10 = .... 2s10 = .... 3s and .... rem.10 = .... 4s and .... rem.10 = .... 5s11 = .... 2s and .... rem.11 = .... 3s and .... rem....89 = .... 9s and .... rem.
These bonds must be formed before short division can be efficient, are useful as a partial help toward selection of the proper quotient figures in long division, and are the chief instruments for one of the important problem series in applied arithmetic,—"How manyxs can I buy forycents atzcents perxand how much will I have left?" That these bonds are at present sadly neglected is shown by Kirby ['13], who found that pupils in the last half of grade 3 and thefirst half of grade 4 could do only about four such examples per minute (in a ten-minute test), and even at that rate made far from perfect records, though they had been taught the regular division tables. Sixty minutes of practice resulted in a gain of nearly 75 percent in number done per minute, with an increase in accuracy as well.
(4)The equation form.—The equation form with an unknown quantity to be determined, or a missing number to be found, should be connected with its meaning and with the problem attitude long before a pupil begins algebra, and in the minds of pupils who never will study algebra.
Children who have just barely learned to add and subtract learn easily to do such work as the following:—
Write the missing numbers:—
4 + 8 = ....5 + .... = 14.... + 3 = 11.... = 5 + 216 = 7 + ....12 = .... + 5
The equation form is the simplest uniform way yet devised to state a quantitative issue. It is capable of indefinite extension if certain easily understood conventions about parentheses and fraction signs are learned. It should be employed widely in accounting and the treatment of commercial problems, and would be except for outworn conventions. It is a leading contribution of algebra to business and industrial life. Arithmetic can make it nearly as well. It saves more time in the case of drills on reducing fractions to higher and lower terms alone than is required to learn its meaning and use. To rewrite a quantitative problemas an equation and then make the easy selection of the necessary technique to solve the equation is one of the most universally useful intellectual devices known to man. The words 'equals,' 'equal,' 'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer equivalents should therefore early give way on many occasions to the '=' which so far surpasses them in ultimate convenience and simplicity.
(5)Addition and subtraction facts in the case of fractions.—In the case of adding and subtracting fractions, certain specific bonds—between the situation of halves and thirds to be added and the responses of thinking of the numbers as equal to so many sixths, between the situation thirds and fourths to be added and thinking of them as so many twelfths, between fourths and eighths to be added and thinking of them as eighths, and the like—should be formed separately. The general rule of thinking of fractions as their equivalents with some convenient denominator should come as an organization and extension of such special habits, not as an edict from the textbook or teacher.
(6)Fractional equivalents.—Efficiency requires that in the end the much used reductions should be firmly connected with the situations where they are needed. They may as well, therefore, be so connected from the beginning, with the gain of making the general process far easier for the dull pupils to master. We shall see later that, for all save the very gifted pupils, the economical way to get an understanding of arithmetical principles is not, usually, to learn a rule and then apply it, but to perform instructive operations and, in the course of performing them, to get insight into the principles.
(7)Protective habits in multiplying and dividing with fractions.—In multiplying and dividing with fractions special bonds should be formed to counteract the now harmfulinfluence of the 'multiply = get a larger number' and 'divide = get a smaller number' bonds which all work with integers has been reënforcing.
For example, at the beginning of the systematic work with multiplication by a fraction, let the following be printed clearly at the top of every relevant page of the textbook and displayed on the blackboard:—
When you multiply a number by anything more than 1 the result is larger than the number.
When you multiply a number by 1 the result is the same as the number.
When you multiply a number by anything less than 1 the result is smaller than the number.
Let the pupils establish the new habit by many such exercises as:—
18 × 4 = ....4 × 4 = ....2 × 4 = ....1 × 4 = ....1⁄2× 4 = ....1⁄4× 4 = ....1⁄8× 4 = ....9 × 2 = ....6 × 2 = ....3 × 2 = ....1 × 2 = ....1⁄3× 2 = ....1⁄6× 2 = ....1⁄9× 2 = ....
In the case of division by a fraction the old harmful habit should be counteracted and refined by similar rules and exercises as follows:—
When you divide a number by anything more than 1 the result is smaller than the number.
When you divide a number by 1 the result is the same as the number.
When you divide a number by anything less than 1 the result is larger than the number.
State the missing numbers:—
8 = .... 4s12 = .... 6s9 = .... 9s8 = .... 2s12 = .... 4s9 = .... 3s8 = .... 1s12 = .... 3s9 = .... 1s8 = ....1⁄2s12 = .... 2s9 = ....1⁄3s8 = ....1⁄4s12 = .... 1s9 = ....1⁄9s8 = ....1⁄8s12 = ....1⁄2s12 = ....1⁄3s12 = ....1⁄4s
16 ÷ 16 =9 ÷ 9 =10 ÷ 10 =12 ÷ 6 =16 ÷ 8 =9 ÷ 3 =10 ÷ 5 =12 ÷ 4 =16 ÷ 4 =9 ÷ 1 =10 ÷ 1 =12 ÷ 3 =16 ÷ 2 =9 ÷1⁄3=10 ÷1⁄5=12 ÷ 2 =16 ÷ 1 =9 ÷1⁄9=10 ÷1⁄10=12 ÷ 1 =16 ÷1⁄2=12 ÷1⁄2=16 ÷1⁄4=12 ÷1⁄3=16 ÷1⁄8=12 ÷1⁄4=12 ÷1⁄6=
(8)'% of' means 'hundredths times.'—In the case of percentage a series of bonds like the following should be formed:—
5percentof= .05 times20" ""= .20 "6" ""= .06 "25%"= .25 ×12%"= .12 ×3%"= .03 ×
Four five-minute drills on such connections between 'xpercent of' and 'its decimal equivalent times' are worth an hour's study of verbal definitions of the meaning of percent as per hundred or the like. The only use of thestudy of such definitions is to facilitate the later formation of the bonds, and, with all save the brighter pupils, the bonds are more needed for an understanding of the definitions than the definitions are needed for the formation of the bonds.
(9)Habits of verifying results.—Bonds should early be formed between certain manipulations of numbers and certain means of checking, or verifying the correctness of, the manipulation in question. The additions to 9 + 9 and the subtractions to 18 − 9 should be verified by objective addition and subtraction and counting until the pupil has sure command; the multiplications to 9 × 9 should be verified by objective multiplication and counting of the result (in piles of tens and a pile of ones) eight or ten times,[4]and by addition eight or ten times;[4]the divisions to 81 ÷ 9 should be verified by multiplication and occasionally objectively until the pupil has sure command; column addition should be checked by adding the columns separately and adding the sums so obtained, and by making two shorter tasks of the given task and adding the two sums; 'short' multiplication should be verified eight or ten times by addition; 'long' multiplication should be checked by reversing multiplier and multiplicand and in other ways; 'short' and 'long' division should be verified by multiplication.
These habits of testing an obtained result are of threefold value. They enable the pupil to find his own errors, and to maintain a standard of accuracy by himself. They give him a sense of the relations of the processes and the reasons why the right ways of adding, subtracting, multiplying, and dividing are right, such as only the very brightpupils can get from verbal explanations. They put his acquisition of a certain power, say multiplication, to a real and intelligible use, in checking the results of his practice of a new power, and so instill a respect for arithmetical power and skill in general. The time spent in such verification produces these results at little cost; for the practice in adding to verify multiplications, in multiplying to verify divisions, and the like is nearly as good for general drill and review of the addition and multiplication themselves as practice devised for that special purpose.
Early work in adding, subtracting, and reducing fractions should be verified by objective aids in the shape of lines and areas divided in suitable fractional parts. Early work with decimal fractions should be verified by the use of the equivalent common fractions for .25, .75, .125, .375, and the like. Multiplication and division with fractions, both common and decimal, should in the early stages be verified by objective aids. The placing of the decimal point in multiplication and division with decimal fractions should be verified by such exercises as:—
201.23 )24.60246It cannot be 200; for 200 × 1.23 is much more than 24.6.It cannot be 2; for 2 × 1.23 is much less than 24.6.
The establishment of habits of verifying results and their use is very greatly needed. The percentage of wrong answers in arithmetical work in schools is now so high that the pupils are often being practiced in error. In many cases they can feel no genuine and effective confidence in the processes, since their own use of the processes brings wrong answers as often as right. In solving problems they often cannot decide whether they have done the right thing or the wrong, since even if they have done the right thing, they mayhave done it inaccurately. A wrong answer to a problem is therefore too often ambiguous and uninstructive to them.[5]
These illustrations of the last few pages are samples of the procedures recommended by a consideration of all the bonds that one might form and of the contribution that each would make toward the abilities that the study of arithmetic should develop and improve. It is by doing more or less at haphazard what psychology teaches us to do deliberately and systematically in this respect that many of the past advances in the teaching of arithmetic have been made.
A scrutiny of the bonds now formed in the teaching of arithmetic with questions concerning the exact service of each, results in a list of bonds of small value or even no value, so far as a psychologist can determine. I present here samples of such psychologically unjustifiable bonds with some of the reasons for their deficiencies.
(1)Arbitrary units.—In drills intended to improve the ability to see and use the meanings of numbers as names for ratios or relative magnitudes, it is unwise to employ entirely arbitrary units. The procedure in II (on page 84) is better than that in I. Inches, half-inches, feet, and centimeters are better as units of length than arbitrary As. Square inches, square centimeters, and square feet are better for areas. Ounces and pounds should be lifted rather than arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and cubic inches are better for volume.
All the real merit in the drills on relative magnitude advocated by Speer, McLellan and Dewey, and others can be secured without spending time in relating magnitudesfor the sake of relative magnitude alone. The use of units of measure in drills which will never be used inbona fidemeasuring is like the use of fractions like sevenths, elevenths, and thirteenths. A very little of it is perhaps desirable to test the appreciation of certain general principles, but for regular training it should give place to the use of units of practical significance.
Fig. 3.Fig. 3.I. IfAis 1 which line is 2? Which line is 4? Which line is 3?AandCtogether equal what line?AandBtogether equal what line? How much longer isBthanA? How much longer isBthanC? How much longer isDthanA?
I. IfAis 1 which line is 2? Which line is 4? Which line is 3?AandCtogether equal what line?AandBtogether equal what line? How much longer isBthanA? How much longer isBthanC? How much longer isDthanA?
Fig. 4.Fig. 4.II.Ais 1 inch long. Which line is 2 inches long? Which line is 4 inches long? Which line is 3 inches long?AandCtogether make ... inches?AandBtogether make ... inches?Bis ... ... longer thanA?Bis ... ... longer thanC?Dis ... ... longer thanA?
II.Ais 1 inch long. Which line is 2 inches long? Which line is 4 inches long? Which line is 3 inches long?AandCtogether make ... inches?AandBtogether make ... inches?Bis ... ... longer thanA?Bis ... ... longer thanC?Dis ... ... longer thanA?
(2)Multiples of 11.—The multiplications of 2 to 12 by 11 and 12 as single connections should be left for the pupil to acquire by himself as he needs them. These connections interfere with the process of learning two-place multiplication. The manipulations of numbers there required can be learned much more easily if 11 and 12 are used as multipliers in just the same way that 78 or 96 would be. Later the 12 × 2, 12 × 3, etc., may be taught. There is less reason for knowing the multiples of 11 than for knowing the multiples of 15, 16, or 25.
(3)Abstract and concrete numbers.—The elaborate emphasis of the supposed fact that we cannot multiply 726 by 8 dollars and the still more elaborate explanations of why nevertheless we find the cost of 726 articles at $8 each by multiplying 726 by 8 and calling the answer dollars are wasteful. The same holds of the corresponding pedantry about division. These imaginary difficulties should not be raised at all. The pupil should not think of multiplying or dividing men or dollars, but simply of the necessary equation and of the sort of thing that the missing number represents. "8 × 726 = .... Answer is dollars," or "8, 726, multiply. Answer is dollars," is all that he needs to think, and is in the best form for his thought. Concerning the distinction between abstract and concrete numbers, both logic and common sense as well as psychology support the contention of McDougle ['14, p. 206f.], who writes:—
"The most elementary counting, even that stage when the counts were not carried in the mind, but merely in notches on a stick or by DeMorgan's stones in a pot, requires some thought; and the most advanced counting implies memory of things. The terms, therefore, abstract and concrete number, have long since ceased to be used by thinking people.
"Recently the writer visited an arithmetic class in aState Normal School and saw a group of practically adult students confused about this very question concerning abstract and concrete numbers, according to their previous training in the conventionalities of the textbook. Their teacher diverted the work of the hour and she and the class spent almost the whole period in reëstablishing the requirements 'that the product must always be the same kind of unit as the multiplicand,' and 'addends must all be alike to be added.' This is not an exceptional case. Throughout the whole range of teaching arithmetic in the public schools pupils are obfuscated by the philosophical encumbrances which have been imposed upon the simplest processes of numerical work. The time is surely ripe, now that we are readjusting our ideas of the subject of arithmetic, to revise some of these wasteful and disheartening practices. Algebra historically grew out of arithmetic, yet it has not been laden with this distinction. No pupil in algebra letsxequal the horses; he letsxequal thenumberof horses, and proceeds to drop the idea of horses out of his consideration. He multiplies, divides, and extracts the root of thenumber, sometimes handling fractions in the process, and finally interprets the result according to the conditions of his problem. Of course, in the early number work there have been the sense-objects from which number has been perceived, but the mind retreats naturally from objectivity to the pure conception of number, and then to the number symbol. The following is taken from the appendix to Horn's thesis, where a seventh grade girl gets the population of the United States in 1820:—
7,862,166233,6341,538,0229,633,822whitesfree negroesslaves
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:—
291544ladiesgentlemencabbage-heads
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 itaccording 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.'"
(4)Least common multiple.—The whole set of bonds involved in learning 'least common multiple' should be left out. In adding and subtracting fractions the pupil shouldnotfind the least common multiple of their denominators but should find any common multiple that he can find quickly and correctly. No intelligent person would ever waste time in searching for the least common multiple of sixths, thirds, and halves except for the unfortunate traditions of an oversystematized arithmetic, but would think of their equivalents in sixths or twelfths or twenty-fourths orany other convenient common multiple. The process of finding the least common multiple is of such exceedingly rare application in science or business or life generally that the textbooks have to resort to purely fantastic problems to give drill in its use.
(5)Greatest common divisor.—The whole set of bonds involved in learning 'greatest common divisor' should also be left out. In reducing fractions to lowest terms the pupil should divide by anything that he sees that he can divide by, favoring large divisors, and continue doing so until he gets the fraction in terms suitable for the purpose in hand. The reader probably never has had occasion to compute a greatest common divisor since he left school. If he has computed any, the chances are that he would have saved time by solving the problem in some other way!
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.[6]Most of these problems are fantastic. The others are trivial, or are better solved by trial and adaptation.
1.A certain school consists of 132 pupils in the high school, 154 in the grammar, and 198 in the primary grades. If each group is divided into sections of the same number containing as many pupils as possible, how many pupils will there be in each section?2.A farmer has 240 bu. of wheat and 920 bu. of oats, which he desires to put into the least number of boxes of the same capacity, without mixing the two kinds of grain. Find how many bushels each box must hold.3.Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, and begin to toll at the same instant. When will they next toll together?4.A, B, C, and D start together, and travel the same way around an island which is 600 mi. in circuit. A goes 20 mi. per day, B 30, C 25, and D 40. How long must their journeying continue, in order that they may all come together again?5.The periods of three planets which move uniformly in circular orbits round the sun, are respectively 200, 250, and 300 da. Supposing their positions relatively to each other and the sun to be given at any moment, determine how many da. must elapse before they again have exactly the same relative positions.
1.A certain school consists of 132 pupils in the high school, 154 in the grammar, and 198 in the primary grades. If each group is divided into sections of the same number containing as many pupils as possible, how many pupils will there be in each section?
2.A farmer has 240 bu. of wheat and 920 bu. of oats, which he desires to put into the least number of boxes of the same capacity, without mixing the two kinds of grain. Find how many bushels each box must hold.
3.Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, and begin to toll at the same instant. When will they next toll together?
4.A, B, C, and D start together, and travel the same way around an island which is 600 mi. in circuit. A goes 20 mi. per day, B 30, C 25, and D 40. How long must their journeying continue, in order that they may all come together again?
5.The periods of three planets which move uniformly in circular orbits round the sun, are respectively 200, 250, and 300 da. Supposing their positions relatively to each other and the sun to be given at any moment, determine how many da. must elapse before they again have exactly the same relative positions.
(6)Rare and unimportant words.—The bonds between rare or unimportant words and their meanings should not be formed for the mere sake of verbal variety in the problems of the textbook. A pupil should not be expected to solve a problem that he cannot read. He should not be expected in grades 2 and 3, or even in grade 4, to read words that he has rarely or never seen before. He should not be given elaborate drill in reading during the time devoted to the treatment of quantitative facts and relations.
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.
To definerareandunimportantconcretely, I will say that for pupils up to the middle of grade 3, such words as the following are rare and unimportant (though each of them occurs in the very first fifty pages of some well-known beginner's book in arithmetic).
absenteesaccountAdeleadmittedAgnesagreedAlbanyAllenallowedalternateAndrewArkansasarrivedassemblyautomobilebaking powderbalancebarleybeggarBertieBessiebinBostonbouquetbronzebuckwheatByroncamphorCarlCarrieCecilCharlottecharityChicagocinnamonClaraclothespinscollectcommacommitteeconcertconfectionercranberriescranecurrantsdairymanDanielDaviddealerdebtdeliveredDenverdepartmentdepositeddictationdischargeddiscoverdiscoverydish-waterdrugdueEdgarEddieEdwinelectionelectricEllaEmilyenrolledentertainmentenvelopeEstherEthelexceedsexplanationexpressiongenerallygentlemenGilbertGracegradingGrahamgrammarHaroldhatchetHeraldshesitationHorace MannimpossibleincomeindicatedinmostinsertsinstallmentsinstantlyinsuranceIowaJackJennieJohnnyJosephjourneyJuliaKatherinelettuce-plantlibraryLottieLulamarginMarthaMatthewMaudmeadowmentallymercurymineralMissourimolassesMortonmovementsmuslinNellieniecesOaklandobservingobtainedofferedofficeonionsoppositeoriginalpackagepacketpalmPatrickPaulpaymentspeepPeterperchphaetonphotographpianopigeonsPilgrimspreservingproprietorpurchasedRachelRalphrapidityratherreadilyreceiptsregisterremandedrespectivelyRobertRogerRuthryeSamuelSan FranciscoseldomshearedshinglesskyrocketssloopsolvespeckledspongessproutstackStephenstrapsuccessfullysuggestedsunnysupplySusanSusie'ssyllabletalcumtermtestthermometerThomastorpedoestradertransactiontreasurytricycletubetwo-seatedunitedusuallyvacantvariousvasevelocipedevoteswalnutsWalterWashingtonwatchedwhistlewoodlandworsted
(7)Misleading facts and procedures.—Bonds should not be formed between articles of commerce and grossly inaccurate prices therefor, between events and grossly improbable consequences, or causes or accompaniments thereof, nor between things, qualities, and events which have no important connections one with another in the real world. In general, things should not be put together in the pupil's mind that do not belong together.
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.
1.If a duck flying3⁄5as fast as a hawk flies 90 miles in an hour, how fast does the hawk fly?2.At5⁄8of a cent apiece how many eggs can I buy for $60?3.At $.68 a pair how many pairs of overshoes can you buy for $816?4.At $.13 a dozen how many dozen bananas can you buy for $3.12?5.How many pecks of beans can be put into a box that will hold just 21 bushels?6.Write answers:537365?361000Beginning at the bottom say 11, 18, and 2 (writing it in its place) are 20. 5, 11, 14, and 6 (writing it) are 20, 5, 10. The number, omitted, is 62.a.58197364?1758b.625?904172050c.752414130?2460d.314429?761000e.?845223952367
1.If a duck flying3⁄5as fast as a hawk flies 90 miles in an hour, how fast does the hawk fly?
2.At5⁄8of a cent apiece how many eggs can I buy for $60?
3.At $.68 a pair how many pairs of overshoes can you buy for $816?
4.At $.13 a dozen how many dozen bananas can you buy for $3.12?
5.How many pecks of beans can be put into a box that will hold just 21 bushels?
6.Write answers:
537365?361000Beginning at the bottom say 11, 18, and 2 (writing it in its place) are 20. 5, 11, 14, and 6 (writing it) are 20, 5, 10. The number, omitted, is 62.
a.58197364?1758b.625?904172050c.752414130?2460d.314429?761000e.?845223952367
(8)Trivialities and absurdities.—Bonds should not be formed between insignificant or foolish questions and the labor of answering them, nor between the general arithmetical work of the school and such insignificant or foolish questions. The following are samples from recent textbooks of excellent standing:—
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?8 times the number of stripes in our flag is the number of years from 1800 until Roosevelt was elected President. In what year was he elected President?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?
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?
8 times the number of stripes in our flag is the number of years from 1800 until Roosevelt was elected President. In what year was he elected President?
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?
(9)Useless methods.—Bonds should not be formed between a described situation and a method of treating the situation which would not be a useful one to follow in the case of the real situation. For example, "If I set 96 trees in rows, sixteen trees in a row, how many rows will I have?" forms the habit of treating by division a problem that in reality would be solved by counting the rows. So also "I wish to give 25 cents to each of a group of boys and find that it will require $2.75. How many boys are in the group?" forms the habit of answering a question by division whose answer must already have been present to give the data of the problem.
(10)Problems whose answers would, in real life, be already known.—The custom of giving problems in textbooks which could not occur in reality because the answer has to be known to frame the problem is a natural result of the lazy author's tendency to work out a problem to fit a certain process and a certain answer. Such bogus problems are very, very common. In a random sampling of a dozen pages of "General Review" problems in one of the most widely used of recent textbooks, I find that about 6 percent of the problems are of this sort. Among the problems extemporized by teachers these bogus problems are probably still more frequent. Such are:—
A clerk in an office addressed letters according to a given list. After she had addressed 2500,4⁄9of 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 furnished20,000 horse power. The canal on the Michigan side furnished 2½ times as much. How many horse power does the latter furnish?
A clerk in an office addressed letters according to a given list. After she had addressed 2500,4⁄9of 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 furnished20,000 horse power. The canal on the Michigan side furnished 2½ 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 $6000. How much did he leave his wife?" The former is better because it makes no false pretenses.
(11)Needless linguistic difficulties.—It should be unnecessary to add that bonds should not be formed between the pupil's general attitude toward arithmetic and needless, useless difficulty in language or needless, useless, wrong reasoning. Our teaching is, however, still tainted by both of these unfortunate connections, which dispose the pupil to think of arithmetic as a mystery and folly.
Consider, for example, the profitless linguistic difficulty of problems 1-6, whose quantitative difficulties are simply those of:—
1.5 + 8 + 3 + 72.64 ÷ 8, and knowledge that 1 peck = 8 quarts3.12 ÷ 44.6 ÷ 25.3 × 26.4 × 41.What amount should you obtain by putting together 5 cents, 8 cents, 3 cents, and 7 cents? Did you find this result by adding or multiplying?2.How many times must you empty a peck measure to fill a basket holding 64 quarts of beans?3.If a girl commits to memory 4 pages of history in one day, in how many days will she commit to memory 12 pages?4.If Fred had 6 chickens how many times could he give away 2 chickens to his companions?5.If a croquet-player drove a ball through 2 arches at each stroke, through how many arches will he drive it by 3 strokes?6.If mamma cut the pie into 4 pieces and gave each person a piece, how many persons did she have for dinner if she used 4 whole pies for dessert?
1.5 + 8 + 3 + 72.64 ÷ 8, and knowledge that 1 peck = 8 quarts3.12 ÷ 44.6 ÷ 25.3 × 26.4 × 4
1.5 + 8 + 3 + 72.64 ÷ 8, and knowledge that 1 peck = 8 quarts3.12 ÷ 44.6 ÷ 25.3 × 26.4 × 4
1.What amount should you obtain by putting together 5 cents, 8 cents, 3 cents, and 7 cents? Did you find this result by adding or multiplying?
2.How many times must you empty a peck measure to fill a basket holding 64 quarts of beans?
3.If a girl commits to memory 4 pages of history in one day, in how many days will she commit to memory 12 pages?
4.If Fred had 6 chickens how many times could he give away 2 chickens to his companions?
5.If a croquet-player drove a ball through 2 arches at each stroke, through how many arches will he drive it by 3 strokes?
6.If mamma cut the pie into 4 pieces and gave each person a piece, how many persons did she have for dinner if she used 4 whole pies for dessert?
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.
Fig. 5.Fig. 5.
LESSON I
1.In this picture, how many girls are in the swing?2.How many girls are pulling the swing?3.If you count both girls together, how many are they?Onegirl andoneother girl are how many?4.How many kittens do you see on the stump?5.How many on the ground?6.How many kittens are in the picture? One kitten and one other kitten are how many?7.If you should ask me how many girls are in the swing, or how many kittens are on the stump, I could answer aloud,One; or I could writeOne; or thus,1.8.If I writeOne, this is called theword One.9.This,1, is named afigure One, because it means the same as the wordOne, and stands forOne.10.Write 1. What is this named? Why?11.A figure 1 may stand foronegirl,onekitten, oroneanything.12.When children first attend school, what do they begin to learn?Ans.Letters and words.13.Could you read or write before you had learned either letters or words?14.If we have all theletterstogether, they are named the Alphabet.15.If we write or speakwords, they are named Language.16.You are commencing to study Arithmetic; and you can read and write in Arithmetic only as you learn the Alphabet and Language of Arithmetic. But little time will be required for this purpose.
1.In this picture, how many girls are in the swing?
2.How many girls are pulling the swing?
3.If you count both girls together, how many are they?Onegirl andoneother girl are how many?
4.How many kittens do you see on the stump?
5.How many on the ground?
6.How many kittens are in the picture? One kitten and one other kitten are how many?
7.If you should ask me how many girls are in the swing, or how many kittens are on the stump, I could answer aloud,One; or I could writeOne; or thus,1.
8.If I writeOne, this is called theword One.
9.This,1, is named afigure One, because it means the same as the wordOne, and stands forOne.
10.Write 1. What is this named? Why?
11.A figure 1 may stand foronegirl,onekitten, oroneanything.
12.When children first attend school, what do they begin to learn?Ans.Letters and words.
13.Could you read or write before you had learned either letters or words?
14.If we have all theletterstogether, they are named the Alphabet.
15.If we write or speakwords, they are named Language.
16.You are commencing to study Arithmetic; and you can read and write in Arithmetic only as you learn the Alphabet and Language of Arithmetic. But little time will be required for this purpose.
Fig. 6.Fig. 6.
LESSON II
1.If we speak or write words, what do we name them, when taken together?2.What are you commencing to study?Ans.Arithmetic.3.What Language must you now learn?4.What do we name this,1? Why?5.This figure,1, is part of the Language of Arithmetic.6.If I should write something to stand forTwo—twogirls,twokittens, ortwothings of any kind—what do you think we would name it?7.Afigure Twois written thus:2.Make afigure two.8.Why do we name this afigure two?9.This figure two (2) is part of the Language of Arithmetic.10.In this picture one boy is sitting, playing a flageolet. What is the other boy doing? If the boy standing should sit down by the other, how many boys would be sitting together? One boy and one other boy are how many boys?11.You see a flageolet and a violin. They are musical instruments. One musical instrument and one other musical instrument are how many?12.I will write thus: 1 1 2. We say that 1 boy and 1 other boy, counted together, are 2 boys; or are equal to 2 boys. We will now write something to show that the first 1 and the other 1 are to be counted together.Plus13.We name a line drawn thus,—, ahorizontal line. Draw such a line. Name it.14.A line drawn thus, | , we name avertical line. Draw such a line. Name it.15.Now I will put two such lines together; thus, +. What kind of a line do we name the first (—)? And what do we name the last? (|)? Are these lines long or short? Where do they cross each other?16.Each of you write thus: —, | , +.17.This, +, is namedPlus.Plusmeansmore; and + also meansmore.18.I will write.One and One More Equal Two.19.Now I will write part of this in the Language of Arithmetic. I write the firstOnethus, 1; then the otherOnethus, 1. Afterward I write, for the wordMore, thus, +, placing the + between 1 and 1, so that the whole stands thus: 1 + 1. As I write, I say,One and One more.20.Each of you write 1 + 1. Read what you have written.21.This +, when written between the 1s, shows that they are to be put together, or counted together, so as to make 2.22.Because + shows what is to be done, it is called aSign. If we take its name,Plus, and the wordSign, and put both words together, we haveSign Plus, orPlus Sign. In speaking of this we may call itSign Plus, orPlus Sign, orPlus.23.1, 2, +, are part of the Language of Arithmetic.Write the following in the Language of Arithmetic:24.One and one more.25.One and two more.26.Two and one more.
1.If we speak or write words, what do we name them, when taken together?
2.What are you commencing to study?Ans.Arithmetic.
3.What Language must you now learn?
4.What do we name this,1? Why?
5.This figure,1, is part of the Language of Arithmetic.
6.If I should write something to stand forTwo—twogirls,twokittens, ortwothings of any kind—what do you think we would name it?
7.Afigure Twois written thus:2.Make afigure two.
8.Why do we name this afigure two?
9.This figure two (2) is part of the Language of Arithmetic.
10.In this picture one boy is sitting, playing a flageolet. What is the other boy doing? If the boy standing should sit down by the other, how many boys would be sitting together? One boy and one other boy are how many boys?
11.You see a flageolet and a violin. They are musical instruments. One musical instrument and one other musical instrument are how many?
12.I will write thus: 1 1 2. We say that 1 boy and 1 other boy, counted together, are 2 boys; or are equal to 2 boys. We will now write something to show that the first 1 and the other 1 are to be counted together.
Plus
13.We name a line drawn thus,—, ahorizontal line. Draw such a line. Name it.
14.A line drawn thus, | , we name avertical line. Draw such a line. Name it.
15.Now I will put two such lines together; thus, +. What kind of a line do we name the first (—)? And what do we name the last? (|)? Are these lines long or short? Where do they cross each other?
16.Each of you write thus: —, | , +.
17.This, +, is namedPlus.Plusmeansmore; and + also meansmore.
18.I will write.
One and One More Equal Two.
19.Now I will write part of this in the Language of Arithmetic. I write the firstOnethus, 1; then the otherOnethus, 1. Afterward I write, for the wordMore, thus, +, placing the + between 1 and 1, so that the whole stands thus: 1 + 1. As I write, I say,One and One more.
20.Each of you write 1 + 1. Read what you have written.
21.This +, when written between the 1s, shows that they are to be put together, or counted together, so as to make 2.
22.Because + shows what is to be done, it is called aSign. If we take its name,Plus, and the wordSign, and put both words together, we haveSign Plus, orPlus Sign. In speaking of this we may call itSign Plus, orPlus Sign, orPlus.
23.1, 2, +, are part of the Language of Arithmetic.
Write the following in the Language of Arithmetic:
24.One and one more.
25.One and two more.
26.Two and one more.
(12)Ambiguities and falsities.—Consider the ambiguities and false reasoning of these problems.
1.If you can earn 4 cents a day, how much can you earn in 6 weeks? (Are Sundays counted? Should a child who earns 4 cents some day expect to repeat the feat daily?)2.How many lines must you make to draw ten triangles and five squares? (I can do this with 8 lines, though the answer the book requires is 50.)3.A runner ran twice around an1⁄8mile track in two minutes. What distance did he run in2⁄3of a minute? (I do not know, but I do know that, save by chance, he did not run exactly2⁄3of1⁄8mile.)4.John earned $4.35 in a week, and Henry earned $1.93. They put their money together and bought a gun. What did it cost? (Maybe $5, maybe $10. Did they pay for the whole of it? Did they use all their earnings, or less, or more?)5.Richard has 12 nickels in his purse. How much more than 50 cents would you give him for them? (Would a wise child give 60 cents to a boy who wanted to swap 12 nickels therefor, or would he suspect a trick and hold on to his own coins?)6.If a horse trots 10 miles in one hour how far will he travel in 9 hours?7.If a girl can pick 3 quarts of berries in 1 hour how many quarts can she pick in 3 hours?(These last two, with a teacher insisting on the 90 and 9, might well deprive a matter-of-fact boy of respect for arithmetic for weeks thereafter.)The economics and physics of the next four problems speak for themselves.8.I lost $15 by selling a horse for $85. What was the value of the horse?9.If floating ice has 7 times as much of it under the surface of the water as above it, what part is above water? If an iceberg is 50 ft. above water, what is the entire height of the iceberg? How high above water would an iceberg 300 ft. high have to be?10.A man's salary is $1000 a year and his expenses $625.How many years will elapse before he is worth $10,000 if he is worth $2500 at the present time?11.Sound travels 1120 ft. a second. How long after a cannon is fired in New York will the report be heard in Philadelphia, a distance of 90 miles?
1.If you can earn 4 cents a day, how much can you earn in 6 weeks? (Are Sundays counted? Should a child who earns 4 cents some day expect to repeat the feat daily?)
2.How many lines must you make to draw ten triangles and five squares? (I can do this with 8 lines, though the answer the book requires is 50.)
3.A runner ran twice around an1⁄8mile track in two minutes. What distance did he run in2⁄3of a minute? (I do not know, but I do know that, save by chance, he did not run exactly2⁄3of1⁄8mile.)
4.John earned $4.35 in a week, and Henry earned $1.93. They put their money together and bought a gun. What did it cost? (Maybe $5, maybe $10. Did they pay for the whole of it? Did they use all their earnings, or less, or more?)
5.Richard has 12 nickels in his purse. How much more than 50 cents would you give him for them? (Would a wise child give 60 cents to a boy who wanted to swap 12 nickels therefor, or would he suspect a trick and hold on to his own coins?)
6.If a horse trots 10 miles in one hour how far will he travel in 9 hours?
7.If a girl can pick 3 quarts of berries in 1 hour how many quarts can she pick in 3 hours?
(These last two, with a teacher insisting on the 90 and 9, might well deprive a matter-of-fact boy of respect for arithmetic for weeks thereafter.)
(These last two, with a teacher insisting on the 90 and 9, might well deprive a matter-of-fact boy of respect for arithmetic for weeks thereafter.)
The economics and physics of the next four problems speak for themselves.
8.I lost $15 by selling a horse for $85. What was the value of the horse?
9.If floating ice has 7 times as much of it under the surface of the water as above it, what part is above water? If an iceberg is 50 ft. above water, what is the entire height of the iceberg? How high above water would an iceberg 300 ft. high have to be?
10.A man's salary is $1000 a year and his expenses $625.How many years will elapse before he is worth $10,000 if he is worth $2500 at the present time?
11.Sound travels 1120 ft. a second. How long after a cannon is fired in New York will the report be heard in Philadelphia, a distance of 90 miles?
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 (and, as will be shown later, such arrangement of them) 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:—
1. Consider the situation the pupil faces.
2. Consider the response you wish to connect with it.
3. Form the bond; do not expect it to come by a miracle.
4. Other things being equal, form no bond that will have to be broken.
5. Other things being equal, do not form two or three bonds when one will serve.
6. Other things being equal, form bonds in the way that they are required later to act.
7. Favor, therefore, the situations which life itself will offer, and the responses which life itself will demand.
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 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 ['91], using such examples as