Read Ebook: The Psychology of Arithmetic by Thorndike Edward L Edward Lee

Font size: 10 px 15 px 20 px 25 px 30 px 35 px

Background color: BlackWhiteGrayLight GrayDark GrayDim Gray

Text color: BlackWhiteGrayLight GrayDark GrayDim Gray

Apply/Save settings

Ebook has 805 lines and 85034 words, and 17 pages

The second general fact is that certain bonds are of service for only a limited time and so need to be formed only to a limited and slight degree of strength. The data of problems set to illustrate a principle or improve some habit of computation are, of course, the clearest cases. The pupil needs to remember that John bought 3 loaves of bread and that they were 5-cent loaves and that he gave 25 cents to the baker only long enough to use the data to decide what change John should receive. The connections between the total described situation and the answer obtained, supposing some considerable computation to intervene, is a bond that we let expire almost as soon as it is born.

To prevent pupils from responding to the form of statement rather than the essential facts, we should then not teach them to forget the form of statement, but rather give them all the common forms of statement to which the response in question is an appropriate response, and only such. If a certain form of statement does in life always signify a certain arithmetical procedure, the bond between it and that procedure may properly be made very strong.

Another case of the formation of bonds to only a slight degree of strength concerns the use of so-called 'crutches' such as writing +, -, and x in copying problems like those below:--

Add Subtract Multiply 23 79 32 61 24 3 -- -- --

or altering the figures when 'borrowing' in subtraction, and the like. Since it is undesirable that the pupil should regard the 'crutch' response as essential to the total procedure, or become so used to having it that he will be disturbed by its absence later, it is supposed that the bond between the situation and the crutch should not be fully formed. There is a better way out of the difficulty, in case crutches are used at all. This is to associate the crutch with a special 'set,' and its non-use with the general set which is to be the permanent one. For example, children may be taught from the start never to write the crutch sign or crutch figure unless the work is accompanied by "Write ... to help you to...."

Write - to help you to Find the differences:-- remember that you must 39 67 78 56 45 subtract in this row. 23 44 36 26 24 -- -- -- -- --

Remember that you must Find the differences:-- subtract in this row. 85 27 96 38 78 63 14 51 45 32 -- -- -- -- --

The bond evoking the use of the crutch may then be formed thoroughly enough so that there is no hesitation, insecurity, or error, without interfering to any harmful extent with the more general bond from the situation to work without the crutch.

THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS

It is argued by many that such bonds are valuable for a short time; namely, while arithmetical procedures in connection with which they serve are learned, but that their value is only to serve as a means for learning these procedures and that thereafter they may be forgotten. "They are formed only as accessory means to certain more purely arithmetical knowledge or discipline; after this is acquired they may be forgotten. Everybody does in fact forget them, relearning them later if life requires." So runs the argument.

In some cases learning such words and facts only to use them in solving a certain sort of problems and then forget them may be profitable. The practice is, however, exceedingly risky. It is true that everybody does in fact forget many such meanings and facts, but this commonly means either that they should not have been learned at all at the time that they were learned, or that they should have been learned more permanently, or that details should have been learned with the expectation that they themselves would be forgotten but that a general fact or attitude would remain. For example, duodecagon should not be learned at all in the elementary school; indorsement should either not be learned at all there, or be learned for permanence of a year or more; the details of the metric system should be so taught as to leave for several years at least knowledge of the facts that there is a system so named that is important, whose tables go by tens, hundreds, or thousands, and a tendency to connect meter, kilogram, and liter with measurement by the metric system and with approximate estimates of their several magnitudes.

If an arithmetical procedure seems to require accessory bonds which are to be forgotten, once the procedure is mastered, we should be suspicious of the value of the procedure itself. If pupils forget what compound interest is, we may be sure that they will usually also have forgotten how to compute it. Surely there is waste if they have learned what it is only to learn how to compute it only to forget how to compute it!

THE STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES

The next case of the formation of bonds to slight strength is the problematic one of forming the bonds involved in understanding the reasons for certain processes only to forget them after the process has become a habit. Should a pupil, that is, learn why he inverts and multiplies, only to forget it as soon as he can be trusted to divide by a fraction? Should he learn why he puts the units figure of each partial product in multiplication under the figure that he multiplies by, only to forget the reason as soon as he has command of the process? Should he learn why he gets the number of square inches in a rectangle by multiplying the length by the width, both being expressed in linear inches, and forget why as soon as he is competent to make computations of the areas of rectangles?

PROPAEDEUTIC BONDS

The formation of bonds to a limited strength because they are to be lost in their first form, being worked over in different ways in other bonds to which they are propaedeutic or contributing is the most important case of low strength, or rather low permanence, in bonds.

The rule for such bonds is, of course, to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them, but not to overlearn them. There is a difference between learning something to be held for a short time, and the same amount of energy spent in learning for long retention. The former sort of learning is, of course, appropriate with many of these propaedeutic bonds.

A. B. C. D. E. 2 x 25 3 x 15 2 x 12 4 x 11 6 x 25 3 x 25 10 x 15 2 x 15 4 x 15 6 x 15 5 x 25 4 x 15 2 x 25 4 x 12 6 x 12 10 x 25 2 x 15 2 x 11 4 x 25 6 x 11 4 x 25 7 x 15 3 x 25 5 x 11 7 x 12 6 x 25 9 x 15 3 x 15 5 x 12 7 x 15 8 x 25 5 x 15 3 x 11 5 x 15 7 x 25 7 x 25 8 x 15 3 x 12 5 x 25 7 x 11 9 x 25 6 x 15 8 x 12 9 x 12 8 x 25

State the missing numbers:--

Do this section again. Do all the first column first. Then do the second column, then the third, and so on.

Consider, from the same point of view, exercises like + 2, + 5, + 6, given as a preparation for written multiplication. The work of

and the like is facilitated if the pupil has easy control of the process of getting a product, and keeping it in mind while he adds a one-place number to it. The practice with + 2 and the like is also good practice intrinsically. So some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication.

For example, as an introduction to long division, a pupil may be given exercises using one-figure divisors in the long form, as:--

The important recommendation concerning these purely propaedeutic bonds, and bonds formed only for later reconstruction, is to be very critical of them, and not indulge in them when, by the exercise of enough ingenuity, some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well. Arithmetical teaching has done very well in this respect, tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones. It is in the teaching of reading that we find the formation of propaedeutic bonds of dubious value often carried to demonstrably wasteful extremes.

THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES

THE AMOUNT OF PRACTICE

It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion.

Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? Every case of the action of each bond is to be counted.

THE FUNDAMENTAL BONDS

If estimating for the entire series is too long a task, it will be sufficient to use eight or ten from each, say:--

TABLE 2

ESTIMATES OF THE AMOUNT OF PRACTICE PROVIDED IN BOOKS I AND II OF THE AVERAGE THREE-BOOK TEXT IN ARITHMETIC; BY 50 EXPERIENCED TEACHERS

Having made his estimates the reader should compare them first with similar estimates made by experienced teachers , and then with the results of actual counts for representative textbooks in arithmetic .

It will be observed in Table 2 that even experienced teachers vary enormously in their estimates of the amount of practice given by an average textbook in arithmetic, and that most of them are in serious error by overestimating the amount of practice. In general it is the fact that we use textbooks in arithmetic with very vague and erroneous ideas of what is in them, and think they give much more practice than they do.

The authors of the textbooks as a rule also probably had only very vague and erroneous ideas of what was in them. If they had known, they would almost certainly have revised their books. Surely no author would intentionally provide nearly four times as much practice on 2 + 2 as on 8 + 8, or eight times as much practice on 2 x 2 as on 9 x 8, or eleven times as much practice on 2 - 2 as on 17 - 8, or over forty times as much practice on 2 ? 2 as on 75 ? 8 and 75 ? 9, both together. Surely no author would have provided intentionally only twenty to thirty occurrences each of 16 - 7, 16 - 8, 16 - 9, 17 - 8, 17 - 9, and 18 - 9 for the entire course through grade 6; or have left the practice on 60 ? 7, 60 ? 8, 60 ? 9, 61 ? 7, 61 ? 8, 61 ? 9, and the like to occur only about once a year!

TABLE 3

AMOUNT OF PRACTICE: ADDITION BONDS IN A RECENT TEXTBOOK OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION

The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times, 22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.

TABLE 4

AMOUNT OF PRACTICE: SUBTRACTION BONDS IN A RECENT TEXTBOOK OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION

TABLE 5

FREQUENCIES OF SUBTRACTIONS NOT INCLUDED IN TABLE 4

These are cases where the pupil would, by reason of his stage of advancement, probably operate 35-30, 46-46, etc., as one bond.

TABLE 6

AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK OF EXCELLENT REPUTE. BOOKS I AND II

TABLE 7

AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B, PARTS I AND II

TABLE 8

DIVISION BONDS, WITH AND WITHOUT REMAINDERS. BOOK B

All work through grade 6, except estimates of quotient figures in long division.

Dividend 2 3 4 5 Divisor 1 2 1 2 3 1 2 3 4 1 2 3 4 5 Number of Occurrences 41 386 27 189 240 26 397 66 185 23 136 43 53 135

Share on: WhatsApp @Hash Facebook Tweet