Without Home Study{Reasoning and Time Expenditure−.01Fundamentals and Time Expenditure.09Including Home Study{Reasoning and Time Expenditure.13Fundamentals and Time Expenditure.49
These correlations, it should be borne in mind, are for school systems, not for individual pupils. It might be that, though the system which devoted the most time to arithmetic did not show corresponding superiority in the product over the system devoting only half as much time, the pupils within the system did achieve in exact proportion to the time they gave to study. Neither correlation would permit inference concerning the effect of different amounts of time spent by the same pupil.
Stone considered also the printed announcements of the courses of study in arithmetic in these twenty-six systems. Nineteen judges rated these announced courses of study for excellence according to the instructions quoted below:—
Judges please read before scoring
I. Some Factors Determining Relative Excellence.
(N. B. The following enumeration is meant to be suggestive rather than complete or exclusive. And each scorer is urged to rely primarily on his own judgment.)
1. Helpfulness to the teacher in teaching the subject matter outlined.2. Social value or concreteness of sources of problems.3. The arrangement of subject matter.4. The provision made for adequate drill.5. A reasonable minimum requirement with suggestions for valuable additional work.6. The relative values of any predominating so-called methods—such as Speer, Grube, etc.7. The place of oral or so-called mental arithmetic.8. The merit of textbook references.
1. Helpfulness to the teacher in teaching the subject matter outlined.
2. Social value or concreteness of sources of problems.
3. The arrangement of subject matter.
4. The provision made for adequate drill.
5. A reasonable minimum requirement with suggestions for valuable additional work.
6. The relative values of any predominating so-called methods—such as Speer, Grube, etc.
7. The place of oral or so-called mental arithmetic.
8. The merit of textbook references.
II. Cautions and Directions.
(Judges please follow as implicitly as possible.)
1. Include references to textbooks as parts of the Course of Study.This necessitates judging the parts of the texts referred to.2. As far as possible become equally familiar with all courses before scoring any.3. When you are ready to begin to score, (1) arrange in serial order according to excellence, (2) starting with the middle one score it 50, then score above and below 50 according as courses are better or poorer, indicating relative differences in excellence by relative differences in scores,i.e.in so far as you find that the courses differ by about equal steps, score those better than the middle one 51, 52, etc., and those poorer 49, 48, etc., but if you find that the courses differ by unequal steps show these inequalities by omitting numbers.4. Write ratings on the slip of paper attached to each course.
1. Include references to textbooks as parts of the Course of Study.
This necessitates judging the parts of the texts referred to.
2. As far as possible become equally familiar with all courses before scoring any.
3. When you are ready to begin to score, (1) arrange in serial order according to excellence, (2) starting with the middle one score it 50, then score above and below 50 according as courses are better or poorer, indicating relative differences in excellence by relative differences in scores,i.e.in so far as you find that the courses differ by about equal steps, score those better than the middle one 51, 52, etc., and those poorer 49, 48, etc., but if you find that the courses differ by unequal steps show these inequalities by omitting numbers.
4. Write ratings on the slip of paper attached to each course.
The systems whose courses of study were thus rated highest did not manifest any greater achievement in Stone's tests than the rest. The thirteen with the most approved announcements of courses of study were in fact a little inferior in achievement to the other thirteen, and the correlation coefficients were slightly negative.
Stone also compared eighteen systems where there was supervision of the work by superintendents or supervisors as well as by principals with four systems where the principals and teachers had no such help. The scores in his tests were very much lower in the four latter cities.
Fig. 26.Fig. 26.—Type too large.
We have already noted that the task of reading and copying numbers is one of the hardest that the eyes have to perform in the elementary school, and that it should be alleviated by arranging much of the work so that only answers need be written by the pupil. The figures to be read and copiedshould obviously be in type of suitable size and style, so arranged and spaced on the page or blackboard as to cause a minimum of effort and strain.
Fig. 27.Fig. 27.—12-point, 11-point, and 10-point type.
Size.—Type may be too large as well as too small, though the latter is the commoner error. If it is too large, as in Fig. 26, which is a duplicate of type actually used in a form of practice pad, the eye has to make too many fixations to take in a given content. All things considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6, and 10-point in grades 7 and 8 seem the most desirable sizes. These are shown in Fig. 27. Too small type occurs oftenest in fractions and in the dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30 are samples from actual school practice. Samples of the desirable size are shown in Figs. 31 and 32. The technique of modern typesetting makes it very difficult and expensive to make fractions of the horizontal type
(14,38,56)
large enough without making the whole-number figures with which they are mingled too large or giving an uncouth appearance to the total. Consequently fractions somewhat smaller than are desirable may have to be used occasionally in textbooks.[19]There is no valid excuse, however, for the excessively small fractions which often are made in blackboard work.
Fig. 28.Fig. 28.—Type of measurements too small.This is a picture of Mary's garden.How many feet is it around the garden?
Fig. 28.Fig. 28.—Type of measurements too small.
This is a picture of Mary's garden.How many feet is it around the garden?
Fig. 29.Fig. 29.—Type too small.
Fig. 30.Fig. 30.—Numbers too small and badly designed.
Fig. 31.Fig. 31.—Figure 28 with suitable numbers.
Fig. 32.Fig. 32.—Figure 30 with suitable numbers.
Style.—The ordinary type forms often have 3 and 8 so made as to require strain to distinguish them. 5 is sometimes easily confused with 3 and even with 8. 1, 4, and 7 may be less easily distinguishable than is desirable. Figure 33 shows a specially good type in which each figure is represented by its essential[20]features without any distracting shading or knobs or turns. Figure 34 shows some of the types in common use. There are no demonstrably great differences amongst these. In fractions there is a notable gain from using the slant form (2⁄3,3⁄4) for exercises in additionand subtraction, and for almost all mixed numbers. This appears clearly to the eye in the comparison of Fig. 35 below, where the same fractions all in 10-point type are displayed in horizontal and in slant form. The figures in the slant form are in general larger and the space between them and the fraction-line is wider. Also the slant form makes it easier for the eye to examine the denominators to see whether reductions are necessary. Except for a few cases to show that the operations can be done just as truly with the horizontal forms, the book and the blackboard should display mixed numbers and fractions to be added or subtracted in the slant form. The slant line should be at an angle of approximately 45 degrees. Pupils should be taught to use this form in their own work of this sort.
Fig. 33.Fig. 33.—Block type; a very desirable type except that it is somewhat too heavy.
Fig. 34.Fig. 34.—Common styles of printed numbers.
When script figures are presented they should be of simple design, showing clearly the essential features of the figure, the line being everywhere of equal or nearly equal width (that is, without shading, and without ornamentation or eccentricity of any sort). The opening of the 3 should be wide to prevent confusion with 8; the top of the 3 should be curved to aid its differentiation from 5; the down stroke of the 9 should be almost or quite straight; the 1, 4, 7, and 9 should be clearly distinguishable. There are many ways of distinguishing them clearly, the best probably being to use the straight line for 1, the open 4 with clear angularity, a wide top to the 7, and a clearly closed curve for the top of the 9.
Fig. 35.Fig. 35.—Diagonal and horizontal fractions compared.
Figs. 36, 37.Fig. 36.—Good vertical spacing.Fig. 37.—Bad vertical spacing.
Figs. 38, 39.Figs.38 (above) and 39 (below).—Good and bad left-right spacing.
The pupil's writing of figures should be clear. He will thereby be saved eyestrain and errors in his school work as well as given a valuable ability for life. Handwriting of figures is used enormously in spite of the development of typewriters; illegible figures are commonly more harmful than illegible letters or words, since the context far less often tells what the figure is intended to be; the habit of making clear figures is not so hard to acquire, since they are written unjoined and require only the automatic action of ten minor acts of skill. The schools have missed a great opportunity in this respect. Whereas the hand writing of words is often better than it needs to be for life's purposes, the writing of figures is usually much worse. The figures presented in books on penmanship are also commonly bad, showing neglect or misunderstanding of the matter on the part of leaders in penmanship.
Spacing.—Spacing up and down the column is rarely too wide, but very often too narrow. The specimens shown in Figs. 36 and 37 show good practice contrasted with the common fault.
Spacing from right to left is generally fairly satisfactory in books, though there is a bad tendency to adopt some one routine throughout and so to miss chances to use reductions and increases of spacing so as to help the eye and the mind in special cases. Specimens of good and bad spacing are shown in Figs. 38 and 39. In the work of the pupils, the spacing from right to left is often too narrow. This crowding of letters, together with unevenness of spacing, adds notably to the task of eye and mind.
The composition or make-up of the page.—Other things being equal, that arrangement of the page is best which helps a child most to keep his place on a page and to find it after having looked away to work on the paper on which he computes, or for other good reasons. A good page and a bad page in this respect are shown in Figs. 40 and 41.
Fig. 40.Fig. 40.—A page well made up to suit the action of the eye.
Fig. 41.Fig. 41.—The same matter as in Fig. 40, much less well made up.
Objective presentations.—Pictures, diagrams, maps, and other presentations should not tax the eye unduly,
(a) by requiring too fine distinctions, or(b) by inconvenient arrangement of the data, preventing easy counting, measuring, comparison, or whatever the task is, or(c) by putting too many facts in one picture so that the eye and mind, when trying to make out any one, are confused by the others.
(a) by requiring too fine distinctions, or
(b) by inconvenient arrangement of the data, preventing easy counting, measuring, comparison, or whatever the task is, or
(c) by putting too many facts in one picture so that the eye and mind, when trying to make out any one, are confused by the others.
Illustrations of bad practices in these respects are shown in Figs. 42 to 52. A few specimens of work well arranged for the eye are shown in Figs. 53 to 56.
Good rules to remember are:—
Other things being equal, make distinctions by the clearest method, fit material to the tendency of the eye to see an 'eyeful' at a time (roughly 1½ inch by ½ inch in a book; 1½ ft. by ½ ft. on the blackboard), and let one picture teach only one fact or relation, or such facts and relations as do not interfere in perception.
The general conditions of seating, illumination, paper, and the like are even more important when the eyes are used with numbers than when they are used with words.
Fig. 42.Fig. 42.—Try to count the rungs on the ladder, or the shocks in the wagon.
Fig. 43.Fig. 43.—How many oars do you see? How many birds? How many fish?
Fig. 44.Fig. 44.—Count the birds in each of the three flocks of birds.
Fig. 45.Fig. 45.—Note the lack of clear division of the hundreds. Consider the difficulty of counting one of these columns of dots.
Fig. 46.Fig. 46.—What do you suppose these pictures are intended to show?
Fig. 47.Fig. 47.—Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate?
Fig. 47.—Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate?
Fig. 48.Fig. 48.—How long did it take you to find out what these pictures mean?
Fig. 49.Fig. 49.—Count the figures in the first row, using your eyes alone; have some one make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc.
Fig. 49.—Count the figures in the first row, using your eyes alone; have some one make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc.
Fig. 50.Fig. 50.—Can you answer the question without measuring? Could a child of seven or eight?
Fig. 51.Fig. 51.—What are these drawings intended to show? Why do they show the facts only obscurely and dubiously?
Fig. 52.Fig. 52.—What are these drawings intended to show? What simple change would make them show the facts much more clearly?
Fig. 53.Fig. 53.—Arranged in convenient "eye-fulls."
Fig. 54.Fig. 54.—Clear, simple, and easy of comparison.Tell which bar has—1.About 5 percent of its length black.2.About 10 percent of its length black.3.About 25 percent of its length black.4.About 75 percent of its length black.5.About 90 percent of its length black.6.About 95 percent of its length black.
Fig. 54.Fig. 54.—Clear, simple, and easy of comparison.
Tell which bar has—1.About 5 percent of its length black.2.About 10 percent of its length black.3.About 25 percent of its length black.4.About 75 percent of its length black.5.About 90 percent of its length black.6.About 95 percent of its length black.
Fig. 55.Fig. 55.—Clear, simple, and well spaced.
Fig. 56.Fig. 56.—Well arranged, though a little wider spacing between the squares would make it even better.
We mean by concrete objects actual things, events, and relations presented to sense, in contrast to words and numbers and symbols which mean or stand for these objects or for more abstract qualities and relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart measures, bank books, and checks are such concrete things. A foot rule put successively along the three thirds of a yard rule, a bell rung five times, and a pound weight balancing sixteen ounce weights are such concrete events. A pint beside a quart, an inch beside a foot, an apple shown cut in halves display such concrete relations to a pupil who is attentive to the issue.
Concrete presentations are obviously useful in arithmetic to teach meanings under the general law that a word or number or sign or symbol acquires meaning by being connected with actual things, events, qualities, and relations.We have also noted their usefulness as means to verifying the results of thinking and computing, as when a pupil, having solved, "How many badges each 5 inches long can be made from 31⁄3yd. of ribbon?" by using 10 ×12⁄5, draws a line 31⁄3yd. long and divides it into 5-inch lengths.
Concrete experiences are useful whenever the meaning of a number, like 9 or7⁄8or .004, or of an operation, like multiplying or dividing or cubing, or of some term, like rectangle or hypothenuse or discount, or some procedure, like voting or insuring property against fire or borrowing money from a bank, is absent or incomplete or faulty. Concrete work thus is by no means confined to the primary grades but may be appropriate at all stages when new facts, relations, and procedures are to be taught.
How much concrete material shall be presented will depend upon the fact or relation or procedure which is to be made intelligible, and the ability and knowledge of the pupil. Thus 'one half' will in general require less concrete illustration than 'five sixths'; and five sixths will require less in the case of a bright child who already knows2⁄3,3⁄4,3⁄8,5⁄8,7⁄8,2⁄5,3⁄5, and4⁄5than in the case of a dull child or one who only knows2⁄3and3⁄4. As a general rule the same topic will require less concrete material the later it appears in the school course. If the meanings of the numbers are taught in grade 2 instead of grade 1, there will be less need of blocks, counters, splints, beans, and the like. If 1½ + ½ = 2 is taught early in grade 3, there will be more gain from the use of 1½ inches and ½ inch on the foot rule than if the same relations were taught in connection with the general addition of like fractions late in grade 4. Sometimes the understanding can be had either by connecting the idea with the reality directly, or by connecting the two indirectlyviasome other idea. The amount of concrete material to beused will depend on its relative advantage per unit of time spent. Thus it might be more economical to connect5⁄12,7⁄12, and11⁄12with real meanings indirectly by calling up the resemblance to the2⁄3,3⁄4,3⁄8,5⁄8,7⁄8,2⁄5,3⁄5,4⁄5, and5⁄6already studied, than by showing5⁄12of an apple,7⁄12of a yard,11⁄12of a foot, and the like.
In general the economical course is to test the understanding of the matter from time to time, using more concrete material if it is needed, but being careful to encourage pupils to proceed to the abstract ideas and general principles as fast as they can. It is wearisome and debauching to pupils' intellects for them to be put through elaborate concrete experiences to get a meaning which they could have got themselves by pure thought. We should also remember that the new idea, say of the meaning of decimal fractions, will be improved and clarified by using it (see page 183 f.), so that the attainment of aperfectconception of decimal fractions before doing anything with them is unnecessary and probably very wasteful.
A few illustrations may make these principles more instructive.
(a) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reënforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers.
(b) Numbers very much smaller than one, such as1⁄32,1⁄64, .04, and .002, also need some concrete aids. A diagram like that of Fig. 57 is useful.
(c)Majorityandpluralityshould be understood by every citizen. They can be understood without concrete aid, but an actual vote is well worth while for the gain in vividness and surety.
Fig. 57.Fig. 57.—Concrete aid to understanding fractions with large denominators.A =1⁄1000sq. ft.; B =1⁄100sq. ft.; C =1⁄50sq. ft.; D =1⁄10sq. ft.
(d) Insurance against loss by fire can be taught by explanation and analogy alone, but it will be economical to have some actual insuring and payment of premiums and a genuine loss which is reimbursed.
(e) Four play banks in the corners of the room, receiving deposits, cashing checks, and later discounting notes will give good educational value for the time spent.
(f) Trade discount, on the contrary, hardly requires more concrete illustration than is found in the very problems to which it is applied.
(g) The process of finding the number of square units in a rectangle by multiplying with the appropriate numbers representing length and width is probably rather hindered than helped by the ordinary objective presentation as an introduction. The usual form of objective introduction is as follows:—
Fig. 58.Fig. 58.
How long is this rectangle? How large is each square? How many square inches are there in the top row? How many rows arethere? How many square inches are there in the whole rectangle? Since there are three rows each containing 4 square inches, we have 3 × 4 square inches = 12 square inches.Draw a rectangle 7 inches long and 2 inches wide. If you divide it into inch squares how many rows will there be? How many inch squares will there be in each row? How many square inches are there in the rectangle?
How long is this rectangle? How large is each square? How many square inches are there in the top row? How many rows arethere? How many square inches are there in the whole rectangle? Since there are three rows each containing 4 square inches, we have 3 × 4 square inches = 12 square inches.
Draw a rectangle 7 inches long and 2 inches wide. If you divide it into inch squares how many rows will there be? How many inch squares will there be in each row? How many square inches are there in the rectangle?
Fig. 59.Fig. 59.
It is better actually to hide the individual square units as in Fig. 59. There are four reasons: (1) The concrete rows and columns rather distract attention from the essential thing to be learned. This is not that "xrows one square wide,ysquares in a row will makexysquares in all," but that "by using proper units and the proper operation the area of any rectangle can be found from its length and width." (2) Children have little difficulty in learning tomultiply rather than add, subtract, or divide when computing area. (3) The habit so formed holds good for areas like 12⁄3by 4½, with fractional dimensions, in which any effort to count up the areas of rows is very troublesome and confusing. (4) The notion that a square inch is an area 1' by 1' rather than ½' by 2' or1⁄3in. by 3 in. or 1½ in. by2⁄3in. is likely to be formed too emphatically if much time is spent upon the sort of concrete presentation shown above. It is then better to use concrete counting of rows of small areas as a means ofverification afterthe procedure is learned, than as a means of deriving it.
There has been, especially in Germany, much argument concerning what sort of number-pictures (that is, arrangement of dots, lines, or the like, as shown in Fig. 60) is best for use in connection with the number names in the early years of the teaching of arithmetic.
Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and others have measured the accuracy of children in estimating the number of dots in arrangements of one or more of these different types.[21]Many writers interpret a difference in favor of estimating, say, the square arrangements of Born or Lay as meaning that such is the best arrangement to use in teaching. The inference is, however, unjustified. That certain number-pictures are easier to estimate numerically does not necessarily mean that they are more instructive in learning. One set may be easier to estimate just because they are more familiar, having been oftener experienced. Even if the favored set was so after equal experience with all sets, accuracy of estimation would be a sign of superiority for use in instruction only if all other things were equal (or in favor of the arrangementin question). Obviously the way to decide which of these is best to use in teaching is by using them in teaching and measuring all relevant results, not by merely recording which of them are most accurately estimated in certain time exposures.
Fig. 60.Fig. 60.—Various proposed arrangements of dots for use in teaching the meanings of the numbers 1 to 10.
It may be noted that the Born, Lay, and Freeman pictures have claims for special consideration on grounds of probable instructiveness. Since they are also superior in the tests in respect to accuracy of estimate, choice should probably be made from these three by any teacher who wishes to connect one set of number-pictures systematically with the number names, as by drills with the blackboard or with cards.
Such drills are probably useful if undertaken with zeal, and if kept as supplementary to more realistic objective work with play money, children marching, material to be distributed, garden-plot lengths to be measured, and the like, and if so administered that the pupils soon get the generalized abstract meaning of the numbers freed from dependence on an inner picture of any sort. This freedom is so important that it may make the use of many types of number-pictures advisable rather than the use of the one which in and of itself is best.
As Meumann says: "Perceptual reckoning can be overdone. It had its chief significance for the surety and clearness of the first foundation of arithmetical instruction. If, however, it is continued after the first operations become familiar to the child, and extended to operations which develop from these elementary ones, it necessarily works as a retarding force and holds back the natural development of arithmetic. This moves on to work with abstract number and with mechanical association and reproduction." ['07, Vol. 2, p. 357.]
Such drills are commonly overdone by those who makeuse of them, being given too often, and continued after their instructiveness has waned, and used instead of more significant, interesting, and varied work in counting and estimating and measuring real things. Consequently, there is now rather a prejudice against them in our better schools. They should probably be reinstated but to a moderate and judicious use.
There has been much dispute over the relative merits of oral and written work in arithmetic—a question which is much confused by the different meanings of 'oral' and 'written.'Oralhas meant (1) work where the situations are presented orally and the pupil's final responses are given orally, or (2) work where the situations are presented orally and the pupils' final responses are written or partly written and partly oral, or (3) work where the situations are presented in writing or print and the final responses are oral.Writtenhas meant (1) work where the situations are presented in writing or print and the final responses are made in writing, or (2) work where also many of the intermediate responses are written, or (3) work where the situations are presented orally but the final responses and a large percentage of the intermediate computational responses are written. There are other meanings than these.
It is better to drop these very ambiguous terms and ask clearly what are the merits and demerits, in the case of any specified arithmetical work, of auditory and of visual presentation of the situations, and of saying and of writing each specified step in the response.
The disputes over mentalversuswritten arithmetic are also confused by ambiguities in the use of 'mental.' Mental has been used to mean "done without pencil and paper"and also "done with few overt responses, either written or spoken, between the setting of the task and the announcement of the answer." In neither case is the wordmentalspecially appropriate as a description of the total fact. As before, we should ask clearly, "What are the merits and demerits of making certain specified intermediate responses in inner speech or imaged sounds or visual images or imageless thought—that is,withoutactual writing or overt speech?"
It may be said at the outset that oral, written, and inner presentations of initial situations, oral, written, and inner announcements of final responses, and oral, written, and inner management of intermediate processes have varying degrees of merit according to the particular arithmetical exercise, pupil, and context. Devotion to oralness or mentalness as such is simply fanatical. Various combinations, such as the written presentation of the situation with inner management of the intermediate responses and oral announcement of the final response have their special merits for particular cases.
These merits the reader can evaluate for himself for any given sort of work for a given class by considering: (1) The amount of practice received by the class per hour spent; (2) the ease of correction of the work; (3) the ease of understanding the tasks; (4) the prevention of cheating; (5) the cheerfulness and sociability of the work; (6) the freedom from eyestrain, and other less important desiderata.
It should be noted that the stock schemes A, B, C, and D below are only a few of the many that are possible and that schemes E, F, G, and H have special merits.
The common practice of either having no use made of pencil and paper or having all computations and even much verbal analysis written out elaborately for examination is unfavorable for learning. The demands which life itselfwill make of arithmetical knowledge and skill will range from tasks done with every percentage of written work from zero up to the case where every main result obtained by thought is recorded for later use by further thought. In school the best way is that which, for the pupils in question, has the best total effect upon quality of product, speed, and ease of production, reënforcement of training already given, and preparation for training to be given. There is nothing intellectually criminal about using a pencil as well as inner thought; on the other hand there is no magical value in writing out for the teacher's inspection figures that the pupil does not need in order to attain, preserve, verify, or correct his result.
Presentation of Initial SituationManagement of Intermediate ProcessesAnnouncement of Final ResponseA. Printed or writtenWrittenWrittenB. " "InnerOral by one pupil, inner by the restC. Oral (by teacher)WrittenWrittenD. " "InnerOral by one pupil, inner by the restE. As in A or CA mixture, the pupil writing what he needsAs in A or B or HF. The real situation itself, in part at leastAs in EAs in A or B or HG. Both read by the pupil and put orally by the teacherAs in EAs in A or B or HH. As in A or C or GAs in EWritten by all pupils, announced orally by one pupil
The common practice of having the final responses of alleasytasks given orally has no sure justification. On the contrary, the great advantage of having all pupils really do the work should be secured in the easy work more than anywhere else. If the time cost of copying the figures is eliminated by the simple plan of having them printed, and if the supervision cost of examining the papers is eliminated by having the pupils correct each other's work in these easy tasks, written answers are often superior to oral except for the elements of sociability and 'go' and freedom from eyestrain of the oral exercise. Such written work provides the gifted pupils with from two to ten times as much practice as they would get in an oral drill on the same material, supposing them to give inner answers to every exercise done by the class as a whole; it makes sure that the dull pupils who would rarely get an inner answer at the rate demanded by the oral exercise, do as much as they are able to do.
Two arguments often made for the oral statement of problems by the teacher are that problems so put are better understood, especially in the grades up through the fifth, and that the problems are more likely to be genuine and related to the life the pupils know. When these statements are true, the first is a still better argument for having the pupils read the problemsaided by the teacher's oral statement of them. For the difficulty is largely that the pupils cannot read well enough; and it is better to help them to surmount the difficulty rather than simply evade it. The second is not an argument for oralnessversuswrittenness, but for good problemsversusbad; the teacher who makes up such good problems should, in fact, take special care to write them down for later use, which may be by voice or by the blackboard or by printed sheet, as is best.
Dewey, and others following him, have emphasized the desirability of having pupils do their work as active seekers, conscious of problems whose solution satisfies some real need of their own natures. Other things being equal, it is unwise, they argue, for pupils to be led along blindfold as it were by the teacher and textbook, not knowing where they are going or why they are going there. They ought rather to have some living purpose, and be zealous for its attainment.
This doctrine is in general sound, as we shall see, but it is often misused as a defense of practices which neglect the formation of fundamental habits, or as a recommendation to practices which are quite unworkable under ordinary classroom conditions. So it seems probable that its nature and limitations are not thoroughly known, even to its followers, and that a rather detailed treatment of it should be given here.
Consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained, or care about attaining the end (well or ill understood) is well spent.
It is well for a pupil who has learned (1) the meanings of the numbers one to ten, (2) how to count a collection of ten or less, and (3) how to measure in inches a magnitude of ten, nine, eight inches, etc., to be confronted with the problem of true adding without counting or measuring, as in 'hidden' addition and measurement by inference. For example, the teacher has three pencils counted and put under a book; has two more counted and put under the book; and asks, "How many pencils are there under the book?" Answers, when obtained, are verified or refuted by actual counting and measuring.
The time here is well spent because the children can do the necessary thinking if the tasks are well chosen; because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding, that is, thinking of the two numbers and inferring their sum; and further, because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it.
The manipulation of short multiplication may be introduced by confronting the pupils with such problems as, "How to tell how many Uneeda biscuit there are in four boxes, by opening only one box." Correct solutions by addition should be accepted. Correct solutions by multiplication, if any gifted children think of this way, should be accepted, even if the children cannot justify their procedure. (Inferring the manipulation from the place-values of numbers is beyond all save the most gifted and probably beyond them.) Correct solution by multiplication by some childwho happens to have learned it elsewhere should be accepted. Let the main proof of the trustworthiness of the manipulation be by measurement and by addition. Proof by the stock arguments from the place-values of numbers may also be used. If no child hits on the manipulation in question, the problem of finding the lengthwithoutadding may be set. If they still fail, the problem may be made easier by being put as "4 times 22 gives the answer. Write down what you think 4 times 22 will be." Other reductions of the difficulty of the problem may be made, or the teacher may give the answer without very great harm being done. The important requirement is that the pupils should be aware of the problem and treat the manipulation as a solution of it, not as a form of educational ceremonial which they learn to satisfy the whims of parents and teachers. In the case of any particular class a situation that is more appealing to the pupils' practical interests than the situation used here can probably be devised.
The time spent in this way is well spent (1) because all but the very dull pupils can solve the problem in some way, (2) because the significance of the manipulation as an economy over addition is worth bringing out, and (3) because there is no way of beginning training in short multiplication that is much better.
In the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square inches in 35 square feet, or the number of days in 32 years, or whatever similar problem can be brought up so as to be felt as a problem.
Suppose that it is the 35 square feet. Solutions by (5 × 144) + (30 × 144), however arranged, or by (10 × 144) + (10 × 144) + (10 × 144) + (5 × 144), or by 3500 + (35 × 40) +(35 × 4), or by 7 × (5 × 144), however arranged, should all be listed for verification or rejection. The pupils need not be required to justify their procedures by a verbal statement. Answers like 432,720, or 720,432, or 1152, or 4220, or 3220 should be listed for verification or rejection. Verification may be by a mixture of short multiplication and objective work, or by a mixture of short multiplication and addition, or by addition abbreviated by taking ten 144s as 1440, or even (for very stupid pupils) by the authority of the teacher. Or the manipulation in cases like 53 × 9 or 84 × 7 may be verified by the reverse short multiplication. The deductive proof of the correctness of the manipulation may be given in whole or in part in connection with exercises like