THE TEACHING OF NUMERATION 337
From the way in which they are arranged it is seen at a glance
which numbers can be -divided by two: all those which have no
cube at the base. These are even numbers because they can be
set out in pairs, that is, two by two; and the division into two is
very easy because all that is needed is to separate the two lines of
cubes which stand one under the other. Counting the cubes of
each row, one gets the quotient. In order to reconstruct the
original number it is enough to bring the two rows together again*
None of this is difficult for children of five years old.
Here again repetition soon induces monotony; why should
we not change the exercises? Let us take the series of the ten
lengths, and instead of putting the one on the nine, let us put it
on the ten; and the two on the nine instead of on the eight; and
the three on the eight instead of on the seven. We can also place
the two on the ten, the three on the nine, and the four on the eight.
In every case the result is a length greater than ten, which has to
be named—eleven, twelve, thirteen, etc. up to twenty. And why
should the cubes be utilized for these games only up nine, that is,
be so few?
The operations learnt on the ten are continued up to twenty
without any difficulty; the only difficulty is that of the tens of
units, for which several lessons are necessary.
LESSONS ON THE TENS OF UNITS
ARITHMETICAL CALCULATIONS BEYOND TEN
The material necessary consists of various rectangular cards
on which is printed the 10 in figures'five or six cms. high, and other
rectangular cards equal in height and half in breadth of the first
and bearing the separate numbers 1 to 9. The simple numbers
are placed in a row—1, 2, 3, 4, 5, 6, 7, 8, 9. Then, as there are
more numbers, it is necessary to start again and to take up the
one again. This one resembles the piece which in the system of
lengths extends beyond the nine in the ten stick. Counting along