is but the inverse of addition we correlate the two in our primary work, placing the emphasis, however, on addition. When a

5

child learns that one of the combinations of 9 is +4 it takes but the question to elicit that 9, take away 5, leaves 4, or 5 and 4 more are 9.

The details of the work which we have just sketched will be discussed in the following reports:

[Receiving class.] After the children could count serially with facility 1 dropped this work and began training them in imaging numbers in groups or wholes. This was intended as work preparatory to systematically analyzing and synthesizing numbers. (Combination work). Great stress was laid on this matter of counting by groups, for I found that otherwise in adding figures the children would count up the sum serially on

their fingers, or else make lines or dots on their paper in this fashion, 1. or 11 count2 .. 2 11 ing as they did so one, two, three, four, five.

It seemed to me this habit of counting by groups could not be established by using unarranged units. So I adopted a symmetric form of arrangement for all numbers under ten. I pasted on sheets of stiff manila paper, 8 by 10 inches, small, bright colored dises (the parquet material of the kindergarten) in the symmetrical forms shown in fig. 3, which I adopted for the reason that the eye grasps them more readily than it does any other form. These papers I flashed, one by one, before the class, taking the papers away before the children had time to count the dots serially. I asked, “How many dots did you see?" The children replied as they were called upon in turn, "I see four dots;" "I see six dots;" "I see eight dots." I varied this objective work frequently by holding up one of the papers, say the one with the six dots on it, and covering one half of them with my hand, I asked, "How many threes do you see?" The reply was, "I see one three." I now removed my hand and again asked, "Now how many threes do you see?" "I see two threes," someone would reply. But how many altogether on the paper?" I asked. "There are six on the paper. "Then two threes are how many?" Or, changing the form of my question, "How many threes in six?" I gave my class a few minutes quick, snappy work of this kind each day for a time. After a few days they

were able to recognize units under 10 very readily and to see that each was made up of groups. No point was made of memorizing these groups at this time. All I wanted was to get the children to have some readiness in seeing and visualizing groups, in breaking these groups into smaller groups, and recombining the parts into wholes.

After the children could recognize these groups readily I dropped this objective work and began memorizing the combinations. The first day I began with the number three as the children all knew that one and one makes two. The paper containing three dots was held before the class and the question put, "Can you tell me one way to make a With the paper before them the answer came readily enough, "Two and one

three?" ,

2 1

more," or "one and two." I then wrote the symbols on the board 1

-+2 associat

3 3

ing both forms. The next number taken up was 4. The same questions were asked and

review, I put the combinations of both 3 and 4 on the board in irregular order in the following manner. I then pointed to the combinations and called for the answers, writ

3

1

2

3

1

2

1

1

1

ing them myself in their proper places. In this manner the work was continued from day to day, new combinations being taken up as soon as the old ones seemed to be thoroughly learned.

For variety in this work I now began sending my class to the board. Sometimes I had the combinations which we had studied written on the board for each one, the answers to be supplied and written in by the children. Again I dictated them in this way: "Write two numbers which make