(_b_) It will be pushed upward by colder and heavier currents of air from the north and south.

(_c_) If the earth did not rotate, there would be constant winds towards the south, north of the equator; and towards the north, south of the equator.

(_d_) These currents of air are travelling from a region of less motion to a region of greater motion, and have a tendency to lag behind the earth's motion as they approach the equator.

(_e_) Hence they will seem to blow in a direction contrary to the earth's rotation, namely, towards the west.

(_f_) These two movements, towards the equator and towards the west, combine to give the currents of air a direction towards the south-west north of the equator, and towards the north-west south of the equator.

4. _Verification_:

Read the geography text to see if our inferences are correct.

THE DEVELOPMENT OF GENERAL KNOWLEDGE

=The Conceptual Lesson.=--As an example of a lesson involving a process of conception, or cla.s.sification, may be taken one in which the pupil might gain the cla.s.s notion _noun_. The pupil would first be presented with particular examples through sentences containing such words as John, Mary, Toronto, desk, boy, etc. Thereupon the pupil is led to examine these in order, noting certain characteristics in each.

Examining the word _John_, for instance, he notes that it is a word; that it is used to name and also, perhaps, that it names a person, and is written with a capital letter. Of the word _Toronto_, he may note much the same except that it names a place; of the word _desk_, he may note especially that it is used to name a thing and is written without a capital letter. By comparing any and all the qualities thus noted, he is supposed, finally, by noting what characteristics are common to all, to form a notion of a cla.s.s of words used to name.

=The Inductive Lesson.=--To exemplify an inductive lesson, there may be noted the process of learning the rule that to multiply the numerator and denominator of any fraction by the same number does not alter the value of the fraction.

_Conversion of fractions to equivalent fractions with different denominators_

The teacher draws on the black-board a series of squares, each representing a square foot. These are divided by vertical lines into a number of equal parts. One or more of these parts are shaded, and pupils are asked to state what fraction of the whole square has been shaded.

The same squares are then further divided into smaller equal parts by horizontal lines, and the pupils are led to discover how many of the smaller equal parts are contained in the shaded parts.

[Ill.u.s.tration: 1/2=3/6 2/3=8/12 3/4=15/20 3/5=18/30]

Examine these equations one by one, treating each after some such manner as follows:

How might we obtain the numerator 18 from the numerator 3? (Multiply by 6.)

The denominator 30 from the denominator 5? (Multiply by 6.)

13 3 24 8 35 15 36 18 --- = -; --- = --; --- = --; --- = --.

23 6 34 12 45 20 56 30

If we multiply both the numerator and the denominator of the fraction 3/5 by 6, what will be the effect upon the value of the fraction? (It will be unchanged.)

What have we done with the numerator and denominator in every case? How has the fraction been affected? What rule may we infer from these examples? (Multiplying the numerator and denominator by the same number does not alter the value of the fraction.)

THE FORMAL STEPS

In describing the process of acquiring either a general notion or a general truth, the psychologist and logician usually divide it into four parts as follows:

1. The person is said to a.n.a.lyse a number of particular cases. In the above examples this would mean, in the conceptual lesson, noting the various characteristics of the several words, John, Toronto, desk, etc.; and in the second lesson, noting the facts involved in the several cases of shading.

2. The mind is said to compare the characteristics of the several particular cases, noting any likenesses and unlikenesses.

3. The mind is said to pick out, or abstract, any quality or quant.i.ties common to all the particular cases.

4. Finally the mind is supposed to synthesise these common characteristics into a general notion, or concept, in the conceptual process, and into a general truth if the process is inductive.

Thus the conceptual and inductive processes are both said to involve the same four steps of:

1. _a.n.a.lysis._--Interpreting a number of individual cases.

2. _Comparison._--Noting likenesses and differences between the several individual examples.

3. _Abstraction._--Selecting the common characteristics.

4. _Generalization._--Synthesis of common characteristics into a general truth or a general notion, as the case may be.

=Criticism.=--Here again it will be found, however, that the steps of the logician do not fully represent what takes place in the pupil's mind as he goes through the learning process in a conceptual or inductive lesson. It is to be noted first that the above outline does not signify the presence of any problem to cause the child to proceed with the a.n.a.lysis of the several particular cases. a.s.suming the existence of the problem, unless this problem involves all the particular examples, the question arises whether the learner will suspend coming to any conclusion until he has a.n.a.lysed and compared all the particular cases before him. It is here that the actual learning process is found to vary somewhat from the outline of the psychologist and logician. As will be seen below, the child really finds his problem in the first particular case presented to him. Moreover, as he a.n.a.lyses out the characteristics of this case, he does not really suspend fully the generalizing process until he has examined a number of other cases, but, as the teacher is fully aware, is much more likely to jump at once to a more or less correct conclusion from the one example. It is true, of course, that it is only by going on to compare this with other cases that he a.s.sures himself that this first conclusion is correct. This slight variation of the actual learning process from the formal outline will become evident if one considers how a child builds up any general notion in ordinary life.

CONCEPTION AS A LEARNING PROCESS

=A. In Ordinary Life.=--Suppose a young child has received a vague impression of a cow from meeting a first and only example; we find that by accepting this as a problem and by applying to it such experience as he then possesses, he is able to read some meaning into it, for instance, that it is a brown, four-footed, hairy object. This idea, once formed, does not remain a mere particular idea, but becomes a general means for interpreting other experiences. At first, indeed, the idea may serve to read meaning, not only into another cow, but also into a horse or a buffalo. In course of time, however, as this first imperfect concept of the animal is used in interpreting cows and perhaps other animals, the first crude concept may in time, by comparison, develop into a relatively true, or logical, concept, applicable to only the actual members of the cla.s.s. Now here, the child did not wait to generalize until such time as the several really essential characteristics were decided upon, but in each succeeding case applied his present knowledge to the particular thing presented. It was, in other words, by a series of regular selecting and relating processes, that his general notion was finally clarified.

=B. In the School.=--Practically the same conditions are noted in the child's study of particular examples in an inductive or conceptual lesson in the school, although the process is much more rapid on account of its being controlled by the teacher. In the lesson outlined above, the pupil finds a problem in the very first word _John_, and adjusts himself thereto in a more or less perfect way by an apperceptive process involving both a selecting and a relating of ideas. With this first more or less perfect notion as a working hypothesis, the pupil goes on to examine the next word. If he gains the true notion from the first example, he merely verifies this through the other particular examples.

If his first notion is not correct, however, he is able to correct it by a further process of a.n.a.lysis and synthesis in connection with other examples. Throughout the formal stages, therefore, the pupil is merely applying his growing general knowledge in a selective, or a.n.a.lytic, way to the interpreting of several particular examples, until such time as a perfect general, or cla.s.s, notion is obtained and verified. It is, indeed, on account of this immediate tendency of the mind to generalize, that care must be taken to present the children with typical examples.

To make them examine a sufficient number of examples is to ensure the correcting of crude notions that may be formed by any of the pupils through their generalizing perhaps from a single particular.

INDUCTION AS A LEARNING PROCESS

In like manner, in an inductive lesson, although the results of the process of the development of a general principle may for convenience be arranged logically under the above four heads, it is evident that the child could not wholly suspend his conclusions until a number of particular cases had been examined and compared. In the lesson on the rule for conversion of fractions to equivalent fractions with different denominators, the pupils could not possibly apperceive, or a.n.a.lyse, the examples as suggested under the head of selection, or a.n.a.lysis, without at the same time implicitly abstracting and generalizing. Also in the lesson below on the predicate adjective, the pupils could not note, in all the examples, all the features given under a.n.a.lysis and fail at the same time to abstract and generalize. The fact is that in such lessons, if the selection, or a.n.a.lysis, is completed in only one example, abstraction and generalization implicitly unfold themselves at the same time and const.i.tute a relating, or synthetic, act of the mind. The fourfold arrangement of the matter, however, may let the teacher see more fully the children's mental att.i.tude, and thus enable him to direct them intelligently through the apperceptive process. It will undoubtedly also impress on the teacher's mind the need of having the pupils compare particular cases until a correct notion is fully organized in experience.

TWO PROCESSES SIMILAR

Notwithstanding the distinction drawn by psychologists between conception as a process of gaining a general notion, and induction as a process of arriving at a general truth, it is evident from the above that the two processes have much in common. In the development of many lesson topics, in fact, the lesson may be viewed as involving both a conceptual and an inductive process. In the subject of grammar, for instance, a first lesson on the p.r.o.noun may be viewed as a conceptual lesson, since the child gains an idea of a cla.s.s of words, as indicated by the new general term p.r.o.noun, this term representing the result of a conceptual process. It may equally be viewed as an inductive lesson, since the child gains from the lesson a general truth, or judgment, as expressed in his new definition--"A p.r.o.noun is a word that represents an object without naming it," the definition representing the result of an inductive process. This fact will be considered more fully, however, in Chapter XXVIII.

FURTHER EXAMPLES OF INDUCTIVE LESSONS

As further ill.u.s.trations of an inductive process, the following outlines of lessons might be noted. The processes are outlined according to the formal steps. The student-teacher should consider how the children are to approach each problem and to what extent they are likely to generalize as the various examples are being interpreted during the a.n.a.lytic stage.

1. THE SUBJECTIVE PREDICATE ADJECTIVE

_a.n.a.lysis, or selection:_

Divide the following sentences into subject and predicate: