AA Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down and $21 a month for only a year and a half.No interestto pay. We ask you to pay only for the piano and allow you plenty of time.BWe offer the well-known D.C.K. Piano for $390. $50 cash and $20 a month thereafter. Regular interest at 6%. The interest soon is reduced to less than $1 a month.CThe D.C.K. Piano. Special Offer, $375, cash. Compare our prices with those of any reliable firm.
A
A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down and $21 a month for only a year and a half.No interestto pay. We ask you to pay only for the piano and allow you plenty of time.
B
We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a month thereafter. Regular interest at 6%. The interest soon is reduced to less than $1 a month.
C
The D.C.K. Piano. Special Offer, $375, cash. Compare our prices with those of any reliable firm.
If you consider chiefly the "only," "No interest to pay," "only," and "plenty of time" in offer A, attaching much weight to them and little to the thought, "How much will $50 plus(18 × $21)be?", you will probably decide wrongly.
The situations of life are often complicated by many elements of little or even of no relevance to the correct solution. The offerer of A may belong to your church; your dearest friend may urge you to accept offer B; you may dislike to talk with the dealer who makes offer C; you may have a prejudice against owing money to a relative; that prejudice may be wise or foolish; you may have a suspicion that the B piano is shopworn; that suspicion may be well-founded or groundless; the salesman for C says, "You don't want your friends to say that you bought on the installment plan. Only low-class persons do that," etc. The statement of arithmetical problems in school usually assists the pupil to the extent of ruling out all save definitely quantitative elements,and of ruling out all quantitative elements except those which should be considered. The first of the two simplifications is very beneficial, on the whole, since otherwise there might be different correct solutions to a problem according to the nature and circumstances of the persons involved. The second simplification is often desirable, since it will often produce greater improvement in the pupils, per hour of time spent, than would be produced by the problems requiring more selection. It should not, however, be a universal custom; for in that case the pupils are tempted to think that in every problem they must use all the quantities given, as one must use all the pieces in a puzzle picture.
It is obvious that the elements selected must not only be right but also be in the right relations to one another. For example, in the problems below, the 6 must be thought of in relation to a dozen and as being half of a dozen, and also as being 6 times 1. 1 must be mentally tied to "each." The 6 as half of a dozen must be related to the $1.00, $1.60, etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc.
Buying in QuantityThese are a grocer's prices for certain things by the dozen and for a single one. He sells a half dozen at half the price of a dozen. Find out how much you save by buying 6 all at one time instead of buying them one at a time.
Buying in Quantity
These are a grocer's prices for certain things by the dozen and for a single one. He sells a half dozen at half the price of a dozen. Find out how much you save by buying 6 all at one time instead of buying them one at a time.
Doz.Each1.Evaporated Milk$1.00$.092.Puffed Rice1.60.143.Puffed Wheat1.10.104.Canned Soup1.90.175.Sardines1.80.166.Beans (No. 2 cans)1.50.137.Pork and Beans1.70.158.Peas (No. 2 cans)1.40.129.Tomatoes (extra cans)3.20.2810.Ripe olives (qt. cans)7.20.65
It is obvious also that in such arithmetical work as wehave been describing, the pupil, to be successful, must 'think things together.' Many bonds must coöperate to determine his final response.
As a preface to reasoning about a problem we often have the discovery of the problem and the classification of just what it is, and as a postscript we have the critical inspection of the answer obtained to make sure that it is verified by experiment or is consistent with known facts. During the process of searching for, selecting, and weighting facts, there may be similar inspection and validation, item by item.
The pedagogy of the past made two notable errors in practice based on two errors about the psychology of reasoning. It considered reasoning as a somewhat magical power or essence which acted to counteract and overrule the ordinary laws of habit in man; and it separated too sharply the 'understanding of principles' by reasoning from the 'mechanical' work of computation, reading problems, remembering facts and the like, done by 'mere' habit and memory.
Reasoning or selective, inferential thinking is not at all opposed to, or independent of, the laws of habit, but really is their necessary result under the conditions imposed by man's nature and training. A closer examination of selective thinking will show that no principles beyond the laws of readiness, exercise, and effect are needed to explain it; that it is only an extreme case of what goes on in associative learning as described under the 'piecemeal' activity of situations; and that attributing certain features of learning to mysterious faculties of abstraction or reasoning gives no real help toward understanding or controlling them.
It is true that man's behavior in meeting novel problems goes beyond, or even against, the habits represented by bonds leading from gross total situations and customarily abstracted elements thereof. One of the two reasons therefor, however, is simply that the finer, subtle, preferential bonds with subtler and less often abstracted elements go beyond, and at times against, the grosser and more usual bonds. One set is as much due to exercise and effect as the other. The other reason is that in meeting novel problems the mental set or attitude is likely to be one which rejects one after another response as their unfitness to satisfy a certain desideratum appears. What remains as the apparent course of thought includes only a few of the many bonds which did operate, but which, for the most part, were unsatisfying to the ruling attitude or adjustment.
Successful responses to novel data, associations by similarity and purposive behavior are in only apparent opposition to the fundamental laws of associative learning. Really they are beautiful examples of it. Man's successful responses to novel data—as when he argues that the diagonal on a right triangle of 796.278 mm. base and 137.294 mm. altitude will be 808.022 mm., or that Mary Jones, born this morning, will sometime die—are due to habits, notably the habits of response to certain elements or features, under the laws of piecemeal activity and assimilation.
Nothing is less like the mysterious operations of a faculty of reasoning transcending the laws of connection-forming, than the behavior of men in response to novel situations. Let children who have hitherto confronted only such arithmetical tasks, in addition and subtraction with one- and two-place numbers and multiplication with one-place numbers, as those exemplified in the first line below, be told to do the examples shown in the second line.
AddAddAddSubt.Subt.MultiplyMultiplyMultiply85—3724—35682319—85—3724—85—97—63—MultiplyMultiplyMultiply3223—4322—3426—
They will add the numbers, or subtract the lower from the upper number, or multiply3 × 2and2 × 3,etc., getting 66, 86, and 624, or respond to the element of 'Multiply' attached to the two-place numbers by "I can't" or "I don't know what to do," or the like; or, if one is a child of great ability, he may consider the 'Multiply' element and the bigness of the numbers, be reminded by these two aspects of the situation of the fact that
'99 multiply'—
gave only 81, and that
'1010 multiply'——
gave only 100, or the like; and so may report an intelligent and justified "I can't," or reject the plan of3 × 2and2 × 3,with 66, 86, and 624 for answers, as unsatisfactory. What the children will do will, in every case, be a product of the elements in the situation that are potent with them, the responses which these evoke, and the further associates which these responses in turn evoke. If the child were one of sufficient genius, he might infer the procedure to be followed as a result of his knowledge of the principles of decimal notation and the meaning of 'Multiply,' responding correctly to the 'place-value' element of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties; but if he did thus invent the shorthand addition of a collection of twenty-three collections, each of 32 units, he would still do it by the operation of bonds, subtle but real.
Association by similarity is, as James showed long ago, simply the tendency of an element to provoke the responses which have been bound to it.abcdeleads tovwxyzbecauseahas been bound tovwxyzby original nature, exercise, or effect.
Purposive behavior is the most important case of the influence of the attitude or set or adjustment of an organism in determining (1) which bonds shall act, and (2) which results shall satisfy. James early described the former fact, showing that the mechanism of habit can give the directedness or purposefulness in thought's products, provided that mechanism includes something paralleling the problem, the aim, or need, in question.
The second fact, that the set or attitude of the man helps to determine which bonds shall satisfy, and which shall annoy, has commonly been somewhat obscured by vague assertions that the selection and retention is of what is "in point," or is "the right one," or is "appropriate," or the like. It is thus asserted, or at least hinted, that "the will," "the voluntary attention," "the consciousness of the problem," and other such entities are endowed with magic power to decide what is the "right" or "useful" bond and to kill off the others. The facts are that in purposive thinking and action, as everywhere else, bonds are selected and retained by the satisfyingness, and are killed off by the discomfort, which they produce; and that the potency of the man's set or attitude to make this satisfy and that annoy—to put certain conduction-units in readiness to act and others in unreadiness—is in every way as important as its potency to set certain conduction-units in actual operation.
Reasoning is not a radically different sort of force operating against habit but the organization and coöperation of many habits, thinking facts together. Reasoning is notthe negation of ordinary bonds, but the action of many of them, especially of bonds with subtle elements of the situation. Some outside power does not enter to select and criticize; the pupil's own total repertory of bonds relevant to the problem is what selects and rejects. An unsuitable idea is not killed off by someactus purusof intellect, but by the ideas which it itself calls up, in connection with the total set of mind of the pupil, and which show it to be inadequate.
Almost nothing in arithmetic need be taught as a matter of mere unreasoning habit or memory, nor need anything, first taught as a principle, ever become a matter of mere habit or memory.5 × 4 = 20should not be learned as an isolated fact, nor remembered as we remember that Jones' telephone number is 648 J 2. Almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come. The use of this organized hierarchy of habits to solve novel problems is reasoning.
The activities essential to acquiring ability in arithmetic can rely on little in man's instinctive equipment beyond the purely intellectual tendencies of curiosity and the satisfyingness of thought for thought's sake, and the general enjoyment of success rather than failure in an enterprise to which one sets oneself. It is only by a certain amount of artifice that we can enlist other vehement inborn interests of childhood in the service of arithmetical knowledge and skill. When this can be done at no cost the gain is great. For example, marching in files of two, in files of three, in files of four, etc., raising the arms once, two times, three times, showing a foot, a yard, an inch with the hands, and the like are admirable because learning the meanings of numbers thus acquires some of the zest of the passion for physical action. Even in late grades chances to make pictures showing the relations of fractional parts, to cut strips, to fold paper, and the like will be useful.
Various social instincts can be utilized in matches after the pattern of the spelling match, contests between rows, certain number games, and the like. The scoring of boththe play and the work of the classroom is a useful field for control by the teacher of arithmetic.
Hunt ['12] has noted the more important games which have some considerable amount of arithmetical training as a by-product and which are more or less suitable for class use. Flynn ['12] has described games, most of them for home use, which give very definite arithmetical drill, though in many cases the drills are rather behind the needs of children old enough to understand and like the game itself.
It is possible to utilize the interests in mystery, tricks, and puzzles so as to arouse a certain form of respect for arithmetic and also to get computational work done. I quote one simple case from Miss Selkin's admirable collection ['12, p. 69 f.]:—
I.ADDITION
"We must admit that there is nothing particularly interesting in a long column of numbers to be added. Let the teacher, however, suggest that he can write the answer at sight, and the task will assume a totally different aspect."A very simple number trick of this kind can be performed by making use of the principle of complementary addition. The arithmetical complement of a number with respect to a larger number is the difference between these two numbers. Most interesting results can be obtained by using complements with respect to 9."The children may be called upon to suggest several numbers of two, three, or more digits. Below these write an equal number of addends and immediately announce the answer. The children, impressed by this apparently rapid addition, will set to work enthusiastically to test the results of this lightning calculation."Example:—
"We must admit that there is nothing particularly interesting in a long column of numbers to be added. Let the teacher, however, suggest that he can write the answer at sight, and the task will assume a totally different aspect.
"A very simple number trick of this kind can be performed by making use of the principle of complementary addition. The arithmetical complement of a number with respect to a larger number is the difference between these two numbers. Most interesting results can be obtained by using complements with respect to 9.
"The children may be called upon to suggest several numbers of two, three, or more digits. Below these write an equal number of addends and immediately announce the answer. The children, impressed by this apparently rapid addition, will set to work enthusiastically to test the results of this lightning calculation.
"Example:—
357}999682A× 37932997642}317B206
"Explanation:—The addends in group A are written down at random or suggested by the class. Those in group B are their complements. To write the first number in group B we look at the first number in group A and, starting at the left write 6, the complement of 3 with respect to 9; 4, the complement of 5; 2, the complement of 7. The second and third addends in group B are derived in the same way. Since we have three addends in each group, the problem reduces itself to multiplying 999 by 3, or to taking 3000 − 3. Any number of addends may be used and each addend may consist of any number of digits."
"Explanation:—The addends in group A are written down at random or suggested by the class. Those in group B are their complements. To write the first number in group B we look at the first number in group A and, starting at the left write 6, the complement of 3 with respect to 9; 4, the complement of 5; 2, the complement of 7. The second and third addends in group B are derived in the same way. Since we have three addends in each group, the problem reduces itself to multiplying 999 by 3, or to taking 3000 − 3. Any number of addends may be used and each addend may consist of any number of digits."
Respect for arithmetic as a source of tricks and magic is very much less important than respect for its everyday services; and computation to test such tricks is likely to be undertaken zealously only by the abler pupils. Consequently this source of interest should probably be used only sparingly, and perhaps the teacher should give such exhibitions only as a reward for efficiency in the regular work. For example, if the work for a week is well done in four days the fifth day might be given up to some semi-arithmetical entertainment, such as the demonstration of an adding machine, the story of primitive methods of counting, team races in computation, an exhibition of lightning calculation and intellectual sleight-of-hand by the teacher, or the voluntary study of arithmetical puzzles.
The interest in achievement, in success, mentioned above is stronger in children than is often realized and makes advisable the systematic use of the practice experiment as a method of teaching much of arithmetic. Children who thus compete with their own past records, keeping an exact score from week to week, make notable progress and enjoy hard work in making it.
Negatively the difficulty of the work that pupils should be expected to do is conditioned by the gradual maturingof their capacities. Other things being equal, the common custom of reserving hard things for late in the elementary school course is, of course, sound. It seems probable that little is gained by using any of the child's time for arithmetic before grade 2, though there are many arithmetical facts that he can learn in grade 1. Postponement of systematic work in arithmetic to grade 3 or even grade 4 is allowable if better things are offered. With proper textbooks and oral and written exercises, however, a child in grades 2 and 3 can spend time profitably on arithmetical work. When all children can be held in school through the eighth grade it does not much matter whether arithmetic is begun early or late. If, however, many children are to leave in grades 5 and 6 as now, we may think it wise to provide somehow that certain minima of arithmetical ability be given them.
There are, so far as is known, no special times and seasons at which the human animal by inner growth is specially ripe for one or another section or aspect of arithmetic, except in so far as the general inner growth of intellectual powers makes the more abstruse and complex tasks suitable to later and later years.
Indeed, very few of even the most enthusiastic devotees of the recapitulation theory or culture-epoch theory have attempted to apply either to the learning of arithmetic, and Branford is the only mathematician, so far as I know, who has advocated such application, even tempered by elaborate shiftings and reversals of the racial order. He says:—
"Thus, for each age of the individual life—infancy, childhood, school, college—may be selected from the racial history the most appropriate form in which mathematical experience can be assimilated. Thus the capacity of the infant and early childhood is comparable with the capacity of animal consciousness and primitive man. The mathematics suitable to later childhood and boyhood (and, of course, girlhood) is comparable with Archæan mathematics passing on through Greek and Hindu to mediæval European mathematics; while the student is become sufficiently mature to begin the assimilation of modern and highly abstract European thought. The filling in of details must necessarily be left to the individual teacher, and also, within some such broadly marked limits, the precise order of the marshalling of the material for each age. For, though, on the whole, mathematical development has gone forward, yet there have been lapses from advances already made. Witness the practical world-loss of much valuable Hindu thought, and, for long centuries, the neglect of Greek thought: witness the world-loss of the invention by the Babylonians of the Zero, until re-invented by the Hindus, passed on by them to the Arabs, and by these to Europe."Moreover, many blunders and false starts and false principles have marked the whole course of development. In a phrase, rivers have their backwaters. But it is precisely the teacher's function to avoid such racial mistakes, to take short cuts ultimately discovered, and to guide the young along the road ultimately found most accessible with such halts and retracings—returns up side-cuts—as the mental peculiarities of the pupils demand."All this, the practical realization of the spirit of the principle, is to be wisely left to the mathematical teacher, familiar with the history of mathematical science and with the particular limitations of his pupils and himself." ['08, p. 245.]
"Thus, for each age of the individual life—infancy, childhood, school, college—may be selected from the racial history the most appropriate form in which mathematical experience can be assimilated. Thus the capacity of the infant and early childhood is comparable with the capacity of animal consciousness and primitive man. The mathematics suitable to later childhood and boyhood (and, of course, girlhood) is comparable with Archæan mathematics passing on through Greek and Hindu to mediæval European mathematics; while the student is become sufficiently mature to begin the assimilation of modern and highly abstract European thought. The filling in of details must necessarily be left to the individual teacher, and also, within some such broadly marked limits, the precise order of the marshalling of the material for each age. For, though, on the whole, mathematical development has gone forward, yet there have been lapses from advances already made. Witness the practical world-loss of much valuable Hindu thought, and, for long centuries, the neglect of Greek thought: witness the world-loss of the invention by the Babylonians of the Zero, until re-invented by the Hindus, passed on by them to the Arabs, and by these to Europe.
"Moreover, many blunders and false starts and false principles have marked the whole course of development. In a phrase, rivers have their backwaters. But it is precisely the teacher's function to avoid such racial mistakes, to take short cuts ultimately discovered, and to guide the young along the road ultimately found most accessible with such halts and retracings—returns up side-cuts—as the mental peculiarities of the pupils demand.
"All this, the practical realization of the spirit of the principle, is to be wisely left to the mathematical teacher, familiar with the history of mathematical science and with the particular limitations of his pupils and himself." ['08, p. 245.]
The latitude of modification suggested by Branford reduces the guidance to be derived from racial history to almostnil. Also it is apparent that the racial history in the case of arithmetical achievement is entirely a matter of acquisition and social transmission. Man's original nature is destitute of all arithmetical ideas. The human germs do not know even that one and one make two!
A scientific plan for teaching arithmetic would begin with an exact inventory of the knowledge and skill which the pupils already possessed. Our ordinary notions of what a child knows at entrance to grade 1, or grade 2, or grade 3,and of what a first-grade child or second-grade child can do, are not adequate. If they were, we should not find reputable textbooks arranging to teach elaborately facts already sufficiently well known to over three quarters of the pupils when they enter school. Nor should we find other textbooks presupposing in their first fifty pages a knowledge of words which not half of the children can read even at the end of the 2 B grade.
We do find just such evidence that ordinary ideas about the abilities of children at the beginning of systematic school training in arithmetic may be in gross error. For example, a reputable and in many ways admirable recent book has fourteen pages of exercises to teach the meaning of two and the fact that one and one make two! As an example of the reverse error, consider putting all these words in the first twenty-five pages of a beginner's book:—absentees, attendance, blanks, continue, copy, during, examples, grouped, memorize, perfect, similar, splints, therefore, total!
Little, almost nothing, has been done toward providing an exact inventory compared with what needs to be done. We may note here (1) the facts relevant to arithmetic found by Stanley Hall, Hartmann, and others in their general investigations of the knowledge possessed by children at entrance to school, (2) the facts concerning the power of children to perceive differences in length, area, size of collection, and organization within a collection such as is shown in Fig. 24, and certain facts and theories about early awareness of number.
In the Berlin inquiry of 1869, knowledge of the meaning of two, three, and four appeared in 74, 74, and 73 percent of the children upon entrance to school. Some of those recorded as ignorant probably really knew, but failed to understand that they were expected to reply or were shy. Only85 percent were recorded as knowing their fathers' names. Seven eighths as many children knew the meanings of two, three, and four as knew their fathers' names. In a similar but more careful experiment with Boston children in September, 1880, Stanley Hall found that 92 percent knew three, 83 percent knew four, and 71½ percent knew five. Three was known about as well as the color red; four was known about as well as the color blue or yellow or green. Hartmann ['90] found that two thirds of the children entering school in Annaberg could count from one to ten. This is about as many as knew money, or the familiar objects of the town, or could repeat words spoken to them.
Fig. 24.Fig. 24.—Objective presentation.
In the Stanford form of the Binet tests counting four pennies is given as an ability of the typical four-year-old. Counting 13 pennies correctly in at least one out of twotrials, and knowing three of the four coins,—penny, nickel, dime, and quarter,—are given as abilities of the typical six-year-old.
We know that educated adults can tell how many lines or dots, etc., they see in a single glance (with an exposure too short for the eye to move) up to four or more, according to the clearness of the objects and their grouping. For example, Nanu ['04] reports that when a number of bright circles on a dark background are shown to educated adults for only .033 second, ten can be counted when arranged to form a parallelogram, but only five when arranged in a row. With certain groupings, of course, their 'perception' involves much inference, even conscious addition and multiplication. Similarly they can tell, up to twenty and beyond, the number of taps, notes, or other sounds in a series too rapid for single counting if the sounds are grouped in a convenient rhythm.
These abilities are, however, the product of a long and elaborate learning, including the learning of arithmetic itself. Elementary psychology and common experience teach us that the mere observation of groups or quantities, no matter how clear their number quality appears to the person who already knows the meanings of numbers, does not of itself create the knowledge of the meanings of numbers in one who does not. The experiments of Messenger ['03] and Burnett ['06] showed that there is no direct intuitive apprehension even of two as distinct from one. We have tolearnto feel the two touches or see the two dots or lines as two.
We do not know by exact measurements the growth in children of this ability to count or infer the number of elements in a collection seen or series heard. Still less do weknow what the growth would be without the influence of school training in counting, grouping, adding, and multiplying. Many textbooks and teachers seem to overestimate it greatly. Not all educated adults can, apart from measurement, decide with surety which of these lines is the longer, or which of these areas is the larger, or whether this is a ninth or a tenth or an eleventh of a circle.
Children upon entering school have not been tested carefully in respect to judgments of length and area, but we know from such studies as Gilbert's ['94] that the difference required in their case is probably over twice that required for children of 13 or 14. In judging weights, for example, a difference of 6 is perceived as easily by children 13 to 15 years of age as a difference of 15 by six-year-olds.
A teacher who has adult powers of estimating length or area or weight and who also knows already which of the twois longer or larger or heavier, may use two lines to illustrate a difference which they really hide from the child. It is unlikely, for example, that the first of these lines ______________ ________________ would be recognized as shorter than the second by every child in a fourth-grade class, and it is extremely unlikely that it would be recognized as being7â„8of the length of the latter, rather than3â„4of it or5â„6of it or9â„10of it or11â„12of it. If the two were shown to a second grade, with the question, "The first line is 7. How long is the other line?" there would be very many answers of 7 or 9; and these might be entirely correct arithmetically, the pupils' errors being all due to their inability to compare the lengths accurately.
The quantities used should be such that their mere discrimination offers no difficulty even to a child of blunted sense powers. If7â„8and 1 are to be compared,AandBare not allowable.C,D, andEare much better.
Teachers probably often underestimate or neglect the sensory difficulties of the tasks they assign and of the material they use to illustrate absolute and relative magnitudes. The result may be more pernicious when the pupils answer correctly than when they fail. For their correctanswering may be due to their divination of what the teacher wants; and they may call a thing an inch larger to suit her which does not really seem larger to them at all. This, of course, is utterly destructive of their respect for arithmetic as an exact and matter-of-fact instrument. For example, if a teacher drew a series of lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this form— _____ ________ and asked, "This is 20 inches long, how long is this?" she might, after some errors and correction thereof, finally secure successful response to all the lines by all the children. But their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be actually damaged by the exercise.
There has been some disagreement concerning the origin of awareness of number in the individual, in particular concerning the relative importance of the perception of how-many-ness and that of how-much-ness, of the perception of a defined aggregate and the perception of a defined ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and Decroly and Degand ['12].)
The chief facts of significance for practice seem to be these: (1) Children with rare exceptions hear the namesone,two,three,four,half,twice,two times,more,less,as many as,again,first,second, andthird, long before they have analyzed out the qualities and relations to which these words refer so as to feel them at all clearly. (2) Their knowledge of the qualities and relations is developed in the main in close association with the use of these words to the child and by the child. (3) The ordinary experiences of the first five years so develop in the child awareness of the 'how many somethings' in various groups, of the relative magnitudes of two groups or quantities of any sort, and of groups and magnitudes asrelated to others in a series. For instance, if fairly gifted, a child comes, by the age of five, to see that a row of four cakes is an aggregate of four, seeing each cake as a part of the four and the four as the sum of its parts, to know that two of them are as many as the other two, that half of them would be two, and to think, when it is useful for him to do so, of four as a step beyond three on the way to five, or to think of hot as a step from warm on the way to very hot. The degree of development of these abilities depends upon the activity of the law of analysis in the individual and the character of his experiences.
(4) He gets certain bad habits of response from the ambiguity of common usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears his parents or older children or others count pennies or rolls or eggs by saying one, two, three, four, and so on. He himself is perhaps misled into so counting. Thus the names properly belonging to a series of aggregations varying in amount come to be to him the names of the positions of the parts in a counted whole. This happens especially with numbers above 3 or 4, where the correct experience of the number as a name for the group has rarely been present. This attaching to the cardinal numbers above three or four the meanings of the ordinal numbers seems to affect many children on entrance to school. The numbering of pages in books, houses, streets, etc., and bad teaching of counting often prolong this error.
(5) He also gets the habit, not necessarily bad, but often indirectly so, of using many names such as eight, nine, ten, eleven, fifteen, a hundred, a million, without any meaning.
(6) The experiences of half, twice, three times as many, three times as long, etc., are rarer; even if they were not, they would still be less easily productive of the analysis of the proper abstract element than are the experiences oftwo, three, four, etc., in connection with aggregates of things each of which is usually called one, such as boys, girls, balls, apples. Experiences of the names, two, three, and four, in connection with two twos, two threes, two fours, are very rare.
Hence, the names, two, three, etc., mean to these children in the main, "one something and one something," "one something usually called one, and one something usually called one, and another something usually called one," and more rarely and imperfectly "two times anything," "three times anything," etc.
With respect to Mr. Phillips' emphasis of the importance of the series-idea in children's minds, the matters of importance are: first, that the knowledge of a series of number names in order is of very little consequence to the teaching of arithmetic and of still less to the origin of awareness of number. Second, the habit of applying this series of words in counting in such a way that 8 is associated with the eighth thing, 9 with the ninth thing, etc., is of consequence because it does so much mischief. Third, the really valuable idea of the number series, the idea of a series of groups or of magnitudes varying by steps, is acquired later, as a result, not a cause, of awareness of numbers.
With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers should be emphasized in schools, not because it is the main origin of the child's awareness of number, but because it isnot, and because the ordinary practical issues of child life donotadequately stimulate its action. It also seems both more economical and more scientific to introduce it through multiplication, division, and fractions rather than to insist that 4 and 5 shall from the start mean 4 or 5 times anything that is called 1, for instance, that 8 inches shall be called 4 two-inches, or 10 cents, 5 two-cents.If I interpret Professor Dewey's writings correctly, he would agree that the use of inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful, spoonful, cent, nickel, dime, and dollar gives a sufficient range of units for the first two school years. Teaching the meanings of ½ of 4, ½ of 6, ½ of 8, ½ of 10, ½ of 20,1â„3of 6,1â„3of 9,1â„3of 30, ¼ of 8, two 2s, five 2s, and the like, in early grades, each in connection with many different units of measure, provides a sufficient assurance that numbers will connect with relationships as well as with collections.
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 (32-1½), and drawing for both (16½-6). Writing (6½-4), arithmetic (14½-13), history (9-6½), reading (8½-8), and singing (6-7½) 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:—
Arithmetic327 liked greatly,96 disliked greatly.History164 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 studieslisted below. He reports the results in the following table of percents:
Top Thirdof Studies for InterestMiddle Thirdof Studies for InterestLowest Thirdof Studies for InterestDrawing78202Manual Subjects66268History642412Reading53389Singing324820Drill205525Arithmetic165331Science233740Nature Study163648Dictation45739Composition182854Scripture43858Recitation92368Geography42472Grammar—694
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.
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.