The criticism was valid and should have been met in part by replacing the deductive explanations by inductive verifications, and in part by using the deductive reasoning as a check after the process itself is mastered. The very same discussions of place-value which are futile as proof that you must do a certain thing before you have done it, often become instructive as an explanation of why the thing that you have learned to do and are familiar with and have verified by other tests works as well as it does. The general deductive theory of arithmetic should not be learned only to be forgotten. Much of it should, by most pupils, not be learned at all. What is learned should be learned much later than now, as a synthesis and rationale of habits, not as their creator. What is learned of such deductive theory should rank among the most rather than least permanent of a pupil's stock of arithmetical knowledge and power. There are bonds which are formed only to be lost, and bonds formed only to be lost _in their first form_, being used in a new organization as material for bonds of a higher order; but the bonds involved in deductive explanations of why certain processes are right are not such: they are not to be formed just to be forgotten, nor as mere propædeutics to routine manipulations. PROPÆDEUTIC BONDS The formation of bonds to a limited strength because they are to be lost in their first form, being worked over in different ways in other bonds to which they are propædeutic or contributing is the most important case of low strength, or rather low permanence, in bonds. The bond between four 5s in a column to be added and the response of thinking '10, 15, 20' is worth forming, but it is displaced later by the multiplication bond or direct connection of 'four 5s to be added' with '20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms serial bonds which as series might well be left to disappear. Their separate steps are kept as permanent bonds for use in column addition, but their serial nature is changed from 2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.; after playing their part in producing the bonds whereby any multiple of 2 by 2 to 9, can be got, the original serial bonds are, as series, needed no longer. The verbal response of saying 'and' in adding, after helping to establish the bonds whereby the general set of the mind toward adding coöperates with the numbers seen or thought of to produce their sum, should disappear; or remain so slurred in inner speech as to offer no bar to speed.