(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.

3.  “There is one more mode which I shall describe:  it is by memory.  Before I explain this mode, I wish to ask you some questions, which I should like to have you answer as quick as you can.

“How much is four times five?  Four and five?

“How much is seven times nine?  Seven and nine?

“Eight times six?  Eight and six?

“Nine times seven?  Nine and seven?”

After asking a few questions of this kind, it was perceived that the pupils could tell much more readily what was the result when the numbers were to be multiplied than when they were to be added.

“The reason is,” said the teacher, “because you committed the multiplication table to memory, and have not committed the addition table.  Now many persons have committed the addition table, so that it is perfectly familiar to them, and when they see any two numbers, the amount which is produced when they are added together comes to mind in an instant.  Adding in this way is the last of the three modes I was to describe.