is much harder than either of the above.
| 43 1 2 13 | is much harder than | 21 33 35. |
| — | — |
It may be said that these differences in difficulty are due to different amounts of practice. This is probably not true, but if it were, it would not change the argument; if the two abilities were identical, the practice of one would improve the other equally.
I shall not undertake here this task of listing and describing the elementary functions which constitute arithmetical learning, partly because what they are is not fully known, partly because in many cases a final ability may be constituted in several different ways whose descriptions become necessarily tedious, and partly because an adequate statement of what is known would far outrun the space limits of this chapter. Instead, I shall illustrate the results by some samples.
KNOWLEDGE OF THE MEANING OF A FRACTION
As a first sample, consider knowledge of the meaning of a fraction. Is the ability in question simply to understand that a fraction is a statement of the number of parts, each of a certain size, the upper number or numerator telling how many parts are taken and the lower number or denominator telling what fraction of unity each part is? And is the educational treatment required simply to describe and illustrate such a statement and have the pupils apply it to the recognition of fractions and the interpretation of each of them? And is the learning process (1) the formation of the notions of part, size of part, number of part, (2) relating the last two to the numbers in a fraction, and, as a necessary consequence, (3) applying these notions adequately whenever one encounters a fraction in operation?
Precisely this was the notion a few generations ago. The nature of fractions was taught as one principle, in one step, and the habits of dealing with fractions were supposed to be deduced from the general law of a fraction's nature. As a result the subject of fractions had to be long delayed, was studied at great cost of time and effort, and, even so, remained a mystery to all save gifted pupils. These gifted pupils probably of their own accord built up the ability piecemeal out of constituent insights and habits.
At all events, scientific teaching now does build up the total ability as a fusion or organization of lesser abilities. What these are will be seen best by examining the means taken to get them. (1) First comes the association of ½ of a pie, ½ of a cake, ½ of an apple, and such like with their concrete meanings so that a pupil can properly name a clearly designated half of an obvious unit like an orange, pear, or piece of chalk. The same degree of understanding of 1⁄4, 1⁄8, 1⁄3, 1⁄6, and 1⁄5 is secured. The pupil is taught that 1 pie = 2 1⁄2s, 3 1⁄3s, 4 1⁄4s, 5 1⁄5s, 6 1⁄6s, and 8 1⁄8s; similarly for 1 cake, 1 apple, and the like.
So far he understands 1⁄x of y in the sense of certain simple parts of obviously unitary ys.