We may err either by accustoming our pupils too much to the
consideration of tangible substances when we teach them arithmetic, or
by turning their attention too much to signs. The art of forming a
sound and active understanding, consists in the due mixture of facts
and reflection. Dr. Reid has, in his "Essay on the Intellectual Powers
of Man," page 297, pointed out, with great ingenuity, the admirable
economy of nature in limiting the powers of reasoning during the first
years of infancy. This is the season for cultivating the senses, and
whoever, at this early age, endeavours to force the tender shoots of
reason, will repent his rashness.

In the chapter "on Toys," we have recommended the use of plain,
regular solids, cubes, globes, &c. made of wood, as playthings for
children, instead of uncouth figures of men, women and animals. For
teaching arithmetic, half inch cubes, which can be easily grasped by
infant fingers, may be employed with great advantage; they can be
easily arranged in various combinations; the eye can easily take in a
sufficient number of them at once, and the mind is insensibly led to
consider the assemblages in which they may be grouped, not only as
they relate to number, but as they relate to quantity or shape;
besides, the terms which are borrowed from some of these shapes, as
squares, cubes, &c. will become familiar. As these children advance in
arithmetic to square or cube, a number will be more intelligible to
them than to a person who has been taught these words merely as the
formula of certain rules. In arithmetic, the first lessons should be
short and simple; two cubes placed _above_ each other, will soon be
called two; if placed in any other situations near each other, they
will still be called two; but it is advantageous to accustom our
little pupils to place the cubes with which they are taught in
succession, either by placing them upon one another, or laying in
columns upon a table, beginning to count from the cube next to them,
as we cast up in addition. For this purpose, a board about six inches
long, and five broad, divided into columns perpendicularly by slips of
wood three eighths of an inch wide, and one eighth of an inch thick,
will be found useful; and if a few cubes of colours _different from
those already mentioned_, with numbers on their six sides, are
procured, they may be of great service. Our cubes should be placed,
from time to time, in a different order, or promiscuousl