To have a really correct apprehension of the principle of decimal notation, for instance, to understand that it is purely arbitrary, and that we might in the same way take any other number than ten as the base of a numerical scale,—that we might increase for instance by fives, or eights, or nines, or twelves, just as well as by tens—all this requires considerable maturity of intellect, and some subtlety of reasoning. Indeed I doubt whether many of the pretentious sciolists, who insist so much on young children giving the rationale of everything, have themselves ever yet made an ultimate analysis of the first step in arithmetical notation. Many of them would open their eyes were you to tell them, for instance, that the number of fingers on your two hands may be just as correctly expressed by the figures 11, 12, 13, 14, or 15, as by the figures 10,—a truism perfectly familiar to every one acquainted with the generalizations of higher arithmetic. Yet it is up-hill work to make the matter quite clear to a beginner. We may wisely therefore give our children at first an arbitrary rule for notation. We give them an equally arbitrary rule for addition. They accept these rules and work upon them, and learn thereby the practical operations of arithmetic. The theory will follow in due time. When perfectly familiar with the practice and the forms of arithmetic, and sufficiently mature in intellect, they awaken gradually and surely, and almost without an effort, to the beautiful logic which underlies the science.

How do we learn language in childhood? Is it not solely on authority and by example? A child who lives in a family where no language is used but that which is logically and grammatically correct, will learn to speak with logical and grammatical correctness long before it is able to give any account of the processes of its own mind in the matter, or indeed to understand those processes when explained by others. In other words, practice in language precedes theory. It should do so in other things. The parent who should take measures to prevent a child from speaking its mother tongue, except just so far and so fast as it could understand and explain the subtle logic which underlies all language, would be quite as wise as the teacher who refuses to let a child become expert in practical reckoning, until it can understand and explain at every step the rationale of the process,—who will not suffer a child to learn the multiplication table until it has mastered the metaphysics of the science of numbers, and can explain with the formalities of syllogism exactly how and why seven times nine make sixty-three.

These illustrations have carried me a little, perhaps, from my subject. But they seemed necessary to show that I am not beating the air. I have feared lest, in our very best schools, in the rebound from the exploded errors of the old system, we have unconsciously run into an error in the opposite extreme.

My positions on the particular point now under consideration may be summed up briefly, as follows:

  1. In developing the faculties, we should follow the order of nature.
  2. The faculties of memory and faith should be largely exercised and cultivated in childhood.
  3. While the judgment and the reasoning faculty should be exercised during every stage of the intellectual development, the appropriate season for their main development and culture is near the close, rather than near the beginning, of an educational course.
  4. The methods of reasoning used with children should be of a simple kind, dealing largely in direct intuitions, rather than formal and syllogistic.
  5. It is a mistake to spend a large amount of time and effort in requiring young children formally to explain the rationale of their intellectual processes, and especially in requiring them to give such explanations before they have become by practice thoroughly familiar with the processes themselves.

VI.
TEACHING CHILDREN WHAT THEY DO NOT UNDERSTAND.

It is not uncommon to hear persons declaim against teaching children what they do not understand. If by this is meant that children should not learn a set of words as parrots do, merely by the ear, and without attaching any idea to what they utter, no one will dissent from the propriety of the rule. But if the meaning is that they should learn nothing except what they fully comprehend, the rule certainly needs to be hedged in by some grave precautions.

There are indeed few things which any one, the oldest or the wisest, fully comprehends. Who knows what matter is? Certainly not the most eminent of philosophers. They do not pretend to know. We pick up a pebble. Who can tell what it is, absolutely? We say that it is something which has certain qualities. But even these we know mainly by negations. The pebble is hard, that is, it does not yield to pressure. It is opaque, that is, it does not transmit light. It is heavy, that is, it does not remain still, but goes towards the centre of the earth unless intercepted by some interposing body.

Who knows the meaning, absolutely, of a single article of the Creed? Certainly not the most eminent of divines. We know certain things about the great mysteries of the Godhead, and even these things we know, not directly, but by certain faint, distant analogies, and we express our knowledge in terms chosen mainly from Scripture and arranged with care by wise and learned men. These venerable formularies, containing the most exact verbal expression which the Church has been able to frame, of what the Scriptures teach about God and his ways, we commit to memory, and we repeat them with comfort and edification. But we do not pretend to penetrate the very essence of their meaning. Who by searching can find out God? One must be God himself to understand him.