(b.) The next mode of counting is to do it mentally, without using your fingers at all; but, as it is necessary for you to have some plan to secure your adding the right number, you divide the units into sets of two each. Thus you remember that eight consists of four twos, and you accordingly say, when adding eight to seven, 'Seven; eight, nine; ten, eleven; twelve, thirteen,' &c.
(c.) "The third mode is to add by threes in the same way. You recollect that eight consists of two threes and a two; so you say, 'Seven; eight, nine, ten; eleven, twelve, thirteen; fourteen, fifteen.'"
The teacher here stops to ascertain how many of the class are accustomed to add in either of these modes. It is a majority.
2. The next general method is calculating; that is, you do not unite one number to another by the dull and tedious method of applying the units, one by one, as in the ways described under the preceding head, but you come to a result more rapidly by some mode of calculating. These modes are several.
(a.) "Doubling a number, and then adding or subtracting, as the case may require. For instance, in the example already specified, in order to add seven and eight, you say, 'Twice seven are fourteen, and one are fifteen'" ("Yes, sir, yes, sir"); "or, 'Twice eight are sixteen, and taking one off leaves fifteen." ("Yes, sir.")
(b.) Another way of calculating is to skip about the column, adding those numbers which you can combine most easily, and then bringing in the rest as you best can. Thus, if you see three eights in one column, you say, 'Three times eight are twenty-four,' and then you try to bring in the other numbers. Often, in such cases, you forget what you have added and what you have not, and get confused ("Yes, sir"), or you omit something in your work, and consequently it is incorrect.
(c.) If nines occur, you sometimes add ten, and then take off one, for it is very easy to add ten.
(d.) Another method of calculating, which is, however, not very common, is this: to take our old case, adding eight to seven, you take as much from the eight to add to the seven as will be sufficient to make ten, and then it will be easy to add the rest. Thus you think in a minute that three from the eight will make the seven a ten, and then there will be five more to add, which will make fifteen. If the next number was seven, you would say five of it will make twenty, and then there will be two left, which will make twenty-two.' This mode, though it may seem more intricate than any of the others, is, in fact, more rapid than any of them, when one is a little accustomed to it.
"These are the four principal modes of calculating which occur to me. Pupils do not generally practice any of them exclusively, but occasionally resort to each, according to the circumstances of the particular case."
The teacher here stopped to inquire how many of the class were accustomed to add by calculating in either of these ways or in any simpler ways.