Information as to arithmetical rules and tables, as to weights and measures, and other arithmetical facts, is easily converted into knowledge of those facts, but it is not easily converted into knowledge of arithmetic.
In each case a _sense_ must be evolved if the information is to be assimilated, and so converted into real knowledge; and though it is true that the sense in question grows, in part at least, by feeding on appropriate information, it is equally true that if, owing to defective training, the sense remains undeveloped, the information supplied will remain unassimilated, and the tacit assumption that the possession of information is equivalent to the possession of real knowledge will delude both the teacher and the taught.

It is possible, as one knows from experience, for a boy to have mastered all the arithmetical rules and tables with which his master has supplied him, and to have all his measures and weights at his fingers' ends, and yet to be so destitute of the arithmetical sense as to give without a moment's misgiving an entirely nonsensical answer to a simple arithmetical problem,--to say, for example, as I have known half a class of boys say, that a _room_ is _five shillings and sixpence wide_.

Such a boy, though his head may be stuffed with arithmetical information, has no knowledge of arithmetic.
The gulf between memorised information and real knowledge becomes deep and wide in proportion as the subject matter is one which demands for its effective apprehension either intellectual effort or emotional insight.

When both these variables are demanded, the gulf widens and deepens at a ratio which is "geometrical" rather than "arithmetical"; and when a high degree of each is demanded, the separation between knowledge and information is complete.
The Art Master who should try to train the aesthetic sense of his pupils by making them learn by heart a string of propositions in which he had set out the artistic merits of sundry masterpieces of painting and sculpture, would expose himself to well-merited ridicule.

So would the teacher who should try to train the scientific sense of his pupils by no other method than that of making them learn scientific formulae by heart.