By means of toys, handwork and games, as well as various private individual experiments, a child touches on most sides of mathematics in the nursery class. In experimenting with bricks he must of necessity have considered relative size, balance and adjustment, form and symmetry; in fitting them back into their boxes some of the most difficult problems of cubic content; in weighing out "pretence" sugar and butter by means of sand and clay new problems are there for consideration; in making a paper-house questions of measurement evolve. This is all in the incidental play of the Nursery School, and yet we might say that a child thus occupied is learning mathematics more than anything else. Here, if he remained till six, he did a certain amount of necessary counting, and he may have acquired skill in recognising groups, he may have unconsciously and incidentally performed achievements in the four rules, but never, of course, in any shortened or technical form. Probably he knows some figures. It is best to give these to a child when he asks for or needs them, as in the case of records of games. On the other hand he may be content with strokes. Various mathematical relationships are made clear in his games or trials of strength, such as distance in relation to time or strength, weight in relation to power and to balance, length and breadth in relation to materials, value of material in relation to money or work. By means of many of his toys the properties of solids have become working knowledge to him. Here, then, is our starting-point for the transition period.

AFTER THE NURSERY STAGE

Undoubtedly the aim of the transition class is partly to continue by means of games and dramatic play the kind of knowledge gained in the Nursery School; but it has also the task of beginning to organise such knowledge, as the grouping into tens and hundreds. This organisation of raw material and the presenting of shortened processes, as occur in the first 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 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.

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 all, a purpose that is apparent and stimulating enough to produce willing practice. A child will do much to be a good shopkeeper, a good tram conductor, a good banker; he will always play the game for all it is worth.