Fig. 25a.Fig.25a.—Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.
Fig.25a.—Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.
Fig. 25b.Fig.25b.—Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.
Fig.25b.—Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.
Write the products:—
A.3 4s =B.5 7s =C.9 2s =5 2s =8 3s =4 4s =7 2s =4 2s =2 7s =1 6 =4 5s =6 4s =1 3 =4 7s =5 5s =3 7s =5 9s =3 6s =4 1s =7 5s =3 2s =6 8s =7 1s =3 9s =9 8s =6 3s =5 1s =4 3s =4 9s =8 6s =2 4s =3 5s =8 4s =2 2s =9 6s =8 5s =8 7s =2 5s =7 9s =5 8s =5 4s =6 2s =7 6s =8 2s =7 4s =7 3s =8 9s =9 3s =D.4 20s =E.9 60s =F.40 × 2 = 804 200s =9 600s =20 × 2 =6 30s =5 30s =30 × 2 =6 300s =5 300s =40 × 2 =7 × 50 =8 × 20 =20 × 3 =7 × 500 =8 × 200 =30 × 3 =3 × 40 =2 × 70 =300 × 3 = 9003 × 400 =2 × 700 =300 × 2 =
A.3 4s =B.5 7s =C.9 2s =5 2s =8 3s =4 4s =7 2s =4 2s =2 7s =1 6 =4 5s =6 4s =1 3 =4 7s =5 5s =3 7s =5 9s =3 6s =4 1s =7 5s =3 2s =6 8s =7 1s =3 9s =9 8s =6 3s =5 1s =4 3s =4 9s =8 6s =2 4s =3 5s =8 4s =2 2s =9 6s =8 5s =8 7s =2 5s =7 9s =5 8s =5 4s =6 2s =7 6s =8 2s =7 4s =7 3s =8 9s =9 3s =
D.4 20s =E.9 60s =F.40 × 2 = 804 200s =9 600s =20 × 2 =6 30s =5 30s =30 × 2 =6 300s =5 300s =40 × 2 =7 × 50 =8 × 20 =20 × 3 =7 × 500 =8 × 200 =30 × 3 =3 × 40 =2 × 70 =300 × 3 = 9003 × 400 =2 × 700 =300 × 2 =
Write the missing numbers: (rstands for remainder.)
25 = .... 3s and ....r.30 = .... 4s and ....r.25 = .... 4s " ....r.30 = .... 5s " ....r.25 = .... 5s " ....r.30 = .... 6s " ....r.25 = .... 6s " ....r.30 = .... 7s " ....r.25 = .... 7s " ....r.30 = .... 8s " ....r.25 = .... 8s " ....r.30 = .... 9s " ....r.25 = .... 9s " ....r.26 = .... 3s and ....r.31 = .... 4s and ....r.26 = .... 4s " ....r.31 = .... 5s " ....r.26 = .... 5s " ....r.31 = .... 6s " ....r.26 = .... 6s " ....r.31 = .... 7s " ....r.26 = .... 7s " ....r.31 = .... 8s " ....r.26 = .... 8s " ....r.31 = .... 9s " ....r.26 = .... 9s " ....r.
Write the whole numbers or mixed numbers which these fractions equal:—
54439542737453118328884639894168114751388566
Write the missing figures:—
68=424=2810=515=1023=6
Write the missing numerators:—
12=1281041661413=12918615242114=121682420283215=1020152540353023=12182161524934=8161220243228
Find the products. Cancel when you can:—
516× 4 =1112× 3 =23× 5 =712× 8 =85× 15 =16× 8 =
The use of bodily action, social games, and the like was discussed in the section on original tendencies. "Significance as a means of securing other desired ends than arithmetical learning itself" is therefore our next topic. Such significance can be given to arithmetical work by using that work as a means to present and future success in problems of sports, housekeeping, shopwork, dressmaking, self-management, other school studies than arithmetic, and general school life and affairs. Significance as a means to future ends alone can also be more clearly and extensively attached to it than it now is.
Whatever is done to supply greater strength of motive in studying arithmetic must be carefully devised so as not to get a strong but wrong motive, so as not to get abundant interest but in something other than arithmetic, and so as not to kill the goose that after all lays the golden eggs—the interest in intellectual activity and achievement itself. It is easy to secure an interest in laying out a baseball diamond, measuring ingredients for a cake, making a balloon of a certain capacity, or deciding the added cost of an extra trimming of ribbon for one's dress. The problem is toattachthat interest to arithmetical learning. Nor should a teacher be satisfied with attaching the interest as a mere tail that steers the kite, so long as it stays on, or as a sugar-coating that deceives the pupil into swallowing the pill, or as an anodyne whose dose must be increased and increased if it is to retain its power. Until the interest permeates the arithmetical activity itself our task is only partly done, and perhaps is made harder for the next time.
One important means of really interfusing the arithmetical learning itself with these derived interests is to lead the pupil to seek the help of arithmetic himself—to lead him, in Dewey's phrase, to 'feel the need'—to take the'problem' attitude—and thus appreciate the technique which he actively hunts for to satisfy the need. In so far as arithmetical learning is organized to satisfy the practical demands of the pupil's life at the time, he should, so to speak, come part way to get its help.
Even if we do not make the most skillful use possible of these interests derived from the quantitative problems of sports, housekeeping, shopwork, dressmaking, self-management, other school studies, and school life and affairs, the gain will still be considerable. To have them in mind will certainly preserve us from giving to children of grades 3 and 4 problems so devoid of relation to their interests as those shown below, all found (in 1910) in thirty successive pages of a book of excellent repute:—
A chair has 4 legs. How many legs have 8 chairs? 5 chairs?A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?(Eight more of the same sort.)In 1890 New York had 1,513,501 inhabitants, Milwaukee had 206,308, Boston had 447,720, San Francisco 297,990. How many had these cities together?(Five more of the same sort.)Milton was born in 1608 and died in 1674. How many years did he live?(Several others of the same sort.)The population of a certain city was 35,629 in 1880 and 106,670 in 1890. Find the increase.(Several others of this sort.)A number of others about the words in various inaugural addresses and the Psalms in the Bible.
A chair has 4 legs. How many legs have 8 chairs? 5 chairs?
A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?
(Eight more of the same sort.)
In 1890 New York had 1,513,501 inhabitants, Milwaukee had 206,308, Boston had 447,720, San Francisco 297,990. How many had these cities together?
(Five more of the same sort.)
Milton was born in 1608 and died in 1674. How many years did he live?
(Several others of the same sort.)
The population of a certain city was 35,629 in 1880 and 106,670 in 1890. Find the increase.
(Several others of this sort.)
A number of others about the words in various inaugural addresses and the Psalms in the Bible.
It also seems probable that with enough care other systematic plans of textbooks can be much improved in this respect. From every point of view, for example, the early work in arithmetic should be adapted to some extent to the healthy childish interests in home affairs, the behavior of other children, and the activities of material things, animals, and plants.
TABLE 9
Frequency of Appearance of Certain Words about Family Life, Play, and Action in Eight Elementary Textbooks in Arithmetic, pp. 1-50.
ABCDEFGHbaby24brother26111family224father13521helphome2442271mother4295517sister122911forkknifeplate4221spoondoll101106109game1355jump4marbles10410101play13run13singtagtoy1car24231cut10628dig2flower14112grow1plant2seed31string11011wheel510
The words used by textbooks give some indication of how far this aim is being realized, or rather of how far short we are of realizing it. Consider, for example, the words home, mother, father, brother, sister, help, plate, knife, fork, spoon, play, game, toy, tag, marbles, doll, run, jump, sing, plant, seed, grow, flower, car, wheel, string, cut, dig. The frequency of appearance in the first fifty pages of eight beginners' arithmetics was as shown in Table 9. The eight columns refer to the eight books (the first fifty pages of each). The numbers refer to the number of times the word in question appeared, the number 10 meaning 10or moretimes in the fifty pages. Plurals, past tenses, and the like were counted.Help,fork,knife,spoon,jump,sing, andtagdid not appear at all!Toyandgrowappeared each once in the 400 pages!Play,run,dig,plant, andseedappeared once in a hundred or more pages.Babydid not appear as often asbuggy.Familyappeared no oftener thanfenceorFriday.Fatherappears about a third as often asfarmer.
Book A shows only 10 of these thirty words in the fifty pages; book B only 4; book C only 12; and books D, E, F, G, and H only 13, 8, 14, 13, 10, respectively. The total number of appearances (counting the 10s as only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for E, 62 for F, 30 for G, and 37 for H. The five words—apple, egg, Mary, milk, and orange—are used oftener than all these thirty together.
If it appeared that this apparent neglect of childish affairs and interests was deliberate to provide for a more systematic treatment of pure arithmetic, a better gradation of problems, and a better preparation for later genuine use than could be attained if the author of the textbook were tied to the child's apron strings, the neglect could be defended. It is not at all certain that children in grade 2 get much more enjoymentor ability from adding the costs of purchases for Christmas or Fourth of July, or multiplying the number of cakes each child is to have at a party by the number of children who are to be there, than from adding gravestones or multiplying the number of hairs of bald-headed men. When, however, there is nothing gained by substituting remote facts for those of familiar concern to children, the safe policy is surely to favor the latter. In general, the neglect of childish data does not seem to be due to provision for some other end, but to the same inertia of tradition which has carried over the problems of laying walls and digging wells into city schools whose children never saw a stone wall or dug well.
I shall not go into details concerning the arrangement of courses of study, textbooks, and lesson-plans to make desirable connections between arithmetical learning and sports, housework, shopwork, and the rest. It may be worth while, however, to explain the termself-management, since this source of genuine problems of real concern to the pupils has been overlooked by most writers.
By self-management is meant the pupil's use of his time, his abilities, his knowledge, and the like. By the time he reaches grade 5, and to some extent before then, a boy should keep some account of himself, of how long it takes him to do specified tasks, of how much he gets done in a specified time at a certain sort of work and with how many errors, of how much improvement he makes month by month, of which things he can do best, and the like. Such objective, matter-of-fact, quantitative study of one's behavior is not a stimulus to morbid introspection or egotism; it is one of the best preventives of these. To treat oneself impersonally is one of the essential elements of mental balance and health. It need not, and should not, encouragepriggishness. On the contrary, this matter-of-fact study of what one is and does may well replace a certain amount of the exhortations and admonitions concerning what one ought to do and be. All this is still truer for a girl.
The demands which such an accounting of one's own activities make of arithmetic have the special value of connecting directly with the advanced work in computation. They involve the use of large numbers, decimals, averaging, percentages, approximations, and other facts and processes which the pupil has to learn for later life, but to which his childish activities as wage-earner, buyer and seller, or shopworker from 10 to 14 do not lead. Children have little money, but they have time in thousands of units! They do not get discounts or bonuses from commercial houses, but they can discount their quantity of examples done for the errors made, and credit themselves with bonuses of all sorts for extra achievements.
There remains the most important increase of interest in arithmetical learning—an increase in the interest directly bound to achievement and success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is a game and all boys and girls are players." It should not be ameregame for them and they should notmerelyplay, but their unpractical interest in doing it because they can do it and can see how well they do do it is one of the school's most precious assets. Any healthy means to give this interest more and better stimulus should therefore be eagerly sought and cherished.
Two such means have been suggested in other connections. The first is the extension of training in checking and verifying work so that the pupil may work to a standard of approximately 100% success, and may know how nearly he is attaining it. The second is the use of standardized practice material and tests, whereby the pupil may measure himself against his own past, and have a clear, vivid, and trustworthy idea of just how much better or faster he can do the same tasks than he could do a month or a year ago, and of just how much harder things he can do now than then.
Another means of stimulating the essential interest in quantitative thinking itself is the arrangement of the work so that real arithmetical thinking is encouraged more than mere imitation and assiduity. This means the avoidance of long series of applied problems all of one type to be solved in the same way, the avoidance of miscellaneous series and review series which are almost verbatim repetitions of past problems, and in general the avoidance of excessive repetition of any one problem-situation. Stimulation to real arithmetical thinking is weak when a whole day's problem work requires no choice of methods, or when a review simply repeats without any step of organization or progress, or when a pupil meets a situation (say the 'buyxthings atyper thing, how much pay' situation) for the five-hundredth time.
Another matter worthy of attention in this connection is the unwise tendency to omit or present in diluted form some of the topics that appeal most to real intellectual interests, just because they are hard. The best illustration, perhaps, is the problem of ratio or "How many times as large (long, heavy, expensive, etc.) asxisy?" Mastery of the 'times as' relation is hard to acquire, but it is well worth acquiring, not only because of its strong intellectual appeal, but also because of its prime importance in the applications of arithmetic to science. In the older arithmetics it was confused by pedantries and verbal difficulties and penalizedby unreal problems about fractions of men doing parts of a job in strange and devious times. Freed from these, it should be reinstated, beginning as early as grade 5 with such simple exercises as those shown below and progressing to the problems of food values, nutritive ratios, gears, speeds, and the like in grade 8.
John is 4 years old.Fred is 6 years old.Mary is 8 years old.Nell is 10 years old.Alice is 12 years old.Bert is 15 years old.Who is twice as old as John?Who is half as old as Alice?Who is three times as old as John?Who is one and one half times as old as Nell?Who is two thirds as old as Fred?etc., etc., etc.Alice is .... times as old as John.John is .... as old as Mary.Fred is .... times as old as John.Alice is .... times as old as Fred.Fred is .... as old as Mary.etc., etc., etc.
John is 4 years old.Fred is 6 years old.Mary is 8 years old.Nell is 10 years old.Alice is 12 years old.Bert is 15 years old.Who is twice as old as John?Who is half as old as Alice?Who is three times as old as John?Who is one and one half times as old as Nell?Who is two thirds as old as Fred?etc., etc., etc.Alice is .... times as old as John.John is .... as old as Mary.Fred is .... times as old as John.Alice is .... times as old as Fred.Fred is .... as old as Mary.etc., etc., etc.
Finally it should be remembered that all improvements in making arithmetic worth learning and helping the pupil to learn it will in the long run add to its interest. Pupils like to learn, to achieve, to gain mastery. Success is interesting. If the measures recommended in the previous chapters are carried out, there will be little need to entice pupils to take arithmetic or to sugar-coat it with illegitimate attractions.
We shall consider in this chapter the influence of time of day, size of class, and amount of time devoted to arithmetic in the school program, the hygiene of the eyes in arithmetical work, the use of concrete objects, and the use of sounds, sights, and thoughts as situations and of speech and writing and thought as responses.[17]
Computation of one or another sort has been used by several investigators as a test of efficiency at different times in the day. When freed from the effects of practice on the one hand and lack of interest due to repetition on the other, the results uniformly show an increase in speed late in the school session with a falling off in accuracy that about balances it.[18]There is no wisdom in putting arithmetic early in the session because of itsdifficulty. Lively and sociable exercises in mental arithmetic with oral answers in fact seem to be admirably fitted for use late in the session. Except for the general principles (1) of starting the day with work that will set a good standard of cheerful, efficientproduction and (2) of getting the least interesting features of the day's work done fairly early in the day, psychology permits practical exigencies to rule the program, so far as present knowledge extends. Adequate measurements of the effect of time of day onimprovementhave not been made, but there is no reason to believe that any one time between 9a.m.and 4p.m.is appreciably more favorable to arithmetical learning than to learning geography, history, spelling, and the like.
The influence of size of class upon progress in school studies is very difficult to measure because (1) within the same city system the average of the six (or more) sizes of class that a pupil has experienced will tend to approximate closely to the corresponding average for any other child; because further (2) there may be a tendency of supervisory officers to assign more pupils to the better teachers; and because (3) separate systems which differ in respect to size of class probably differ in other respects also so that their differences in achievement may be referable to totally different differences.
Elliott ['14] has made a beginning by noting size of class during the year of test in connection with his own measures of the achievements of seventeen hundred pupils, supplemented by records from over four hundred other classes. As might be expected from the facts just stated, he finds no appreciable difference between classes of different sizes within the same school system, the effect of the few months in a small class being swamped by the antecedents or concomitants thereof.
The effect of the amount of time devoted to arithmetic in the school program has been studied extensively by Rice ['02 and '03] and Stone ['08].
Dr. Rice ['02] measured the arithmetical ability of some6000 children in 18 different schools in 7 different cities. The results of these measurements are summarized in Table 10. This table "gives two averages for each grade as well as for each school as a whole. Thus, the school at the top shows averages of 80.0 and 83.1, and the one at the bottom, 25.3 and 31.5. The first represents the percentage of answers which were absolutely correct; the second shows what per cent of the problems were correct in principle,i.e.the average that would have been received if no mechanical errors had been made."
The facts of Dr. Rice's table show that there is a positive relation between the general standing of a school system in the tests and the amount of time devoted to arithmetic by its program. The relation is not close, however, being that expressed by a correlation coefficient of .36½. Within any one school system there is no relation between the standing of a particular school and the amount of time devoted to arithmetic in that school's program. It must be kept in mind that the amount of time given in the school program may be counterbalanced by emphasizing work at home and during study periods, or, on the other hand, may be a symptom of correspondingly small or great emphasis on arithmetic in work set for the study periods at home.
A still more elaborate investigation of this same topic was made by Stone ['08]. I quote somewhat fully from it, since it is an instructive sample of the sort of studies that will doubtless soon be made in the case of every elementary school subject. He found that school systems differed notably in the achievements made by their sixth-grade pupils in his tests of computation (the so-called 'fundamentals') and of the solution of verbally described problems (the so-called 'reasoning'). The facts were as shown in Table 11.
TABLE 10
Averages for Individual Schools in Arithmetic
CitySchool6th Year7th Year8th YearSchool AverageResultPrincipleResultPrincipleResultPrincipleResultPrinciplePercent ofMechanical ErrorsMinutesDailyIII179.380.381.182.391.793.980.083.13.753I180.481.564.267.280.982.876.680.34.660I280.983.443.550.972.779.169.375.17.725I372.274.063.566.274.576.667.872.26.145I469.972.254.657.866.569.164.370.38.545II171.275.333.635.736.840.060.264.87.160III243.745.053.956.751.153.154.558.97.460IV158.960.431.234.141.643.555.158.45.660IV259.863.1——22.522.553.958.88.3—IV354.958.135.238.643.545.051.557.610.560IV442.345.116.119.248.748.742.848.211.2—V144.148.729.232.551.158.345.951.310.540VI168.371.333.536.626.930.739.042.99.033VI246.149.519.524.230.240.636.543.616.230VI334.536.430.535.123.324.136.042.515.248VII135.237.729.132.525.127.240.545.911.742VII235.238.715.016.419.621.236.540.610.175VII327.633.78.910.111.311.325.331.519.645
High achievement by a system in computation went with high achievement in solving the problems, the correlation being about .50; and the system that scored high in addition or subtraction or multiplication or division usually showed closely similar excellence in the other three, the correlations being about .90.
TABLE 11
Scores Made by the Sixth-Grade Pupils of Each of Twenty-Six School Systems
SystemScore in Testswith ProblemsScore in Testsin Computing233561841244293513174443042446435632546421672246823111646937072049121681850937581553227793533284585382747655031731552293510601274926152958216272951136363049146613561969134047734378212736341011759326126791368219848409959143569
Of the conditions under which arithmetical learning took place, the one most elaborately studied was the amount of time devoted to arithmetic. On the basis of replies by principals of schools to certain questions, he gave each ofthe twenty-six school systems a measure for the probable time spent on arithmetic up through grade 6. Leaving home study out of account, there seems to be little or no correlation between the amount of time a system devotes to arithmetic and its score in problem-solving, and not much more between time expenditure and score in computation. With home study included there is little relation tothe achievement of the system in solving problems, but there is a clear effect on achievement in computation. The facts as given by Stone are:—
TABLE 12
Correlation of Time Expenditures with Abilities