more or less arbitrary restrictions are placed upon the meanings of the technical terms used, in order that they may always have the same significance. The following terms illustrate the contrast between the loose, inclusive meaning of a given term in ordinary conversation and the restricted meaning of the same term in a given science: In geometry: line, surface, square, circle. In psychology: sensation, feeling. In economics: wealth. In physics: work, sound, heat, temperature, mass. In order that the student may think clearly and progress surely in these subjects, it is almost absolutely necessary that he learn the exact definitions of these words. Thus, instead of confusing mass with volume he must learn that mass means the quantity of matter that a body contains." In psychology, instead of using sensation vaguely for almost any kind of experience, he must confine it to "consciousness of qualities or conditions either of things or of one's own body."

Clearness and precision are attributes of good definitions. -In formulating a definition it is important to phrase it in such language that it can be easily understood and at the same time be precise or true to the facts. That is, clearness and precision are two of the most important qualities of good definitions. A good example of lack of clearness is the definition of network, quoted above on page 221. It is often difficult to secure a definition that is clear and at the same time precise. For example, consider the following definition: "To multiply one number, called the multiplicand, by another, called the multiplier, is to use the multiplicand as we must use unity to obtain the multiplier." This definition may be precise, but its meaning must be puzzled out by the highschool pupils. In reaching this definition the author of the algebra from which it is quoted criticizes the ordinary simpler definition in the following words :

Multiplication has been defined in arithmetic as the process of taking one number, called the multiplicand, as many times as there are units in the other, called the multiplier. It is evident that this

definition holds only when the multiplier is a whole number, and fails when it is a fraction.

Thus, to multiply 7 by 2 would mean to take 7 as many times as there are units in 21, that is, 2 times. This is impossible. One cannot do a thing 2 times. (22: 3)

Get the meaning first, then add the symbol. One further point in connection with definitions remains to be noted. This is the possibility of reversing the usual form of statement and giving the meaning, or a meaning, first and adding the term at the end. To take an example that is easily understood, consider the first teaching about nouns in the elementary school. At first only names of common objects are taken up. The children get an idea of this class of words, that is, names of objects. The teacher now wishes to introduce the technical term noun, in order to have a symbol for talking about this class of words and to help the children in thinking about it as a class. She ordinarily does this by giving the definition in this form: "Nouns are names of things." But this is objectionable, since the statement is not precise or true. The difficulty would be obviated, however, if she began with a statement of the general meaning which the children have acquired, that is, "names of things," and then added the new symbol nouns. The statement would then stand Names of things are nouns." In this form it is simple; it represents adequately the meaning which the children have acquired; it is true; and it can be expanded to include other kinds of nouns as these are learned. Unfortunately the form of statement which we find in a dictionary has influenced our practice so much that in many simple cases teachers will struggle to construct a complete definition, which is necessary when the term to be defined comes at the beginning of the statement, instead of being satisfied for the time being with an incomplete statement in the form suggested. Moreover, the ordinary practice distorts the purpose of the term, or symbol, at the stage of learning that has been reached. Its purpose

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is to make definite and portable a meaning that has been worked out. The stress should be placed on this meaning rather than on the more complete meaning which may be developed later. If the practice advocated here is followed, there will be less memorizing of definitions as mere words without meanings.

Summary of principles of teaching abstract and general meanings. The above discussion has been concerned almost entirely with abstract and general terms and very little with abstract and general propositions, of which examples were given above on page 206. The same general principles, however, would apply in the latter case. We may summarize our main points as far as they concern abstract and general terms as follows:

1. In practical life a new abstract or general meaning is commonly learned by having attention called to it at first in connection with some practical issue in a single situation. Later other experiences modify and expand the meaning derived from this first experience.

2. In teaching such meanings in school it is important to begin with particular personal experiences in the same way, and usually to study intensively at first one example of the general class in question.

3. In many cases students already possess a wealth of familiar personal experiences which can be recalled, and it is not necessary to provide new ones. The old ones may be analyzed and the abstract aspect under consideration brought to light.

4. For poor or mediocre students active analytical thought and discovery by the student is desirable. Superior students may learn readily from an exposition by the teacher or in a textbook.

5. While considerable effective thinking can be done in terms that the thinker cannot define exactly, in scientific thinking it is necessary to have terms defined in a precise way.

6. In teaching, a definition should consist of a summary formulation of the meaning which the student has developed up to date. This meaning may keep expanding as thought progresses.

Same principles apply in teaching abstract and general propositions. The above principles, together with some of those set forth in the preceding section on reflective thinking,

would also govern the teaching of abstract and general propositions; for example, rules in mathematics and physics, laws of growth in biology, the laws of climate in geography, propositions concerning the distribution of population in sociology, etc.

Generalizing is a constant process, not a final step. — In this connection it is desirable to add only one further qualification. This is practically an elaboration of paragraph 6 on page 225, and serves as a corrective to the notion often implied in books on methods of teaching, that generalization is almost the last step in the process of reaching a refined and exact general meaning or conclusion. The opposite and true notion is expressed by Dewey as follows:

Generalization is not a separate and single act; it is rather a constant tendency and function of the entire discussion or recitation. Every step forward toward an idea that comprehends, that explains, that unites what was isolated and therefore puzzling, generalizes. .. The factor of formulation, of conscious stating, involved in generalization should also be a constant function, not a single formal act. Definition means essentially the growth of a meaning out of vagueness into definiteness. Such final verbal definition as takes place should be only the culmination of a steady growth in distinctness. In the reaction against ready-made verbal definitions and rules the pendulum should never swing to the opposite extreme, that of neglecting to summarize the net meaning that emerges from dealing with particular facts. Only as general summaries are made from time to time does the mind reach a conclusion or a resting place, and only as conclusions are reached is there an intellectual deposit available in future understanding. (5: 211)

Generalizations aid in solving personal and social problems. It is desirable to keep in mind the relation of the discussion in this section of the chapter to the whole chapter and to the first section. The whole topic of the chapter is Reflective Thinking. The aspect of this topic which was discussed in section one was the solution of problems. The aspect discussed

in this, the second section, is the acquiring of abstract and general meanings. The chief practical reason for acquiring these meanings is that they may help in solving problems. As James says, "The whole function of conceiving, of fixing and holding fast to meanings, has no significance apart from the fact that the conceiver is a creature with partial purposes and private ends." (6 Vol. I: 482.) The ways in which general meanings may help in solving problems or attaining the private ends mentioned by James will be discussed briefly in the following paragraph. This is partly a repetition of the discussion on pages 190-192, but from a slightly different point of view.

By identifying a particular problem as an example of a known class. An abstraction, or general notion, may be used in two ways. First, when confronted by some particular difficulty or problem, if it can be identified as an example of some general class which is already understood, the methods of dealing with it are easily determined. For example, recently it has been discovered that many diseases are transmitted by insects. In connection with this discovery a technique of guarding against and destroying insects has been developed, and the public is being educated to believe in it and use it. Now, in investigating the cause of some particular disease the medical investigator has the advantage of knowing that it may be spread by insects, and he keeps this possibility in mind as he makes his experiments. If he succeeds in discovering and proving that it is carried by a certain insect in a certain way, then the methods that have been developed for handling similar cases may be used. That is, as soon as he can isolate and identify the abstract phase of the disease, "carried by insects in certain ways," he can take advantage of all the general knowledge that has been developed about the class "diseases spread by insects." The notable examples of this class are yellow fever spread by the Stegomyia mosquito; malaria spread by the Anopheles mosquito; typhoid spread partially by house flies; plague spread by rats and fleas.