Chapter 48

process as a mode of acquiring knowledge. An examination will show that the deductive process follows the ordinary process of learning, or of selecting certain elements of old knowledge, and organizing them into a new particular experience in order to meet a certain problem.

=Deduction as Formal Reasoning.=--It is usually stated by psychologists and logicians that in this process the person starts with the general truth and ends with the particular inference, or conclusion, for example:

Winds coming from the ocean are saturated with moisture.

The prevailing winds in British Columbia come from the Pacific.

Therefore these winds are saturated with moisture.

All winds become colder as they rise.

The winds of British Columbia rise as they go inland.

Therefore, the winds (atmosphere) in British Columbia become colder as they go inland.

The atmosphere gives out moisture as it becomes colder.

The atmosphere in British Columbia becomes colder as it goes inland.

Therefore, the atmosphere gives out moisture in British Columbia.

=Steps in Process.=--The various elements involved in a deductive process are often analysed into four parts in the following order:

1. _Principles._ The general laws which are to be applied in the solution of the problem. These, in the above deductions, constitute the first sentence in each, as,

The air becomes colder as it rises.

Air gives out its moisture as it becomes colder, etc.

2. _Data._ This includes the particular facts already known relative to the problem. In this lesson, the data are set forth in the second sentences, as follows:

The prevailing winds in British Columbia come from the Pacific; the wind rises as it goes inland, etc.

3. _Inferences._ These are the conclusions arrived at as a result of noting relations between data and principles. In the above lesson, the inferences are:

The atmosphere, or trade-winds, coming from the Pacific rise, become colder, and give out much moisture.

4. _Verification._ In some cases at least the learner may use other means to verify his conclusions. In the above lesson, for example, he may look it up in the geography or ask some one who has had actual experience.

=Deduction Involves a Problem.=--It is to be noted, however, that in a deductive learning process, the young child does not really begin with the general principle. On the contrary, as noted in the study of the learning process, the child always begins with a particular unsolved problem. In the case just cited, for instance, the child starts with the problem, "What is the condition of the rainfall in British Columbia?" It is owing to the presence of this problem, moreover, that the mind calls up the principles and data. These, of course, are already possessed as old knowledge, and are called up because the mind feels a connection between them and the problem with which it is confronted. The principles and data are thus both involved in the selecting process, or step of analysis. What the learner really does, therefore, in a deductive lesson is to interpret a new problem by selecting as interpreting ideas the principles and data. The third division, inference, is in reality the third step of our learning process, since the inference is a new experience organized out of the selected principles and data. Moreover, the verification is often found to take the form of ordinary expression. As a process of learning, therefore, deduction does not exactly follow the formal outline of the psychologists and logicians of (1) principles, (2) data, (3) inference, and (_4_) verification; but rather that of the learning process, namely, (1) problem, (2) selecting activity, including principles and data, (3) relating activity=inference, (4) expression=verification.

=Example of Deduction as Learning Process.=--A simple and interesting lesson, showing how the pupil actually goes through the deductive process, is found in paper cutting of forms balanced about a centre, say the letter X.

1. _Problem._ The pupil starts with the problem of discovering a way of cutting this letter by balancing about a centre.

2. _Selection._ Principles and Data. The pupil calls up as data what he knows of this letter, and as principles, the laws of balance he has learned from such letters as, A, B, etc.

3. _Organization or Inference._ The pupil infers from the principle involved in cutting the letter A, that the letter X (Fig. A) may be balanced about a vertical diameter, as in Fig. B.

Repeating the process, he infers further from the principle involved in cutting the letter B, that this result may again be balanced about a horizontal diameter, as in Fig. C.

4. _Expression or Verification._ By cutting Figure D and unfolding Figures E and F, he is able to verify his conclusion by noting the shape of the form as it unfolds, thus:

FURTHER EXAMPLES FOR STUDY

The following are given as further examples of deductive processes.

The materials are here arranged in the formal or logical way. The student-teacher should rearrange them as they would occur in the child's learning process.

I. DIVISION OF DECIMALS

1. _Principles_:

(_a_) Multiplying the dividend and divisor by the same number does not alter the quotient.

(_b_) To multiply a decimal by 10, 100, 1000, etc., move the decimal point 1, 2, 3, etc., places respectively to the right.

2. _Data_:

Present knowledge of facts contained in such an example as .0027 divided by .05.

3. _Inferences_:

(_a_) The divisor (.05) may be converted into a whole number by multiplying it by 100.

(_b_) If the divisor is multiplied by 100, the dividend must also be multiplied by 100 if the quotient is to be unchanged.

(_c_) The problem thus becomes .27 divided by 5, for which the answer is .054.

4. _Verification_:

Check the work to see that no mistakes have been made in the calculation. Multiply the quotient by the divisor to see if the result is equal to the dividend.

II. TRADE-WINDS

1. _Principles_:

(_a_) Heated air expands, becomes lighter, and is pushed upward by cooler and heavier currents of air.

(_b_) Air currents travelling towards a region of more rapid motion have a tendency to "lag behind," and so appear to travel in a direction opposite to that of the earth's rotation.

2. _Data_:

(_a_) The most heated portion of the earth is the tropical region.

(_b_) The rapidity of the earth's motion is greatest at the equator and least at the poles.

(_c_) The earth rotates on its axis from west to east.

3. _Inferences_:

(_a_) The heated air in equatorial regions will be constantly rising.

(_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 classification, may be taken one in which the pupil might gain the class 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 class 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.

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.)

1×3 3 2×4 8 3×5 15 3×6 18 --- = -; --- = --; --- = --; --- = --. 2×3 6 3×4 12 4×5 20 5×6 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 analyse 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 quantities 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. _Analysis._--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 analysis of the several particular cases. Assuming 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 analysed 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 analyses 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 assures 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 class. 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 analysis 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 analytic, way to the interpreting of several particular examples, until such time as a perfect general, or class, 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 analyse, the examples as suggested under the head of selection, or analysis, 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 analysis and fail at the same time to abstract and generalize. The fact is that in such lessons, if the selection, or analysis, is completed in only one example, abstraction and generalization implicitly unfold themselves at the same time and constitute 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 attitude, 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 pronoun may be viewed as a conceptual lesson, since the child gains an idea of a class of words, as indicated by the new general term pronoun, 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 pronoun 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