"In a rough way we may summarize the conclusions of the writers to whom reference has chiefly been made, as follows: 1. Let the child learn to count things, thus getting the notion of number. These things are, for the purpose of counting, considered alike, and they may be single objects or groups. 2. Let him acquire the number series, exercising with it beyond the circle of actually counted things. 3. In the learning of symbols it does not seem to be a matter of moment as to whether these are given with the first presentation of number or not. They must, however, be acquired soon. 4. Unconsciously and gradually the child will acquire the idea (never expressed to him in words) of the one-to-one correspondence of number, name, symbol, and thereafter the pure concept of number will play a small part in his arithmetical calculations. 5. The ratio idea of number should be introduced early, and applied in the work with fractions."