The translation of the abstract statement of such problems into terms of the concrete may often be facilitated by requiring the pupil to supplement, or make explicit, his image by means of a drawing. His chief difficulty has been that he actually could not mentally see how carpet looked upon the floor, or paper on the wall, or boards in a fence around a field. From the point of view, however, of training in the power to think through rapidly and skilfully this class of problems, the permanent necessity of translating the abstract formulation into concrete terms would be far from ideal. Ultimately these problems should be quickly and readily analyzed into their essential elements, their relation to some fundamental principle, or general method, seen, and the work done by the appropriate rule.

It is not alone in mathematics that thinking power often depends upon the ability to translate quickly from the abstract formulation of the problem over into the concrete. Many a problem of sociology, of economics, or of practical life, notably in matters of conduct, can be adequately grasped only by repeated translations into terms of concrete and specific situations. The man who can make these translations with the greatest facility is often the man who arrives most quickly and most certainly at the right conclusions.

(2) Practice in translation must be given.

In all training to think, the functional relation between concrete and abstract imagery must be kept unbroken. There must be the adequate background of concrete images derived from abundant first-hand experiences of the right sort to serve as the basis of translation of the abstract into the concrete; and there must be sufficient practice, at the time of training, in the habit of making such translations at critical points in the thought process. This habit is an important asset to thinking power. It often marks the

critical difference in the mode of attack of a new problem by the successful thinker and the unsuccessful.

(3) Practice in translation makes abstract images more efficient.

A still further reason for giving practice in the translation of abstract terms over into the concrete is to be found in the fact that it makes the abstract images themselves more fluent and facile as tools of thinking. There is a feeling of warmth and relevancy to the concrete images which gradually gets transferred to the abstract. The meaning of the abstract becomes an indissoluble part of it, immediately felt and appreciated, needing not to be consciously and reflectively unfolded in order to perform its function readily, accurately, and adequately. In other words, practice in translation does away with the need of translation through the transformation of the abstract which has been effected by the process.

It is doubtful if teachers fully appreciate the value and importance, especially in the early stages of the pursuit of a new subject, of abundant drill in the matter of translating the abstract in terms of the concrete, or of the more abstract in terms of the less abstract and more familiar. This applies particularly to mathematics where so many important general principles and rules are expressed in terms of formulæ. These formulæ cannot become real tools of the mind, fully adapted to the solution of new problems, except as they have been fully appropriated and assimilated and worked over into the very texture of the mind itself. This means that the functional relation between these abstract elements and their concrete equivalents has become so very close that it is immediately felt and appreciated rather than being explicitly thought. Now, every branch of study moves in the direction of more or less abstract formulation of thought. Its formulæ may not be so compact as those of mathematics, but they are often more complex, as in economics, sociology, psychology,

ethics, and religion; hence flexibility, freedom, and power in their use demand abundant practice in their translation into concrete terms.

3. TRAINING IN THINKING DEMANDS THAT THE TRANSITION BE EFFECTED FROM THE USE OF CONCRETE IMAGES TO THE USE OF ABSTRACT IMAGES.

The preceding paragraphs have emphasized the importance of the concrete. But from the point of view of thinking power the abstract image is the more powerful tool of thought. While it is necessary to emphasize the concrete for the sake of giving fullness and richness of meaning to the abstract, yet it is also necessary to pass on to the free and flexible use of the abstract. Along with the development of the habit of translating the abstract in terms of the concrete should go the training of the power to translate and sum up the concrete in terms of the abstract. It may be all right for the child of five to count upon his fingers, but it cramps his mental growth for him to be allowed to continue to do so after this practice has performed its function of getting him started. In the case of the problems already mentioned of papering, plastering, carpet laying, etc., the teacher would fail in the performance of his full duty if he did not lead the children on beyond the necessity of working out all the problems in the concrete.

For the small child it may be all right and even necessary for him to think of law in the concrete terms of the policeman and of God in terms of a big benevolent man; but growth in the power of intelligent thought along these lines demands the development of the power to think in more and more abstract terms. If we fix the child in the habit of thinking in terms of the concrete by giving him practice in the concrete beyond the necessity of furnishing a sufficient background for the attainment of fullness and richness of meaning for his abstract symbols, we are limiting his possibilities of becoming strong and powerful in

his thought processes. Particularly as the period of adolescence draws near should more attention be given to the cultivation of thought in more abstract terms.

4. THE CHILD'S ABILITY TO GIVE FORMAL DEFINITIONS OF THINGS IS NOT A PROPER TEST OF HIS KNOWLEDGE OF THEIR MEANING.

We have already seen that meaning is primarily functional in nature. The child of the lower grades has not passed over into that stage of development in which meanings take the more structural and descriptive forms of science. If we wish to know whether he understands the meanings of words used in his reading lesson or in any other subject, we should be reasonably well satisfied with answers in functional and concrete terms. It is more or less waste of time for the teacher to be too insistent upon definitions in the abstract and descriptive terms which would be required to satisfy the adult mind.

5. THERE IS A REAL DANGER THAT EDUCATION MAY BECOME A PROCESS OF JUGGLING WITH SYMBOLS.

(1) Value of learning dependent on grasp of meaning. We have emphasized the fact that symbol and meaning are correlative. When the child is taught in such a way that he gets a symbol without meaning, psychologically it is no symbol for him, even though it be for us. The value of the symbol for the teacher is not the sole basis for teaching it to the child. It is necessary to develop along with it the correlative meaning. If the child is not ready for this, or if it cannot be done, then the symbol should not be taught. The learning of beautiful memory gems, of catechisms, of Biblical verses, etc., is of very doubtful value, unless much attention is given to the problem of developing their meanings in terms comprehensible to the child. The doctrine here laid down is only another phase

of the discussion centering in the functional relation between abstract and concrete imagery.

(2) Application to reading.

Fluency in reading,—the ability to pronounce hard words and to control the other elements of technique involved in the reading process, is no guarantee of the assimilation of thought by the reader. Some children have a fatal facility in the handling of symbols. This facility must not be taken as a measure of the grasp of the meaning, or content, of what is read. It is not safe to assume that children understand without taking some trouble to test. And the verbal test is not always sufficient. Dramatization, drawing, and constructive work are more certain tests where they can be applied. Reading means intellectual death if content is ignored and the process is reduced to one of drill upon the manipulation of technique. The writer has had opportunity to observe quite widely, and he believes that there is a very real danger that reading may become too much a process of juggling with symbols.

(3) Danger of loss of interest.

The process of learning symbols without sufficient attention to meaning leads either to loss of interest or to artificiality. A certain child who had natural facility in reading undertook to read Shakespeare at an early age. He found little difficulty in handling the technique of the reading process and took a certain pride in his achievement. But after a time, the whole thing began to pall upon him because it all seemed such senseless stuff. The result was that he studiously ignored Shakespeare until he became a young man and was required to take up the reading in higher English courses. Another young man said that he thought that he might be able to enjoy Tennyson's "The Princess," if he had not been put through such a stiff drill in analysis of the poem. Mastery of technique at the expense of meaning was deadening in both of these cases,