In the blank space below, work as many of the following examples as possible in the time allowed. Work them in order as numbered, entering each answer in the "answer" column before commencing a new example. Do not work on any other paper.Answer1. The children in a certain school gave a Christmas party. One of the presents was a box of candy. In filling the boxes, one grade used 16 pounds of candy, another 17 pounds, a third 12 pounds, and a fourth 13 pounds. What did the candy cost at 26¢. a pound?2. A school in a certain city used 2516 pieces of chalk in 37 school days. Three new rooms were opened, each room holding 50 children, and the school was then found to use 84 sticks of chalk per day. How many more sticks of chalk were used per day than at first?3. Several boys went on a bicycle trip of 1500 miles. The first week they rode 374 miles, the second week 264 miles, the third 423 miles, the fourth 401 miles. They finished the trip the next week. How many miles did they ride the last week?4. Forty-five boys were hired to pick apples from 15 trees in an apple orchard. In 50 minutes each boy had picked 48 choice apples. If all the apples picked were packed away carefully in 8 boxes of equal size, how many apples were put in each box?5. In a certain school 216 children gave a sleigh-ride party. They rented 7 sleighs at a cost of $30.00 and paid $24.00 for the refreshments. The party travelled 15 miles in 2½ hours and had a very pleasant time. What was each child's share of the expense?6. A girl found, by careful counting, that there were 2400 letters on one page of her history, and only 2295 letters on a page of her reader. How many more letters had she read in one book than in the other if she had read 47 pages in each of the books?7. Each of 59 rooms in the schools of a certain city contributed 25 presents to a Christmas entertainment for poor children. The stores of the city gave 1986 other articles for presents. What was the total number of presents given away at the entertainment?8. Forty-eight children from a certain school paid 10¢. apiece to ride 7 miles on the cars to a woods. There in a few hours they gathered 2765 nuts. 605 of these were bad, but the rest were shared equally among the children. How many good nuts did each one get?Total
In the blank space below, work as many of the following examples as possible in the time allowed. Work them in order as numbered, entering each answer in the "answer" column before commencing a new example. Do not work on any other paper.
Answer1. The children in a certain school gave a Christmas party. One of the presents was a box of candy. In filling the boxes, one grade used 16 pounds of candy, another 17 pounds, a third 12 pounds, and a fourth 13 pounds. What did the candy cost at 26¢. a pound?2. A school in a certain city used 2516 pieces of chalk in 37 school days. Three new rooms were opened, each room holding 50 children, and the school was then found to use 84 sticks of chalk per day. How many more sticks of chalk were used per day than at first?3. Several boys went on a bicycle trip of 1500 miles. The first week they rode 374 miles, the second week 264 miles, the third 423 miles, the fourth 401 miles. They finished the trip the next week. How many miles did they ride the last week?4. Forty-five boys were hired to pick apples from 15 trees in an apple orchard. In 50 minutes each boy had picked 48 choice apples. If all the apples picked were packed away carefully in 8 boxes of equal size, how many apples were put in each box?5. In a certain school 216 children gave a sleigh-ride party. They rented 7 sleighs at a cost of $30.00 and paid $24.00 for the refreshments. The party travelled 15 miles in 2½ hours and had a very pleasant time. What was each child's share of the expense?6. A girl found, by careful counting, that there were 2400 letters on one page of her history, and only 2295 letters on a page of her reader. How many more letters had she read in one book than in the other if she had read 47 pages in each of the books?7. Each of 59 rooms in the schools of a certain city contributed 25 presents to a Christmas entertainment for poor children. The stores of the city gave 1986 other articles for presents. What was the total number of presents given away at the entertainment?8. Forty-eight children from a certain school paid 10¢. apiece to ride 7 miles on the cars to a woods. There in a few hours they gathered 2765 nuts. 605 of these were bad, but the rest were shared equally among the children. How many good nuts did each one get?Total
These proposed measures of ability to apply arithmetic illustrate very nicely the differences of opinion concerning what applied arithmetic and arithmetical reasoning should be. The thinker who emphasizes the fact that in life out of school the situation demanding quantitative treatment is usually real rather than described, will condemn a test all of whose constituents aredescribedproblems. Unless we are excessively hopeful concerning the transfer of ideas of method and procedure from one mental function to another we shall protest against the artificiality of No. 3 of the Stone series, and of the entire Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is a striking example of the mixture of ability to understand quantitative relations with the ability to understand words. Consider these five, for example, in comparison with the revised versions attached.[3]
1. The children of a school gave a sleigh-ride party. There were 9 sleighs, and each sleigh held 30 children. How many children were there in the party?Revision.If one sleigh holds 30 children, 9 sleighs hold .... children.2. Two school-girls played a number-game. The score of the girl that lost was 57 points and she was beaten by 16 points. What was the score of the girl that won?Revision.Mary and Nell played a game. Mary had a score of 57. Nell beat Mary by 16. Nell had a score of....3. A girl counted the automobiles that passed a school. The total was 60 in two hours. If the girl saw 27 pass the first hour how many did she see the second?Revision.In two hours a girl saw 60 automobiles. She saw 27 the first hour. She saw .... the second hour.4. On a playground there were five equal groups of children each playing a different game. If there were 75 children all together, how many were there in each group?Revision.75 pounds of salt just filled five boxes. The boxes were exactly alike. There were .... pounds in a box.5. A teacher weighed all the children in a certain grade. One girl weighed 70 pounds. Her older sister was 49 pounds heavier. How many pounds did the sister weigh?Revision.Mary weighs 70 lb. Jane weighs 49 pounds more than Mary. Jane weighs .... pounds.
1. The children of a school gave a sleigh-ride party. There were 9 sleighs, and each sleigh held 30 children. How many children were there in the party?
Revision.If one sleigh holds 30 children, 9 sleighs hold .... children.
2. Two school-girls played a number-game. The score of the girl that lost was 57 points and she was beaten by 16 points. What was the score of the girl that won?
Revision.Mary and Nell played a game. Mary had a score of 57. Nell beat Mary by 16. Nell had a score of....
3. A girl counted the automobiles that passed a school. The total was 60 in two hours. If the girl saw 27 pass the first hour how many did she see the second?
Revision.In two hours a girl saw 60 automobiles. She saw 27 the first hour. She saw .... the second hour.
4. On a playground there were five equal groups of children each playing a different game. If there were 75 children all together, how many were there in each group?
Revision.75 pounds of salt just filled five boxes. The boxes were exactly alike. There were .... pounds in a box.
5. A teacher weighed all the children in a certain grade. One girl weighed 70 pounds. Her older sister was 49 pounds heavier. How many pounds did the sister weigh?
Revision.Mary weighs 70 lb. Jane weighs 49 pounds more than Mary. Jane weighs .... pounds.
The distinction between a problem described as clearly and simply as possible and the same problem put awkwardly or in ill-known words or willfully obscured should be regarded; and as a rule measurements of ability to apply arithmetic should eschew all needless obscurity or purely linguistic difficulty. For example,
A boy bought a two-cent stamp. He gave the man in the store 10 cents. The right change was .... cents.
A boy bought a two-cent stamp. He gave the man in the store 10 cents. The right change was .... cents.
is better as a test than
If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in payment, what change should he be expected to receive in return?
If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in payment, what change should he be expected to receive in return?
The distinction between the description of abona fideproblem that a human being might be called on to solve out of school and the description of imaginary possibilities or puzzles should also be considered. Nos. 3 and 9 of Stone are bad because to frame the problems one must first know the answers, so that in reality there could never be any point in solving them. It is probably safe to say that nobody in the world ever did or ever will or ever should find the number of apples in a box by the task of No. 4 of the Courtis Test 8.
This attaches no blame to Dr. Stone or to Mr. Courtis. Until very recently we were all so used to the artificial problems of the traditional sort that we did not expect anything better; and so blind to the language demands of described problems that we did not see their very great influence. Courtis himself has been active in reform and has pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.
"Tests Nos. 6 and 8, the so-called reasoning tests, have proved the least satisfactory of the series. The judgments of various teachers and superintendents as to the inequalities of the units in any one test, and of the differences between the different editions of the same test, have proved the need of investigating these questions. Tests of adults in many lines of commercial work have yielded in many cases lower scores than those of the average eighth grade children. At the same time the scores of certain individuals of marked ability have been high, and there appears to be a general relation between ability in these tests and accuracy in the abstract work. The most significant facts, however, have been the difficulties experienced by teachers in attempting to remedy the defects in reasoning. It is certain that the tests measure abilities of value but the abilities are probably not what they seem to be. In an attempt to measure the value of different units, for instance, as many problems as possible were constructed based upon a single situation. Twenty-one varieties were secured by varying the relative form of the question and the relative position of the different phrases. One of these proved nineteen times as hard as another as measured by the number of mistakes made by the children; yet the cause of the difference was merely the changes in the phrasing. This and other facts of the same kind seem to show that Tests 6 and 8 measure mainly the ability to read."
The scientific measurement of the abilities and achievements concerned with applied arithmetic or problem-solving is thus a matter for the future. In the case of described problems a beginning has been made in the series which form a part of the National Intelligence Tests ['20], one of which is shown on page 49 f. In the case of problems with real situations, nothing in systematic form is yet available.
Systematic tests and scales, besides defining the abilities we are to establish and improve, are of very great service in measuring the status and improvement of individuals and of classes, and the effects of various methods of instruction and of study. They are thus helpful to pupils, teachers, supervisors, and scientific investigators; and are being more and more widely used every year. Information concerning the merits of the different tests, the procedure to follow in giving and scoring them, the age and grade standards to be used in interpreting results, and the like, is available in the manuals of Educational Measurement, such as Courtis,Manual of Instructions for Giving and Scoring the Courtis Standard Tests in the Three R's['14]; Starch,Educational Measurements['16]; Chapman and Rush,Scientific Measurement of Classroom Products['17]; Monroe, DeVoss, and Kelly,Educational Tests and Measurements['17]; Wilson and Hoke,How to Measure['20]; and McCall,How to Measure in Education['21].
National Intelligence Tests.Scale A. Form 1, Edition 1
Find the answers as quickly as you can.Write the answers on the dotted lines.Use the side of the page to figure on.
Begin here
1Five cents make 1 nickel. How many nickels make a dime?Answer......2John paid 5 dollars for a watch and 3 dollars for a chain. How many dollars did he pay for the watch and chain?Answer......3Nell is 13 years old. Mary is 9 years old. How much younger is Mary than Nell?Answer......4One quart of ice cream is enough for 5 persons. How many quarts of ice cream are needed for 25 persons?Answer......5John's grandmother is 86 years old. If she lives, in how many years will she be 100 years old?Answer......6If a man gets $2.50 a day, what will he be paid for six days' work?Answer......7How many inches are there in a foot and a half?Answer......8What is the cost of 12 cakes at 6 for 5 cents?Answer......9The uniforms for a baseball team of nine boys cost $2.50 each. The shoes cost $2 a pair. What was the total cost of uniforms and shoes for the nine?Answer......10A train that usually arrives at half-past ten was 17 minutes late. When did it arrive?Answer......11At 10¢ a yard, what is the cost of a piece 10½ ft. long?Answer......12A man earns $6 a day half the time, $4.50 a day one fourth of the time, and nothing on the remaining days for a total period of 40 days. What did he earn in all in the 40 days?Answer......13What per cent of $800 is 4% of $1000?Answer......14If 60 men need 1500 lb. flour per month, what is the requirement per man per day counting a month as 30 days?Answer......15A car goes at the rate of a mile a minute. A truck goes 20 miles an hour. How many times as far will the car go as the truck in 10 seconds?Answer......16The area of the base (inside measure) of a cylindrical tank is 90 square feet. How tall must it be to hold 100 cubic yards?Answer......
From National Intelligence Tests by National Research Council.Copyright, 1920, by The World Book Company, Yonkers-on-Hudson, New York.Used by permission of the publishers.
It would be a useful work for some one to try to analyze arithmetical learning into the unitary abilities which compose it, showing just what, in detail, the mind has to do in order to be prepared to pass a thorough test on the whole of arithmetic. These unitary abilities would make a very long list. Examination of a well-planned textbook will show that such an ability as multiplication is treated as a composite of the following: knowledge of the multiplications up to 9 × 9; ability to multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is not required and no zeros occur in the multiplicand; ability to multiply by 2, 3, ... 9, with carrying; the ability to handle zeros in the multiplicand; the ability to multiply with two-place numbers not ending in zero; the ability to handle zero in the multiplier as last number; the ability to multiply with three (or more)-place numbers not including a zero; the ability to multiply with three- and four-place numbers with zero in second or third, or second and third, as well as in last place; the ability to save time by annexing zeros; and so on and on through a long list of further abilities required to multiply with United States money, decimal fractions, common fractions, mixed numbers, and denominate numbers.
The units or 'steps' thus recognized by careful teaching would make a long list, but it is probable that a still more careful study of arithmetical ability as a hierarchy of mental habits or connections would greatly increase the list. Consider, for example, ordinary column addition. The majority of teachers probably treat this as a simple application of the knowledge of the additions to 9 + 9, plus understanding of 'carrying.' On the contrary there are at least seven processes or minor functions involved in two-place column addition, each of which is psychologically distinct and requires distinct educational treatment.
These are:—
A. Learning to keep one's place in the column as one adds.
B. Learning to keep in mind the result of each addition until the next number is added to it.
C. Learning to add a seen to a thought-of number.
D. Learning to neglect an empty space in the columns.
E. Learning to neglect 0s in the columns.
F. Learning the application of the combinations to higher decades may for the less gifted pupils involve as much time and labor as learning all the original addition tables. And even for the most gifted child the formation of the connection '8 and 7 = 15' probably never quite insures the presence of the connections '38 and 7 = 45' and '18 + 7 = 25.'
G. Learning to write the figure signifying units rather than the total sum of a column. In particular, learning to write 0 in the cases where the sum of the column is 10, 20, etc. Learning to 'carry' also involves in itself at least two distinct processes, by whatever way it is taught.
We find evidence of such specialization of functions in the results with such tests as Woody's. For example,2 + 5 + 1 = .... surely involves abilities in part different from
243—
because only 77 percent of children in grade 3 do the former correctly, whereas 95 percent of children in that grade do the latter correctly. In grade 2 the difference is even more marked. In the case of subtraction
44—
involves abilities different from those involved in
93—,
being much less often solved correctly in grades 2 and 4.
60—
is much harder than either of the above.
It may be said that these differences in difficulty are due to different amounts of practice. This is probably not true, but if it were, it would not change the argument; if the two abilities were identical, the practice of one would improve the other equally.
I shall not undertake here this task of listing and describing the elementary functions which constitute arithmetical learning, partly because what they are is not fully known, partly because in many cases a final ability may be constituted in several different ways whose descriptions become necessarily tedious, and partly because an adequate statement of what is known would far outrun the space limits of this chapter. Instead, I shall illustrate the results by some samples.
As a first sample, consider knowledge of the meaning of a fraction. Is the ability in question simply to understand that a fraction is a statement of the number of parts, each of a certain size, the upper number or numerator telling how many parts are taken and the lower number or denominator telling what fraction of unity each part is? And is the educational treatment required simply to describe and illustrate such a statement and have the pupils apply it to the recognition of fractions and the interpretation of each of them? And is the learning process (1) the formation of the notions of part, size of part, number of part, (2) relating the last two to the numbers in a fraction, and, as a necessary consequence, (3) applying these notions adequately whenever one encounters a fraction in operation?
Precisely this was the notion a few generations ago. The nature of fractions was taught as one principle, in one step, and the habits of dealing with fractions were supposed to be deduced from the general law of a fraction's nature. As a result the subject of fractions had to be long delayed, was studied at great cost of time and effort, and, even so, remained a mystery to all save gifted pupils. These gifted pupils probably of their own accord built up the ability piecemeal out of constituent insights and habits.
At all events, scientific teaching now does build up the total ability as a fusion or organization of lesser abilities. What these are will be seen best by examining the means taken to get them. (1) First comes the association of ½ of a pie, ½ of a cake, ½ of an apple, and such like with their concrete meanings so that a pupil can properly name a clearly designated half of an obvious unit like an orange, pear, or piece of chalk. The same degree of understandingof1⁄4,1⁄8,1⁄3,1⁄6, and1⁄5is secured. The pupil is taught that 1 pie = 21⁄2s, 31⁄3s, 41⁄4s, 51⁄5s, 61⁄6s, and 81⁄8s; similarly for 1 cake, 1 apple, and the like.
So far he understands1⁄xofyin the sense of certain simple parts of obviously unitaryys.
(2) Next comes the association with ½ of an inch, ½ of a foot, ½ of a glassful and other cases whereyis not so obviously a unitary object whose pieces still show their derivation from it. Similarly for1⁄4,1⁄3, etc.
(3) Next comes the association with1⁄2of a collection of eight pieces of candy,1⁄3of a dozen eggs,1⁄5of a squad of ten soldiers, etc., until1⁄2,1⁄3,1⁄4,1⁄5,1⁄6, and1⁄8are understood as names of certain parts of a collection of objects.
(4) Next comes the similar association when the nature of the collection is left undefined, the pupil responding to1⁄2of 6 is ...,1⁄4of 8 is ..., 2 is1⁄5of ...,1⁄3of 6 is ...,1⁄3of 9 is ..., 2 is1⁄3of ..., and the like.
Each of these abilities is justified in teaching by its intrinsic merits, irrespective of its later service in helping to constitute the general understanding of the meaning of a fraction. The habits thus formed in grades 3 or 4 are of constant service then and thereafter in and out of school.
(5) With these comes the use of1⁄5of 10, 15, 20, etc.,1⁄6of 12, 18, 42, etc., as a useful variety of drill on the division tables, valuable in itself, and a means of making the notion of a unit fraction more general by adding1⁄7and1⁄9to the scheme.
(6) Next comes the connection of3⁄4,2⁄5,3⁄5,4⁄5,2⁄3,1⁄6,5⁄6,3⁄8,5⁄8,7⁄8,3⁄10,7⁄10, and9⁄10, each with its meaning as a certain part of some conveniently divisible unit, and, (7) and (8), connections between these fractions and their meanings as parts of certain magnitudes (7) and collections (8) of convenientsize, and (9) connections between these fractions and their meanings when the nature of the magnitude or collection is unstated, as in4⁄5of 15 = ...,5⁄8of 32 = ....
(10) That the relation is general is shown by using it with numbers requiring written division and multiplication, such as7⁄8of 1736 = ..., and with United States money.
Elements (6) to (10) again are useful even if the pupil never goes farther in arithmetic. One of the commonest uses of fractions is in calculating the cost of fractions of yards of cloth, and fractions of pounds of meat, cheese, etc.
The next step (11) is to understand to some extent the principle that the value of any of these fractions is unaltered by multiplying or dividing the numerator and denominator by the same number. The drills in expressing fractions in lower and higher terms which accomplish this are paralleled by (12) and (13) simple exercises in adding and subtracting fractions to show that fractions are quantities that can be operated on like any quantities, and by (14) simple work with mixed numbers (addition and subtraction and reductions), and (15) improper fractions. All that is done with improper fractions is (a) to have the pupil use a few of them as he would any fractions and (b) to note their equivalent mixed numbers. In (12), (13), and (14) only fractions of the same denominators are added or subtracted, and in (12) (13), (14), and (15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are used. As hitherto, the work of (11) to (15) is useful in and of itself. (16) Definitions are given of the following type:—
Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers.
Numbers like7⁄8,1⁄5,2⁄3,3⁄4,11⁄8,7⁄6,1⁄3,4⁄3,1⁄8,1⁄6are called fractions.
Numbers like 5¼, 73⁄8, 9½, 164⁄5, 3157⁄8, 11⁄3, 12⁄3are called mixed numbers.
(17) The terms numerator and denominator are connected with the upper and lower numbers composing a fraction.
Building this somewhat elaborate series of minor abilities seems to be a very roundabout way of getting knowledge of the meaning of a fraction, and is, if we take no account of what is got along with this knowledge. Taking account of the intrinsically useful habits that are built up, one might retort that the pupil gets his knowledge of the meaning of a fraction at zero cost.
Consider next the knowledge of the subtraction and division 'Tables.' The usual treatment presupposes that learning them consists of forming independently the bonds:—
3 − 1 = 24 ÷ 2 = 23 − 2 = 16 ÷ 2 = 34 − 1 = 36 ÷ 3 = 2......18 − 9 = 981 ÷ 9 = 9
In fact, however, these 126 bonds are not formed independently. Except perhaps in the case of the dullest twentieth of pupils, they are somewhat facilitated by the already learned additions and multiplications. And by proper arrangement of the learning they may be enormously facilitated thereby. Indeed, we may replace the independent memorizing of these facts by a set of instructiveexercises wherein the pupil derives the subtractions from the corresponding additions by simple acts of reasoning or selective thinking. As soon as the additions giving sums of 9 or less are learned, let the pupil attack an exercise like the following:—
Write the missing numbers:—
ABCD3 and ... are 5.5 and ... are 8.4 and ... are 5.4 and ... are 8.3 and ... are 9.3 and ... are 6.5 and ... are 6.1 and ... are 7.4 and ... are 7.4 and ... are 9.6 and ... are 9.6 and ... are 7.5 and ... are 7.2 and ... = 6.1 and ... are 8.8 and ... are 9.6 and ... are 8.5 and ... = 9.3 and ... are 7.3 + ... are 4.4 and ... are 6.2 and ... = 7.1 + ... are 3.7 + ... are 8.2 and ... are 5.3 and ... = 8.1 + ... are 5.4 + ... are 9.2 and ... = 8.1 and ... = 4.4 + ... are 8.2 + ... are 3.3 and ... = 6.2 and ... = 4.7 + ... are 9.1 + ... are 9.6 and ... = 9.3 and ... = 8.2 + ... = 4.3 + ... = 6.4 and ... = 6.6 and ... = 7.3 + ... = 8.5 + ... = 9.4 and ... = 7.2 and ... = 5.4 + ... = 5.1 + ... = 3.
The task for reasoning is only to try, one after another, numbers that seem promising and to select the right one when found. With a little stimulus and direction children can thus derive the subtractions up to those with 9 as the larger number. Let them then be taught to do the same with the printed forms:—
Subtract
978586356224etc.——————
and 9 − 7 = ..., 9 − 5 = ..., 7 − 5 = ..., etc.
In the case of the divisions, suppose that the pupil has learned his first table and gained surety in such exercises as:—
4 5s = ....6 × 5 = ....9 nickels = .... cents.8 5s = ....4 × 5 = ....6 " = .... "3 5s = ....2 × 5 = ....5 " = .... "7 5s = ....9 × 5 = ....7 " = .... "
If one ball costs 5 cents,two balls cost .... cents,three balls cost .... cents, etc.
He may then be set at once to work at the answers to exercises like the following:—
Write the answers and the missing numbers:—
ABCD.... 5s = 1540 = .... 5s.... × 5 = 2520 cents = .... nickels..... 5s = 2020 = .... 5s.... × 5 = 5030 cents = .... nickels..... 5s = 4015 = .... 5s.... × 5 = 3515 cents = .... nickels..... 5s = 2545 = .... 5s.... × 5 = 1040 cents = .... nickels..... 5s = 3050 = .... 5s.... × 5 = 40.... 5s = 3525 = .... 5s.... × 5 = 45
EFor 5 cents you can buy 1 small loaf of bread.For 10 cents you can buy 2 small loaves of bread.For 25 cents you can buy .... small loaves of bread.For 45 cents you can buy .... small loaves of bread.For 35 cents you can buy .... small loaves of bread.
F5 cents pays 1 car fare.15 cents pays .... car fares.10 cents pays .... car fares.20 cents pays .... car fares.
GHow many 5 cent balls can you buy with 30 cents? ....How many 5 cent balls can you buy with 35 cents? ....How many 5 cent balls can you buy with 25 cents? ....How many 5 cent balls can you buy with 15 cents? ....
In the case of the meaning of a fraction, the ability, and so the learning, is much more elaborate than commonpractice has assumed; in the case of the subtraction and division tables the learning is much less so. In neither case is the learning either mere memorizing of facts or the mere understanding of a principlein abstractofollowed by its application to concrete cases. It is (and this we shall find true of almost all efficient learning in arithmetic) the formation of connections and their use in such an order that each helps the others to the maximum degree, and so that each will do the maximum amount for arithmetical abilities other than the one specially concerned, and for the general competence of the learner.
As another instructive topic in the constitution of arithmetical abilities, we may take the case of the reasoning involved in understanding the manipulations of figures in two (or more)-place addition and subtraction, multiplication and division involving a two (or more)-place number, and the manipulations of decimals in all four operations. The psychology of these is of special interest and importance. For there are two opposite explanations possible here, leading to two opposite theories of teaching.
The common explanation is that these methods of manipulation, if understood at all, are understood as deductions from the properties of our system of decimal notation. The other is that they are understood partly as inductions from the experience that they always give the right answer. The first explanation leads to the common preliminary deductive explanations of the textbooks. The other leads to explanations by verification;e.g., of addition by counting, of subtraction by addition, of multiplication by addition, of division by multiplication. Samples of these two sorts of explanation are given below.
SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATIONMultiplicationis the process of taking one number as many times as there are units in another number.TheProductis the result of the multiplication.TheMultiplicandis the number to be taken.TheMultiplieris the number denoting how many times the multiplicand is to be taken.The multiplier and multiplicand are theFactors.Multiply 623 by 3OPERATIONMultiplicand623Multiplier3Product1869Explanation.—For convenience we write the multiplier under the multiplicand, and begin with units to multiply. 3 times 3 units are 9 units. We write the nine units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION1.The children of the third grade are to have a picnic. 32 are going. How many sandwiches will they need if each of the 32 children has four sandwiches?324Here is a quick way to find out:—Think "4 × 2," write 8 under the 2 in the ones column.Think "4 × 3," write 12 under the 3 in the tens column.2.How many bananas will they need if each of the 32 children has two bananas? 32 × 2 or 2 × 32 will give the answer.3.How many little cakes will they need if each child has three cakes? 32 × 3 or 3 × 32 will give the answer.3233 × 2 = .... Where do you write the 6?3 × 3 = .... Where do you write the 9?4.Prove that 128, 64, and 96 are right by adding four 32s, two 32s, and three 32s.323232323232323232MultiplicationYoumultiplywhen you find the answers to questions likeHow many are 9 × 3?How many are 3 × 32?How many are 8 × 5?How many are 4 × 42?1.Read these lines. Say the right numbers where the dots are:If youadd3 to 32, you have .... 35 is thesum.If yousubtract3 from 32, the result is .... 29 is thedifferenceorremainder.If youmultiply3 by 32 or 32 by 3, you have .... 96 is theproduct.Find the products. Check your answers to the first line by adding.2.3.4.5.6.7.8.9.41334244534334243242322210.11.12.13.14.15.16.43523223415114333324217.2133Write the 9 in the ones column.Write the 6 in the hundreds column.Write the 3 in the tens column.Check your answer by adding.Add21321321318.19.20.21.22.23.24.2143124322311323142432323322SHORT DIVISION: DEDUCTIVE EXPLANATIONDivide 1825 by 4Divisor 4 |1825Dividend456¼QuotientExplanation.—For convenience we write the divisor at the left of the dividend, and the quotient below it, and begin at the left to divide.4is not contained in 1 thousand any thousand times, therefore the quotient contains no unit of any order higher than hundreds. Consequently we find how many times 4 is contained in the hundreds of the dividend. 1 thousand and 8 hundreds are 18 hundreds. 4 is contained in 18 hundreds 4 hundred times and 2 hundreds remaining. We write the 4 hundreds in the quotient. The 2 hundreds we consider as united with the 2 tens, making 22 tens. 4 is contained in 22 tens 5 tens times, and 2 tens remaining. We write the 5 tens in the quotient, and the remaining 2 tens we consider as united with the 5 units, making 25 units. 4 is contained in 25 units 6 units times and 1 unit remaining. We write the 6 units in the quotient and indicate the division of the remainder, 1 unit, by the divisor 4.Therefore the quotient of 1825 divided by 4 is 456¼, or 456 and 1 remainder.SHORT DIVISION: INDUCTIVE EXPLANATIONDividing Large Numbers1.Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent bag of marbles. There are 128 marbles in it. How many should each boy have, if they divide the marbles equally among the four boys?4 |128Think "12 = three 4s." Write the 3 over the 2 in the tens column.Think "8 = two 4s." Write the 2 over the 8 in the ones column.32 is right, because 4 × 32 = 128.2.Mary, Nell, and Alice are going to buy a book as a present for their Sunday-school teacher. The present costs 69 cents. How much should each girl pay, if they divide the cost equally among the three girls?3 |69Think "6 = .... 3s." Write the 2 over the 6 in the tens column.Think "9 = .... 3s." Write the 3 over the 9 in the ones column.23 is right, for 3 × 23 = 69.3.Divide the cost of a 96-cent present equally among three girls. How much should each girl pay? girls. How much should each girl pay? 3 |964.Divide the cost of an 84-cent present equally among 4 girls. How much should each girl pay?5.Learn this: (Read ÷ as "divided by.")12 + 4 = 16.16 is the sum.12 − 4 = 8.8 is the difference or remainder.12 × 4 = 48.48 is the product.12 ÷ 4 = 3.3 is the quotient.6.Find the quotients. Check your answers by multiplying.3 |992 |865 |1556 |2464 |1683 |219[Uneven division is taught by the same general plan, extended.]LONG DIVISION: DEDUCTIVE EXPLANATIONTo Divide by Long Division1. Let it be required to divide 34531 by 15.OperationDivisorDivided15 ) 34531 (3045453130123021⁄15QuotientRemainderFor convenience we write the divisor at the left and the quotient at the right of the dividend, and begin to divide as in Short Division.15 is contained in 3 ten-thousands 0 ten-thousands times; therefore, there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is contained in 34 thousands 2 thousands times; we write the 2 thousands in the quotient. 15 × 2 thousands = 30 thousands, which, subtracted from 34 thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we have 45 hundreds.15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the quotient. 15 × 3 hundreds = 45 hundreds, which subtracted from 45 hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to the three tens, which equal 30 units, the 1 unit, we have 31 units.15 in 31 units 2 units times; we write the 2 units in the quotient. 15 × 2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a remainder. Indicating the division of the 1 unit, we annex the fractional expression,1⁄15unit, to the integral part of the quotient.Therefore, 34531 divided by 15 is equal to 23021⁄15.[B. Greenleaf,Practical Arithmetic, '73, p. 49.]LONG DIVISION: INDUCTIVE EXPLANATIONDividing by Large Numbers1. Just before Christmas Frank's father sent 360 oranges to be divided among the children in Frank's class. There are 29 children. How many oranges should each child receive? How many oranges will be left over?Here is the best way to find out:1229|36029705812and 12 remainderThink how many 29s there are in 36. 1 is right.Write 1 over the 6 of 36. Multiply 29 by 1.Write the 29 under the 36. Subtract 29 from 36.Write the 0 of 360 after the 7.Think how many 29s there are in 70. 2 is right.Write 2 over the 0 of 360. Multiply 29 by 2.Write the 58 under 70. Subtract 58 from 70.There is 12 remainder.Each child gets 12 oranges, and there are 12 left over. This is right, for 12 multiplied by 29 = 348, and 348 + 12 = 360.8.31 |99,587In No. 8, keep on dividing by 31 until you have used the 5, the 8, and the 7, and have four figures in the quotient.9.22 |25310.22 |289511.21 |889112.22 |29013.32 |16,368Check your results for 9, 10, 11, 12, and 13.1. The boys and girls of the Welfare Club plan to earn money to buy a victrola. There are 23 boys and girls. They can get a good second-hand victrola for $5.75. How much must each earn if they divide the cost equally?Here is the best way to find out:$.2523|$5.7546115115Think how many 23s there are in 57. 2 is right.Write 2 over the 7 of 57. Multiply 23 by 2.Write 46 under 57 and subtract. Write the 5 of 575 after the 11.Think how many 23s there are in 115. 5 is right.Write 5 over the 5 of 575. Multiply 23 by 5.Write the 115 under the 115 that is there and subtract.There is no remainder.Put $ and the decimal point where they belong.Each child must earn 25 cents. This is right, for $.25 multiplied by 23 = $5.75.2. Divide $71.76 equally among 23 persons. How much is each person's share?3. Check your result for No. 2 by multiplying the quotient by the divisor.Find the quotients. Check each quotient by multiplying it by the divisor.4.23 |$99.135.25 |$18.506.21 |$129.157.13 |$29.258.32 |$73.921 bushel = 32 qt.9. How many bushels are there in 288 qt.?10. In 192 qt.?11. In 416 qt.?
SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION
Multiplicationis the process of taking one number as many times as there are units in another number.
TheProductis the result of the multiplication.
TheMultiplicandis the number to be taken.
TheMultiplieris the number denoting how many times the multiplicand is to be taken.
The multiplier and multiplicand are theFactors.
Multiply 623 by 3OPERATIONMultiplicand623Multiplier3Product1869
Explanation.—For convenience we write the multiplier under the multiplicand, and begin with units to multiply. 3 times 3 units are 9 units. We write the nine units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.
SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION
1.The children of the third grade are to have a picnic. 32 are going. How many sandwiches will they need if each of the 32 children has four sandwiches?
2.How many bananas will they need if each of the 32 children has two bananas? 32 × 2 or 2 × 32 will give the answer.
3.How many little cakes will they need if each child has three cakes? 32 × 3 or 3 × 32 will give the answer.
4.Prove that 128, 64, and 96 are right by adding four 32s, two 32s, and three 32s.
Multiplication
Youmultiplywhen you find the answers to questions like
How many are 9 × 3?How many are 3 × 32?How many are 8 × 5?How many are 4 × 42?
1.Read these lines. Say the right numbers where the dots are:If youadd3 to 32, you have .... 35 is thesum.If yousubtract3 from 32, the result is .... 29 is thedifferenceorremainder.If youmultiply3 by 32 or 32 by 3, you have .... 96 is theproduct.
Find the products. Check your answers to the first line by adding.
2.3.4.5.6.7.8.9.41334244534334243242322210.11.12.13.14.15.16.435232234151143333242
18.19.20.21.22.23.24.2143124322311323142432323322
SHORT DIVISION: DEDUCTIVE EXPLANATION
Divide 1825 by 4
Divisor 4 |1825Dividend456¼Quotient
Explanation.—For convenience we write the divisor at the left of the dividend, and the quotient below it, and begin at the left to divide.4is not contained in 1 thousand any thousand times, therefore the quotient contains no unit of any order higher than hundreds. Consequently we find how many times 4 is contained in the hundreds of the dividend. 1 thousand and 8 hundreds are 18 hundreds. 4 is contained in 18 hundreds 4 hundred times and 2 hundreds remaining. We write the 4 hundreds in the quotient. The 2 hundreds we consider as united with the 2 tens, making 22 tens. 4 is contained in 22 tens 5 tens times, and 2 tens remaining. We write the 5 tens in the quotient, and the remaining 2 tens we consider as united with the 5 units, making 25 units. 4 is contained in 25 units 6 units times and 1 unit remaining. We write the 6 units in the quotient and indicate the division of the remainder, 1 unit, by the divisor 4.
Therefore the quotient of 1825 divided by 4 is 456¼, or 456 and 1 remainder.
SHORT DIVISION: INDUCTIVE EXPLANATION
Dividing Large Numbers
1.Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent bag of marbles. There are 128 marbles in it. How many should each boy have, if they divide the marbles equally among the four boys?
4 |128
Think "12 = three 4s." Write the 3 over the 2 in the tens column.
Think "8 = two 4s." Write the 2 over the 8 in the ones column.
32 is right, because 4 × 32 = 128.
2.Mary, Nell, and Alice are going to buy a book as a present for their Sunday-school teacher. The present costs 69 cents. How much should each girl pay, if they divide the cost equally among the three girls?
3 |69
Think "6 = .... 3s." Write the 2 over the 6 in the tens column.
Think "9 = .... 3s." Write the 3 over the 9 in the ones column.
23 is right, for 3 × 23 = 69.
3.Divide the cost of a 96-cent present equally among three girls. How much should each girl pay? girls. How much should each girl pay? 3 |96
4.Divide the cost of an 84-cent present equally among 4 girls. How much should each girl pay?
5.Learn this: (Read ÷ as "divided by.")
12 + 4 = 16.16 is the sum.12 − 4 = 8.8 is the difference or remainder.12 × 4 = 48.48 is the product.12 ÷ 4 = 3.3 is the quotient.
6.Find the quotients. Check your answers by multiplying.
3 |992 |865 |1556 |2464 |1683 |219
[Uneven division is taught by the same general plan, extended.]
LONG DIVISION: DEDUCTIVE EXPLANATION
To Divide by Long Division
1. Let it be required to divide 34531 by 15.
Operation
DivisorDivided15 ) 34531 (3045453130123021⁄15QuotientRemainder
For convenience we write the divisor at the left and the quotient at the right of the dividend, and begin to divide as in Short Division.
15 is contained in 3 ten-thousands 0 ten-thousands times; therefore, there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is contained in 34 thousands 2 thousands times; we write the 2 thousands in the quotient. 15 × 2 thousands = 30 thousands, which, subtracted from 34 thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we have 45 hundreds.
15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the quotient. 15 × 3 hundreds = 45 hundreds, which subtracted from 45 hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.
15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to the three tens, which equal 30 units, the 1 unit, we have 31 units.
15 in 31 units 2 units times; we write the 2 units in the quotient. 15 × 2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a remainder. Indicating the division of the 1 unit, we annex the fractional expression,1⁄15unit, to the integral part of the quotient.
Therefore, 34531 divided by 15 is equal to 23021⁄15.
[B. Greenleaf,Practical Arithmetic, '73, p. 49.]
LONG DIVISION: INDUCTIVE EXPLANATION
Dividing by Large Numbers
1. Just before Christmas Frank's father sent 360 oranges to be divided among the children in Frank's class. There are 29 children. How many oranges should each child receive? How many oranges will be left over?
Here is the best way to find out:
1229|36029705812and 12 remainderThink how many 29s there are in 36. 1 is right.Write 1 over the 6 of 36. Multiply 29 by 1.Write the 29 under the 36. Subtract 29 from 36.Write the 0 of 360 after the 7.Think how many 29s there are in 70. 2 is right.Write 2 over the 0 of 360. Multiply 29 by 2.Write the 58 under 70. Subtract 58 from 70.There is 12 remainder.Each child gets 12 oranges, and there are 12 left over. This is right, for 12 multiplied by 29 = 348, and 348 + 12 = 360.
8.31 |99,587In No. 8, keep on dividing by 31 until you have used the 5, the 8, and the 7, and have four figures in the quotient.
9.22 |25310.22 |289511.21 |889112.22 |29013.32 |16,368
Check your results for 9, 10, 11, 12, and 13.
1. The boys and girls of the Welfare Club plan to earn money to buy a victrola. There are 23 boys and girls. They can get a good second-hand victrola for $5.75. How much must each earn if they divide the cost equally?
Here is the best way to find out:
$.2523|$5.7546115115Think how many 23s there are in 57. 2 is right.Write 2 over the 7 of 57. Multiply 23 by 2.Write 46 under 57 and subtract. Write the 5 of 575 after the 11.Think how many 23s there are in 115. 5 is right.Write 5 over the 5 of 575. Multiply 23 by 5.Write the 115 under the 115 that is there and subtract.There is no remainder.Put $ and the decimal point where they belong.Each child must earn 25 cents. This is right, for $.25 multiplied by 23 = $5.75.
2. Divide $71.76 equally among 23 persons. How much is each person's share?
3. Check your result for No. 2 by multiplying the quotient by the divisor.
Find the quotients. Check each quotient by multiplying it by the divisor.
4.23 |$99.135.25 |$18.506.21 |$129.157.13 |$29.258.32 |$73.92
1 bushel = 32 qt.
9. How many bushels are there in 288 qt.?10. In 192 qt.?11. In 416 qt.?
Crucial experiments are lacking, but there are several lines of well-attested evidence. First of all, there can be no doubt that the great majority of pupils learn these manipulations at the start from the placing of units under units, tens under tens, etc., in adding, to the placing of the decimal point in division with decimals, by imitation and blindfollowing of specific instructions, and that a very large proportion of the pupils do not to the end, that is to the fifth school-year, understand them as necessary deductions from decimal notation. It also seems probable that this proportion would not be much reduced no matter how ingeniously and carefully the deductions were explained by textbooks and teachers. Evidence of this fact will appear abundantly to any one who will observe schoolroom life. It also appears in the fact that after the properties of the decimal notation have been thus used again and again;e.g., for deducing 'carrying' in addition, 'borrowing' in subtraction, 'carrying' in multiplication, the value of the digits in the partial product, the value of each remainder in short division, the value of the quotient figures in division, the addition, subtraction, multiplication, and division of United States money, and the placing of the decimal point in multiplication, no competent teacher dares to rely upon the pupil, even though he now has four or more years' experience with decimal notation, to deduce the placing of the decimal point in division with decimals. It may be an illusion, but one seems to sense in the better textbooks a recognition of the futility of the attempt to secure deductive derivations of those manipulations. I refer to the brevity of the explanations and their insertion in such a form that they will influence the pupils' thinking as little as possible. At any rate the fact is sure that most pupils do not learn the manipulations by deductive reasoning, or understand them as necessary consequences of abstract principles.
It is a common opinion that the only alternative is knowing them by rote. This, of course, is one common alternative, but the other explanation suggests that understanding the manipulations by inductive reasoning from their results is another and an important alternative. The manipulations of 'long' multiplication, for instance, learned by imitation or mechanical drill, are found to give for 25 ×Aa result about twice as large as for 13 ×A, for 38 or 39 ×Aa result about three times as large; for 115 ×Aa result about ten times as large as for 11 ×A. With even the very dull pupils the procedure is verified at least to the extent that it gives a result which the scientific expert in the case—the teacher—calls right. With even the very bright pupils, who can appreciate the relation of the procedure to decimal notation, this relation may be used not as the sole deduction of the procedure beforehand, but as one partial means of verifying it afterward. Or there may be the condition of half-appreciation of the relation in which the pupil uses knowledge of the decimal notation to convince himself that the proceduredoes, but not that itmustgive the right answer, the answer being 'right' because the teacher, the answer-list, and collateral evidence assure him of it.
I have taken the manipulation of the partial products as an illustration because it is one of the least favored cases for the explanation I am presenting. If we take the first case where a manipulation may be deduced from decimal notation, known merely by rote, or verified inductively, namely, the addition of two-place numbers, it seems sure that the mental processes just described are almost the universal rule.
Surely in our schools at present children add the 3 of 23 to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten because they see the teacher do so and are told to do so. They are protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but because they do not know how to add 8 and 5, because they have been taught the habit of adding figures that stand one above the other, or with a + between them; and because they areshown or told what they are to do. They are protected from adding 3 + 5 and 2 + 3, again, by no deductive reasoning but for the second and third reasons just given. In nine cases out of ten they do not even think of the possibility of adding in any other way than the '3 + 3, 2 + 5' way, much less do they select that way on account of the facts that 53 = 50 + 3 and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that (a+b) + (c+d) = (a+c) + (b+d)!
Just as surely all but the very dullest twentieth or so of children come in the end to something more than rote knowledge,—tounderstand, toknowthat the procedure in question is right.
Whether they knowwhy76 is right depends upon what is meant bywhy. If it means that 76 is the result which competent people agree upon, they do. If it means that 76 is the result which would come from accurate counting they perhaps know why as well as they would have, had they been given full explanations of the relation of the procedure in two-place addition to decimal notation. Ifwhymeans because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and (a+b) + (c+d) = (a+c) + (b+d), they do not. Nor, I am tempted to add, would most of them by any sort of teaching whatever.
I conclude, therefore, that school children may and do reason about and understand the manipulations of numbers in this inductive, verifying way without being able to, or at least without, under present conditions, finding it profitable to derive them deductively. I believe, in fact, that pure arithmeticas it is learned and knownis largely aninductive science. At one extreme is a minority to whom it is a series of deductions from principles; at the other extreme is a minority to whom it is a series of blind habits; between the two is the great majority, representing every gradation but centering about the type of the inductive thinker.
When the analysis of the mental functions involved in arithmetical learning is made thorough it turns into the question, 'What are the elementary bonds or connections that constitute these functions?' and when the problem of teaching arithmetic is regarded, as it should be in the light of present psychology, as a problem in the development of a hierarchy of intellectual habits, it becomes in large measure a problem of the choice of the bonds to be formed and of the discovery of the best order in which to form them and the best means of forming each in that order.
The importance of habit-formation or connection-making has been grossly underestimated by the majority of teachers and writers of textbooks. For, in the first place, mastery by deductive reasoning of such matters as 'carrying' in addition, 'borrowing' in subtraction, the value of the digits in the partial products in multiplication, the manipulation of the figures in division, the placing of the decimal point after multiplication or division with decimals, or the manipulation of the figures in the multiplication and division offractions, is impossible or extremely unlikely in the case of children of the ages and experience in question. They do not as a rule deduce the method of manipulation from their knowledge of decimal notation. Rather they learn about decimal notation by carrying, borrowing, writing the last figure of each partial product under the multiplier which gives that product, etc. They learn the method of manipulating numbers by seeing them employed, and by more or less blindly acquiring them as associative habits.
In the second place, we, who have already formed and long used the right habits and are thereby protected against the casual misleadings of unfortunate mental connections, can hardly realize the force of mere association. When a child writes sixteen as 61, or finds 428 as the sum of
15191618
or gives 642 as an answer to 27 × 36, or says that 4 divided by ¼ = 1, we are tempted to consider him mentally perverse, forgetting or perhaps never having understood that he goes wrong for exactly the same general reason that we go right; namely, the general law of habit-formation. If we study the cases of 61 for 16, we shall find them occurring in the work of pupils who after having been drilled in writing 26, 36, 46, 62, 63, and so on, in which the order of the six in writing is the same as it is in speech, return to writing the 'teen numbers. If our language said onety-one for eleven and onety-six for sixteen, we should probably never find such errors except as 'lapses' or as the results of misperception or lack of memory. They would then be more frequentbeforethe 20s, 30s, etc., were learned.
If pupils are given much drill on written single column addition involving the higher decades (each time writingthe two-figure sum), they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5 is reached; and it should not surprise us if the pupil still occasionally writes the two-figure sum for the first column though a second column is to be added also. On the contrary, unless some counter force influences him, he is absolutely sure to make this mistake.
The last mistake quoted (4 ÷ ¼ = 1) is interesting because here we have possibly one of the cases where deduction from psychology alone can give constructive aid to teaching. Multiplication and division by fractions have been notorious for their difficulty. The former is now alleviated by usingofinstead of × until the new habit is fixed. The latter is still approached with elaborate caution and with various means of showing why one must 'invert and multiply' or 'multiply by the reciprocal.'
But in the author's opinion it seems clear that the difficulty in multiplying and dividing by a fraction was not that children felt any logical objections to canceling or inverting. I fancy that the majority of them would cheerfully invert any fraction three times over or cancel numbers at random in a column if they were shown how to do so. But if you are a youngster inexperienced in numerical abstractions and if you have haddivideconnected with 'make smaller' three thousand times and never once connected with 'make bigger,' you are sure to be somewhat impelled to make the number smaller the three thousand and first time you are asked to divide it. Some of my readers will probably confess that even now they feel a slight irritation or doubt in saying or writing that16⁄1÷1⁄8= 128.
The habits that have been confirmed by every multiplication and division by integers are, in this particular of 'the ratio of result to number operated upon,' directly opposed tothe formation of the habits required with fractions. And that is, I believe, the main cause of the difficulty. Its treatment then becomes easy, as will be shown later.
These illustrations could be added to almost indefinitely, especially in the case of the responses made to the so-called 'catch' problems. The fact is that the learner rarely can, and almost never does, survey and analyze an arithmetical situation and justify what he is going to do by articulate deductions from principles. He usually feels the situation more or less vaguely and responds to it as he has responded to it or some situation like it in the past. Arithmetic is to him not a logical doctrine which he applies to various special instances, but a set of rather specialized habits of behavior toward certain sorts of quantities and relations. And in so far as he does come to know the doctrine it is chiefly by doing the will of the master. This is true even with the clearest expositions, the wisest use of objective aids, and full encouragement of originality on the pupil's part.
Lest the last few paragraphs be misunderstood, I hasten to add that the psychologists of to-day do not wish to make the learning of arithmetic a mere matter of acquiring thousands of disconnected habits, nor to decrease by one jot the pupil's genuine comprehension of its general truths. They wish him to reason not less than he has in the past, but more. They find, however, that you do not secure reasoning in a pupil by demanding it, and that his learning of a general truth without the proper development of organized habits back of it is likely to be, not a rational learning of that general truth, but only a mechanical memorizing of a verbal statement of it. They have come to know that reasoning is not a magic force working in independence of ordinary habits of thought, but an organization and coöperation of those very habits on a higher level.
The older pedagogy of arithmetic stated a general law or truth or principle, ordered the pupil to learn it, and gave him tasks to do which he could not do profitably unless he understood the principle. It left him to build up himself the particular habits needed to give him understanding and mastery of the principle. The newer pedagogy is careful to help him build up these connections or bonds ahead of and along with the general truth or principle, so that he can understand it better. The older pedagogy commanded the pupil to reason and let him suffer the penalty of small profit from the work if he did not. The newer provides instructive experiences with numbers which will stimulate the pupil to reason so far as he has the capacity, but will still be profitable to him in concrete knowledge and skill, even if he lacks the ability to develop the experiences into a general understanding of the principles of numbers. The newer pedagogy secures more reasoning in reality by not pretending to secure so much.
The newer pedagogy of arithmetic, then, scrutinizes every element of knowledge, every connection made in the mind of the learner, so as to choose those which provide the most instructive experiences, those which will grow together into an orderly, rational system of thinking about numbers and quantitative facts. It is not enough for a problem to be a test of understanding of a principle; it must also be helpful in and of itself. It is not enough for an example to be a case of some rule; it must help review and consolidate habits already acquired or lead up to and facilitate habits to be acquired. Every detail of the pupil's work must do the maximum service in arithmetical learning.
As hitherto, I shall not try to list completely the elementary bonds that the course of study in arithmetic should provide for. The best means of preparing the student of this topic for sound criticism and helpful invention is to let him examine representative cases of bonds now often neglected which should be formed and representative cases of useless, or even harmful, bonds now often formed at considerable waste of time and effort.
(1)Numbers as measures of continuous quantities.—The numbers one, two, three, 1, 2, 3, etc., should be connected soon after the beginning of arithmetic each with the appropriate amount of some continuous quantity like length or volume or weight, as well as with the appropriate sized collection of apples, counters, blocks, and the like. Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two inches, three inches, etc.; weights should be lifted and called one pound, two pounds, etc.; things should be measured in glassfuls, handfuls, pints, and quarts. Otherwise the pupil is likely to limit the meaning of, say,fourto four sensibly discrete things and to have difficulty in multiplication and division. Measuring, or counting by insensibly marked off repetitions of a unit, binds each number name to its meaning as ——times whatever 1 is, more surely than mere counting of the units in a collection can, and should reënforce the latter.
(2)Additions in the higher decades.—In the case of all save the very gifted children, the additions with higher decades—that is, the bonds, 16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the like—need to be specifically practiced until the tendency becomes generalized. 'Counting' by 2s beginning with 1, and with 2,counting by 3s beginning with 1, with 2, and with 3, counting by 4s beginning with 1, with 2, with 3, and with 4, and so on, make easy beginnings in the formation of the decade connections. Practice with isolated bonds should soon be added to get freer use of the bonds. The work of column addition should be checked for accuracy so that a pupil will continually get beneficial practice rather than 'practice in error.'
(3)The uneven divisions.—The quotients with remainders for the divisions of every number to 19 by 2, every number to 29 by 3, every number to 39 by 4, and so on should be taught as well as the even divisions. A table like the following will be found a convenient means of making these connections:—