| 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
As another instructive topic in the constitution of arithmetical abilities, we may take the case of the reasoning involved in understanding the manipulations of figures in two (or more)-place addition and subtraction, multiplication and division involving a two (or more)-place number, and the manipulations of decimals in all four operations. The psychology of these is of special interest and importance. For there are two opposite explanations possible here, leading to two opposite theories of teaching.
The common explanation is that these methods of manipulation, if understood at all, are understood as deductions from the properties of our system of decimal notation. The other is that they are understood partly as inductions from the experience that they always give the right answer. The first explanation leads to the common preliminary deductive explanations of the textbooks. The other leads to explanations by verification; e.g., of addition by counting, of subtraction by addition, of multiplication by addition, of division by multiplication. Samples of these two sorts of explanation are given below.
SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION
Multiplication is the process of taking one number as many times as there are units in another number.