to one person.

So, if we add 7 to 8 (7 + 8), we add a _number to a number_, and these
numbers for a _definite_ reason represent in themselves groups of
homogeneous units.

Again, when the child shows us the 9, he is handling a rod which is
inflexible--an object complete in itself, yet composed of _nine equal
parts_ which can be counted. And when he comes to add 8 to 2, he will
place next to one another, two rods, two objects, one of which has
eight equal lengths and the other two. When, on the other hand, in
ordinary schools, to make the calculation easier, they present the
child with different objects to count, such as beans, marbles, etc.,
and when, to take the case I have quoted (8 + 2), he takes a group of
eight marbles and adds two more marbles to it, the natural impression
in his mind is not that he has added 8 to 2, but that he has added 1
+ 1 + 1 + 1 + 1 + 1 + 1 + 1 to 1 + 1. The result is not so clear, and
the child is required to make the effort of holding in his mind the
idea of a group of eight objects as _one united whole_, corresponding
to a single number, 8.

This effort often puts the child back, and delays his understanding of
number by months or even years.

The addition and subtraction of numbers under ten are made very much
simpler by the use of the didactic material for teaching lengths. Let
the child be presented with the attractive problem of arranging the
pieces in such a way as to have a set of rods, all as long as the
longest. He first arranges the rods in their right order (the long
stair); he then takes the last rod (1) and lays it next to the 9.
Similarly, he takes the last rod but one (2) and lays it next to the
8, and so on up to the 5.

This very simple game represents the addition of numbers within the
ten: 9 + 1, 8 + 2, 7 + 3, 6 + 4. Then, when he puts the rods back in
their places, he must first take away the 4 and put it back under the
5, and then take away in their turn the 3, the 2, the 1. By this
action he has put the rods back again in their right gradation, but
he has also performed a series of arithmetical subtractions, 10 - 4,
10 - 3, 10 - 2, 10 - 1.

The teaching of the actual figures marks an advance from the rods to
the process of counting with separate units. When the figures are
known, they will serve the very purpose in the abstract which the rods
serve in the concrete; that is, they will stand for the _uniting into
one whole_ of a certain number of separate unit