Finally, we should bear in mind that working with qualities and relations that are only partly understood or even misunderstood does under certain conditions give control over them. The general process of analytic learning in life is to respond as well as one can; to get a clearer idea thereby; to respond better the next time; and so on. For instance, one gets some sort of notion of what 1/5 means; he then answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?; by being told when he is right and when he is wrong, he gets from these experiences a better idea of 1/5; again he does his best with 1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2, etc., and thereby gains still further, and so on. What begins as a blind habit of manipulation started by imitation may thus grow into the power of correct response to the essential element. The pupil who has at the start no notion at all of 'multiplying' may learn what multiplying is by his experience that '4 6 multiplying gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing something with numbers and differentiates right results, he will often reach in the end the abstractions which he is supposed to need in the beginning. It may even be the case with some of the abstractions required in arithmetic that elaborate provision for comprehension beforehand is not so efficient as the same amount of energy devoted partly to provision for analysis itself beforehand and partly to practice in response to the element in question without full comprehension. It certainly is not the best psychology and not the best educational theory to think that the pupil first masters a principle and then merely applies it--first does some thinking and then computes by mere routine. On the contrary, the applications should help to establish, extend, and refine the principle--the work a pupil does with numbers should be a main means of increasing his understanding of the principles of arithmetic as a science. CHAPTER X THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC THE ESSENTIALS OF ARITHMETICAL REASONING