Building this somewhat elaborate series of minor abilities seems to be a very roundabout way of getting knowledge of the meaning of a fraction, and is, if we take no account of what is got along with this knowledge. Taking account of the intrinsically useful habits that are built up, one might retort that the pupil gets his knowledge of the meaning of a fraction at zero cost. KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES Consider next the knowledge of the subtraction and division 'Tables.' The usual treatment presupposes that learning them consists of forming independently the bonds:-- 3 - 1 = 2 4 ÷ 2 = 2 3 - 2 = 1 6 ÷ 2 = 3 4 - 1 = 3 6 ÷ 3 = 2 . . . . . . . . . . . . 18 - 9 = 9 81 ÷ 9 = 9 In fact, however, these 126 bonds are not formed independently. Except perhaps in the case of the dullest twentieth of pupils, they are somewhat facilitated by the already learned additions and multiplications. And by proper arrangement of the learning they may be enormously facilitated thereby. Indeed, we may replace the independent memorizing of these facts by a set of instructive exercises wherein the pupil derives the subtractions from the corresponding additions by simple acts of reasoning or selective thinking. As soon as the additions giving sums of 9 or less are learned, let the pupil attack an exercise like the following:-- Write the missing numbers:-- A B C D 3 and ... are 5. 5 and ... are 8. 4 and ... are 5. 4 and ... are 8. 3 and ... are 9. 3 and ... are 6. 5 and ... are 6. 1 and ... are 7. 4 and ... are 7. 4 and ... are 9. 6 and ... are 9. 6 and ... are 7. 5 and ... are 7. 2 and ... = 6. 1 and ... are 8. 8 and ... are 9. 6 and ... are 8. 5 and ... = 9. 3 and ... are 7. 3 + ... are 4. 4 and ... are 6. 2 and ... = 7. 1 + ... are 3. 7 + ... are 8. 2 and ... are 5. 3 and ... = 8. 1 + ... are 5. 4 + ... are 9. 2 and ... = 8. 1 and ... = 4. 4 + ... are 8. 2 + ... are 3. 3 and ... = 6. 2 and ... = 4. 7 + ... are 9. 1 + ... are 9. 6 and ... = 9. 3 and ... = 8. 2 + ... = 4. 3 + ... = 6. 4 and ... = 6. 6 and ... = 7. 3 + ... = 8. 5 + ... = 9. 4 and ... = 7. 2 and ... = 5. 4 + ... = 5. 1 + ... = 3. The task for reasoning is only to try, one after another, numbers that seem promising and to select the right one when found. With a little stimulus and direction children can thus derive the subtractions up to those with 9 as the larger number. Let them then be taught to do the same with the printed forms:-- Subtract 9 7 8 5 8 6 3 5 6 2 2 4 etc. - - - - - -