(5) _Addition and subtraction facts in the case of fractions._--In the case of adding and subtracting fractions, certain specific bonds--between the situation of halves and thirds to be added and the responses of thinking of the numbers as equal to so many sixths, between the situation thirds and fourths to be added and thinking of them as so many twelfths, between fourths and eighths to be added and thinking of them as eighths, and the like--should be formed separately. The general rule of thinking of fractions as their equivalents with some convenient denominator should come as an organization and extension of such special habits, not as an edict from the textbook or teacher. (6) _Fractional equivalents._--Efficiency requires that in the end the much used reductions should be firmly connected with the situations where they are needed. They may as well, therefore, be so connected from the beginning, with the gain of making the general process far easier for the dull pupils to master. We shall see later that, for all save the very gifted pupils, the economical way to get an understanding of arithmetical principles is not, usually, to learn a rule and then apply it, but to perform instructive operations and, in the course of performing them, to get insight into the principles. (7) _Protective habits in multiplying and dividing with fractions._--In multiplying and dividing with fractions special bonds should be formed to counteract the now harmful influence of the 'multiply = get a larger number' and 'divide = get a smaller number' bonds which all work with integers has been reënforcing. For example, at the beginning of the systematic work with multiplication by a fraction, let the following be printed clearly at the top of every relevant page of the textbook and displayed on the blackboard:-- _When you multiply a number by anything more than 1 the result is larger than the number._ _When you multiply a number by 1 the result is the same as the number._ _When you multiply a number by anything less than 1 the result is smaller than the number._ Let the pupils establish the new habit by many such exercises as:-- 18 × 4 = .... 9 × 2 = .... 4 × 4 = .... 6 × 2 = .... 2 × 4 = .... 3 × 2 = .... 1 × 4 = .... 1 × 2 = .... 1/2 × 4 = .... 1/3 × 2 = .... 1/4 × 4 = .... 1/6 × 2 = .... 1/8 × 4 = .... 1/9 × 2 = .... In the case of division by a fraction the old harmful habit should be counteracted and refined by similar rules and exercises as follows:-- _When you divide a number by anything more than 1 the result is smaller than the number._ _When you divide a number by 1 the result is the same as the number._ _When you divide a number by anything less than 1 the result is larger than the number._ State the missing numbers:--