six;'! “two numbers which make seven,” etc. For seat work I prepared hectograph papers to be filled in in a similar way.

After we had gotten well started on the combinations, say as far along as 8, I began combining each with the ten series, as for example, 15 25 35

105 +-3 +3 -3

+ 3

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This work I had written as well as oral. The rhythm of the series enabled the children to learn them readily.

I used the vertical form in writing combinations rather than the horizontal, that is 6 +4 instead of 6+4 - 10. In the every day problems of a child's life the vertical form

is used and it seems desirable to use only those forms which are really practical.

As I used the addition method of subtraction it was a very easy matter to associate this further step with the work which I have already described. To illustrate: When I held up this paper I asked, “How many dots?” The pupil called upon replied, “There are nine.” “What two numbers then make nine?” Answer, “Five and four are nine." I now placed one hand over the four dots and asked, “How many and five are nine?”

9 The answer was obvious. I then put the form - 5 on the board reading

it, “Five and how many more make nine?" (Miss MARY MAYBERRY.)

FIG. 4.

(First grade). One of the greatest difficulties which confronted me in

taking up systematic work with the combinations was the habit which my children had of counting up their sums serially. The first step I took had for its object the habit of seeing groups in the small combinations. I arranged dots on paper in symmetrical forms. For example, I wanted to teach the combinations of 9. I grouped the following numbers: 5, 4; 6, 3; 7, 2; 8, 1 in the following way:

FIG. 3.

I flashed these papers before the class one at a time and then asked, “How many dots do you see?” Then holding one hand over four spots I asked, “How many spots do you see now!" The child called upon replied, “I see five spots.” “How many were under my hand then?” “Now, four and five are how many?"

In a similar way I took up subtraction. Covering four dots I asked, “How many do you see now?'' Answer, "Five." "Then nine take away four leaves how many?" I then placed the symbols for the process on the board. In a similar way I ran through the other combinations which make nine. After illustrating the combinations in this way I began memorizing them.

This work, however, gave me only a few of the possible combinations of 9, only those which were formed by adding and subtracting two numbers. I showed the children that each of the two numbers which make 9 can be broken up into smaller numbers.

When they realized this I changed my work a little and asked the children to give me any com

3 4 6 2

1 1 binations which put together made 9. The answers began to come 4, 3, 3, 2, 3.


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They soon got so quick in making these combinations that they gave me in a ten minute recitation 55 different sets of numbers, each equalling 9. Some of these were: 1


1 1

1 12


etc. 3



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The brightest of my class without any suggestion from me began giving such numbers as “200,009 take away 200,000 leaves 9." I rejected these, for they seemed too difficult for the majority of the class. In this way and by holding to quick responses I found that the children had little tendency to drop back into the old habit of counting on their fingers.

As I broke the digits into their complementary parts I also began systematic work in forming higher combinations. The first step in this work was that of combining each of the digits with the 10 series for sight reading, as10 20 30 40

100 + 2



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Now, since all combinations of numbers up including 10 are known, the higher combinations may be formed by adding the digits to a new series ending in 2, then in 3, as 23, 43, 53,

103, until all possible combinations are exhausted. It will be observed that in practice there in an accompanying rhythm which makes it easy for the children. I took up these higher combinations, both additively and subtractively. For example,


9 when the child learned that +4, he also learned -4. Again, when the child knew

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throughout the teaching of combinations. (Miss NELLIE O'Connor.)

[First grade). In establishing the 10 series (counting by 10's) I first used objects for illustrative purposes. I gave each child a number of toothpicks and some rubber bands. I instructed each one to count out all their sticks into bundles of 10 and place a band about each. “Now, children," I said, “hold up one bundle. How many toothpicks in your bundle?" "Ten,” one replied. “All right, hold up another ten. How many tens have you now?''

“Two tens,

one replied. “We have a better name for the two bundles than two ten,” I said. “We say twenty.” So with three ten, thirty; four ten, forty, etc. Along with these language forms I took the symbols. (Miss Julia Lemon.)

[Second grade.) In all my work with figures my aim has been to get accurate and rapid work. With pupils of the second grade little or no work involving reasoning can be given, consequently the work that is given should be almost solely of the reflex type. If a child can learn that c-a-t spells cat, he can as readily and accurately learn that

2 +2. This being so, I insisted on perfect and quick work. I made the first impressions


of all reflexes as strong as I could, had many repetitions, and aroused as many emotions a3 possible to aid in forming a strong reflex.

When I began teaching I took up the combinations above 9 one by one objectively. The problem, “How far could objects be used in learning these combinations?'' immediately confronted me. I found, taking the number ten for example, that when I gave the pupils each 10 objects, beans, toothpicks, etc., by which to learn the combinations that the result was they immediately began to count them serially and thus did the very thing which defeated my attempt to build combination reflexes. This matter of serial counting was a grave problem as the children had gotten into the habit of counting their fingers when adding, and anything which lead them on in this I saw must be stopped at once. But, barring the fault of counting serially, would the process of working with beans

6 lead to the reflex desired, that 4? From experience I found that it would not. The

10 concrete is one kind of work; that with pure number is another. The only realconnection lies in the application of one to the other. A boy may handle 6 marbles and 4 marbles

6 every day for a month and yet never establish the reflex that + 4. His attention is

10 focused on the marbles, not on their numerical relation. Because of these reasons I decided to use but little objective work in connection with the addition and subtraction of numbers.

On the board before the class I wrote out the combinations of every number as I took it up. These I left for study and reference until they were mastered. Taking the number

6 7 10 I put up the following combinations, -4 +3 +2 +1. I drilled on these



every day by flashing a card before. the class on which one of these combinations was written. After they knew these combinations pretty well I took the subtraction forms both orally and in writing, as 10 10 10 10 10

10 10 4 -7 --6 -3 -8 -2 -5

Combining the two processes of addition and subtraction, I gave for quick oral work such


exercises as, “Ten, take away three, add add five,” etc. When the pupils had the answer they raised their hands and dropped them when someone gave the result which they had gotten. I also by way of variety had the children fill in blanks as,


9 ( ) ( ) 10 10 ( ) O) +O) + 8 + 9 - 6

-O) 3 etc.

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While this was going on I paid much attention to those combinations above 20, as-




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(Second grade). I found on taking my class that they seemed to be thorough on the combinations of all numbers as far as 13 so I began my agvance work at this point. I began with numbers immediately without any preliminary objective work. I began by


7 putting the combinations of 13 on the board. After learning +4 +5 and +6 I began


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In a similar way I applied all the combinations of 13 to the larger numbers.

After I had drilled on these small combinations for a time I wrote them on the board in a place where they would not be erased. In this way, if any child had not learned them perfectly, he could have them for study. This plan of putting the combinations on the board I found valuable in another way. When a child in solving an exercise forgot the answer, it was better for him to look at the board and get the correct answer than to guess at it, putting down in consequence a wrong answer and thus forming a wrong association which rendered more difficult the building of the proper reflex.

Each day I devoted about ten minutes to a quick review of the combinations which the class were supposed to know. In doing this I sometimes wrote on cards combinationg 9 8 6 29 56 98 5 7

7 5 etc. one on each card. These I held before the class,


one at a time, for an instant and called for an answer.

In all my work with the combinations I used the vertical arrangement ( + 4



11 rather than the horizontal form (74 -11 ) because it is the more practical, being the form, of course, used in column addition. If, when a child begins to add columns of gures, he has been used to the horizontal form, it will be necessary for him to made an

entirely new association. This would be unnecessary had he always used the vertical arrangement, and thus time would be saved. I took care when teaching the combinations of 13, as well as with other numbers, to

13 13 associate the subtractive forms as, 5 - (). These forms are only modifications of

3 8 + 5 and should be associated, otherwise three distinct reflexes must be established with

13 a considerable loss of time. (Miss CAROLYN HORTOP.)

(Second grade). In my work with combinations I found that much written as well as oral work was needed to strenghen the reflexes. Though the class could answer perfectly any of the combinations which they had had when given orally, they could not always do so when written. The fact that but one association, the auditory, had been established, explains the condition. To emphasize both the auditory and visual forms the

4 24 55 following plan was used. A series of combinations as, + 6 + 6 + 6 were written on

the board. I pointed to one of these and the result was given orally, or sometimes I had the child called upon step to the board and write his answer. It is a good plan when first taking up a given combination to have the children repeat the combination and the answer, thus strengthening the auditory reflex as well as associating it with the visual. Profitable work along this line was done by dictating the questions and requiring the children to write instantly the answers only. When such work was done quickly I felt certain that the proper reflexes were being well established. (Miss Mary C. O'CONNELL.)

[Third grade). As the learning of combinations is pure reflex work the children need constant drill and repetition. I found that, although the pupils had had all the combinations before reaching the third grade, a systematic review was necessary to secure accuracy and rapidity in their use. Sometimes I placed on the board a large number of the combinations paying, as I did so, especial attention to the higher combinations upon which the children were weak. In conducting this review I pointed quickly to each combination calling on the pupils to give ån answer. Sometimes for a change I allowed one to rise and answer as many as he could. Throughout the work of my grade I made it a point to give each day a few minutes drill of this sort, also applying the combinations in column addition. (Miss L. Ray JACOBS.)

It will be seen from the extracts just given that in our work in teaching combinations of figures we make no use of the abacus or of objects such as toothpicks and shoe pegs for the reason that they can not conveniently be arranged in the symmetrical forms indicated in fig. 3. It is impossible for a child to distinguish at a glance a group of 5 sticks arranged serially from a group of 6, or one of 7. To determine their number he is forced to count each. In doing this he loses the feeling of the unity of the group. Instead of seeing a whole made up of parts he sees the parts as wholes. If he is asked to find the sum of five and four with his objects he arranges

1111 them usually in this manner: 11111

111111. In practice he counts up, one, two, three, four, five, six, seven, eight, nine. To be sure he gets the

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