If you reverse this, which has generally been the system of instruction pursued--if you set a child to learn its multiplication, pence, and other tables, before you have shewn it by _realities_, the combinations of unity which these tables express in words--you are rendering the whole abstruse, difficult, and uninteresting; and, in short, are giving it knowledge which it is unable to apply.
As far as regards the general principles of numerical tuition, it may be sufficient to state, that we should begin with unity, and proceed very gradually, by slow and sure steps, through the simplest forms of combinations to the more comprehensive.

Trace and retrace your first steps--the children can never be too thoroughly familiar with the first principles or facts of number.
We have various ways of teaching arithmetic, in use in the schools; I shall speak of them all, beginning with a description of the arithmeticon, which is of great utility.
[Illustration] I have thought it necessary in this edition to give the original woodcut of the arithmeticon, which it will be seen contains twelve wires, with one ball on the first wire, two on the second, and so progressing up to twelve.

The improvement is, that each wire should contain twelve balls, so that the whole of the multiplication table may be done by it, up to 12 times 12 are 144.

The next step was having the balls painted black and white alternately, to assist the sense of seeing, it being certain that an uneducated eye cannot distinguish the combinations of colour, any more than an uneducated ear can distinguish the combinations of sounds.

So far the thing succeeded with respect to the sense of seeing; but there was yet another thing to be legislated for, and that was to prevent the children's attention being drawn off from the objects to which it was to be directed, viz.
the smaller number of balls as separated from the greater.