irst four rules, forms the work also of the junior school. To give to a
child shortened processes which he would be very unlikely to discover in
less than a lifetime, is simply giving him the experience of the race,
as primitive man did to his son. But the important point is to decide
when a child's discovery should end and the teacher's demonstration
begin.

This is the period when we are accustomed to speak of beginning
"abstract" work; it is well to be clear what it means, and how it stands
related to a child's need for experience. When we leave the problems of
life, such as shopping, keeping records of games and making measurements
for construction; and when we begin to work with pure number, we are
said to be dealing with the abstract. Formerly dealing with pure number
was called "simple," and dealing with actual things, such as money and
measures, "compound," and they were taken in this order. But experience
has reversed the process, and a child comes to see the need of abstract
practice when he finds he is not quick enough or accurate enough, or his
setting out seems clumsy, in actual problems. This was discussed at
greater length in the chapter on Play.

For instance, he might set down the points of a game by strokes, each
line representing a different opponent:

John ||||||||||||||||

Henry |||||||||||

Tom |||

He will see how difficult it is to estimate at a glance the exact score,
and how easy it is to be inaccurate. It seems the moment to show him
that the idea of grouping or enclosing a certain number, and always
keeping to the same grouping, is helpful:

John |||||||||| |||||| = 1 ten and 6 singles.

Henry |||||||||| | = 1 ten and 1 single.

Tom ||| = 3 singles.

After doing this a good many times he could be told that this is a
universal method, and he would doubtless enjoy the purely puzzle
pleasure in working long sums to perfect practice. This pleasure is very
common in children at this stage, but too often it comes to them merely
through being shown the "trick" of carrying tens. They have reached a
purely abstract point, but they cannot get through it without some more
material help. The following is an example of the kind of help that can
be given in getting clear the concept of the ten grouping and the
processes it involves:

[Illustration: Board with hooks, in ranks of nine, and rings]

The whole apparatus is a rectangular pi