Fractions II Based upon the Speer Arithmetic L. W. COLWELL, Principal Linne School Chicago Ill. O evident is it that arithmetical processes are needed in daily life to enable us to adapt means to end and husband our resources, that it has become customary, on beginning the study of mathematics, to turn at once to its abstract technique, as these processes may be termed. So long has this practice obtained and so justifiable does it seem that we have accepted it heretofore unquestioningly and styled the processes fundamental. Regarding them in this light, we are thereby predisposed to look with suspicion upon an attempt to substitute for the study of operations with figures, that which is really fundamental, viz.: relations of quantity. Yet we know very well that the things of prime importance are not letters, figures, or signs, but ideas; moreover it is written in the words of those we dare not dispute that thinking is comparing, that numerical symbols are purely representative, and that number cannot furnish a basis upon which to develop the science of mathematics. (See quotations in Primary Book.) unlar. Fig. 1 m Direct study of quantity reveals the real fundamentals of mathematics as products of comparative magnitudes. Convenient processes of reckoning quantity with figures are needed, but their use should develop, like all habits, from internal pressure for expression, not from external imposition nor dictation. Calculating can be done by machines; education deals with brains. Few magnitudes can be perceived entire, yet upon the few ratios that can be directly sensed, depends the estimate of all those vast or subtle units that the hand cannot reach for nor the eye take in. (See article in May issue.) Upon the seen rests the unseen. "The importance of bringing simple basic ratios definitely into the consciousness is better understood when we look beyond them. By means of perceived relations we must pass to the inferring of relations." Primary Book "The higher processes in mathematics lie at the very foundation of the subject." Sylvester "Equations constitute the true starting point of arithmetic." Comte "Upon these equations made known by the activity of mind upon the magnitudes themselves, all mathematical deduction depends." Primary Book 7 Fig. 2 Sums and Differences equals the sum of what units? (Fig. 1.) (Give several pairs.) 8 The sum of n and b (Fig. 1) equals the sum of what? 9 Two times / equals the sum of what? (Fig. 3.) 10 c and what unit equal p? (Fig. 1) d and what nnit equal 3 times c? (Fig. 2.) 6 and what unit equal the sum of 10 and 2? (Fig. 5.) II is how much more than c? (Fig. 1) What unit equals the difference between (Fig. 5.) What unit equals m less b? Names of Terms How much more than ? and p? between 6 and 10? 12 Use the integral names in Fig. 5 like the letter names in the other figures. 13 Draw Fig. 5 on blackboard; instead of writing in the names 2, 4, 6, etc., write in the names, 3, 6, 9, etc. Give ratios, find missing terms, sums, differences, etc., as above. 14 Use other sets of names, similarly; 4, 8, 12, etc. 20, 40, 60, etc. 25, 50, 75, etc. 32, 64, 96, etc. Fix these names by applying them to other magnitudes showing the same ratio. Constantly vary the forms but preserve the same ratios. Return over and over with new sets of units to be named by the same series of names until the names become significant of certain relations of magnitude. 8 Observe the relative magnitude of and i. e., Compare with . How many 's in? Or, better, what part of a & equals a? Applications 9 A bottle holds a pt. It is full of water. How much water is there in the bottle? What is the quantity relation of the water to pint? is the ratio of what to pt.? Then what part of a pint of water is there in the bottle? 10 If $25 is of the cost of my horse, what is the cost? What is the size relation of to? 2 is the ratio of what to $25? II Represent by drawings or by blocks units that may be named I, }, 1, 1, 1, 1, §, 1. Give their ratios as in problem 3 of Exercise II. 12 is the ratio of what to? is the ratio of what? 13 At the rate of yd. for 27c what is the cost of yd. of ribbon? How much is as compared with? (Compare the magnitudes.) Keep distinctly in mind the fact that the terms, †, I, I, etc., must not unvaryingly call up any particular size or special units. It is imperative that the experience of the children should connect them with such a variety of forms that the particular forms shall sink out of sight and the terms shall stand for relations that can be represented in new forms freely selected or created. I 2 What is the ratio of m to n (Fig. 1)? of n to m? 3 What is the magnitude of as compared with the magnitude of 1, (Fig. 6)? If the names in Fig. 6 be changed so as to make become 1, what must I become? What must it become if becomes 1? 4 What is the ratio of 1 to ? What is the quantity I as compared with ? with ? with? with 2? with 4? with 7? 5 What is the ratio of a yard to of a yard. If yard cloth cost $1.05, what cost I yard? Solution: I yard bears what relation to of a yard? It should cost what fraction as much? 6 When 2 dozen eggs are worth 40c, how much are eggs a dozen? Solution: How much is I as compared to 23 or? What part of 40c then is the value of a dozen eggs? 7 At $ a pound, how much citron can be bought for $2 for $61? Solution: What is the ratio of $1 to $? How much can be bought for $1? Then how much for $2? for $63 or $? 8 How many in 2? 3 and what = 4? Then 3 and how much = Exercise III 41? 2 Call I and name the units in Fig. I as above. Call 61. , tc. 3 Deal with the other magnitudes similarly. If 8 in Fig. 5 represents a square yard of oilcloth what do th other units represent? 21 4 What other way of expressing {, 1, 1, 4, 4, 12, 41, 18, etc.; 1, 13, 23, 74, 251}, etc.? Fig. 6 What is the ratio of y tot (Fig, 3)? of of y to1⁄2 of ? of of y to of ? of of y tot of t? of of y to an of t? of 17 of y to 27 oft? AB What is true of the comparative value of all these ratios? Ans. They are equal. 2 Express in simplest form the ratio of 10 to 25. Compare the ratio of of 10 to of 25 to the ratio of 10 to 25. Are these ratios equal? What is the ratio of 2 to 5? of 10 to 25? 3 What units are exact measures of n? Ans. b, r, and c. (Fig. 1.) 4 What units are common measures of m and n? Ans. b and r. 5 What units will measure both 8 and 12 (Fig 5)? What is the greatest common measure of 8 and 12? It is hoped that the foregoing exercises indicate in some degree not only the sufficiency of observing ratio for solving all ordinary problems in fractions, but also the necessity of some primary and immediate knowledge of quantity as a basis for inference and operations with figures. Though the reader may miss the familiar nomenclature such as improper fractions, mixed numbers, least common denominator, inverting divisor, etc., it will be remarked that these things appear above, all involved in simple observations of quantity and too closely intermingled to be proper subjects for separate treatment. Child-Study in the Home The average American child needs tranquilizing rather than stimulating, and through the ear he may be soothed. The lullaby of the mother has a physiological significance, the soft rhythmical cadences seem to excite pleasurable vibrations in the nerves that by their monotonous recurrence soothe and quiet the infant although he can have no real consciousness of music. As he grows older and the sense of hearing can bear more intense sounds, he seems to delight in them for their very intensity. He enjoys noise for the sake of noise. Now is the time that he can be trained to the appreciation of musical sounds. If left to himself he will be noisy only, if directed he may be made musical. Northwestern Monthly. HERE AND THERE A Klondike School The newspapers say a Boston teacher has gone to Dawson city. She has carried her school-house and all necessary equipments with her, intending to teach all the winter through. They tell us the name is Mrs. L. C. Howland, and that she is a graduate of Harvard Annex. The schoolhouse is a portable building made in sections that can be easily fitted together. If there are no children at the Klondike then she will teach the miners. When a teacher once gets the fever, it never dies out. Teach she must in some form or other, to the end of her days. The Public School Art League It was was organized in the city of Boston in 1892. composed of artists and lovers of art who believed in the educational value of an association with works of art. With the aid of the Boston school committee the League began its labors by decorating two school-rooms, and has pushed steadily forward ever since with a broadening and beneficient influence. Mr. Ernest Fenallosa says of the basic idea of the League: "It seems to me the term art is too often identified by the public with the more special professional mark of making pictures and sculpture. No doubt this is an important portion of art, but it does not give the key to the work of the league, which aims to train the faculty and taste, and educate the young to the uses, the public function, all the discipline even, of art as a whole. The purpose of art education in general is no more to turn the whole nature into professional makers of oil paintings than the general education in music is to train a world of composers. "It is to make more cultivated and well-rounded citizens by stimulating a most important part of their nature, and a desire for a knowledge of beauty. The development of the faculty of taste throughout the community would help to bring about right and helpful living, and is second only in importance to religion." The Beginning of the Movement brightening the rooms of the dingy school, crowded with children, representing the decided foreign element characteristic of that section of the city. From the nature of the environment of the children it seemed imperative to the teachers of the school that something should be done to give them a little glimpse of the beautiful. With the cooperation of Miss Starr and others who became interested in the work, the school-room walls were beautifully tinted, and in every room a picture or a cast was placed. The Woman's Club of Chicago, becoming alive to the import of the subject, formed an association of the members of the educational and literature sections. Other agencies are at work and many of the Chicago schools are tastefully decorated Enterprise and Interest of Chicago Teachers During the first week of last September over three thousand five hundred teachers voluntarily took a week out of their vacations, and contributed money besides, to maintain and attend a series of institutes lasting for five days. At these institutes were lecturers from all parts of the country, and the attendance was large and enthusiastic at every session. In what other city could this scheme be duplicated? The Committee of Sixty appointed last year by the Chicago Institute of Education,. for the promotion of outdoor study of nature, did wonders. Committee on Maps told the teachers where to go. Committee on Syllabi outlined the localities for field work. Committee on Libraries secured a nature study alcove in the John Crerar Library, in which books will be placed for the benefit of students. Committee on Instruction and School Exhibits has presented to the monthly meetings specimens of work done in schools the city nature study in and supplied schools with materials from the suburbs. It has also made it possible for the schools to secure a cheap air pump and aquarium. Committee on City Industries has prepared a list of the industrial and commercial interests in the city. Committee on Transportation looked to secure cheap rates for the children to go to the suburbs. Public Information Committee put newsy accounts in the daily papers as to what was being done in the schools along these lines. of New York City The movement to introduce art into the school-room by bringing children into daily familiarity with examples of The Free Lecture System in the Public Schools good art originated in Manchester, England. The first attempt in this country was made some twenty-odd years ago, in Boston, by a committee of the American Social Science Association. The girls' high school was selected as affording perhaps the most promising conditions for success in such an experiment. The walls of the school hall were painted in terra cotta, and more than fifteen hundred dollars was raised for the purchase of casts of the Parthenon frieze, together with statues, busts and pictures. From time to time additions have been made to the collection, and the graduating classes of the school leave behind them as a memento of their interest various beautiful and appropriate examples of art. Though for nearly twenty years the Girls' High school of Boston stood as the only representative of such an experiment in school-room decoration, it is certain that the subtle influence exerted on the minds and hearts, manners and morals of the young people who have gathered there from year to year has created an atmosphere which is making it possible for similar enterprises to prosper elsewhere. The Chicago Public School Art Society This society has the same object in view as the Art League that of making the school-rooms beautiful in color and filling them with the right sort of pictures and casts. It began with the efforts of a principal of one of Hull house district. She appealed to Miss Ellen Gates Starr, one of the workers of the community, for help in "For some years past the Board of Education of the city of New York has carried on under its immediate auspices a systematic series of free lectures upon various branches of Literature, Science and Art. These lectures have been given in the public school buildings chiefly, although in one or two instances private halls, which had been placed at the disposal of the Board of Education, have been utilized for this purpose. During the last season, which was the seventh winter's work, 1065 lectures were delivered at thirty-four different centres in the city of New York, with an attendance of 426,927. This represents an average of over thirty lectures at each centre, with an average attendance of 400 at each lecture. Of the 1065 lectures, 772 were illustrated by stereopticon views, 52 by scientific experiments and 241 were not illustrated. The subjects upon which lectures were delivered were grouped under the following six heads: Physiology and Hygiene; Natural Science; Travel and Geography; American History and Civics; General History; Art, Literature and Social Science. Most of the lectures were single lectures, but many of the individual lectures were grouped together in such a way as to constitute a series relating to the same general subject. The success of the New York experiment has been so marked that school authorities in all parts of the country have been raising the question whether it is not possible to carry out in the territory subject to their jurisdiction, a somewhat similar plan." Geology in a Snow-storm There is a beautiful and fundamental geology in a snowstorm; we are admitted into Nature's oldest laboratory, and see the working of the law by which the foundations of the material universe were laid - the law or mystery of crystallization. The earth is built upon crystals; the granite rock is only a denser and more compact snow, or a kind of ice that was vapor once and may be vapor again. "Every stone is nothing else but a congealed lump of frozen earth," says Plutarch. By cold and pressure air can be liquified, perhaps solidified. A little more time, a little more heat, and the hills are but April snow-banks. Nature has but forms, the cell and the crystal- the crystal first, the cell last. All organic nature is built up of the cell; all inorganic of the crystal. Cell cell rises the vegetable, rises the animal; upon crystal wedded to and compacted with crystal stretches the earth beneath them. See in the falling snow the old cooling and precipitation, and the shooting, radiating forms that are the architects of planet and globe. We love the sight of the brown and ruddy earth; it is the color of life, while a snow-covered plain is the face of death; yet snow is but the mask of the life-giving rain; it, too, is the friend of the tender, sculpturesque, immaculate, warming, fertilizing snow.-John Burroughs, A Year in the Fields. man Our Defective Alphabet Our vowels are named for their long sounds and are a, e. i, o, and u, which last has a consonant y preceding the long sound. Our consonants are named be, se, de, ef, je, aitch, ja, ka, el, em, en, pe, kyu or cue, ar, es, te, ve, double-yu, eks wi, ze. The more unfortunate names are of h, q, w, and y. Call them he, kwe, we and ye then their force will be shown in their names. Since the spelling books and school dictionaries do not give the names of the letters of the alphabet we see no reason why teachers should not rename at least those four consonants, and the fifth vowel, u should be named oo, as the continental languages have it. Call it yu or you, we add a consonant to it which sticks to it in so many places that reference to a huge dictionary is necessary, too frequently, when we meet it. That some ignoramus, centuries ago, should attach y to u is no reason for our doing so now. The names and uses of all the letters had their origin in the dark ages when mysticism and superfluity were the proofs of wisdom. The result is that no spoken word can be spelled nor written word pronounced with any degree of accuracy without consulting authority. Worcester gives one hundred forty-six observations on the use of the vowels and sixty-one on consonants. The latter includes a very small number of combinations which, were they all counted, would increase the number of rules governing the use of the letters of the alphabet to over three hundred. He did not discover near all of their uses. His own name illustrates his oversight, for orces in English names is pronounced as oos. The long i sound is obtained by no less than nine unnatural spellings, illustrated in the words, my, buy, rye, sigh, die, isle, climb, might, sign, also aisle and knight, but these latter modifications may be to distinguish words rather than the sound of i. Long a is represented by at least ten useless variations; aye, day, date, eight, sail, great, bass, feign, veil and guage are words for illustration. Short e sound is had in nine useless variations as in fair, fare, pear, there, prayer, heir, bury, leopard and less. The words with final are pronounced as though spelled with both short e and u, as, feu, peu, etc. Short u is obtained in the syllables ending with the letters er, or, ir, ur, and yr, the r having no force as a consonant in such cases. These are a few illustrations out of one hundred thirty or more which could be given. For others and for rules for consonants the reader is referred to the introductory articles of any large dictionary. One observation of importance should be noted here, as ortheopists have, until quite recently, overlooked it. The letter r, as a consonant, can never end a syllable except by combining with other consonants, the words more, roar, door, and thousands of others, are pronounced as if written mo-u, ro-u, do-u, etc., giving two syllables. Some people pronounce some such words by rolling or prolonging the first vowel sound, leaving off the short u sound. When combining with other consonants has not the same sound as at the beginning of syllables. It is more nearly a vowel in the former than the latter. Besides w and r we have l, m, and n as the only consonants which generally demand position nearest the vowel, These three admit of only one exception at the end of syllables by taking a preceding r. L always demands the nearest position in beginning syllables while m and n will admit a consonant y or an r to follow, as, myn, pronounced as mu in music, nyu or mra and nra, the latter never appearing in English. Our alphabet requires twenty-six small type letters, twenty-six small script, twenty-six capital type and twentysix capital script or one hundred four characters. Grammar adds to the confusion of the student of English by adding six hundred rules, exceptions, and special applications. The formation of words pronounced alike, spelled differently perhaps, but with strangely contrasted meanings, and then, the ideas with several words each are further complications of our tongue. To read a newspaper with accuracy we must know about ten thousand facts which do not readily yield to classification. It takes child or man hours of study to fix in mind the relations of each entirely new fact coming under his observation.- Sel. Lines showing action TOGETHER The Editor's Page Address the editor at 237 Langdon St., Madison, Wisconsin. The New Year If the proverbial "good resolution" is to be a part of the year's beginning, let it take the form of a determination to judge the motives of the children very slowly and with the greatest care. How can we know why they do these exasperating things "that wear the life out"? We look upon things from one point of view, with one pair of eyes. They see everything from another focus and with other eyes. There is little chance to judge of their reasoning or conclusions that result in seeming neglect and disobedience If it is not a deliberate intention to disobey, but the result of some counter influence we do not see or feel, then it is not a wilful disobedience that need to surprise and irritate us. But whatever it is or is not, be very sure the offense is not against the teacher, as teacher, as often as she may conclude. How it would "clear up" the whole situation when trouble arises, if we could be mind readers or see into the hearts of our children with an X ray accuracy and impartiality. In place of impatience and resentment, the womanly teacher would feel her heart go out to these "trying" children with a great pity and kindliness. Judge slowly, teachers, judge kindly. If there is any real reason why these children are irritating- and there generally is find it, and try to understand them this coming year. try to Have you tested the sight and hearing of your children this year? Don't neglect it. You may be doing injustice to some child every day. The January number of PRIMARY EDUCATION is always a snow number. The following information is given for any teachers who have not been able to find the music to which Miss Allen has set her delightful motion songs. "Swinging 'Neath the Old Apple-Tree." Page 67, Franklin Square Song Collection No I. (Per Biglow and Main, N. Y. City.) "Comin' Thro' the Rye." Page 51, The Song Budget. (C. W. Bardeen, Syracuse, N. Y.) In this book, as in some others, this music is published under the title: "If a Body Finds a Lesson." Encourage the children to feed the birds this winter. Miss Mann's article in December number told you how. The Plan Book Miss Minnie M. George (Oak Park Ill.) edits a new monthly desk book. Its purpose is to give suitable material for each month's work, with suggestions for associating the different forms of work done in modern school-rooms. The contents consists mainly of stories, poems, and gems of literature work, programs for special days, songs new and old, science or nature lessons, blackboard reading lessons, drawing lessons, blackboard illustrations, and seat work. Cold Luncheons How many teachers are allowing themselves to eat cold luncheons in the school-room at the noon-hour? Don't do it. No matter what the seeming necessity may be for taking a cold hasty meal in the school-room, there is a more urgent need for you to abandon it. You have talked all the morning, you are tired, keyed-up in a nerve tension, and in no condition to eat anything for a half-hour. Take that half-hour and give yourself a rest, and don't mark slates or papers, or clean blackboards while you are resting. You are alone in your school-room, (of course you do not do so inhuman a thing as to "keep in " children at noon), and take some position to rest you. Miss Call's "Power Through Repose" will suggest many ways in which you can rest physically, by taking certain positions; you know yourselves how to do many of these things. Now if you are so situated that you cannot - absolutely cannot - get a warm noon meal, you must learn how to manage to get one in the school-room. May I suggest? A little oil stove can be bought for seventy-five cents, and with this you are independent. With a utensil or two, tea, cocoa, or cereal coffee, eggs, beef tea, warmed bread, oysters, and other similar things can be easily prepared and your health be saved. Your health is your capital and you imperil it every time you eat a cold, hasty luncheon at school with your mind full of school anxiety. Yo may go on for a while and not become painfully aware of this fact, but nature is relentless and your reminder will come by and by in a broken-down, deranged nervous system that will make you unfit for school or anything else. Indigestion is the foe to every virtue, every grace, and to Christianity itself. Keep in mind the reflex influence of body on mind and mind on body. No wonder school boards are actually discussing the advisability of protecting children from teachers with dyspepsia. The "pews" have rights. Pardon this little sermon, but it is preached from a personal experience of half a life in school, and from a deadly knowledge of the evils of cold, hasty meals in the schoolroom. |