Symbols should now be introduced, in order that the child may begin to construct his own tables. Having written down naught and the first 9 digits, you say, “Now we have no more signs to use. How then can we write the larger numbers, such as two and ten, three and ten, four and ten? We must repeat the old signs. For example, to write three and ten, or thirteen, we can set down 3 for the three, and 1 on the left hand side of the 3, to represent a single ten (13). Similarly to represent two and ten (or twelve) set down 2 for the two, and 1 on the left-hand side of the 2, to represent a single ten (12); and to represent one and ten (or eleven) set down 1 for the one, and 1, on the left-hand side of the one, to represent a single ten (11).

"How do we know, in the number 11, which of the two 1's represents a ten? By remembering that the figure on the right hand always stands for ones, and the figure on the left of it for tens.

"But now how shall we represent ten itself? If we put down 1 by itself and say that shall stand for ten,' we shall not be able to remember when we see it afterwards whether it means one or one-ten. How then can we distinguish between them? Thus, by calling it naught and ten,' and writing down (just as we did ‘one and ten,' 'two and ten'), 0 on the right hand for naught, or nothing, and 1 on the left hand for the ten."

The explanation of 10 will be the only point that need present any difficulty; the subsequent explanations of 23, "as meaning three and two tens," 34"as meaning four and three tens," etc., will be found comparatively easy.

As for the hundreds, the pupil will not find it hard to see that we must "begin again" a second time when we reach ten tens, writing down 0 for the ones, and 10 on the left-hand of the 0, for the tens (100). He may then be taught to write down "ten tens and one" (101), ten tens and two (102), etc., and finally be told that ten tens are called one hundred; so that 100, instead of being described as "no ones and ten tens," may be described as "no ones, no tens, one hundred."

The child must now be practiced in reading numbers of three figures forwards and backwards, thus: 234, two hundreds, three tens, four ones (or units), or four ones, three tens, two hundreds. And, in the following lessons, this exercise of reading small and large numbers forwards and backwards must be constantly recapitulated.

Hitherto the child must not have been allowed to write figures for himself, but must watch the teacher make them at the pupil's dictation, and the teacher must take great pains to write them in precisely the way in which he would like the pupil to write them. For the purpose of uniformity in figure writing, the paper should be divided into

equal squares of a good size; for it is no less important for good figures than for good writing that a child should begin with "large hand." At this stage the child may be permitted to write down a few figures for himself, under close supervision, that he may construct his own "Tables."

He should begin with Tables of Addition of numbers under 10; and Tables of Subtraction should be constructed at the same time, thus: 7 and 6, 13; 6 from 13, 7; 7 from 13, 6.

The teacher should note what parts of these Tables appear to be most difficult for the pupil to recollect, and should practice him specially in these, making him impress them upon himself by repetition and writing, so that he may learn them by heart. He ought not to be allowed to go far in Arithmetic till he can add without pausing to think, and of course he must not now be allowed to use fingers, or the assistance of the abacus. The latter may still be allowed in experimenting and making discoveries with numbers, but not for the ordinary purposes of calculation. In order that finger-counting may be discouraged, the pupil should for some time calculate aloud, and in the presence of the teacher. And just as in reading the pupil was not allowed to spell to himself, so neither must he count to himself; he must calculate, as he reads, on the "look and say" principle; and in answer to the question, "8 and 7?" he must reply at once, without either counting or thinking, “15.”

and 7 make? From the abacus What do 28 and 7 make? From What 38 and 7? 45. What 48

Having committed to memory the statement that 8 and 7 make 15, the pupil must be asked, What do 18 he ascertains, and writes down 25. the same source he writes down 35. and 7? 55. The teacher must continue these questions till he forces the pupil to discover for himself that his formula, "8 and 7 make 15" will always help him to determine the unit figure of the result when two figures are added together of which one ends in 8, the other in 7. The same process must be repeated with 7 and 6, 17 and 6, 27 and 6, 6 and 7, 16 and 7, 26 and 7, etc. Thus the pupil will learn to add with rapidity numbers under 100 to numbers under 10.

He is now in a position to construct for himself Tables of Multiplication. But first he should receive a little stimulus to urge him to undertake his new labor with zeal. Tell him to make ten heaps of marbles, 7 in each heap. And how many do they make altogether? "I must count." "Well, count, then; but I will write down the number on a piece of paper, which I will fold up and give to you; and see whether I am not right." It makes 70. "Yes, you are right. Then ten heaps of 7 marbles make-?" Seventy. "Seventy what?" Seventy marbles. "And ten heaps of 7 nuts would make?" Seventy. "Seventy what?" Nuts. "And ten heaps of 7 ones, or units, make-?" Seventy units. "Then we will say that ten sevens always make?" Seventy. Having repeated the process with ten

heaps of 8, of 9, of 6, of 5, etc., you force the pupil at last to discover the law, which he would express in his own way by saying that "ten times a number make that number with ―ty at the end."

This short cut is so charming to a child that it is well to leave him to enjoy it for a time without further observation; but in the next lesson, asking him what 70 means, and receiving the answer, 70 units or 7 tens, you thereby show him that the new rule tells us that ten sevens are the same as seven tens; and this he may verify at once for himself by his marbles. In the same lesson you may teach him "eleven times" in the same way, by experiment. The advantage of thus beginning with "10 times and 11 times" is that you at once show a boy the manifest utility of his new knowledge, and at the same time give him something to learn which he cannot fail to remember.

After this stimulating foretaste you must now proceed methodically to show him how to construct tables of Multiplication by means of Addition. And here the main business is that the pupil may not be discouraged by the prospect of the burden of committing so great a mass to memory. For this purpose it is expedient not to form the whole of the Tables at once. And before he begins to learn any portion by heart, very often a few remarks of the teacher may help to lighten the labor. For example, in learning "twice," you may show him that he is only repeating in a new form what he has said before in his Addition Tables; for "twice 9 are 18" is the same thing as saying that "9 and 9 make 18." Again, when he comes to learn the more advanced Tables, e. g.: “7 times," the child may be shown that he has already learned 7 times 2, 7 times 3, etc., up to 7 times 6, in the previous Tables, so that a good deal of the apparently new work is really repetition of old work.

But when all is done that is possible in the way of help, the task of committing the whole to memory has to be faced; and the truest kindness is to see that the child learns the whole at last, without trusting in any kind of external aid, such as Memoria Technica, or anything else. There is no reason, however, why the teacher should not resort to any devices that may facilitate the process without impairing the result. "Children," says Preceptor, "are so constructed that they (and perhaps their elders as well) more easily remember what they take in indirectly with unconscious interest, than what they try to remember with a conscious strain. Very often a child will remember 8 times or 9 times better if he is allowed to write it out or print it in large colored figures; or should he find a difficulty in remembering some particular formula, e. g. 8x9= 72, very often you may stamp it on his memory by some irrational jingle, such as:

"This rhyme is mine, and strictly true,

That 8 times 9 are 72.'"

But our object is that the child should repeat the Tables without stopping to think about rhymes—especially when, as in this instance,

the rhyme will mislead, if one number, e. g. 7, be substituted for another, e. g. 8. As a rule, the pupil must depend upon practice and repetition, oral and on paper, for the mastery of the Tables. But Preceptor's hint about writing out and embellishing those Tables which present most difficulty, may very likely be found useful.

As soon as the pupil is pretty familiar with the Multiplication Table he should be taught to repeat the corresponding Division Tables, e. g. five times six is thirty; fives into thirty, six; sixes into thirty, five. But it may be well not to teach the Division Tables at first, lest they should break the "swing" of the Multiplication table, and increase the difficulty of learning it.

33. THE FIRST FOUR RULES APPLIED TO NUMBERS ABOVE A

HUNDRED.

Before proceeding to apply the "four rules" to numbers above a hundred, the pupil must be practiced still more in reading symbols into units, tens, and hundreds, or hundreds, tens, and units (as above, p. 42), and he must now be introduced to thousands.

Coming now to the "first four Rules" applied to large numbers, we have to speak of the reasons for those Rules, or rather of the methods by which the pupil can be led to the Rules, as the result of his own experience. In every case, if possible, the pupil should be helped to discover a Rule for himself; but great care is necessary to avoid confusing him by proceeding too fast; or by using terms or phrases that he does not understand; or by assuming, as axiomatic, truths which he is not at present prepared to accept. If we can succeed in leading him to the Rules for Addition, Subtraction, and Multiplication, we may perhaps dispense with the process in Division, merely indicating it to him in the case of small numbers, and leaving him to take the rest on trust.

(i.) Addition.—The first lesson may be somewhat after this fashion: "If we have two heaps of fruit, the first containing 5 currants, 4 strawberries, 3 plums, and 2 pears; and the second containing 4 currants, 3 strawberries, 2 plums, and 1 pear; and if we wish to add them together, so as to make the two heaps into one, tell me, what must we say the one large heap will contain? You cannot at once answer. Write down, then, in a line what the first heap contains, putting the fruits in order of size, the smallest fruit to the right, and the largest to the left. Write down what the second heap contains in another line exactly under the first line. Now draw a straight line below these, and below this straight line write down what the large heap will contain, beginning from the small fruit on the right."

2 pears, 3 plums, 4 strawberries, 5 currants.
2 plums, 3 strawberries, 4 currants.

1 pear,
pears, 5 plums, 7 strawberries, 9 currants.

66 Now suppose we wish to add together the numbers 2345 and 1234. Read out the first number, beginning with the ones": 5 ones, 4 tens, 3 hundreds, 2 thousands. "Now the second": 4 ones, 3 tens, 2 hundreds, 1 thousand. "Write them in two lines, in the same way in which you wrote down the heaps of fruit, putting the ones to the right, and add them together, beginning from the ones."

2 thousands, 3 hundreds, 4 tens, 5 ones.
1 thousand, 2 hundreds, 3 tens,

4 ones.

"Now read out the result you have written down, beginning from the right": 9 ones, 7 tens, 5 hundreds, 3 thousands. "Now read it

out, beginning from the left": 3 thousands, 5 hundreds, 7 tens, 9 "Write it down in the ordinary way": 3579.

ones.

Having had a little practice in sums of this kind, in which the totals of tens, hundreds, etc., do not exceed nine, the pupil must now be told to add two numbers in which the totals exceed nine, e. g. 237 and 958.

2 hundreds, 3 tens, 7 ones.

9 hundreds, 5 tens, 8 ones.

11 hundreds, 8 tens, 15 ones.

"Read out the result, beginning from the right": 15 ones, 8 tens, 11 hundreds. "But 15 ones are the same as 5 ones and-how many tens?" One ten. "Then we can set down 5 in the column of ones, and carry the one ten to the column of tens, thus making 9 tens instead of 8 tens. Again the 11 hundreds are the same as 1 hundred and ?" 1 thousand. "We can therefore set down 1 hundred in the column of hundreds, and carry the thousand to the thousand column. Thus the result, beginning from the right, is?" 5 ones, 9 tens, 1 hundred, 1 thousand. "Read it out from the left": 1 thousand, 1 hundred, 9 tens, 5 ones. "Write it down." 1195. The working may now be repeated more briefly thus, after writing "thousand," "hundred," "ten," over the different columns:

"8 ones and 7 ones are 15 ones; set down 5 ones and carry 1 ten; 1 ten and 5 tens are 6 tens; 6 tens and 3 tens are 9 tens; set down 9 tens; 9 hundreds and 2 hundreds are 11 hundreds; set down 1 hundred and carry 1 thousand."

After a little practice in sums of this kind, with the headings of the columns thus set down, the headings may be dispensed with. But for some time it will be useful for the pupil to work sums aloud, the teacher setting down the figures, so that the pupil may unconsciously