It is a common opinion that the only alternative is knowing them by rote. This, of course, is one common alternative, but the other explanation suggests that understanding the manipulations by inductive reasoning from their results is another and an important alternative. The manipulations of 'long' multiplication, for instance, learned by imitation or mechanical drill, are found to give for 25 × A a result about twice as large as for 13 × A, for 38 or 39 × A a result about three times as large; for 115 × A a result about ten times as large as for 11 × A. With even the very dull pupils the procedure is verified at least to the extent that it gives a result which the scientific expert in the case—the teacher—calls right. With even the very bright pupils, who can appreciate the relation of the procedure to decimal notation, this relation may be used not as the sole deduction of the procedure beforehand, but as one partial means of verifying it afterward. Or there may be the condition of half-appreciation of the relation in which the pupil uses knowledge of the decimal notation to convince himself that the procedure does, but not that it must give the right answer, the answer being 'right' because the teacher, the answer-list, and collateral evidence assure him of it.
I have taken the manipulation of the partial products as an illustration because it is one of the least favored cases for the explanation I am presenting. If we take the first case where a manipulation may be deduced from decimal notation, known merely by rote, or verified inductively, namely, the addition of two-place numbers, it seems sure that the mental processes just described are almost the universal rule.
Surely in our schools at present children add the 3 of 23 to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten because they see the teacher do so and are told to do so. They are protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but because they do not know how to add 8 and 5, because they have been taught the habit of adding figures that stand one above the other, or with a + between them; and because they are shown or told what they are to do. They are protected from adding 3 + 5 and 2 + 3, again, by no deductive reasoning but for the second and third reasons just given. In nine cases out of ten they do not even think of the possibility of adding in any other way than the '3 + 3, 2 + 5' way, much less do they select that way on account of the facts that 53 = 50 + 3 and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that (a + b) + (c + d) = (a + c) + (b + d)!
Just as surely all but the very dullest twentieth or so of children come in the end to something more than rote knowledge,—to understand, to know that the procedure in question is right.
Whether they know why 76 is right depends upon what is meant by why. If it means that 76 is the result which competent people agree upon, they do. If it means that 76 is the result which would come from accurate counting they perhaps know why as well as they would have, had they been given full explanations of the relation of the procedure in two-place addition to decimal notation. If why means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and (a + b) + (c + d) = (a + c) + (b + d), they do not. Nor, I am tempted to add, would most of them by any sort of teaching whatever.
I conclude, therefore, that school children may and do reason about and understand the manipulations of numbers in this inductive, verifying way without being able to, or at least without, under present conditions, finding it profitable to derive them deductively. I believe, in fact, that pure arithmetic as it is learned and known is largely an inductive science. At one extreme is a minority to whom it is a series of deductions from principles; at the other extreme is a minority to whom it is a series of blind habits; between the two is the great majority, representing every gradation but centering about the type of the inductive thinker.