For example, as an introduction to long division, a pupil may be given exercises using one-figure divisors in the long form, as:-- 773 and 5 remainder ______ 7)5416 49 -- 51 49 -- 26 21 -- 5 The important recommendation concerning these purely propædeutic bonds, and bonds formed only for later reconstruction, is to be very critical of them, and not indulge in them when, by the exercise of enough ingenuity, some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well. Arithmetical teaching has done very well in this respect, tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones. It is in the teaching of reading that we find the formation of propædeutic bonds of dubious value (with letters, phonograms, diacritical marks, and the like) often carried to demonstrably wasteful extremes. CHAPTER VI THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES THE AMOUNT OF PRACTICE It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion. Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive (that is, in the first two books of a three-book series) in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? Every case of the action of each bond is to be counted. THE FUNDAMENTAL BONDS 1 + 1 2 - 1 1 × 1 2 ÷ 1 1 + 2 2 - 2 2 × 1 2 ÷ 2 1 + 3 3 × 1 1 + 4 4 × 1 1 + 5 3 - 1 5 × 1 3 ÷ 1 1 + 6 3 - 2 6 × 1 3 ÷ 2 1 + 7 3 - 3 7 × 1 3 ÷ 3 1 + 8 8 × 1 1 + 9 9 × 1