To suppose that the child from such a lesson as this shall come to understand the difference between noise and sound is ridiculous. The child will probably get the impression that the teacher wished to play a joke, and that she is rather foolish, because she lost the thread of her discourse when she was interrupted by noise, and because she mistook a mandolin for a baby. Most certainly, it is the figure of the teacher herself that is impressed upon the child's mind through such a lesson, and not the object for which the lesson was given.
To obtain a _simple lesson_ from a teacher who has been prepared according to the ordinary methods, is a very difficult task. I remember that, after having explained the material fully and in detail, I called upon one of my teachers to teach, by means of the geometric insets, the difference between a square and a triangle. The task of the teacher was simply to fit a square and a triangle of wood into the empty spaces made to receive them. She should then have shown the child how to follow with his finger the contours of the wooden pieces and of the frames into which they fit, saying, meanwhile, "This is a square--this is a triangle." The teacher whom I had called upon began by having the child touch the square, saying, "This is a line,--another,--another,--and another. There are four lines: count them with your little finger and tell me how many there are. And the corners,--count the corners, feel them with your little finger. See, there are four corners too. Look at this piece well. It is a square." I corrected the teacher, telling her that in this way she was not teaching the child to recognise a form, but was giving him an idea of sides, of angles, of number, and that this was a very different thing from that which she was to teach in this lesson. "But," she said, trying to justify herself, "it is the same thing." It is not, however, the same thing. It is the geometric analysis and the mathematics of the thing. It would be possible to have an idea of the form of the quadrilateral without knowing how to count to four, and, therefore, without appreciating the number of sides and angles. The sides and the angles are abstractions which in themselves do not exist; that which does exist is this piece of wood of a determined form. The elaborate explanations of the teacher not only confused the child's mind, but bridged over the distance that lies between the concrete and the abstract, between the form of an object and the mathematics of the form.