THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE
Dewey, and others following him, have emphasized the desirability of having pupils do their work as active seekers, conscious of problems whose solution satisfies some real need of their own natures. Other things being equal, it is unwise, they argue, for pupils to be led along blindfold as it were by the teacher and textbook, not knowing where they are going or why they are going there. They ought rather to have some living purpose, and be zealous for its attainment.
This doctrine is in general sound, as we shall see, but it is often misused as a defense of practices which neglect the formation of fundamental habits, or as a recommendation to practices which are quite unworkable under ordinary classroom conditions. So it seems probable that its nature and limitations are not thoroughly known, even to its followers, and that a rather detailed treatment of it should be given here.
ILLUSTRATIVE CASES
Consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained, or care about attaining the end (well or ill understood) is well spent.
It is well for a pupil who has learned (1) the meanings of the numbers one to ten, (2) how to count a collection of ten or less, and (3) how to measure in inches a magnitude of ten, nine, eight inches, etc., to be confronted with the problem of true adding without counting or measuring, as in 'hidden' addition and measurement by inference. For example, the teacher has three pencils counted and put under a book; has two more counted and put under the book; and asks, "How many pencils are there under the book?" Answers, when obtained, are verified or refuted by actual counting and measuring.
The time here is well spent because the children can do the necessary thinking if the tasks are well chosen; because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding, that is, thinking of the two numbers and inferring their sum; and further, because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it.
The manipulation of short multiplication may be introduced by confronting the pupils with such problems as, "How to tell how many Uneeda biscuit there are in four boxes, by opening only one box." Correct solutions by addition should be accepted. Correct solutions by multiplication, if any gifted children think of this way, should be accepted, even if the children cannot justify their procedure. (Inferring the manipulation from the place-values of numbers is beyond all save the most gifted and probably beyond them.) Correct solution by multiplication by some child who happens to have learned it elsewhere should be accepted. Let the main proof of the trustworthiness of the manipulation be by measurement and by addition. Proof by the stock arguments from the place-values of numbers may also be used. If no child hits on the manipulation in question, the problem of finding the length without adding may be set. If they still fail, the problem may be made easier by being put as "4 times 22 gives the answer. Write down what you think 4 times 22 will be." Other reductions of the difficulty of the problem may be made, or the teacher may give the answer without very great harm being done. The important requirement is that the pupils should be aware of the problem and treat the manipulation as a solution of it, not as a form of educational ceremonial which they learn to satisfy the whims of parents and teachers. In the case of any particular class a situation that is more appealing to the pupils' practical interests than the situation used here can probably be devised.
The time spent in this way is well spent (1) because all but the very dull pupils can solve the problem in some way, (2) because the significance of the manipulation as an economy over addition is worth bringing out, and (3) because there is no way of beginning training in short multiplication that is much better.
In the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square inches in 35 square feet, or the number of days in 32 years, or whatever similar problem can be brought up so as to be felt as a problem.