The Psychology of Arithmetic

by Edward L. Thorndike

We need not be timid. The pupil will have no difficulty in adding, subtracting, multiplying, and dividing with United States money—unless we create it by our explanations! If we simply form the two bonds described above and show by proper verification that the procedure always gives the right answer, the early teaching of the four operations with United States money will in fact actually show a learning profit! It will save more time in the work with integers than was spent in teaching it! For, in the first place, it will help to make work with four-place and five-place numbers more intelligible and vital. A pupil can understand $16.75 or $28.79 more easily than 1675 or 2879. The former may be the prices of a suit or sewing machine or bicycle. In the second place, it permits the use of a large stock of genuine problems about spending, saving, sharing, and the like with advertisements and catalogues and school enterprises. In the third place, it permits the use of common-sense checks. A boy may find one fourth of 3000 as 7050 or 75 and not be disturbed, but he will much more easily realize that one fourth of $30.00 is not over $70 or less than $1. Even the decimal point of which we used to be so afraid may actually help the eye to keep its place in adding.

INTEREST

So far, the illustrations of improvements in the order of bonds so as to get less interference and more facilitation than the customary orders secure have sought chiefly to improve the mechanical organization of the bonds. Any gain in interest which the changes described effected would be largely due to the greater achievement itself. Dewey and others have emphasized a very different principle of improving the order of formation of bonds—the principle of determination of the bonds to be formed by some vital, engaging problem which arouses interest enough to lighten the labor and which goes beyond or even against cut-and-dried plans for sequences in order to get effective problems. For example, the work of the first month in grade 2B might sacrifice facilitations of the mechanical sort in order to put arithmetic to use in deciding what dimensions a rabbit's cage should have to give him 12 square feet of floor space, how much bread he should have per meal to get 6 ounces a day, how long a ten-cent loaf would last, how many loaves should be bought per week, how much it costs to feed the rabbit, how much he has gained in weight since he was brought to the school, and so on.

Such sacrifices of the optimal order if interest were equal, in order to get greater interest or a healthier interest, are justifiable. Vital problems as nuclei around which to organize arithmetical learning are of prime importance. It is even safe probably to insist that some genuine problem-situation requiring a new process, such as addition with carrying, multiplication by two-place numbers, or division with decimals, be provided in every case as a part of the introduction to that process. The sacrifice should not be too great, however; the search for vital problems that fit an economical order of subject matter is as much needed as the amendment of that order to fit known interests; and the assurance that a problem helps the pupil to learn arithmetic is as important as the assurance that arithmetic is used to help the pupil solve his personal problems.

Much ingenuity and experimentation will be required to find the order that is satisfactory in both quality and quantity of interest or motive and helpfulness of the bonds one to another. The difficulty of organizing arithmetic around attractive problems is much increased by the fact of class instruction. For any one pupil vital, personal problems or projects could be found to provide for many arithmetical abilities; and any necessary knowledge and technique which these projects did not develop could be somehow fitted in along with them. But thirty children, half boys and half girls, varying by five years in age, coming from different homes, with different native capacities, will not, in September, 1920, unanimously feel a vital need to solve any one problem, and then conveniently feel another on, say, October 15! In the mechanical laws of learning children are much alike, and the gain we may hope to make from reducing inhibitions and increasing facilitations is, for ordinary class-teaching, probably greater than that to be made from the discovery of attractive central problems. We should, however, get as much as possible of both.

GENERAL PRINCIPLES

The reader may by now feel rather helpless before the problem of the arrangement of arithmetical subject matter. "Sometimes you complete a topic, sometimes you take it piecemeal months or years apart, often you make queer twists and shifts to get a strategic advantage over the enemy," he may think, "but are there no guiding principles, no general rules?" There is only one that is absolutely general, to take the order that works best for arithmetical learning. There are particular rules, but there are so many and they are so limited by an 'other things being equal' clause, that probably a general eagerness to think out the pros and cons for any given proposal is better than a stiff attempt to adhere to these rules. I will state and illustrate some of them, and let the reader judge.

Other things being equal, one new sort of bonds should not be started until the previous set is fairly established, and two different sets should not be started at once. Thus, multiplication of two- and three-place numbers by 2, 3, 4, and 5 will first use numbers such that no carrying is required, and no zero difficulties are encountered, then introduce carrying, then introduce multiplicands like 206 and 320. If other things were equal, the carrying would be split into two steps—first drills with (4 × 6) + 2, (3 × 7) + 3, (5 × 4) + 1, and the like, and second the actual use of these habits in the multiplication. The objection to this separation of the double habit is that the first part of it in isolation is too artificial—that it may be better to suffer the extra difficulty of forming the two together than to teach so rarely used habits as the (a × b) + c series. Experimental tests are needed to decide this point.

Other things being equal, bonds should be formed in such order that none will have to be broken later. For example, there is a strong argument for teaching long division first, or very early, with remainders, letting the case of zero remainder come in as one of many. If the pupils have been familiarized with the remainder notion by the drills recommended as preparation for short division,[9] the use of remainders in long division will offer little difficulty. The exclusive use of examples without remainders may form the habit of not being exact in computation, of trusting to 'coming out even' as a sole check, and even of writing down a number to fit the final number to be divided instead of obtaining it by honest multiplication.

For similar reasons additions with 2 and 3 as well as 1 to be 'carried' have much to recommend them in the very first stages of column addition with carrying. There is here the added advantage that a pupil will be more likely to remember to carry if he has to think what to carry. The present common practice of using small numbers for ease in the addition itself teaches many children to think of carrying as adding one.