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 piece 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.