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

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In this problem three different kinds of addends are combined, if we accept the usual distinction. Some may say that this is a mistake,--that the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a common unit, such as 'people' of 'population' and then added these common units. But this 'explanation' is entirely gratuitous, as one will find if he questions the pupil about the process. It will be found that the child simply added the figures as numbers only and then interpreted the result, according to the statement of the problem, without so much mental gymnastics. The writer has questioned hundreds of students in Normal School work on this point, and he believes that the ordinary mind-movement is correctly set forth here, no matter how well one may maintain as an academic proposition that this is not logical. Many classes in the Eastern Kentucky State Normal have been given this problem to solve, and they invariably get the same result:--

'In a garden on the Summit are as many cabbage-heads as the total number of ladies and gentlemen in this class. How many cabbage-heads in the garden?'

And the blackboard solution looks like this each time:--

So, also, one may say: I have 6 times as many sheep as you have cows. If you have 5 cows, how many sheep have I? Here we would multiply the number of cows, which is 5, by 6 and call the result 30, which must be linked with the idea of sheep because the conditions imposed by the problem demand it. The mind naturally in this work separates the pure number from its situation, as in algebra, handles it according to the laws governing arithmetical combinations, and labels the result as the statement of the problem demands. This is expressed in the following, which is tacitly accepted in algebra, and should be accepted equally in arithmetic:

'In all computations and operations in arithmetic, all numbers are essentially abstract and should be so treated. They are concrete only in the thought process that attends the operation and interprets the result.'"

The following problems are taken at random from those given by one of the best of the textbooks that make the attempt to apply the facts of Greatest Common Divisor and Least Common Multiple to problems. Most of these problems are fantastic. The others are trivial, or are better solved by trial and adaptation.

All this is so obvious that it may seem needless to relate. It is not. With many textbooks it is now necessary to give definite drill in reading the words in the printed problems intended for grades 2, 3, and 4, or to replace them by oral statements, or to leave the pupils in confusion concerning what the problems are that they are to solve. Many good teachers make a regular reading-lesson out of every page of problems before having them solved. There should be no such necessity.

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If the reader doubts the need of this warning let him examine problems 1 to 5, all from reputable books that are in common use, or have been within a few years, and consider how addition, subtraction, and the habits belonging with each are confused by exercise 6.

On one side of George's slate there are 32 words, and on the other side 26 words. If he erases 6 words from one side, and 8 from the other, how many words remain on his slate?

A certain school has 14 rooms, and an average of 40 children in a room. If every one in the school should make 500 straight marks on each side of his slate, how many would be made in all?

From the Declaration of Independence to the World's Fair in Chicago was 9 times as many years as there are stripes in the flag. How many years was it?

A clerk in an office addressed letters according to a given list. After she had addressed 2500, 4/9 of the names on the list had not been used; how many names were in the entire list?

The Canadian power canal at Sault Ste. Marie furnished 20,000 horse power. The canal on the Michigan side furnished 2-1/2 times as much. How many horse power does the latter furnish?

It may be asserted that the ideal of giving as described problems only problems that might occur and demand the same sort of process for solution with a real situation, is too exacting. If a problem is comprehensible and serves to illustrate a principle or give useful drill, that is enough, teachers may say. For really scientific teaching it is not enough. Moreover, if problems are given merely as tests of knowledge of a principle or as means to make some fact or principle clear or emphatic, and are not expected to be of direct service in the quantitative work of life, it is better to let the fact be known. For example, "I am thinking of a number. Half of this number is twice six. What is the number?" is better than "A man left his wife a certain sum of money. Half of what he left her was twice as much as he left to his son, who receives 00. How much did he leave his wife?" The former is better because it makes no false pretenses.

Consider, for example, the profitless linguistic difficulty of problems 1-6, whose quantitative difficulties are simply those of:--

Arithmetically this work belongs in the first or second years of learning. But children of grades 2 and 3, save a few, would be utterly at a loss to understand the language.

We are not yet free from the follies illustrated in the lessons of pages 96 to 99, which mystified our parents.

LESSON I

LESSON II

The economics and physics of the next four problems speak for themselves.

GUIDING PRINCIPLES

The reader may be wearied of these special details concerning bonds now neglected that should be formed and useless or harmful bonds formed for no valid reason. Any one of them by itself is perhaps a minor matter, but when we have cured all our faults in this respect and found all the possibilities for wiser selection of bonds, we shall have enormously improved the teaching of arithmetic. The ideal is such choice of bonds as will most improve the functions in question at the least cost of time and effort. The guiding principles may be kept in mind in the form of seven simple but golden rules:--

THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS

An inventory of the bonds to be formed in learning arithmetic should be accompanied by a statement of how strong each bond is to be made and kept year by year. Since, however, the inventory itself has been presented here only in samples, the detailed statement of desired strength for each bond cannot be made. Only certain general facts will be noted here.

THE NEED OF STRONGER ELEMENTARY BONDS

The constituent bonds involved in the fundamental operations with numbers need to be much stronger than they now are. Inaccuracy in these operations means weakness of the constituent bonds. Inaccuracy exists, and to a degree that deprives the subject of much of its possible disciplinary value, makes the pupil's achievements of slight value for use in business or industry, and prevents the pupil from verifying his work with new processes by some previously acquired process.

The inaccuracy that exists may be seen in the measurements made by the many investigators who have used arithmetical tasks as tests of fatigue, practice, individual differences and the like, and in the special studies of arithmetical achievements for their own sake made by Courtis and others.

Burgerstein , using such examples as

and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred answer-figures, or 3/5 of an error per example. The children were 9-1/2 to 15 years old. Laser , using the same sort of addition and multiplication, found somewhat over 3 errors per hundred answer-figures in the case of boys and girls averaging 11-1/2 years, during the period of their most accurate work. Holmes , using addition of the sort just described, found 346 errors in 23,713 answer-figures or about 1-1/2 per hundred. The children were from all grades from the third to the eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained per minute. Friedrich with similar examples, giving the very long time of 20 minutes for obtaining about 200 answer-figures, found from 1 to 2 per hundred wrong. King had children in grade 5 do sums, each consisting of 5 two-place numbers. In the most accurate work-period, they made 1 error per 20 columns. In multiplying a four-place by a four-place number they had less than one total answer right out of three. In New York City Courtis found with his Test 7 that in 12 minutes the average achievement of fourth-grade children is 8.8 units attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with 5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right. These results are near enough to those obtained from the country at large to serve as a text here.

The following were set as official standards, in an excellent school system, Courtis Series B being used:--

SPEED PERCENT OF GRADE. ATTEMPTS. CORRECT ANSWERS. Addition 8 12 80 7 11 80 6 10 70 5 9 70 4 8 70

Subtraction 8 12 90 7 11 90 6 10 90 5 9 80 4 7 80

Multiplication 8 11 80 7 10 80 6 9 80 5 7 70 4 6 60

Division 8 11 90 7 10 90 6 8 80 5 6 70 4 4 60

It is clear that numerical work as inaccurate as this has little or no commercial or industrial value. If clerks got only six answers out of ten right as in the Courtis tests, one would need to have at least four clerks make each computation and would even then have to check many of their discrepancies by the work of still other clerks, if he wanted his accounts to show less than one error per hundred accounting units of the Courtis size.

It is also clear that the "habits of ... absolute accuracy, and satisfaction in truth as a result" which arithmetic is supposed to further must be largely mythical in pupils who get right answers only from three to nine times out of ten!

EARLY MASTERY

The bonds in question clearly must be made far stronger than they now are. They should in fact be strong enough to abolish errors in computation, except for those due to temporary lapses. It is much better for a child to know half of the multiplication tables, and to know that he does not know the rest, than to half-know them all; and this holds good of all the elementary bonds required for computation. Any bond should be made to work perfectly, though slowly, very soon after its formation is begun. Speed can easily be added by proper practice.

The chief reasons why this is not done now seem to be the following: Certain important bonds are not given enough attention when they are first used. The special training necessary when a bond is used in a different connection is neglected. The pupil is not taught to check his work. He is not made responsible for substantially accurate results. Furthermore, the requirement of without the training of , , and will involve either a fruitless failure on the part of many pupils, or an utterly unjust requirement of time. The common error of supposing that the task of computation with integers consists merely in learning the additions to 9 + 9, the subtractions to 18 - 9, the multiplications to 8 x 9, and the divisions to 81 ? 9, and in applying this knowledge in connection with the principles of decimal notation, has had a large share in permitting the gross inaccuracy of arithmetical work. The bonds involved in 'knowing the tables' do not make up one fourth of the bonds involved in real adding, subtracting, multiplying, and dividing .

It should be noted that if the training mentioned in and is well cared for, the checking of results as recommended in becomes enormously more valuable than it is under present conditions, though even now it is one of our soundest practices. If a child knows the additions to higher decades so that he can add a seen one-place number to a thought-of two-place number in three seconds or less with a correct answer 199 times out of 200, there is only an infinitesimal chance that a ten-figure column twice added a few minutes apart with identical answers will be wrong. Suppose that, in long multiplication, a pupil can multiply to 9 x 9 while keeping his place and keeping track of what he is 'carrying' and of where to write the figure he writes, and can add what he carries without losing track of what he is to add it to, where he is to write the unit figure, what he is to multiply next and by what, and what he will then have to carry, in each case to a surety of 99 percent of correct responses. Then two identical answers got by multiplying one three-place number by another a few minutes apart, and with reversal of the numbers, will not be wrong more than twice in his entire school career. Checks approach proofs when the constituent bonds are strong.

If, on the contrary, the fundamental bonds are so weak that they do not work accurately, checking becomes much less trustworthy and also very much more laborious. In fact, it is possible to show that below a certain point of strength of the fundamental bonds, the time required for checking is so great that part of it might better be spent in improving the fundamental bonds.

For example, suppose that a pupil has to find the sum of five numbers like .49, .25, .50, .89, and .75. Counting each act of holding in mind the number to be carried and each writing of a column's result as equivalent in difficulty to one addition, such a sum equals nineteen single additions. On this basis and with certain additional estimates we can compute the practical consequences for a pupil's use of addition in life according to the mastery of it that he has gained in school.

These concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work.

I have so computed the amount of checking a pupil will have to do to reach two agreeing numbers , according to his mastery of the elementary processes. The facts appear in Table 1.

It is obvious that a pupil whose mastery of the elements is that denoted by getting them right 96 times out of 100 will require so much time for checking that, even if he were never to use this ability for anything save a few thousand sums in addition, he would do well to improve this ability before he tried to do the sums. An ability of 199 out of 200, or 995 out of 1000, seems likely to save much more time than would be taken to acquire it, and a reasonable defense could be made for requiring 996 or 997 out of 1000.

A precision of from 995 to 997 out of 1000 being required, and ordinary sagacity being used in the teaching, speed will substantially take care of itself. Counting on the fingers or in words will not give that precision. Slow recourse to memory of serial addition tables will not give that precision. Nothing save sure memory of the facts operating under the conditions of actual examples will give it. And such memories will operate with sufficient speed.

TABLE 1

THE EFFECT OF MASTERY OF THE ELEMENTARY FACTS OF ADDITION UPON THE LABOR REQUIRED TO SECURE TWO AGREEING ANSWERS WHEN ADDING FIVE THREE-FIGURE NUMBERS

THE STRENGTH OF BONDS FOR TEMPORARY SERVICE

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.

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