of these, and, as I describe them, I wish you
would notice them carefully, and tell me which you practice."

There are, then, three modes of adding up a column of figures, which I
shall describe.

1. I shall call the first _counting_. You take the first figure, and
then add the next to it by counting up regularly. There are three
distinct ways of doing this.

(a.) "Counting by your fingers." ("Yes, sir.") "You take the first
figure--suppose it is seven--and the one above it, eight. Now you
recollect that to add eight, you must count all the fingers of one
hand, and all but two again. So you say 'seven--eight, nine, ten,
eleven, twelve, thirteen, fourteen, fifteen.'"

"Yes, sir, yes, sir," said the scholars.

(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 co