The chief point is, that no course of instruction which claims at all to give completeness of culture can be regarded as concluded before it has introduced the pupil to the pragmatic study of history, and has taught him to look for causes and effects. This applies preëminently to modern history, on account of its direct connection with the present; but mediæval and ancient history, too, have to be worked over once more from this point of view. History should be the teacher of mankind; if it does not become so, the blame rests largely with those who teach history in schools.
[251.] A well-compiled and well-proportioned brief history of inventions, arts, and sciences should conclude the teaching of history, not only in gymnasia, but also and especially in higher burgher schools, because their courses of study are not supplemented by the university.
Moreover, the whole course in history is properly accompanied by illustrative poetical selections, which, although perhaps not produced during the different epochs, yet stand in some relation to them; and which in some measure, even if only by illustrating ages very far apart, exhibit the vast differences in the freest activities of the human mind.
Note.—National history is not the same for each land, nor everywhere of equal interest, and, owing to its connection with larger events, often unintelligible to young minds when torn out of its place and presented by itself. If its early introduction is desired in order to kindle the heart, special pains must be taken to select that which is intelligible and which appeals to boyhood.
[ CHAPTER III
Mathematics and Nature Study]
[252.] Aptitude for mathematics is not rarer than aptitude for other studies. That the contrary seems true, is owing to a belated and slighted beginning. But that mathematicians are seldom inclined to give as much time to children as they ought is only natural. The elementary lessons in combination and geometry are neglected in favor of arithmetic, and demonstration is attempted where no mathematical imagination has been awakened.
The first essential is attention to magnitudes, and their changes, where they occur. Hence, counting, measuring, weighing, where possible; where impossible, at least the estimating of magnitudes to determine, however vaguely at first, the more and the less, the larger and the smaller, the nearer and the farther.
Special consideration should be given, on the one hand, to the number of permutations, variations, and combinations; and, on the other hand, to the quadratic and cubic relations, where similar planes and bodies are determined by analogous lines.