CHAPTER XII

INTEREST IN ARITHMETIC

CENSUSES OF PUPILS' INTERESTS

Arithmetic, although it makes little or no appeal to collecting, muscular manipulation, sensory curiosity, or the potent original interests in things and their mechanisms and people and their passions, is fairly well liked by children. The censuses of pupils' likes and dislikes that have been made are not models of scientific investigation, and the resulting percentages should not be used uncritically. They are, however, probably not on the average over-favorable to arithmetic in any unfair way. Some of their results are summarized below. In general they show arithmetic to be surpassed in interest clearly by only the manual arts (shopwork and manual training for boys, cooking and sewing for girls), drawing, certain forms of gymnastics, and history. It is about on a level with reading and science. It clearly surpasses grammar, language, spelling, geography, and religion.

Lobsien ['03], who asked one hundred children in each of the first five grades (_Stufen_) of the elementary schools of Kiel, "Which part of the school work (literally, 'which instruction period') do you like best?" found arithmetic led only by drawing and gymnastics in the case of the boys, and only by handwork in the case of the girls.

This is an exaggerated picture of the facts, since no count is made of those who especially dislike arithmetic. Arithmetic is as unpopular with some as it is popular with others. When full allowance is made for this, arithmetic still has popularity above the average. Stern ['05] asked, "Which subject do you like most?" and "Which subject do you like least?" The balance was greatly in favor of gymnastics for boys (28--1), handwork for girls (32--1-1/2), and drawing for both (16-1/2--6). Writing (6-1/2--4), arithmetic (14-1/2--13), history (9--6-1/2), reading (8-1/2--8), and singing (6--7-1/2) come next. Religion, nature study, physiology, geography, geometry, chemistry, language, and grammar are low.

McKnight ['07] found with boys and girls in grades 7 and 8 of certain American cities that arithmetic was liked better than any of the school subjects except gymnastics and manual training. The vote as compared with history was:--

Arithmetic 327 liked greatly, 96 disliked greatly. History 164 liked greatly, 113 disliked greatly.

In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old representing all grades of the elementary school report, so far as they could, the subject most disliked, the subject most liked, the subject next most liked, and the subject next in order. No child was forced to report all of these four judgments, or even any of them. Lobsien counts the likes and the dislikes for each subject. Gymnastics, handwork, and cooking are by far the most popular. History and drawing are next, followed by arithmetic and reading. Below these are geography, writing, singing, nature study, biblical history, catechism, and three minor subjects.

Lewis ['13] secured records from English children in elementary schools of the order of preference of all the studies listed below. He reports the results in the following table of percents:

=================================================================== | TOP THIRD OF | MIDDLE THIRD OF | LOWEST THIRD OF | STUDIES FOR | STUDIES FOR | STUDIES FOR | INTEREST | INTEREST | INTEREST ----------------+--------------+-----------------+----------------- Drawing | 78 | 20 | 2 Manual Subjects | 66 | 26 | 8 History | 64 | 24 | 12 Reading | 53 | 38 | 9 Singing | 32 | 48 | 20 | | | Drill | 20 | 55 | 25 Arithmetic | 16 | 53 | 31 Science | 23 | 37 | 40 Nature Study | 16 | 36 | 48 Dictation | 4 | 57 | 39 | | | Composition | 18 | 28 | 54 Scripture | 4 | 38 | 58 Recitation | 9 | 23 | 68 Geography | 4 | 24 | 72 Grammar | -- | 6 | 94 ===================================================================

Brandell ['13] obtained data from 2137 Swedish children in Stockholm (327), Norrköping (870), and Gothenburg (940).

In general he found, as others have, that handwork, shopwork for boys and household work for girls, and drawing were reported as much better liked than arithmetic. So also was history, and (in this he differs from most students of this matter) so were reading and nature study. Gymnastics he finds less liked than arithmetic. Religion, geography, language, spelling, and writing are, as in other studies, much less popular than arithmetic.

Other studies are by Lilius ['11] in Finland, Walsemann ['07], Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14], in Germany. They confirm the general results stated.

The reasons for the good showing that arithmetic makes are probably the strength of its appeal to the interest in definite achievement, success, doing what one attempts to do; and of its appeal, in grades 5 to 8, to the practical interest of getting on in the world, acquiring abilities that the world pays for. Of these, the former is in my opinion much the more potent interest. Arithmetic satisfies it especially well, because, more than any other of the 'intellectual' studies of the elementary school, it permits the pupil to see his own progress and determine his own success or failure.

The most important applications of the psychology of satisfiers and annoyers to arithmetic will therefore be in the direction of utilizing still more effectively this interest in achievement. Next in importance come the plans to attach to arithmetical learning the satisfyingness of bodily action, play, sociability, cheerfulness, and the like, and of significance as a means of securing other desired ends than arithmetical abilities themselves. Next come plans to relieve arithmetical learning from certain discomforts such as the eyestrain of some computations and excessive copying of figures. These will be discussed here in the inverse order.

RELIEVING EYESTRAIN

At present arithmetical work is, hour for hour, probably more of a tax upon the eyes than reading. The task of copying numbers from a book to a sheet of paper is one of the very hardest tasks that the eyes of a pupil in the elementary schools have to perform. A certain amount of such work is desirable to teach a child to write numbers, to copy exactly, and to organize material in shape for computation. But beyond that, there is no more reason for a pupil to copy every number with which he is to compute than for him to copy every word he is to read. The meaningless drudgery of copying figures should be mitigated by arranging much work in the form of exercises like those shown on pages 216, 217, and 218, and by having many of the textbook examples in addition, subtraction, and multiplication done with a slip of paper laid below the numbers, the answers being written on it. There is not only a resulting gain in interest, but also a very great saving of time for the pupil (very often copying an example more than quadruples the time required to get its answer), and a much greater efficiency in supervision. Arithmetical errors are not confused with errors of copying,[16] and the teacher's task of following a pupil's work on the page is reduced to a minimum, each pupil having put the same part of the day's work in just the same place. The use of well-printed and well-spaced pages of exercises relieves the eyestrain of working with badly made gray figures, unevenly and too closely or too widely spaced. I reproduce in Fig. 25 specimens taken at random from one hundred random samples of arithmetical work by pupils in grade 8. Contrast the task of the eyes in working with these and their task in working with pages 216 to 218. The customary method of always copying the numbers to be used in computation from blackboard or book to a sheet of paper is an utterly unjustifiable cruelty and waste.

[16] Courtis finds in the case of addition that "of all the individuals making mistakes at any given time in a class, at least one third, and usually two thirds, will be making mistakes in carrying or copying."

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 = 80 4 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 = 900 3 × 400 = 2 × 700 = 300 × 2 =

Write the missing numbers: (_r_ stands for remainder.)

25 = .... 3s and .... _r_. 25 = .... 4s " .... _r_. 25 = .... 5s " .... _r_. 25 = .... 6s " .... _r_. 25 = .... 7s " .... _r_. 25 = .... 8s " .... _r_. 25 = .... 9s " .... _r_.

26 = .... 3s and .... _r_. 26 = .... 4s " .... _r_. 26 = .... 5s " .... _r_. 26 = .... 6s " .... _r_. 26 = .... 7s " .... _r_. 26 = .... 8s " .... _r_. 26 = .... 9s " .... _r_.

30 = .... 4s and .... _r_. 30 = .... 5s " .... _r_. 30 = .... 6s " .... _r_. 30 = .... 7s " .... _r_. 30 = .... 8s " .... _r_. 30 = .... 9s " .... _r_.

31 = .... 4s and .... _r_. 31 = .... 5s " .... _r_. 31 = .... 6s " .... _r_. 31 = .... 7s " .... _r_. 31 = .... 8s " .... _r_. 31 = .... 9s " .... _r_.

Write the whole numbers or mixed numbers which these fractions equal:--

5 4 9 4 7 - - - - - 4 3 5 2 3

7 5 11 3 8 - - -- - - 4 3 8 2 8

8 6 9 9 16 - - - - -- 4 3 8 4 8

11 7 13 8 6 -- - -- - - 4 5 8 5 6

Write the missing figures:--

6 2 8 1 2 - = - - = - -- = - - = -- - = - 8 4 4 2 10 5 5 10 3 6

Write the missing numerators:--

1 - = -- - -- - -- - -- 2 12 8 10 4 16 6 14

1 - = -- - -- - -- -- -- 3 12 9 18 6 15 24 21

1 - = -- -- - -- -- -- -- 4 12 16 8 24 20 28 32

1 - = -- -- -- -- -- -- -- 5 10 20 15 25 40 35 30

2 - = -- -- -- - -- -- - 3 12 18 21 6 15 24 9

3 - = - -- -- -- -- -- -- 4 8 16 12 20 24 32 28

Find the products. Cancel when you can:--

5 11 2 -- × 4 = -- × 3 = - × 5 = 16 12 3

7 8 1 -- × 8 = - × 15 = - × 8 = 12 5 6

SIGNIFICANCE FOR RELATED ACTIVITIES

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 to _attach_ that 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.

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.

================================================================ | A | B | C | D | E | F | G | H ----------------+-----+-----+-----+-----+-----+-----+-----+----- baby | | | | 2 | | 4 | | brother | 2 | | 6 | 1 | 1 | | 1 | family | | | 2 | | 2 | | 4 | father | 1 | | 3 | 5 | | 2 | 1 | help | | | | | | | | home | 2 | | 4 | 4 | 2 | 2 | 7 | 1 mother | 4 | 2 | 9 | 5 | | 5 | 1 | 7 sister | | | 1 | 2 | 2 | 9 | 1 | 1 | | | | | | | | fork | | | | | | | | knife | | | | | | | | plate | 4 | 2 | | 2 | | 1 | | spoon | | | | | | | | | | | | | | | | doll | 10 | 1 | 10 | 6 | | 10 | | 9 game | 1 | | | 3 | | | 5 | 5 jump | | | | | | | | 4 marbles | 10 | 4 | 10 | | 10 | | 1 | play | | | 1 | | | 3 | | run | | | | | | 1 | | 3 sing | | | | | | | | tag | | | | | | | | toy | | | | | | | | 1 | | | | | | | | car | | | 2 | 4 | | 2 | 3 | 1 cut | | | 10 | | 6 | 2 | | 8 dig | | | | | | | 2 | flower | 1 | | | 4 | 1 | 1 | 2 | grow | | | | 1 | | | | plant | | | 2 | | | | | seed | | | | 3 | | | 1 | string | | | | | 1 | 10 | 1 | 1 wheel | 5 | | | | | 10 | | ================================================================

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 10 _or more_ times in the fifty pages. Plurals, past tenses, and the like were counted. _Help_, _fork_, _knife_, _spoon_, _jump_, _sing_, and _tag_ did not appear at all! _Toy_ and _grow_ appeared each once in the 400 pages! _Play_, _run_, _dig_, _plant_, and _seed_ appeared once in a hundred or more pages. _Baby_ did not appear as often as _buggy_. _Family_ appeared no oftener than _fence_ or _Friday_. _Father_ appears about a third as often as _farmer_.

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 enjoyment or 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 term _self-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, encourage priggishness. 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.

INTRINSIC INTEREST IN ARITHMETICAL LEARNING

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 a _mere_ game for them and they should not _merely_ play, 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 'buy _x_ things at _y_ per 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.) as _x_ is _y_?" 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 penalized by 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.

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.