The method is essentially communal not competitive. Hardly ever are there two children working on quite the same problem; therefore there can be no competition as to who answered best or quickest. At the end of the lesson, the whole class are in possession of some information which no member of it possessed before. It is not something told them by the teacher: it is often quite new to the teacher herself. The skill of the teacher is shown, not by the knowledge which she imparts, but by the manner in which she utilises the thinking power of the children for the purpose of finding out what she does not yet know.
1.2. Two elements in teaching
Suppose I am teaching, say, the process of multiplication. There are two things which the pupils can get out of my instruction: (A) skill in performing the operation of multiplication itself; and (B) a little of the power to find out for themselves how to do other arithmetical operations. Every process that I teach ought to be so taught as to add something to the pupil's chance of some day making out a rule for himself without the aid of a teacher. If we add together all the A's of a child's arithmetical career, they constitute what I called, in a former article, the body of his arithmetical knowledge; if we sum all the B's, they constitute what is called its life. The sum of the combined A elements constitutes the ability to reckon the bulk or number of dead material and to keep accounts according to any system chosen by an employer. The sum of the B elements gives the extra power of bringing one's knowledge to bear in forming a sound judgment on problems connected with living forces - e.g., on the probable behaviour of a charge of electricity under certain conditions, or the probable honesty and stability of a certain commercial enterprise. Now the A element in any mathematical lesson can be imparted while the class is alert and eager; the B element cannot be imparted except under the peculiar condition called by some mystic writers "Silence in the soul" awaiting further Light. The two states, the alert and the passive, alternate in any good educational regime; the alert phases being very much the longest, the passively recipient ones short but quite undisturbed. But under stress of competition the passive mystic phases of study are being crowded out. The reason is that England is so saturated with the spirit of advertisement that, in any given committee, the majority are almost sure to be against the teaching of anything for which there is nothing to show at the next forthcoming examination.
Many a life of intellectual muddle and intellectual dishonesty begins at the point where some teacher explains the rule for Greatest Common Measure to a child who has not had the proper basis of sub-conscious knowledge laid in actual experiences. Therefore, if you value your child's future clearness in science, trust no teacher to tell him anything about G.C.M. or L.C.M. till you have ascertained that he is able to find, easily and accurately, by means of compasses, the longest length that will repeat exactly into each of two unequal given lengths, and the shortest length into which each of two given unequal lengths will fit.
There seems to be evidence that in ancient times all people in good society were expected to know simple truths about geometric forms in the same way as we all know simple facts in natural history. The elementary properties of the triangle, parallelogram, circle, ellipse, and spiral seem to have been familiar to ordinary people. The}/ were not expected to know much about geometry, but they were expected to have and to use ordinary faculties of observation on facts within every one's reach. Euclid, was in. his day, a sort of Darwin of geometry. He wrote not a geometry for beginners, but a book about the logical concatenation of geometric facts for men already geometers; just as our Darwin wrote a book about the concatenation of biologic forms for people already biologists, to the extent at least of knowing that horses prance and dogs bark and wag tails; that worms creep and birds fly; that some flowers have scent; that some fruits are sweet and others are sour. Euclid's book was a type and model of ail that a good book on logical concatenation should be. The use which was made of it till lately is the type and model of how such a book should not be used. Teachers assumed that the excellence of the book gave them the right to use it in defiance of all the laws of psychology, The result of such misuse is always the same: loss of natural instinct. Textbooks are written expressly on purpose to inform the consciousness. A good textbook should explain everything step by step, and should assume nothing which it does not actually state. Euclid does this in perfection, He wrote, as I have said, for men for whom the words triangle, circle, parallelogram were already charged with associations; and he gave definitions intended for the purpose, not of telling something fresh, but of clearing up and settling conceptions which were hazy from long familiarity ... Now when it became customary to give to boys of ten or twelve what Euclid wrote for grown men, that was not far wrong; boys now can quite well assimilate what was grown-up food two thousand years ago. But if children of twelve are to learn what Euclid wrote for advanced men, children of three should be acquiring the subconscious physical experiences which lads in Greece picked up in the course of nature and by the accidental help of architecture and statuary. This precaution our grandfathers entirely omitted. The effect was somewhat similar to that which would be produced if it ever became the fashion to make children learn theoretic natural history from books illustrated by flat diagrams, before allowing them to see any real animal or plant. Europe has lost geometric instinct and the habit of geometric observation. All of us at this time are in a condition of artificially paralysed geometric faculty; and now the aim and study of all true mathematicians is to restore the vitality of geometric instinct ... Lastly - and this is probably the most important preparation for future living comprehension of mathematical ideas - there is the cultivation of the geometric imagination. At the same age at which the child begins to realize that a tadpole grows into a frog, a boy into a man, a seedling into a flowering plant, let him have the opportunity of watching also how one geometrical type-form grows out of, or flows into, another. A common night-light placed in the bottom of a deep round jar in a dark room throws on a sheet of cardboard held over it patterns of conic-sections, which pass into each other as you change the position of the cardboard. Children very early learn to love watching figures thrown in light; and there is no age at which this amusement can hurt them, provided that the motion is slow, and that no one excites them by trying to explain things. A variety of other methods for training the geometric imagination at a later stage will be dealt with in a future chapter.
Teachers of such subjects as Electricity complain of the difficulty of getting pupils to apply what they know of Mathematics (at what ever level) to the analysis and manipulation of real forces. It is not that the pupil does not know enough (of Arithmetic, or Algebra, or the Calculus, as the case may be), but he too often does not see, and cannot be got to see, how to apply what he knows. Some faculty has been paralysed during his school-life; he lacks something of what should constitute a living mathematical intelligence. In truth he usually lacks several things. In the first place, though he knows a good deal about antithesis of operations (e.g. he knows that subtraction is the opposite of addition, division of multiplication, movement in the direction minus -x of movement in the direction x, and so on), he has not the habit of observing in what respects antithetic operations neutralize each other, and in what respects they are cumulative; and surely no habit is more needed than this as preparation for making calculations in electricity or mechanics. In the next place, he too often knows, about the idea of relevance, only enough to be foggy about it. The reason for this is that his study of the idea of relevance itself began where it ought to have ended; his attention was never called to it till the stage was reached when it would have been right that he should direct his action in regard to it subconsciously from long habit, leaving conscious attention free for dealing with the actual elements of some question which is difficult enough to need thinking about. He was not made to grasp the fundamental idea that a statement may be relevant to one question and irrelevant to another, till some knotty problem occurred involving consideration of which statements are relevant to the special question in hand. So he had to try to grasp at once the idea of relevance and the question, what is relevant to what, in a special problem. Such thrusting on the young brain of two difficulties of different kinds at once, is contrary to all accepted canons of Psychology. Examples are here suggested (Lesson XIII) in which there can be no doubt as to what is relevant to the question at issue; the child's attention is therefore free to focus itself on the idea that there can be facts concerning a thing which in no way concern the particular question which is just now being asked about the thing. Then again, whatever skill he may have acquired in the manipulation of those notations and formulae which he has been taught to use, he knows hardly anything about the manner in which such things come into being. Now an applier of Mathematics to real forces should be able, when occasion requires, to modify his notation, or invent a new formula, for himself. He cannot begin to learn to do this, straight away, while his mind is struggling with problems of electricity or mechanics; he should have had, from the first, the habit of seeing through formulae and notations; of watching them coming into being, of helping to construct them.
The mathematical formulae which some people speak of contemptuously as "dry" are in reality as beautiful as microscopes, or any other well-adjusted and well-finished machinery intended to extend the scope of men's powers; and as for mathematical ideas themselves, they are as grand as any expressed in poetry. Comparatively few people get any real enjoyment out of either; and the reason obviously is that their faculties have been stamped into confusion by a method which I can only compare to that of making a child use a telescope before it can see properly with its eyes; using a complicated machine for extending and refining certain work before he has learned to do, without it, the simpler kinds of that work. To make clearer what it is that is wrong, let us think of an orchard, at harvest time, which contains, besides the trees and fruits, the natural human limbs with which man picks the fruits he can reach; the stools, steps, and ladders of various lengths, which are an extension of his own legs and by means of which he rises to the level of the fruits which he cannot naturally reach; the sickles or fruit-scissors which improve on the action of the hand and enable us to bring down the choicest fruit without risk of spoiling its delicate bloom; the room fitted with shelves on which the fruit is stored for future use; and the baskets in which it is temporarily packed for safe conveyance to the store-house. It is no exaggeration to say that all these various items have analogues in science, especially in mathematics. A mathematical textbook contains truths valuable in themselves because throwing light on the nature of human thought; other truths valuable for use in commerce or physical science; natural processes of reasoning by which the student can gain direct perception of some of these truths; artificial devices for arriving at others: refinements of various sorts; and formulae and tables in which truths are classified, stored for easy access, and preserved in the memory. It is pitiable to see how far are many students, even advanced ones, from any clear realization which of these various things is which. Many have no conception of the difference between direct and inverse in mathematics. They make no clear distinction between the truth itself, the ladder of devices by which they reached it, and the formulae in which they stored. it. And can we wonder at this ? No human being, I suppose, ever attempted to teach a child to climb a ladder, to use a fruit-sickle, or to store fruits on shelves, in the same summer in which it first was able to stand on its legs and grasp a low-growing apple, in which it first experienced the delight of eating fruit. It would probably be an under-statement of the case to affirm that a mode of treatment analogous to this has been inflicted on ninety-nine per cent of the young people who have learned Algebra and Trigonometry even within the last ten years. The methods of doing this stupid thing have no doubt improved enormously since the early days of De Morgan; all praise is due to the patient workers who have done so much to improve them; but the stupid thing is still done; and the parents and the public still insist on its being done. The questions so often put by parents, 'At what age do you think my child had better begin Algebra?' (or Trigonometry) and, 'Can you recommend me to a good teacher?' really mean something analogous to this:- 'I intend to keep my child ignorant of all experiences concerning fruit, and all processes connected with it , till he is old enough to begin receiving straight away, in one continuous series of lessons, information, conveyed by verbal explanations, about how to stand on one's legs, how to climb ladders, how to use sickles, how fruits taste, their hygienic and economic value, their botanical classification, and the best means of preserving them. At what age do you consider this series of lessons should begin, and whom do you recommend me to employ to give it?' The only answer one could make to such a question would be that there is no age at which any such course should begin, and no person who ought to be asked to give it. The question asked by a parent should be, 'At what age would you recommend me to let my child begin learning such portions of Algebra (or Trigonometry) as can only be learned by the aid of complicated devices invented, centuries after the science itself was an actual working possession of our race, for the sake of projecting its action into fields which would be inaccessible to it if only natural and simple tools were used?' The answer should be, 'When the process of learning by the more direct means has become so familiar as to be performed sub-consciously.'
3.2. Handling objects
Each object (in school) is catalogued as intended to teach this, or to prove that, or to illustrate so-and-so; many ... seem to have no idea that it may be well to let a child have things and handle them, without anyone talking, and find out what things have to say.
It is desirable that children should sometimes be free to experiment under varied and accidental conditions ... and learn by making mistakes with some one about ... to whom they can apply when puzzled or discouraged.
3.4. New ideas
Receptivity cannot be generated by early teaching of a subject mixed up with the use of its appropriate technical machinery; but only by suggesting the new ideas by means of objects already familiar to the child's eye and touch.
When he begins to do sums on paper, let him still, for a considerable period, do each addition, subtraction, etc., first in counters; and then, while these are still on the table, work out the same sum on paper.
3.6. Finding results
Anything which he (the teacher) intends to prove should never be stated; children should be led up to find it out for them selves by successive questions.
By training the hand to trace out Nature's actions, we train the unconscious mind to act spontaneously in accordance with Natural Law; and the unconscious mind, so trained, is the best teacher of the conscious mind.
3.8. Geometric objects
First comes the education of the senses. From the time when an infant begins to stroke the cat ... have geometric solids as ornaments or toys, so that the senses of sight and touch may actually develop in contact with true type-form.
3.9. Needle and thread
At early stages the needle and thread has many advantages over any other implement yet devised; a child can ornament cards by setting long straight stitches in a way which causes beautiful curves to grow under his hands without his knowing why or how, and without any pattern being set for him. ... The beauty of some of the designs is unquestionable; and there can be no second opinion about the value of the method, as training, from the point of view of geometry as well as from that of art. What is not quite so obvious at first sight is its bearing on the training of the unconscious mind for science. Without the slightest intellectual strain it puts the children through that normal sequence of orderly attention to classification and detail interspersed with nodal points of synthesis which may be called the very breathing-rhythm of the scientific discoverer.