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. CHAPTER XXIV EXPERIENCES BY MEANS OF DOING