and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc. In the case of the divisions, suppose that the pupil has learned his first table and gained surety in such exercises as:-- 4 5s = .... 6 × 5 = .... 9 nickels = .... cents. 8 5s = .... 4 × 5 = .... 6 " = .... " 3 5s = .... 2 × 5 = .... 5 " = .... " 7 5s = .... 9 × 5 = .... 7 " = .... " If one ball costs 5 cents, two balls cost .... cents, three balls cost .... cents, etc. He may then be set at once to work at the answers to exercises like the following:-- Write the answers and the missing numbers:-- A B C D .... 5s = 15 40 = .... 5s .... × 5 = 25 20 cents = .... nickels. .... 5s = 20 20 = .... 5s .... × 5 = 50 30 cents = .... nickels. .... 5s = 40 15 = .... 5s .... × 5 = 35 15 cents = .... nickels. .... 5s = 25 45 = .... 5s .... × 5 = 10 40 cents = .... nickels. .... 5s = 30 50 = .... 5s .... × 5 = 40 .... 5s = 35 25 = .... 5s .... × 5 = 45 E For 5 cents you can buy 1 small loaf of bread. For 10 cents you can buy 2 small loaves of bread. For 25 cents you can buy .... small loaves of bread. For 45 cents you can buy .... small loaves of bread. For 35 cents you can buy .... small loaves of bread. F 5 cents pays 1 car fare. 15 cents pays .... car fares. 10 cents pays .... car fares. 20 cents pays .... car fares. G How many 5 cent balls can you buy with 30 cents? .... How many 5 cent balls can you buy with 35 cents? .... How many 5 cent balls can you buy with 25 cents? .... How many 5 cent balls can you buy with 15 cents? .... In the case of the meaning of a fraction, the ability, and so the learning, is much more elaborate than common practice has assumed; in the case of the subtraction and division tables the learning is much less so. In neither case is the learning either mere memorizing of facts or the mere understanding of a principle _in abstracto_ followed by its application to concrete cases. It is (and this we shall find true of almost all efficient learning in arithmetic) the formation of connections and their use in such an order that each helps the others to the maximum degree, and so that each will do the maximum amount for arithmetical abilities other than the one specially concerned, and for the general competence of the learner. LEARNING THE PROCESSES OF COMPUTATION