CHAPTER XV

10 × 2 =30 × 14 =10 × 3 =3 × 44 =10 × 4 =30 × 44 =10 × 14 =3 × 144 =10 × 44 =20 × 144 =10 × 144 =40 × 144 =20 × 2 =30 × 144 =20 × 3 =5 × 144 =30 × 3 =35 = 30 + ....30 × 4 =30 × 144 added to 5 × 144 =3 × 14 =

Certain wrong answers may be shown to be wrong in many ways;e.g., 432,720 is too big, for 35 times a thousand square inches is only 35,000; 1152 is too small, for 35 times a hundred square inches would be 3500, or more than 1152.

The time spent in realizing the problem here is fairly well spent because (1) any successful original manipulation inthis case represents an excellent exercise of thought, because (2) failures show that it is useless to juggle the figures at random, and because (3) the previous experience with short multiplication makes it possible for the pupils to realize the problem in a very few minutes. It may, however, be still better to give the pupils the right method just as soon as the problem is realized, without having them spend more time in trying to solve it. Thus:—

1 square foot has 144 square inches. How many square inches are there in 35 square feet (marked out in chalk on the floor as a piece 10 ft. × 3 ft. plus a piece 5 ft. × 1 ft.)?

1 yard = 36 inches. How many inches long is this wall (found by measure to be 13 yards)?

Here is a quick way to find the answers:—

14435——720432——5040 sq. inches in 35 sq. ft.3613——10836——468 inches in 13 yd.

Consider now the following introduction to dividing by a decimal:—

Dividing by a Decimal

1.How many minutes will it take a motorcycle, to go 12.675 miles at the rate of .75 mi. per minute?

16.9.75 |12.6757 55 174 50675675

2.Check by multiplying 16.9 by .75.3.How do you know that the quotient cannot be as little as 1.69?4.How do you know that the quotient cannot be as large as 169?5.Find the quotient for 3.75 ÷ 1.5.6.Check your result by multiplying the quotient by the divisor.7.How do you know that the quotient cannot be .25 or 25 ?8.Look at this problem. .25|7.5How do you know that 3.0 is wrong for the quotient?How do you know that 300 is wrong for the quotient?State which quotient is right for each of these:—9..021 or .21 or 2.1 or 21 or 2101.8|3.7810..021 or .21 or 21 or 2101.8|37.811..03 or .3 or 3 or 30 or 3001.25|37.512..03 or .3 or 3 or 30 or 30012.5|37.513..05 or .5 or 5 or 50 or 5001.25|6.2514..05 or .5 or 5 or 50 or 50012.5|6.2515.Is this rule true? If it is true, learn it.In a correct result, the number of decimal places in the divisor and quotient together equals the number of decimal places in the dividend.

2.Check by multiplying 16.9 by .75.

3.How do you know that the quotient cannot be as little as 1.69?

4.How do you know that the quotient cannot be as large as 169?

5.Find the quotient for 3.75 ÷ 1.5.

6.Check your result by multiplying the quotient by the divisor.

7.How do you know that the quotient cannot be .25 or 25 ?

8.Look at this problem. .25|7.5

How do you know that 3.0 is wrong for the quotient?

How do you know that 300 is wrong for the quotient?

State which quotient is right for each of these:—

9..021 or .21 or 2.1 or 21 or 2101.8|3.78

10..021 or .21 or 21 or 2101.8|37.8

11..03 or .3 or 3 or 30 or 3001.25|37.5

12..03 or .3 or 3 or 30 or 30012.5|37.5

13..05 or .5 or 5 or 50 or 5001.25|6.25

14..05 or .5 or 5 or 50 or 50012.5|6.25

15.Is this rule true? If it is true, learn it.

In a correct result, the number of decimal places in the divisor and quotient together equals the number of decimal places in the dividend.

These and similar exercises excite the problem attitude in childrenwho have a general interest in getting right answers. Such a series carefully arranged is a desirable introduction to a statement of the rule for placing the decimal point in division with decimals. For it attracts attention to the general principle (divisor × quotient should equal dividend), which is more important than the rule for convenient location of the decimal point, and it gives training in placing the point by inspection of the divisor, quotient, and dividend, which suffices for nineteen out of twenty cases that the pupil will ever encounter outside of school. He is likely to remember this method by inspection long after he has forgotten the fixed rule.

It is well for the pupil to be introduced to many arithmetical facts by way of problems about their common uses. The clockface, the railroad distance table in hundredths of a mile, the cyclometer and speedometer, the recipe, and the like offer problems which enlist his interest and energy and also connect the resulting arithmetical learning with the activities where it is needed. There is no time cost, but a time-saving, for the learning as a means to the solution of the problems is quicker than the mere learning of the arithmetical facts by themselves alone. A few samples of such procedure are shown below:—

GRADE 3To be Done at HomeLook at a watch. Has it any hands besides the hour hand and the minute hand? Find out all that you can about how a watch tells seconds, how long a second is, and how many seconds make a minute.GRADE 5Measuring RainfallRainfall per Week(cu. in. per sq. in. of area)June1-71.0568-141.10315-211.04022-28.96029-July 5.915July6-12.78213-19.79020-26.67027-Aug. 2.503Aug.3-9.51210-16.24017-23.21524-30.8111.In which weeks was the rainfall 1 or more?2.Which week of August had the largest rainfall for that month?3.Which was the driest week of the summer? (Driest means with the least rainfall.)4.Which week was the next to the driest?5.In which weeks was the rainfall between .800 and 1.000?6.Look down the table and estimate whether the average rainfall for one week was about .5, or about .6, or about .7, or about .8, or about .9.Dairy RecordsRecord of Star ElsiePounds ofMilkButter-Fatper Poundof MilkJan.1742.0461Feb.1690.0485Mar.1574.0504Apr.1226.0490May1202.0466June1251.0481Read this record of the milk given by the cow Star Elsie. The first column tells the number of pounds of milk given by Star Elsie each month. The second column tells what fraction of a pound of butter-fat each pound of milk contained.1.Read the first line, saying, "In January this cow gave 1742 pounds of milk. There were 461 ten thousandths of a pound of butter-fat per pound of milk." Read the other lines in the same way.2.How many pounds of butter-fat did the cow produce in Jan.?3.In Feb.?4.In Mar.?5.In Apr.?6.In May?7.In June?GRADE 5 OR LATERUsing Recipes to Make Larger or Smaller QuantitiesI. State how much you would use of each material in the following recipes: (a) To make double the quantity. (b) To make half the quantity. (c) To make 1½ times the quantity. You may use pencil and paper when you cannot find the right amount mentally.1.Peanut Penuche2.Molasses Candy1 tablespoon butter½ cup butter2 cups brown sugar2 cups sugar1⁄3cup milk or cream1 cup molasses¾ cup chopped peanuts1½ cups boiling water1⁄3teaspoon salt3.Raisin Opera Caramels4.Walnut Molasses Squares2 cups light brown sugar2 tablespoons butter7⁄8cup thin cream1 cup molasses½ cup raisins1⁄3cup sugar½ cup walnut meats5.Reception Rolls6.Graham Raised Loaf1 cup scalded milk2 cups milk1½ tablespoons sugar6 tablespoons molasses1 teaspoon salt1½ teaspoons salt¼ cup lard1⁄3yeast cake1 yeast cake¼ cup lukewarm water¼ cup lukewarm water2 cups sifted Graham flourWhite of 1 egg½ cup Graham bran3½ cups flour¾ cup flour (to knead)II. How much would you use of each material in the following recipes: (a) To make2⁄3as large a quantity? (b) To make 11⁄3times as much? (c) To make 2½ times as much?1.English Dumplings2.White Mountain Angel Cake½ pound beef suet1½ cups egg whites1¼ cups flour1½ cups sugar3 teaspoons baking powder1 teaspoon cream of tartar1 teaspoon salt1 cup bread flour½ teaspoon pepper¼ teaspoon salt1 teaspoon minced parsley1 teaspoon vanilla¼ cup cold water

GRADE 3

To be Done at Home

Look at a watch. Has it any hands besides the hour hand and the minute hand? Find out all that you can about how a watch tells seconds, how long a second is, and how many seconds make a minute.

GRADE 5

Measuring Rainfall

Rainfall per Week(cu. in. per sq. in. of area)June1-71.0568-141.10315-211.04022-28.96029-July 5.915July6-12.78213-19.79020-26.67027-Aug. 2.503Aug.3-9.51210-16.24017-23.21524-30.811

1.In which weeks was the rainfall 1 or more?

2.Which week of August had the largest rainfall for that month?

3.Which was the driest week of the summer? (Driest means with the least rainfall.)

4.Which week was the next to the driest?

5.In which weeks was the rainfall between .800 and 1.000?

6.Look down the table and estimate whether the average rainfall for one week was about .5, or about .6, or about .7, or about .8, or about .9.

Dairy Records

Record of Star ElsiePounds ofMilkButter-Fatper Poundof MilkJan.1742.0461Feb.1690.0485Mar.1574.0504Apr.1226.0490May1202.0466June1251.0481

Read this record of the milk given by the cow Star Elsie. The first column tells the number of pounds of milk given by Star Elsie each month. The second column tells what fraction of a pound of butter-fat each pound of milk contained.

1.Read the first line, saying, "In January this cow gave 1742 pounds of milk. There were 461 ten thousandths of a pound of butter-fat per pound of milk." Read the other lines in the same way.

2.How many pounds of butter-fat did the cow produce in Jan.?3.In Feb.?4.In Mar.?5.In Apr.?6.In May?7.In June?

GRADE 5 OR LATER

Using Recipes to Make Larger or Smaller Quantities

I. State how much you would use of each material in the following recipes: (a) To make double the quantity. (b) To make half the quantity. (c) To make 1½ times the quantity. You may use pencil and paper when you cannot find the right amount mentally.

1.Peanut Penuche2.Molasses Candy1 tablespoon butter½ cup butter2 cups brown sugar2 cups sugar1⁄3cup milk or cream1 cup molasses¾ cup chopped peanuts1½ cups boiling water1⁄3teaspoon salt3.Raisin Opera Caramels4.Walnut Molasses Squares2 cups light brown sugar2 tablespoons butter7⁄8cup thin cream1 cup molasses½ cup raisins1⁄3cup sugar½ cup walnut meats5.Reception Rolls6.Graham Raised Loaf1 cup scalded milk2 cups milk1½ tablespoons sugar6 tablespoons molasses1 teaspoon salt1½ teaspoons salt¼ cup lard1⁄3yeast cake1 yeast cake¼ cup lukewarm water¼ cup lukewarm water2 cups sifted Graham flourWhite of 1 egg½ cup Graham bran3½ cups flour¾ cup flour (to knead)

II. How much would you use of each material in the following recipes: (a) To make2⁄3as large a quantity? (b) To make 11⁄3times as much? (c) To make 2½ times as much?

1.English Dumplings2.White Mountain Angel Cake½ pound beef suet1½ cups egg whites1¼ cups flour1½ cups sugar3 teaspoons baking powder1 teaspoon cream of tartar1 teaspoon salt1 cup bread flour½ teaspoon pepper¼ teaspoon salt1 teaspoon minced parsley1 teaspoon vanilla¼ cup cold water

In many cases arithmetical facts and principles can be well taught in connection with some problem or project which is not arithmetical, but which has special potency to arouse an intellectual activity in the pupil which by some ingenuity can be directed to arithmetical learning. Playing store is the most fundamental case. Planning for a party, seeing who wins a game of bean bag, understanding the calendar for a month, selecting Christmas presents, planning a picnic, arranging a garden, the clock, the watch with second hand, and drawing very simple maps are situations suggesting problems which may bring a living purpose into arithmetical learning in grade 2. These are all available under ordinary conditions of class instruction. A sample of such problems for a higher grade (6) is shown below.

Estimating AreasThe children in the geography class had a contest in estimating the areas of different surfaces. Each child wrote his estimatesfor each of these maps, A, B, C, D, and E. (Only C and D are shown here.) In the arithmetic class they learned how to find the exact areas. Then they compared their estimates with the exact areas to find who came nearest.Estimating AreasWrite your estimates for A, B, C, D, and E. Then study the next 6 pages and learn how to find the exact areas.(The next 6 pages comprise training in the mensuration of parallelograms and triangles.)

Estimating Areas

The children in the geography class had a contest in estimating the areas of different surfaces. Each child wrote his estimatesfor each of these maps, A, B, C, D, and E. (Only C and D are shown here.) In the arithmetic class they learned how to find the exact areas. Then they compared their estimates with the exact areas to find who came nearest.

Estimating Areas

Write your estimates for A, B, C, D, and E. Then study the next 6 pages and learn how to find the exact areas.

(The next 6 pages comprise training in the mensuration of parallelograms and triangles.)

In some cases the affairs of individual pupils include problems which may be used to guide the individual in question to a zealous study of arithmetic as a means of achieving his purpose—of making a canoe, surveying an island, keeping the accounts of a Girls' Canning Club, or the like. It requires much time and very great skill to direct the work of thirty or more pupils each busy with a special type of his own, so as to make the work instructive for each, but in some cases the expense of time and skill is justified.

In general what should be meant when one says that it is desirable to have pupils in the problem-attitude when they are studying arithmetic is substantially as follows:—

First.—Information that comes as an answer to questionsis better attended to, understood, and remembered than information that just comes.

Second.—Similarly, movements that come as a step toward achieving an end that the pupil has in view are better connected with their appropriate situations, and such connections are longer retained, than is the case with movements that just happen.

Third.—The more the pupil is set toward getting the question answered or getting the end achieved, the greater is the satisfyingness attached to the bonds of knowledge or skill which mean progress thereto.

Fourth.—It is bad policy to rely exclusively on the purely intellectualistic problems of "How can I do this?" "How can I get the right answer?" "What is the reason for this?" "Is there a better way to do that?" and the like. It is bad policy to supplement these intellectualistic problems by only the remote problems of "How can I be fitted to earn a higher wage?" "How can I make sure of graduating?" "How can I please my parents?" and the like. The purely intellectualistic problems have too weak an appeal for many pupils; the remote problems are weak so long as they are remote and, what is worse, may be deprived of the strength that they would have in due time if we attempt to use them too soon. It is the extreme of bad policy to neglect those personal and practical problems furnished by life outside the class in arithmetic the solution of which can really be furthered by arithmetic then and there. It is good policy to spend time in establishing certain mental sets—stimulating, or even creating, certain needs—setting up problems themselves—when the time so spent brings a sufficient improvement in the quality and quantity of the pupils' interest in arithmetical learning.

Fifth.—It would be still worse policy to rely exclusivelyon problems arising outside arithmetic. To learn arithmetic is itself a series of problems of intrinsic interest and worth to healthy-minded children. The need for ability to multiply with United States money or to add fractions or to compute percents may be as truly vital and engaging as the need for skill to make a party dress or for money to buy it or for time to play baseball. The intellectualistic needs and problems should be considered along with all others, and given whatever weight their educational value deserves.

There are certain misconceptions of the doctrine of the problem-attitude. The most noteworthy is that difficulty—temporary failure—an inadequacy of already existing bonds—is the essential and necessary stimulus to thinking and learning. Dewey himself does not, as I understand him, mean this, but he has been interpreted as meaning it by some of his followers.[22]

Difficulty—temporary failure, inadequacy of existing bonds—on the contrary does nothing whatsoever in and of itself; and what is done by the annoying lack of success which sometimes accompanies difficulty sometimes hinders thinking and learning.

Mere difficulty, mere failure, mere inadequacy of existing bonds, does nothing. It is hard for me to add three eight-place numbers at a glance; I have failed to find as effective illustrations for pages 276 to 277 as I wished; my existing sensori-motor connections are inadequate to playing a golf course in 65. But these events and conditions have done nothing to stimulate me in respect to the behavior in question. In the first of the three there is no annoying lack and no dynamic influence at all; in the second there was to somedegree an annoying lack—a slight irritation at not getting just what I wanted,—and this might have impelled me to further thinking (though it did not, and getting one tiptop illustration would as a rule stimulate me to hunt for others more than failing to get such). In the third case the lack of the 65 does not annoy me or have any noteworthy dynamic effect. The lack of 90 instead of 95-100 is annoying and is at times a stimulus to further learning, though not nearly so strong a stimulus as the attainment of the 90 would be! At other times this annoying lack is distinctly inhibitory—a stimulus to ceasing to learn. In the intellectual life the inhibitory effect seems far the commoner of the two. Not getting answers seems as a rule to make us stop trying to get them. The annoying lack of success with a theoretical problem most often makes us desert it for problems to whose solution the existing bonds promise to be more adequate.

The real issue in all this concerns the relative strength, in the pupil's intellectual life, of the "negative reaction" of behavior in general. An animal whose life processes are interfered with so that an annoying state of affairs is set up, changes his behavior, making one after another responses as his instincts and learned tendencies prescribe, until the annoying state of affairs is terminated, or the animal dies, or suffers the annoyance as less than the alternatives which his responses have produced. When the annoying state of affairs is characterized by the failure of things as they are to minister to a craving—as in cases of hunger, loneliness, sex-pursuit, and the like,—we have stimulus to action by an annoying lack or need, with relief from action by the satisfaction of the need.

Such is in some measure true of man's intellectual life. In recalling a forgotten name, in solving certain puzzles, or in simplifying an algebraic complex, there is an annoyinglack of the name, solution, or factor, a trial of one after another response, until the annoyance is relieved by success or made less potent by fatigue or distraction. Even here thedifficultydoes not do anything—but only the annoying interference with our intellectual peace by the problem. Further, although for the particular problem, the annoying lack stimulates, and the successful attainment stops thinking, the later and more important general effect on thinking is the reverse. Successful attainment stops our thinkingon that problembut makes us more predisposed later to thinkingin general.

Overt negative reaction, however, plays a relatively small part in man's intellectual life. Filling intellectual voids or relieving intellectual strains in this way is much less frequent than being stimulated positively by things seen, words read, and past connections acting under modified circumstances. The notion of thinking as coming to a lack, filling it, meeting an obstacle, dodging it, being held up by a difficulty and overcoming it, is so one-sided as to verge on phantasy. The overt lacks, strains, and difficulties come perhaps once in five hours of smooth straightforward use and adaptation of existing connections, and they might as truly be called hindrances to thought—barriers which past successes help the thinker to surmount. Problems themselves come more often as cherished issues which new facts reveal, and whose contemplation the thinker enjoys, than as strains or lacks or 'problems which I need to solve.' It is just as true that the thinker gets many of his problems as results from, or bonuses along with, his information, as that he gets much of his information as results of his efforts to solve problems.

As between difficulty and success, success is in the long run more productive of thinking. Necessity is not the mother of invention. Knowledge of previous inventions isthe mother; original ability is the father. The solutions of previous problems are more potent in producing both new problems and their solutions than is the mere awareness of problems and desire to have them solved.

In the case of arithmetic, learning to cancel instead of getting the product of the dividends and the product of the divisors and dividing the former by the latter, is a clear case of very valuable learning, with ease emphasized rather than difficulty, with the adequacy of existing bonds (when slightly redirected) as the prime feature of the process rather than their inadequacy, and with no sense of failure or lack or conflict. It would be absurd to spend time in arousing in the pupil, before beginning cancellation, a sense of a difficulty—viz., that the full multiplying and dividing takes longer than one would like. A pupil in grade 4 or 5 might well contemplate that difficulty for years to no advantage. He should at once begin to cancel and prove by checking that errorless cancellation always gives the right answer. To emphasize before teaching cancellation the inadequacy of the old full multiplying and dividing would, moreover, not only be uneconomical as a means to teaching cancellation; it would amount to casting needless slurs on valuable past acquisitions, and it would, scientifically, be false. For, until a pupil has learned to cancel, the old full multiplying is not inadequate; it is admirable in every respect. The issue of its inadequacy does not truly appear until the new method is found. It is the best way until the better way is mastered.

In the same way it is unwise to spend time in making pupils aware of the annoying lacks to be supplied by the multiplication tables, the division tables, the casting out of nines, or the use of the product of the length and breadth of a rectangle as its area, the unit being changed to thesquare erected on the linear unit as base. The annoying lack will be unproductive, while the learning takes place readily as a modification of existing habits, and is sufficiently appreciated as soon as it does take place. The multiplication tables come when instead of merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21, four sevens make 28," etc. The division tables come as easy selections from the known multiplications; the casting out of nines comes as an easy device. The computation of the area of a rectangle is best facilitated, not by awareness of the lack of a process for doing it, but by awareness of the success of the process as verified objectively.

In all these cases, too, the pupil would be misled if we aroused first a sense of the inadequacy of counting, adding, and objective division, an awareness of the difficulties which the multiplication and division tables and nines device and area theorem relieve. The displaced processes are admirable and no unnecessary fault should be found with them, and they arenotinadequate until the shorter ways have been learned.

One false inference about the problem-attitude is that the pupil should always understand the aim or issue before beginning to form the bonds which give the method or process that provides the solution. On the contrary, he will often get the process more easily and value it more highly if he is taught what it isforgradually while he is learning it. The system of decimal notation, for example, may better be taken first as a mere fact, just as we teach a child to talk without trying first to have him understand the value of verbal intercourse, or to keep clean without trying first to have him understand the bacteriological consequences of filth.

A second inference—that the pupil should always be taught to care about an issue and crave a process for managing it before beginning to learn the process—is equally false. On the contrary, the best way to become interested in certain issues and the ways of handling them is to learn the process—even to learn it by sheer habituation—and then note what it does for us. Such is the case with ".16662⁄3× = divide by 6," ".3331⁄3× = divide by 3," "multiply by .875 = divide the number by 8 and subtract the quotient from the number."

A third unwise tendency is to degrade the mere giving of information—to belittle the value of facts acquired in any other way than in the course of deliberate effort by the pupil to relieve a problem or conflict or difficulty. As a protest against merely verbal knowledge, and merely memoriter knowledge, and neglect of the active, questioning search for knowledge, this tendency to belittle mere facts has been healthy, but as a general doctrine it is itself equally one-sided. Mere facts not got by the pupil's thinking are often of enormous value. They may stimulate to active thinking just as truly as that may stimulate to the reception of facts. In arithmetic, for example, the names of the numbers, the use of the fractional form to signify that the upper number is divided by the lower number, the early use of the decimal point in U. S. money to distinguish dollars from cents, and the meanings of "each," "whole," "part," "together," "in all," "sum," "difference," "product," "quotient," and the like are self-justifying facts.

A fourth false inference is that whatever teaching makes the pupil face a question and think out its answer is thereby justified. This is not necessarily so unless the question is a worthy one and the answer that is thought out an intrinsically valuable one and the process of thinking used one that isappropriate for that pupil for that question. Merely to think may be of little value. To rely much on formal discipline is just as pernicious here as elsewhere. The tendency to emphasize the methods of learning arithmetic at the expense of what is learned is likely to lead to abuses different in nature but as bad in effect as that to which the emphasis on disciplinary rather than content value has led in the study of languages and grammar, or in the old puzzle problems of arithmetic.

The last false inference that I shall discuss here is the inference that most of the problems by which arithmetical learning is stimulated had better be external to arithmetic itself—problems about Noah's Ark or Easter Flowers or the Merry Go Round or A Trip down the Rhine.

Outside interests should be kept in mind, as has been abundantly illustrated in this volume, but it is folly to neglect the power, even for very young or for very stupid children, of the problem "How can I get the right answer?" Children do have intellectual interests. They do like dominoes, checkers, anagrams, and riddles as truly as playing tag, picking flowers, and baking cake. With carefully graded work that is within their powers they like to learn to add, subtract, multiply, and divide with integers, fractions, and decimals, and to work out quantitative relations.

In some measure, learning arithmetic is like learning to typewrite. The learner of the latter has little desire to present attractive-looking excuses for being late, or to save expense for paper. He has no desire to hoard copies of such and such literary gems. He may gain zeal from the fact that a school party is to be given and invitations are to be sent out, but the problem "To typewrite better" is after all his main problem. Learning arithmetic is in some measure a game whose moves are motivated by the generalset of the mind toward victory—winning right answers. As a ball-player learns to throw the ball accurately to first-base, not primarily because of any particular problem concerning getting rid of the ball, or having the man at first-base possess it, or putting out an opponent against whom he has a grudge, but because that skill is required by the game as a whole, so the pupil, in some measure, learns the technique of arithmetic, not because of particular concrete problems whose solutions it furnishes, but because that technique is required by the game of arithmetic—a game that has intrinsic worth and many general recommendations.

The general facts concerning individual variations in abilities—that the variations are large, that they are continuous, and that for children of the same age they usually cluster around one typical or modal ability, becoming less and less frequent as we pass to very high or very low degrees of the ability—are all well illustrated by arithmetical abilities.

The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In these diagrams each space along the baseline represents a certain score or degree of ability, and the height of the surface above it represents the number of individuals obtaining that score. Thus in Fig. 61, 63 out of 1000 soldiers had no correct answer, 36 out of 1000 had one correct answer, 49 had two, 55 had three, 67 had four, and so on, in a test with problems (stated in words).

Figure 61 shows that these adults varied from no problems solved correctly to eighteen, around eight as a central tendency. Figure 62 shows that children of the same year-age (they were also from the same neighborhood and in the same school) varied from under 40 to over 200 figures correct. Figure 63 shows that even among children who have all reached the same school grade and so had rathersimilar educational opportunities in arithmetic, the variation is still very great. It requires a range from 15 to over 30 examples right to include even nine tenths of them.

Fig. 61.Fig. 61.—The scores of 1000 soldiers in the National Army born in English-speaking countries, in Test 2 of the Army Alpha. The score is the number of correct answers obtained in five minutes. Probably 10 to 15 percent of these men were unable to read or able to read only very easy sentences at a very slow rate. Data furnished by the Division of Psychology in the office of the Surgeon General.

Fig. 61.—The scores of 1000 soldiers in the National Army born in English-speaking countries, in Test 2 of the Army Alpha. The score is the number of correct answers obtained in five minutes. Probably 10 to 15 percent of these men were unable to read or able to read only very easy sentences at a very slow rate. Data furnished by the Division of Psychology in the office of the Surgeon General.

It should, however, be noted that if each individual had been scored by the average of his work on eight or ten different days instead of by his work in just one test, the variability would have been somewhat less than appears in Figs. 61, 62, and 63.

Fig. 62.Fig. 62.—The scores of 100 11-year-old pupils in a test of computation. Estimated from the data given by Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the number of correct figures.

Fig. 62.—The scores of 100 11-year-old pupils in a test of computation. Estimated from the data given by Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the number of correct figures.

It is also the case that if each individual had been scored, not in problem-solving alone or division alone, but in an elaborate examination on the whole field of arithmetic, the variability would have been somewhat less than appears in Figs. 61, 62, and 63. On the other hand, if the officers andthe soldiers rejected for feeblemindedness had been included in Fig. 61, if the 11-year-olds in special classes for the very dull had been included in Fig. 62, and if all children who had been to school six years had been included in Fig. 63, no matter what grade they had reached, the effect would have been toincreasethe variability.

Fig. 63.Fig. 63.—The scores of pupils in grade 6 in city schools in the Woody Division Test A. The score is the number of correct answers obtained in 20 minutes. From Woody ['16, p. 61].

Fig. 63.—The scores of pupils in grade 6 in city schools in the Woody Division Test A. The score is the number of correct answers obtained in 20 minutes. From Woody ['16, p. 61].

In spite of the effort by school officers to collect in any one school grade those somewhat equal in ability or in achievement or in a mixture of the two, the population of the same grades in the same school system shows a very wide range in any arithmetical ability. This is partly because promotion is on a more general basis than arithmetical ability so that some very able arithmeticians are deliberately held back on account of other deficiencies, and some very incompetent arithmeticians are advanced on account of other excellencies. It is partly because of general inaccuracy in classifying and promoting pupils.

In a composite score made up of the sum of the scores in Woody tests,—Add. A, Subt. A, Mult. A, and Div. A, and two tests in problem-solving (ten and six graded problems,with maximum attainable credits of 30 and 18), Kruse ['18] found facts from which I compute those of Table 13, and Figs. 64 to 66, for pupils all having the training of the same city system, one which sought to grade its pupils very carefully.

Figs. 64-66.Figs.64, 65, and 66.—The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving. The time was about 120 minutes. The maximum score attainable was 196.

Figs.64, 65, and 66.—The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving. The time was about 120 minutes. The maximum score attainable was 196.

The overlapping of grade upon grade should be noted. Of the pupils in grade 6 about 18 percent do better than the average pupil in grade 7, and about 7 percent do better than the average pupil in grade 8. Of the pupils in grade 8 about 33 percent do worse than the average pupil in grade 7 and about 12 percent do worse than the average pupil in grade 6.

TABLE 13

Relative Frequencies of Scores in an Extensive Team of Arithmetical Tests.[23]In Percents

The variation within a single class for which a single teacher has to provide is great. Even when teaching is departmental and promotion is by subjects, and when also the school is a large one and classification within a grade is by ability—there may be a wide range for any given special component ability. Under ordinary circumstances the range is so great as to be one of the chief limiting conditions for the teaching of arithmetic. Many methods appropriateto the top quarter of the class will be almost useless for the bottom quarter, andvice versa.

Fig. 67.Fig. 67.

Fig. 68.Fig. 68.Figs.67 and 68.—The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7). The score is the number of units done. Certain long tasks are counted as two units.

Figs.67 and 68.—The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7). The score is the number of units done. Certain long tasks are counted as two units.

Figures 67 and 68 show the scores of ten classes taken at random from ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of computation done in 12 minutes. Observe the very wide variation present in the case of everyclass. The variation within a class would be somewhat reduced if each pupil were measured by his average in eight or ten such tests given on different days. If a rather generous allowance is made for this we still have a variation in speed as great as that shown in Fig. 69, as the fact to be expected for a class of thirty-two 6 B pupils.

Fig. 69.Fig. 69.—A conservative estimate of the amount of variation to be expected within a single class of 32 pupils in grade 6, in the number of units done in Courtis Test 7 when all chance variations are eliminated.

Fig. 69.—A conservative estimate of the amount of variation to be expected within a single class of 32 pupils in grade 6, in the number of units done in Courtis Test 7 when all chance variations are eliminated.

The variations within a class in respect to what processes are understood so as to be done with only occasional errors may be illustrated further as follows:—A teacher in grade 4 at or near the middle of the year in a city doing the customary work in arithmetic will probably find some pupil in her class who cannot do column addition even without carrying, or the easiest written subtraction

(8978),53or37

who does not know his multiplication tables or how to derive them, or understand the meanings of + − × and ÷, or have any useful ideas whatever about division.

There will probably be some child in the class who can do such work as that shown below, and with very few errors.

Add

3⁄8+5⁄8+7⁄8+1⁄82½63⁄83¾1⁄6+3⁄8

Subtract

10.003.494 yd. 1 ft. 6 in.2 yd. 2 ft. 3 in.

Multiply

1¼ × 81625⁄8145206————

Divide

2 )13.5025 )9750

The invention of means of teaching thirty so different children at once with the maximum help and minimum hindrance from their different capacities and acquisitions is one of the great opportunities for applied science.

Courtis, emphasizing the social demand for a certain moderate arithmetical attainment in the case of nearly all elementary school children of, say, grade 6, has urged that definite special means be taken to bring the deficient children up to certain standards, without causing undesirable 'overlearning' by the more gifted children. Certain experimental work to this end has been carried out by him and others, but probably much more must be done before an authoritative program for securing certain minimum standards for all or nearly all pupils can be arranged.

The differences found among children of the same grade in the same city are due in large measure to inborn differences in their original natures. If, by a miracle, the children studied by Courtis, or by Woody, or by Kruse had all received exactly the same nurture from birth to date, they would still have varied greatly in arithmetical ability, perhaps almost as much as they now do vary.

The evidence for this is the general evidence that variation in original nature is responsible for much of the eventual variation found in intellectual and moral traits, plus certain special evidence in the case of arithmetical abilities themselves.

Thorndike found ['05] that in tests with addition and multiplication twins were very much more alike than siblings[24]two or three years apart in age, though the resemblance in home and school training in arithmetic should be nearly as great for the latter as for the former. Also the young twins (9-11) showed as close a resemblance in addition and multiplication as the older twins (12-15), although the similarities of training in arithmetic have had twice as long to operate in the latter case.

If the differences found, say among children in grade 6 in addition, were due to differences in the quantity and quality of training in addition which they have had, then by giving each of them 200 minutes of additional identical training the differences should be reduced. For the 200 minutes of identical training is a step toward equalizing training. It has been found in many investigations of the matter that when we make training in arithmetic more nearly equal for any group the variation within the group is not reduced.

On the contrary, equalizing training seems rather to increase differences. The superior individual seems to have attained his superiority by his own superiority of nature rather than by superior past training, for, during a period of equal training for all, he increases his lead. For example, compare the gains of different individuals due toabout 300 minutes of practice in mental multiplication of a three-place number by a three-place number shown in Table 14 below, from data obtained by the author ['08].[25]

TABLE 14

The Effect of Equal Amounts of Practice upon Individual Difference in the Multiplication Of Three-Place Numbers

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