Boy. Twice as long as the side of the first square.
(Here we have brought out for us the error—a very common one, as you
all know:—now for the teaching-skill in making the boy detect the
error.)
Socrates. So you say the square on a double line will be the double of the first square? Now let us fit to one end of the first square a second square which is equal to it; and let us fit two other squares of the same size to the sides of those two squares; then what figure have we?
Boy. A square.
Socrates. And how many times as great as the first square is it?
Boy. Four times as great.
Socrates. Not twice as great, as you said?
Boy. No, it is four times as great.
(Thus is the error exposed, the boy being thoroughly convinced:—now for the teacher's guidance in the discovery of the new truth.)
Socrates. If in this new square, which is made up of four of the old squares, we draw four diagonals, so as to cut of the four outside corners, each of these diagonals cut each of these squares, how?
Boy. Into halves.
Socrates. And you already know that these four diagonals will be equal, and will form another square?
Boy. Yes, I know.
Socrates. And of what parts of the four squares is this inside square made up?
Boy. Of the four inside halves.
Socrates. And four halves are equal to what?
Boy. To two wholes.
Socrates. Then we have got a square that is equal to how many of the original squares?
Boy. To two of them.
Socrates. And it is a square upon what line?
Boy. Upon the line that divides the original square into two halves.
Socrates. That is, upon its diagonal?
Boy. Yes.
Surely no one can have failed to see that in eliciting the error, in correcting it, in discovering the truth, the boy's mind was being put through a course of discipline most salutary, and it will be hardly possible to doubt that the boy thus taught would be ready of himself to go over the steps of the proof again by himself, and to turn at his leisure to any other form of proof of the propositions that might fall within his reach. At the same time the practised teacher will have suggested to his mind many other useful hints which this lesson could be made to furnish—that this is a special case of the celebrated 47th proposition, the right-angled triangle here being isosceles—that the square of a half is a fourth—the square on the double of a line or of the double of a number is four times the square on that line or of that number—that (2a)2 is not 4 a nor 2 a2 but 4 a2 &c., &c., and he will perceive also that connecting together these similar instances will give the boy a power of remembering them too, such as mere rote-work can never confer.[1]
I do not think that any remarks of mine could impress upon parents so forcibly as the above extract from the lecture of a
- ↑ Extracted from a Lecture by W. A. Shields, of the Peckham Birkbeck Schools, delivered at St. Martin's Hall in 1853. I have given the extract with Mr. Shields' verbal alterations, which chiefly consist in avoiding leading questions to which the answer is "yes," his object being different from that of the author in the original; but the substance is the same, as a sample of teaching.
