Such sacrifices of the optimal order if interest were equal, in order to get greater interest or a healthier interest, are justifiable. Vital problems as nuclei around which to organize arithmetical learning are of prime importance. It is even safe probably to insist that some genuine problem-situation requiring a new process, such as addition with carrying, multiplication by two-place numbers, or division with decimals, be provided in every case as a part of the introduction to that process. The sacrifice should not be too great, however; the search for vital problems that fit an economical order of subject matter is as much needed as the amendment of that order to fit known interests; and the assurance that a problem helps the pupil to learn arithmetic is as important as the assurance that arithmetic is used to help the pupil solve his personal problems. Much ingenuity and experimentation will be required to find the order that is satisfactory in both quality and quantity of interest or motive and helpfulness of the bonds one to another. The difficulty of organizing arithmetic around attractive problems is much increased by the fact of class instruction. For any one pupil vital, personal problems or projects could be found to provide for many arithmetical abilities; and any necessary knowledge and technique which these projects did not develop could be somehow fitted in along with them. But thirty children, half boys and half girls, varying by five years in age, coming from different homes, with different native capacities, will not, in September, 1920, unanimously feel a vital need to solve any one problem, and then conveniently feel another on, say, October 15! In the mechanical laws of learning children are much alike, and the gain we may hope to make from reducing inhibitions and increasing facilitations is, for ordinary class-teaching, probably greater than that to be made from the discovery of attractive central problems. We should, however, get as much as possible of both. GENERAL PRINCIPLES The reader may by now feel rather helpless before the problem of the arrangement of arithmetical subject matter. "Sometimes you complete a topic, sometimes you take it piecemeal months or years apart, often you make queer twists and shifts to get a strategic advantage over the enemy," he may think, "but are there no guiding principles, no general rules?" There is only one that is absolutely general, to _take the order that works best for arithmetical learning_. There are particular rules, but there are so many and they are so limited by an 'other things being equal' clause, that probably a general eagerness to think out the _pros_ and _cons_ for any given proposal is better than a stiff attempt to adhere to these rules. I will state and illustrate some of them, and let the reader judge.