ece of wood about 3/4 of an inch
thick, and about 3x1-1/2 feet of surface. It is painted white, and the
horizontal bars are green, so that the divisions may be apparent at a
distance; it has perpendicular divisions breaking it up into three
columns, each of which contains rows of nine small dresser hooks. It can
be hung on an easel or supported by its own hinge on a table. Each of
the divisions represents a numerical grouping, the one on the right is
for singles or units, the central one for tens, and the left side one
for hundreds: the counters used are button moulds, dipped in red ink,
with small loops of string to hang on the hooks: it is easily seen by a
child that, after nine is reached, the units can no longer remain in
their division or "house," but must be gathered together into a bunch
(fastened by a safety pin) and fixed on one of the hooks of the middle
division.

Sums of two or three lines can thus be set out on the horizontal bars,
and in processes of addition the answer can be on the bottom line. It is
very easy, by this concrete means, to see the process in subtraction,
and indeed the whole difficulty of dealing with ten is made concrete.
The whole of a sum can be gone through on this board with the
button-moulds, and on boards and chalk with figures, side by side, thus
interpreting symbol by material; but the whole process is abstract.

The piece of apparatus is less abstract only in degree than the figures
on the blackboard, because neither represents real life or its problems:
in abstract working we are merely going off at a side issue for the sake
of practice, to make us more competent to deal with the economic affairs
of life. There is a place for sticks and counters, and there is a place
for money and measures, but they are not the same: the former represents
the abstract and the latter the concrete problem if used as in real
life: the bridge between the abstract and the concrete is largely the
work of the transition class and junior school, in respect of the
foundations of arithmetic known as the first four rules.

Games of skill, very thorough shopping or keeping a bankbook, or selling
tickets for tram or train, represent the kind of everyday problem that
should be the centre of the arithmetic work at this transition stage;
and out of the necessities of these problems the abstract and
semi-abstract work should come, but it should _never_ precede the real
work. A real purpose should underlie it a