CHAPTER XIV

THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE

Dewey, and others following him, have emphasized the desirability of having pupils do their work as active seekers, conscious of problems whose solution satisfies some real need of their own natures. Other things being equal, it is unwise, they argue, for pupils to be led along blindfold as it were by the teacher and textbook, not knowing where they are going or why they are going there. They ought rather to have some living purpose, and be zealous for its attainment.

This doctrine is in general sound, as we shall see, but it is often misused as a defense of practices which neglect the formation of fundamental habits, or as a recommendation to practices which are quite unworkable under ordinary classroom conditions. So it seems probable that its nature and limitations are not thoroughly known, even to its followers, and that a rather detailed treatment of it should be given here.

ILLUSTRATIVE CASES

Consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained, or care about attaining the end (well or ill understood) is well spent.

It is well for a pupil who has learned (1) the meanings of the numbers one to ten, (2) how to count a collection of ten or less, and (3) how to measure in inches a magnitude of ten, nine, eight inches, etc., to be confronted with the problem of true adding without counting or measuring, as in 'hidden' addition and measurement by inference. For example, the teacher has three pencils counted and put under a book; has two more counted and put under the book; and asks, "How many pencils are there under the book?" Answers, when obtained, are verified or refuted by actual counting and measuring.

The time here is well spent because the children can do the necessary thinking if the tasks are well chosen; because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding, that is, thinking of the two numbers and inferring their sum; and further, because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it.

The manipulation of short multiplication may be introduced by confronting the pupils with such problems as, "How to tell how many Uneeda biscuit there are in four boxes, by opening only one box." Correct solutions by addition should be accepted. Correct solutions by multiplication, if any gifted children think of this way, should be accepted, even if the children cannot justify their procedure. (Inferring the manipulation from the place-values of numbers is beyond all save the most gifted and probably beyond them.) Correct solution by multiplication by some child who happens to have learned it elsewhere should be accepted. Let the main proof of the trustworthiness of the manipulation be by measurement and by addition. Proof by the stock arguments from the place-values of numbers may also be used. If no child hits on the manipulation in question, the problem of finding the length _without_ adding may be set. If they still fail, the problem may be made easier by being put as "4 times 22 gives the answer. Write down what you think 4 times 22 will be." Other reductions of the difficulty of the problem may be made, or the teacher may give the answer without very great harm being done. The important requirement is that the pupils should be aware of the problem and treat the manipulation as a solution of it, not as a form of educational ceremonial which they learn to satisfy the whims of parents and teachers. In the case of any particular class a situation that is more appealing to the pupils' practical interests than the situation used here can probably be devised.

The time spent in this way is well spent (1) because all but the very dull pupils can solve the problem in some way, (2) because the significance of the manipulation as an economy over addition is worth bringing out, and (3) because there is no way of beginning training in short multiplication that is much better.

In the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square inches in 35 square feet, or the number of days in 32 years, or whatever similar problem can be brought up so as to be felt as a problem.

Suppose that it is the 35 square feet. Solutions by (5 × 144) + (30 × 144), however arranged, or by (10 × 144) + (10 × 144) + (10 × 144) + (5 × 144), or by 3500 + (35 × 40) + (35 × 4), or by 7 × (5 × 144), however arranged, should all be listed for verification or rejection. The pupils need not be required to justify their procedures by a verbal statement. Answers like 432,720, or 720,432, or 1152, or 4220, or 3220 should be listed for verification or rejection. Verification may be by a mixture of short multiplication and objective work, or by a mixture of short multiplication and addition, or by addition abbreviated by taking ten 144s as 1440, or even (for very stupid pupils) by the authority of the teacher. Or the manipulation in cases like 53 × 9 or 84 × 7 may be verified by the reverse short multiplication. The deductive proof of the correctness of the manipulation may be given in whole or in part in connection with exercises like

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 in this 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:--

144 35 ---- 720 432 ---- 5040 sq. inches in 35 sq. ft.

36 13 --- 108 36 --- 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|12.675 7 5 --- 5 17 4 50 ---- 675 675 ---

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:--

.021 or .21 or 2.1 or 21 or 210 ______ 9. 1.8|3.78

.021 or .21 or 21 or 210 ______ 10. 1.8|37.8

.03 or .3 or 3 or 30 or 300 ______ 11. 1.25|37.5

.03 or .3 or 3 or 30 or 300 ______ 12. 12.5|37.5

.05 or .5 or 5 or 50 or 500 ______ 13. 1.25|6.25

.05 or .5 or 5 or 50 or 500 ______ 14. 12.5|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 children _who 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 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=) June 1-7 1.056 8-14 1.103 15-21 1.040 22-28 .960 29-July 5 .915 July 6-12 .782 13-19 .790 20-26 .670 27-Aug. 2 .503 Aug. 3-9 .512 10-16 .240 17-23 .215 24-30 .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 Elsie=

Pounds of Milk Butter-Fat per Pound of Milk Jan. 1742 .0461 Feb. 1690 .0485 Mar. 1574 .0504 Apr. 1226 .0490 May 1202 .0466 June 1251 .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 1-1/2 times the quantity. You may use pencil and paper when you cannot find the right amount mentally.

1. PEANUT PENUCHE

1 tablespoon butter 2 cups brown sugar 1/3 cup milk or cream 3/4 cup chopped peanuts 1/3 teaspoon salt

2. MOLASSES CANDY

1/2 cup butter 2 cups sugar 1 cup molasses 1-1/2 cups boiling water

3. RAISIN OPERA CARAMELS

2 cups light brown sugar 7/8 cup thin cream 1/2 cup raisins

4. WALNUT MOLASSES SQUARES

2 tablespoons butter 1 cup molasses 1/3 cup sugar 1/2 cup walnut meats

5. RECEPTION ROLLS

1 cup scalded milk 1-1/2 tablespoons sugar 1 teaspoon salt 1/4 cup lard 1 yeast cake 1/4 cup lukewarm water White of 1 egg 3-1/2 cups flour

6. GRAHAM RAISED LOAF

2 cups milk 6 tablespoons molasses 1-1/2 teaspoons salt 1/3 yeast cake 1/4 cup lukewarm water 2 cups sifted Graham flour 1/2 cup Graham bran 3/4 cup flour (to knead)

II. How much would you use of each material in the following recipes: (_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as much? (_c_) To make 2-1/2 times as much?

1. ENGLISH DUMPLINGS

1/2 pound beef suet 1-1/4 cups flour 3 teaspoons baking powder 1 teaspoon salt 1/2 teaspoon pepper 1 teaspoon minced parsley 1/2 cup cold water

2. WHITE MOUNTAIN ANGEL CAKE

1-1/2 cups egg whites 1-1/2 cups sugar 1 teaspoon cream of tartar 1 cup bread flour 1/4 teaspoon salt 1 teaspoon vanilla

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 Areas

The children in the geography class had a contest in estimating the areas of different surfaces. Each child wrote his estimates for 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.

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.

GENERAL PRINCIPLES

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 questions is 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 exclusively on 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.

DIFFICULTY AND SUCCESS AS STIMULI

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]

[22] In his _How We Think_.

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 some degree 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 annoying lack 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 the _difficulty_ does 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 thinking _on that problem_ but makes us more predisposed later to thinking _in 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 is the 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 the square 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 are _not_ inadequate until the shorter ways have been learned.

FALSE INFERENCES

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 is _for_ gradually 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 ".1666-2/3 × = divide by 6," ".333-1/3 × = 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 is appropriate 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 general set 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.