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Les diagrammas suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 32X mt^ ■ 'iJ .. -^ '-v. ' i V ■r- n> y u / / 1)^ ■{? A O yv i^ \) A .\. V \" 'I 1 '■ I i I EDITED BY WILLIAM T. HARRIS, A. M., LL. D. Volume XXX] II. hi :^jy ; W- THE INTERNATIONAL EDUCATION SERIES. 12ino, cloth, uniform binding. (t / rilHE INTERNATIONAL EDUCATION SERIES was projected for the pur- -L poee of hrinsinj^ to^other in orderly arranpeineut the best writingH, new and old, upon educational subjt^cts, and i)resentiii<^ a complete courne of reading and training for teachers generally. It irt edited by W. T. IIakris, LL. U., United States Commissioner of Education, who has contributed for tlie different volumes in the way of introductions, analysis, and commentary. The volumes are taste- fully and substantially bound in iniiform style. VOLUMES NOW READY. Vol. I.— THE PHILOSOPHY OF EDUCATION. By Joiiann K. F. Rosen- KHANZ, Doctor of Theology and Professor of Philosophy, University of K6nigsl)erg. Translated by Anna C. Bkackett. Second editicm, revised, with Commentary and complete Analysis. $1.50. Vol. II.-A HISTORY OF EDUCATION. By F. V. N. Painter, A.M., Pro- fessor of Modern Languages and Literature, Roanoke College, Va. $1.50. Vol. III. -THE RISE AND EARLY CONSTITUTION OF UNIVERSITIES. With a Survey ok Medieval KorrATiox. By S. S. Laurie, LL D.,, Professor of the Institutes and History of Education, University of Edin- burgh. $1.50. Vol. IV.-THE VENTILATION AND WARMING OF SCHOOL BI^ILDINOR. By Gilbert B. Morrison, Teacher of Physics and Chemistrv, Kansas City High School. $1.00. Vol. v.— THE EDUCATION OF MAN. By Friedrich Froebel. Translated and annotated by W. N. Hail.mann, A. M., Superintendent of Public Schools, La Porte, Ind. $1.50. VoL VI.— ELEMENTARY PSYCHOLOGY AND EDUCATION. Bv Joseph Baldwin, A. M., LL. D., author of " The Art of School Management." $1.50. Vol. VIL-THE SENSES AND THE WILL. (Part I of ''The Mind op the Child.") By W. Preyer, Professor of Physiology in Jena. Translated by H. W, Brown, Teacher in the State Normal School at Worcester, Mass. $1.50. Vol. VIII.— MEMORY : What it is and how to Improve it. By David Kay, F. R. G S., author of " Education and Educators," etc. $1.50. Vol. IX.- THE DEVELOPMENT OF THE INTELLECT. (Part II of " The Mind op the Child.") Bv W. Preyer, Piofessor of Physiology in Jena. Translated by H. W. Brown. $1.50. Vol. X.— HOW TO STUDY GEOGRAPHY. A Practical Exposition of Methods and Devices in Teaching Geography which ai)ply the Principles and Plana of Ritter anG>o 4 Copyright, 1895, By D. APPLETON AND C03IPANY. Electrotyped and Printed AT THE APPLETON PbESS, U. S. A. EDITOR'S PREFACE. In presenting this book on the Psychology of Num- ber it is believed that a special want is supplied. There is no subject taught in th^' ^kjmpivtary schools that taxes the teacher's resources as to methods and devices to a greater extent than arithmetic. There is no subject taught that is more dangerous to the pupil in the way of deadening his mind and arresting its development, if bad methods are used. The mechanical side of training must be joined to the intellectual in such a form as to prevent the fixing of the mind in thoughtless habits. While the mere processes become mechanical, the mind should by ever-deepening insight continually increase its powder to grasp details in more extensive combina- tions. Methods must be chosen and justified, if they can be justified at all, on psychological grounds. The con- cept of number will at first be grasped by the pupil im- perfectly, lie will see some phases of it and neglect others. Later on he will arrive at operations w^hich demand a view of all that number implies. Each and every number is an implied ratio, but it does not ex- press the ratio as simple number. The German lan- guage is fortunate in having terms that cjxpress the two aspects of numerical quantity. Anzahl expresses the vi KhlToli'S IMIKKACR multiplicity uiul Kinlwlt the unity. Any uuihIkm', sjiy six, fur oxanipio, has these t\v more general aspects of (piantity in general, namely, to dis- creteness aud continuity. There is such a thing as (pialitative unity, or indi- viduality. Quantitative unity, unlike individuality, is always divisible into constituent units. All quantity is a unity of units. It is composed of constituent units, and it is itself a constituent unit of a real or possible larger unity. Every pound contains within it ounces ; every pound is a constituent unit of some hundred- weight or ton. The simj)le number implies both })hases, the multi- plicity and the unity, but does not express them aeethoven or a fugue of J3ach, and soon be- come familiar with it. (^n the plan of the old lessons in counterpoint, the pupil found himself helpless before such a composition. His phrases furnished 110 key to the compositions of Bach or Beethoven, because the latter are constructed on a different counterpoint. VIU EDITOR'S PRKFACK. So the methods of teaching arithmetic by a '^ fixed- unit " system do not lend towards the higlier mathe- matics, but away from it. They furnish little, if any, training in thinking the ratio invoU'ed in the very idea of number. The psychology of number recjuires that the meth- ods be chosen with reference to their power to train the mind of the pupil into this consciousness of the ratio idea. The steps should be short and the ascent gradual ; but it should be continuous, so that the pupil constantly gains in his ability to hold in consciousness the unity of the two aspects of quantity, the unity of the discrete and the continuous, the unity of the nmlti- plex and the simple unit. Measurement is a process that makes these elements clear. The constituent unit becomes the including unit, and vice versa, throuo-h beino; measured and beinc: made the measure of others. This, too, is involved in using the decimal system of numeration, and in understand- ing the different orders of units, each of which both in- cludes constituent units and is included as a constituent of a higher unit. The hint is obtained from this that the first lessons in arithmetic should be based on the practice of meas- uring in its varied applications. Again, since ratio is the fundamental idea, one sees how fallacious are those theories which seek to lay a basis for mathematics by at first producing a clear and vivid idea of unity — as though the idea of quantity were to be built up on this idea. It is shown that sucli abstract unit is not yet quantity nor an element of (jujin- tity, but sinqily the idea of individuality, which is still EDITOR'S PIIKFACE. IX a qualitative idea, and does not become quantitative until it is conceived as composite and made up of con- stituent units homogeneous with itself. The true psychological theory of number is the panacea for that exaggeration of the importance of arithmetic which prevails in our elementary schools. As if it were not enough that the science of number is indispensable for the con(|uest of Nature in time and space, these qualitative-unit teachers make the mistake of supposing that arithmetic deals with spiritual being as much as with matter ; they confound quality with quantity, and conse(iuently mathematics with meta- physics. Mental arithmetic becomes in their psy- 1 chology " the discipline for the pure reason," although ) as a matter of fact the three figures of the regular \ syllogism are neither of them employed in mathematical j reasoning. W. T. Harris. Washington, D. C, June ;25, 1895. m f PEEFACE V It is perliaps natural that a growing impatience with the meagre results of the time given to arithmetic in tlie traditional course of the schools should result in attacks upon that study. While not all educational ex- perts w^ould agree that it is the " most useless of all sub- jects " taught, there is an increasing tendency to think of it and speak of it as a necessary evil, and therefore to be kei)t within the smallest possible bounds. How- ever natural this reaction, it is none the less unwise when turned against arithmetic itself, and not against stupid and stupefying ways of teaching it. So con- ceived, the movement stands only for an aimless swincr of the scholastic pendulum, sure to be followed by an equally unreasonable swing to the other extreme. If methods which cut across the natural grain of the men- tal structure and resist the straixditforward workings of the mental machinery, waste time , create apathy and disgust, dull the ])ower of quick peiception, and culti- vate habits of inaccurate and disconnected attention, what occasion for surprise ? Because wrong methods breed bad results, it hardly follows that education can be made synnnetrical by omitting a subject which stands par eirrellence for clear and clean cut methods xi Xll PREFACE. of tlioiiglit, wliicli forms the introduction to all eilective interpretation of >y"ature, and is a powerful instrument in the reij^ulation of social intercourse. It is custi)marj now to divide studies into "form" studies and '"content" studies, and to de])reciate arith- metic on the ground that it is merely formal. But how are we to sej)arate form and content, and regard one as good in itself and the other as, at best, a necessaiy evil ? If we may para})lirase a celebrated saying of Kant's, while form without content is bai'ren, content without form is nnishv. An education which ne<»:lects the for- mal reIationshi])s constituting the framework of the subject-matter taught is inert and supine. The ])eda- gogicid problem is not solved by railing at " form,"' but in discovering what kind of form we are dealing with, how it is related to its own content, and in work- ing out the educational methods which answer to this relationship. Because, in the case of number, " form " represents the measured adjustment of means to an end, the rhythmical balancing of parts in a whole, the mas- tery of lorni represents directness, accuracy, and econ- omy of ])erce]ition. the power to discriminate the rele- vant from the irrelevant, and ability to mass and con- verge relevant material upon a destined end — repre- sents, in short, precisely what we understand by good sense, by good judgment, the power to put two and two together. AVhen taught as this sort of form, arithmetic aft'ords in its own j)lace an unrivalled means of men- tal disc' ^''le. It is, perhaps, more than a coincidence that til ^articular school of educational thought which is most active in urging the merely '' formaP' quality of arithmetic is also the one which stands most sys- w PJiEFACE. Xlll teiiiatieally for what is coiiflcinned in the following pages as the "fixed unit" method of teaching. As foi* the counterpart objection that nnmber work is lacking in ethical substance and stimulus, much may be learned from a study of Greek civilization, from the recognition of the part which Greek theory and prac- tice assigned to the ideas of rhythm, of balance, of measure, in moral and aesthetic culture. That the Greeks also kept their arithmetical training in closest connection with the study of spatial forms, with meas- urement, may again be more than a coincidence. Even u])on its merely formal side, a study which requires exactitude, continuity, patience, which automatically re- jects all falsification of data, all slovenly manipulation, which sets u^ a controlling standard of balance at every point, cai- ...dly be condemned as lacking in the ethical element. But this idea of balance, of compensation, is more than formal. Xumber represents, as is shown in the following pages, vdhiatioii ; nuTiber is the tool wherein' modern society in its vast and intricate pro- cesses of exchange introduces system, balance and econ- omy into those relationships upon which our daily life depends. Properly conceived and presented, neither geography nor history is a more effective mode of bj'inging home to the ])upil the realities of the social environment in which he lives than is arithmetic. So- ciety has its form also, and it is found in the processes of fixing standards of value and methods of valuation, the ])rocesses of weighing and counting, whether dis- tance, size, or quality ; of measuring and fixing bounds, whether in space or time, and of balancing the various resulting values against one anotlier. Arithmetic can XIV PREFACE. not be properly taught without being an introduction to this form. Thanks are due to Mr. AYilliam Scott, of tho Toronto Normal School, for some assistance with tlic proofs ; and to ^Ir. Alfred T. De Lury, of University College, Lecturer on Methods in Mathematics in Ontario School of Pedagogy, for valued practical assistance. Aiigitst 13, IS'Jo. nsRs CONTENTS. Editor's Preface Author's Preface PAGE V xi CHAPTER l._\ViiAT Psychology can do for the Ifacher . 1I._The Psychical Nature of Numrfr .... in.— The Origix of Number : Dependence of Number on Measurement, and of Measurement on Adjust- ment of Activity IV._The Origin of Number : Summary and Applica- tions v.— The Definition, Aspects, and Factors of Numerical Ideas VI.— The Development of Number; or, the Arithmetical Operations VII.— Numerical Operations as External and as Intrin- sic to Number Vin.— On Primary Number Teaching IX.— On Primary Number Teaching X.~NoTATioN, Addition, Subtraction . . . . XI.— Multiplication and Division . . . • • XII. —Measures and Multiples [XIII. — Fractions XIV.—Decimals XV.— Percentage and its Applications . . . . XVI.— Evolution 2 1 23 35 52 08 93 119 144 1G6 190 207 227 241 261 279 297 THE PSYCHOLOGY OF NUMBER. CHAPTER I. WHAT PSYCHOLOGY CAN DO FOR THE TP:ACHER. The value of any fcKit or theory as bearing on hu- man activity is, in tl)e long run, determined by practi- cal application — that is, by using it for accomplishing some definite purpose. If it works well — if it removes friction, frees activity, economizes effort, makes for richer results — it is valuable as contributing to a perfect adjustment of means to end. If it makes no such con- tribution it is practically useless, no matter what claims may be theoretically urged in its behalf. To this the question of the relation between psychology and educa- tion presents no exception. The value of a knowledge of psychology in general, or of the psychology of a par- ticular subject, will be best made known by its fruits. No amount of argument can settle the question once for all and in advance of any experimental work. But, since education is a rational process, that is a process in harmony with the laws of psychical development, it is plain that the educator need not and should not depend upon vague inductions from a practice not grounded upon principles. Psychology can not dispense with ex- i TIIK l\SVCllOLO(iY OF NUMUKR. ])(.'riciicc, nor can experience, if it is to be rational, dis- pense with psycliolo^^y. It is ])ossil)lo to make actual practice less a matter of mere experiment and more a matter of I'cason ; to make it contribute directly and economically to a rich and rij)e, because rational, experi- ence. And this the educjitional psychologist attempts to do by indicating in what directions help is likely to be found ; by indicating what kind of psychology is likely to hel[) and what is not likely; and, finally, by indicating what valid reasons there are for anticipating any help at all. I. As to the last point suggested, that psychology ought to help the educator, there can be no disagreement. In i\\Q,Jirie of native capacity — for the fullest develo])ment and most fruitful application of endowments of heart and brain. Train- ing and native outfit, culture and nature, are never op- posed to eacii other. It is always a question, not of suppressing or superseding, l)ut of cuhivating native instinct, of training natural ecpiipment to its ripest de- velopment and its richest use. A Pheidias does not despise learning the principles necessary to the mastery of his art, nor a Beethoven disregard the knowledge requisite for the complete technical skill through which he gives expression to his genius. In a sense it is true that the great artist is born, not made ; but it is ecpially true that a scientific insight into the technics of his art heljys to make him. And so it is with the artist teacher. The greater and more scientilic his knowleduje of human nature, the more ready and skilful will be his ap])li- cation of principles to varying circumstances, and the larger and more perfect will be t\n}. })roduct of his artii^tic skill. But the ixenius in education is as rare as the irenius in other realms of human activitv. Education is, and forever will be, in the hands of ordinary men and women; and if psychology — as the basis of scientific WHAT PSYCHOLOGY CAN DO FOR THE TFJACIIER. 7 iiibiglit into Inuiian nature — is of liigli value to tlie few Avlio possess genius, it is indispensable to the many wlio Lave not genius. Fortunately for the race, most }ier- sons, though not " born " teachers, are endowed with some ''genial impulse," some native instinct and skill for education ; for the cardinal requisite in this en- dowment is, after all, sympathy with human life and its aspirations. We are all born to be educators, to be parents, as we are not born to be engineers, or sculp- tors, or musicians, or painters. J^ative capacity for education is therefore much more common than native capacity for any other calling. Were it not so, human society could not hold together at all. But in most people this native sympathy is either dormant or blind and irregular in its action ; it needs to be awakened, to be cultivated, and al)ovo all to be intelligently directed. The instinct to walk, to speak, and the like are imperi- ous instincts, and yet they are not wholly left to " na- ture " ; we do not assume that they will take care of themselves ; we stimulate and guide, we su]">ply them with j)roper conditions and material for their develop- ment. 80 it must be with this instinct, so common yet at present so comparatively inelfectiv^e, which lies at the heart of all educational efforts, the instinct to help others in their struggle for self-mastery and self-expres- sion. The very fact that this instinct is so strong, and all but universal, and that the happiness of the individ- ual and of the race so largely depends upon its develop- ment and intelligent guidance, gives greater force to the demand that its growth may be fostered by favour- able conditions ; and that it may be made certain and reasonable in its action, instead of being left blind and w 8 TlIK PSYCHOLOGY OF NUMBER. falteriiis:, as it surely Avill be with out rational culti- vation. To this it luav be added that native endowment can work itself out in the best i)ossible results only when it works under riu'ht conditions. Even if scientiiic in- sight were not a necessity for the true educator himself, it would still remain a necessity for others in oi'der that they might not obstruct and possibly drive from the profession the teacher possessed of the iid)orn divine light, and restrict or ])aralyze the efforts of the teacher less richly endowed. It is the mediocre and the buuii^ler who can most readilv accommodate himself to the conditions imposed by ignorance and routine; it is the higher type of mind and heart which suffers most from its encounter with incapacity and ignorance. One of the greatest hindrances to true educational progress is the reluctance of the best class of minds to en<];:a<>:e in educational work precisely because the ii'en- eral standard of ethical and psvcholoii'ical knowledire is so low that too often high id(nils are belittled and efforts to realize them even vigorously opposinl. The educa- tional genius, the earnest teacher of any class, has little to expect from an indifference, or a stolidity, which is proof alike against the facts of experience and the dem- onstrations of science. 2. The Secoiul Resource is {'.vj}en'erwe. This, again, is necessary. Psvcholoirv is not a short and easy path that renders personal experience su])erliuous. The real question is : AVhat kind of experience shall it be ? It is in a way perfectly true that only by teachiuir can one become a teacher. J>ut not any and every sort of thiuir which passes for teaching or for "experience" will WHAT PSYCHOLOGY CAN DO FOR THE TEACHER. make a teaclier any more than .dimply sawing a buw across violin strings will make a violinist. It is a cer- tain quality of practice, not mere practice, which pro- duces the expert and the artist. Unless the practice is based upon rational princi})les, upon insight into facts and their meaning, " experience " simply fixes incorrect acts into wrong habits. Nonscientific practice, even if it iinally reaches sane and reasonable results — which is very unlikely — does so by unnecessarily long and cir- cuitous routes ; time and enera;v are wasted that might easily be saved by wise insight and direction at the outset. The worst thing about empiricism in every depart- ment of human activity is that it leads to a blind ob- servance of rule and routine. The mark of the empiric is that he is helpless in the face of uew circumstances ; the mark of the scientitic worker is that he has power in grappling with the uew and the untried ; he is mas- ter of principles which he can effectively apply under novel conditions. The one is a slave of the past, the other is a director of the future. This attachment to routine, this subservience to empiric formula, always reacts into the character of the empiric; he becomes hour by hour more and more a mere routinist aud less and less an artist. Even that which he has once learned and applied with some interest and intelligence tends to become more and more mechanical, and its a])})lica-- tion more and more an unintelligent and unemotional procedure. It is never brightened and (piickened by adaptation to new ends. The machine teacher, like the em])iric in every profession, thus becomes a stupefying and corrupting inlluence in his surroundings ; he him- 10 THE PSYCHOLOGY OF NUMBER. self becomes a mere tradesman, and makes his scliool a mere maebine sbo[). 3. The Tbird Resource is authoritative iihstruction in methods and devices. At present, tbe real opposi- tion is not ])et\veen native skill and experience on tbe one side, and psycbological metbods on tbe otber ; it is ratber between devices picked up no one knows how, metbods inherited from a crude ])ast, or else in- vented, ad hoc, by echicational quackery — and metbods which can be rationally justitied — devices which are tbe natural fruit of knowing the mind's powers and tbe ways in which it works and grows in assimilating its proper nutriment. The mere fact that there are co many metbods current, and constantly pressed upon tbe teacher as tbe acme of tbe educational experience of the past, or as tbe latest and best discovery in peda- gogy, makes an absolute demand for some standard by which they nuiy be tested. Only knowledge of tbe principles upon which all methods are based can free the teacher from dependence upon the educational nos- trums which are recommended like patent medicines, as panaceas for all educational ills. If a teacher is one fairly initiated into tbe real workings of tbe mind, if be realizes its iu)rmal aims and metbods, false devices and schemes can have no attraction for him ; be will not swallow them "as silly people swallow empirics' pills" ; he will reiect them as if bv instinct. All new suixires- tions, new metbods, be will submit to tbe infallible test of science ; and those which will further bis work be can ado})t and rationally aj^ply, seeing clearly their place and bearings, and the conditions under which they can be most effectively employed. Tbe difference be- mmmm WHAT PSYCIIOLOGY CAN DO FOR THE TEACHER. H a ■m tween being overpowered and used by machinery and being al>le to use the machinery is ])recisely the dilier- ence between methods externally inculcated and meth- ods freely adopted, because of insight into the psycho- logical principles from which they spring. Summing u}), we may say that the teaclier requires a sound knowledge of ethical and psychological prin- ciples — first, because such knowledge, besides its indi- rect value as forming logical habits of mind, is necessary to secure the full use of native skill ; secondly, because it is necessary in order to attain a perfected experience with the least expenditure of time and energy ; and thirdly, in order that the educator may not be at the mercy of every sort of doctrine and device, but may have his own standard by which to test the many meth- ods and expedients constantly urged upon him, select- ing those which stand the test and rejecting those which do not, no matter by what authority or influence they may be supported. II. We may now consider more positively how psy- cliology is to perform this function of developing and directing native skill, making experience rational and hence prolific of the best results, and providing a cri- terion for ::.uij::<2:ested devices. Education has two nuiin phases which are never separated from each other, but which it is convenient to distin<>;uish. One is concerned with the organization and workings of the school as part of an organic whole ; the other, with the adaptation of this school structure to the individual pupil. This difference may be illustra- ted by the dift'erence in the attitude of the school board or minister of education or superintendent, whether ■)i| 12 THE PSYCHOLOGY OF NUMBER. state, county, or local, to the school, and that of the individual teacher within the school. The former (the administrators of an organized system) are concerned more with the constitution of the school as a whole ; their survey takes in a wide field, extending in some cases from the kindergarten to the university through- out an entire country, in other cases from the primary school to the high or academic school in a given town or city. Their chief husiness is with the organization and management of the school, or sj'stem of schools, upon certain general principles. What shall he the end and means of the entire institution 'i \Vhat suhjects shall he studied ? At what stage shall they he intro- duced, and in what sequence shall they follow one an- other — that is, what shall be the ari-angement of the school as to its various parts in time ( Again, what shall be the correlation of studies and methods at every period 'i Shall they be taught as dilferent subjects ? in departments ? or shall methods be sought which shall work them into an organic whole? All this lies, in a large measure, outside the purview of the individual teacher ; once within the institution he iinds its purpose, its general lines of work, its constitutional structure, as it were, fixed for him. An individual mav choose to live in France, or (ireat ]>ritain, or the United States, or Canada; but afler he has made his choice, the gen- eral ccndiiions under which he shall exercise his citizen- Si»i- i) L .VciMed for him. So it is, in the main, with tb ni'.i/I.^ial tcachei'. ih\t Aiv "'.tizen who lives within a given system of institutions and laws hnds himself constantly called upon to act. He must adjust his interests and activities to WHAT PSYCIlOLO(iV CAN 1)0 FOR THE TEACHER. 13 a -\^, tliose of otliers in the same coiiiitrv. There is, at the same time, seope for purely individual seleetion and application of means to ends, for unfettered action of strong personality, as well as opportunity and stimulus for the free ])layand realization of individual equipment and acquisition. The hetter the constitution, the system which he can not directly control, the wider and freer and more potent will he this sphere of individual action. ]Now, the individuiil teacher Unds his duties within the school as an entire institution ; he has to adapt this or- ganism, the suhjects taught, the modes of discipline, etc, to the individual pupil. Apart from this personal adaptation on the part of the individual teacher, and the personal assimilation on the part of the individual pupil, the general arrangement of the school is purely meaningless ; it has its object and its justification in this individual realm. Geography, arithmetic, liter- ature, etc., may be provided in the curriculum, and their order, both of sequence and coexistence, laid down ; but this is all dead and formal until it comes to the intelligence and character of the individual pupil, and the individual teacher is tlic Diediiun through which it co?nes. Now, the bearing of this upon the point in hand is that psychology and ethics have to subserve these two functions. These functions, as alreadv intimated, can not be separated from each other ; they are simply the general and the individual aspects of school life ; but for purposes of study, it is convenient and even impor- tant to distinguish them. AVe may consider psychology and ethics from the standpoint of the light they throw upon the organization of the school. as a whole — its end, i ' 14 THE PSYCHOLOGY OF NTMBKR. its chief nietliods, the order and correlation of studies — and we niav consider them from the staninph\i\ invo lv ing tJ TO_f, fi(^tni\^ ; the dilference which makes the individuality of each obiect must be noted, and yet the different individuals must 1)0 grasped as one whole — a sum. It recpdres, then, considerable power of intellectual abstraction even to count three. Unlike objects, in spite of differences in 1 1 m I --h THE PSYCHOLOGY OF NUMBER. quality, must be recognised as forming one grcnp ; while a group of like objects, in spite of their simi- larities in quality, is to be recognised as made up of separate things. Three dilferently coloured cubes, for example, must be apprehended as one group, while a group of three cid)es exactly alike must be apprehended as three individurls. \\ other words, the objects count- ed, whatever be their physical resemblances or differ- ences, are numerically alike in this : they are parts of one u'Jiole — they are rniif'^ constituting a defined miity. The delight whfci. h vl.ild four or five years old often manifests in the apjc,\;ntly mechanical operation of counting chairs, books, ^'a^e-marks, playthings, or even in simply saying ove l"l'e • i^jics of number sym- bols is really delight in his novvl} if piired or rapidly growing power of abstraction and generalization. There is abstraction because the child now knows, in a definite, objective way, that one chair, although a different chair from every other, is, nevertheless, in some particular identical with every other — it is a chair, lie is able to neglect all that sensuous qualitative difference which previously so claimed his attention as to prevent his conscious or objective recognition of the common qual- ity or use through which the things may be classed as one whole. Xow, abiHty to neglect certain features of things in view of another considered more important, is of course of the essence of abstraction in its hii>:liest as well as in this rudimentary form. Generalization, on the other hand, is simj^ly the obverse of abstraction ; they are correlative phases of one activity. In leaving out of account the qualities now seen to be unimportant tu the end in view, though sensuously they may be very THE PSYCHICAL NATURE OF NUMBER. 27 prominent and attractive, tlie mind grasps in one wliole the objects that liave a coimHon quality or use, tliougli the objects are decidedly nidike as regards other quali- ties or uses. If from a collection of ol)jects of different colours a cliild is required to select all the red ones, he not only neglects all that are not red ; he neglects also all the other qualities — shape, size, material, etc. — of the red objects themselves ; and when this abstraction is completed, there is the conception of the group of red things as the result of the other side of the mental process — viz., generalization. The manifestation of the conscious tendency in a cliild to count coincides, then, with the awakening in his mind of conscious power to abstract and generalize. This power can show itself only when there is ability to resist the immediate solicitations of colour, sound, etc., ability to hold the mind from being absorbed in the delight of mere seeing, hearing, handling ; and this means power of abstraction. But this very power to resist the stimulus of some sense qualities and to attend to others means also the power to group the different objects together on the basis of some principle not di- rectly apprehended by the senses — some use or function which all the different objects have — and this is, again, generalization. Discrimination and Kelation. — This power to form a whole out of different objects may be studied in some- what more detail. It includes the two correlative pow- ers of discrimination and relation. 1. Discrimination. — As adults we are constantly deceiving ourselves in regard to the nature and genesis of our mental experiences. Because an object presents i:; M! i ,! \ i 'J ¥ H 1 28 THE PSYCHOLOGY OF NUMIiKU. a certain quality directly to us, we are apt to assume that the (quality is inherent in the object itself, and is ])resented to everybody (juite apart from any intellectual operation. AV^e forget that the objects now have certain qualities for us KiiKply because of analyses j^/v^ij^-* ?/*/// l^erformed. We see in an object just what we ha\e learned to see in it. The contents of the concept re- sultino^ from an elaborate process of analysis-synthesis arc at last given in the percept. An expert geometri- cian's percept of a triangle is quite a different thing from that of a mere tyro in cjeometrv. A man may become such a chemist as never to see water without being conscious that it is composed of oxygen and hydro- gen ; or such a botanist that a passing glance at a liower instantly recalls the name orchid, or ranunculus, and all the differential qualities which belong to this class of plant life. In like manner all of us have become suffi- ciently familiar with numerical ideas to know at a glance that a tree has a great many lefives, a chair a certain number of parts, a cube a definite numl)er of faces. Althouijh this knowledo-e is now direct and "intuitive," it is the result of past discriminations. AYe may be perfectly sure that they are not "intuitions" to the child ; to him the tree, the house, the cube, the black- board, the group of six objects, is one undefined whole, not a whole of parts. The recognition of separate or distinct parts always implies an act of analysis or discrimination definitely performed at some period ^ and such definite analysis has always been preceded by a vague synthesis — that is, the idea of a whole of as yet undistinguished parts. There is perhaps no point at which the teacher is ^ THE PSYCIIK'Ah NATURE OF NUMBER. '20 more likely to c;o astray than in assuming that objects liave for a child the deiiniteness or concreteness of qualities which they have for us. In the application of the pedagogical maxim " from the concrete to the abstract," he is very apt to overlook the necessity of making sure that the "concrete" is really present to the child's mind. lie too easily assumes as already ex- isting in tlie consciousness of the learner what can really exist only as the product of the mind's own activity in the process of deiinition — of discriminating and relating. It is a grave error to suppose that a ti-i- angle, a circle, a written word, a collection of five ob- jects, are concrete wholes, that is, definitely grasped mental wholes to the child, simply because there are certain physical wholes present to his senses. Definite ideas are thus assumed as the basis of later work when there is absolutely nothing corresponding to them in the child's mind, in wliich, indeed, there is only a pano- rama of vague shifting imagery, with a penumbra of all sorts of irrelevant emotions and ideas. Thus, this noted maxim, when translated to inean co7icTete tJiings hi fore the senses^ thei^efore concrete hnoudedge in the mind., becomes really a mischievous fallacy. Or, again, the teacher, mislead by the formula — first, the isolated definite particular ; second, the interconnec- tion ; third, the organic whole — introduces distinction and definition where normally the child should deal only with wholes in vague outline ; and thus substitutes for the poetic and spontaneous character of mental ac- tion a forced mechanical analysis all out of harmoTiy with his existiiig stage of development. Of this we have an example in the prevailing methods of primary t ' 1.1 "\ h I i '} 30 THE PSYCHOLOGY OF NUMBER. M ,y number teaching. The child is from the beginning drilled in the "analysis" of numbers till he knows or is supposed to know '• all that can be done with num- bers." It appears to be forgotten tliat he may and should perform many 0})e rations and reach definite re- sults by implicitly 'iising the ideas they involve long before these ideas can be explicitly developed in con- sciousness. If facts are presented in their proper con- nection as stimulating and directing the primary mental activities, the child is slowly but surely feeling his way towards a conscious recognition of the nature of the process. This unconscious growth towards a reflective grasp of number relations is seriously retarded by un- timely analysis — untimely because it appeals to a power of reflection wdiicli is as yet undeveloped. It is obvious that these two errors are logically op- posed to each other. One overlooks the need of the process of discrimination, of careful analysis ; the other does nothing but analyze and deflne. But while log- ically opposed to each other they are often practically combined. They both arise from one fundamental error — the failure to grasp clearly the place which dis- crimination occupies as the transitional step in the change of a vague whole into a coherent whole. In the ordinary methods of teaching number, for example, both mistakes are found in combination. There is no attention, or too little attention, paid to the essential process of discrimination when it is taken for granted that definite ideas of number will be formed from the hearing and memorizing of numerical tables, or even from the perception of certain objects apart f rani the child's own activity in conceiving a whole of pai'ts and \ THE PSYCHICAL NATURE OF NUMBER. 31 W V relating j)arts in a definite whole. On the other htuul, there is altogether too much definition, definition car- ried to the point of isolation, when, in number teach- ing, a start is made with one thing — endless changes being rung with single objects in order " to develop the number one " — then another object is introduced, then another, and so on. Here the preliminary ac- tivity that resolves a whole into parts is omitted, as well as the connecting link that makes a lohole of all the parts. 2. Relation or Rational Cou7itln(/.--T\i\s involves the putting of units (parts) in a certain ordered relation to one another, as well as marking them off or dis- criminating them. If, when the child discriminates one thing from another, he loses sight of the identity, the link which connects them, he gains no idea of a group, and hence there is no counting. There is, to him, simply a lot of unrelated things. When we reach '' two " in counting, we must still keep m 7nind ^'one^' ; if we do not we have not " two," but merely another one. Two things may be before us, and the word *' two " may be uttered but the concept two is absent. The concept two involves the act of putting together and holding together the two discriminated ones. It is this tension between opposites which is largely the basis of the childish delight in counting. Number is a con- tinued paradox, a continued reconciliation of contradic- tions. If two things are simply fused in each other, forming a sort of vague 07ieness, or if they are 6im})ly kept apart from each other, there is no counting, no " two." It is the correlative differentiation and identi- fication, the holding apart and at the same time bringing 4 I -li , : , i: 71 if I ; k s 32 TI1I<: PSVCIlOLOCiY OF NUMBiai. together, wliicli imparts to the operation of coiniting its faHciiuition. This activity is simply the normal exercise of what are always the fundamental rational functions ; and thus it gives to the child the same sense of power, of ease and mastery in mental movement, tliat an adult may realize from some magnificent generalization through which a vast, disorderly lield of experience is reduced to unity and system. In the simple one, two, three, four of the child, as he counts the familiar ob- jects around liim, there is presented the form of the highest operations of discrimination and identification. Educational Summary. — 21\e idea of 'niimher is not impressed upon the mind hy objects even when these are pt'(^sented under the most feivovredde circumstances. N'und)er is a ji't'oduct of the tcay in ichlch the mind deals imtli objects in the operation of maMng a vagne whole defnite. This operation involves («) discrimi- 7iatio?i or the recognition of the objects as distinct indi- viduals (units) ; (h) generalisation, this latter activity involving two subprocesses ; (J) abstraction, the neg- lecting of all characteristic qualities save just enough to limit each object as (me ; and (2) grovping, the gather- ing together the like objects (units) into a whole or class, the sum. Hence : 1. Kumber can not be taught by the mere presen- tation of things, but only by snch presentation as will stimulate and aid the mental movement of discriminat- ing, abstracting, and grouping which leads to definite numerical ideas. 2. In this process there must be sufficient qualitative difference among the objects used to facilitate the recog- nition of individuals as distinct, but not enough to resist Till-: PSYCHICAL NATURE OF NUMIJHU. '.^ tlio power of <>^roupini^ nil the iiidividiuils, of gnispiiiij them us j^Jirts of one whole or sum. The applieution of this principle will depend largely upon circumstanees (sensory aptitudes, ete.) and the tact of the teacher. In some cases it may he well at the outset to use differently coloured cubes, the different colours serving to individualize each ol)ject or group of objects as a unit, while the common cubical quality facilitates relation. In other cases the diiferen.'e in colour might divert attention from the relating process, and hinder the grasping of the different units as one Slim I the mere difference of position in space would be enough for the necessary discrimination. 3. In any case the aim must be to enable the pupil to get along with the minimum of actual sense dift'erence, and thus further the power of mathematical abstrac- tion and relation. For discrimination must operate just enough for the recognition of the individuality or sin- gleness of eacli object or part, and no further. The end is the facile recognition of groups as (jrouj)s^ the individuals, the single, component parts being consid- ered not for their own sake, but simply as giving defi- nite value to the group. That is to say, the recognition, for instance, of three, or four, or five, must be as nearly as possible an intuition ; a perception of the parts in the whole or a whole of parts, and not a conscious recog- nition of each part by itself, and then a conscious unit- ing it to other parts separately recognised. 4. It is clear that to promote the natural action of the mind in constructing number, the starting point should be not a single thing or an unmeasured whole, but a group of things or a measured whole. Attention . I 11 I. 34 THE PSYCHOLOGY OP KUiMBER. fixed upon a single nnmeasured object will discriminate and unify the qualities which make the thing a qualita- tive whole, but can not discriminate and relate the parts which make the thing a definite quantitative whole. It is equally clear that with groups of things the move- ment in numerical abstracting and relating may be greatly assisted by the arrangement of the things in analytical forms, as is the case, e. g., with the points on dominoes. (I t CHAPTER III. THE ORIGIN OF NUMBER : DEPENDENCE OF NUMBER ON MEASUREMENT, AND OF MEASUREMENT ON ADJUSTMENT OF ACTIVITY. Admitting, then, the psychical iiaturo of number, we are now prepared to deal with its psychological origin. It does not arise, as we have seen, from mere sense perception, but from certain rational processes in construing, in defining and relating the material of sense perception. But we are not to suppose that these processes — numerical abstraction and generalization — account for themselves. They give rise to number, but tiierc IS some reason why we perform them. This rea- son we must now discover, for it lies at the root of the problem of the origin of numher. The Idea of Limit. — If every human being could use at his pleasure all the land he wanted, it is probable that no one would ever measure land with mathematical exactness. There might be, of course — Crusoe-like — a crude estimate of the quantity required for a given purpose ; but there would be no definite numerical val- uation in acres, rods, yards, feet. There would be no need for such accuracy. If food could be had without trouble or care, and in sufficiency for everybody, we should never put our berries in (puirt measures, count 85 ;; \\ l!] r'- ; i 36 THE PSYCHOLOGY OP NUxMBKR. off eggs and oranges by the dozen, and weigh out flonr by the pound. If everything tliat ministers to liunian wants could be had by everybody just when wanted, we should never have to concern ourselves about quantity. If everything with which human activity is in any way concerned were unlimited, thei-e would of course be no need to inquire respecting anything whatever : What are its limits ? How much is there of it? Even if a thing were not actually unlimited, if there were always enough of it to be had with little or no expenditure of energy, it would be practicalJij unlimited, and hence would never be measured. It is because we have to - put forth effort, because we have to take trwible to get things, that tliey are limited for us, and that it becomes worth while to determine their limits, to iind out the quantity of anything with which human energy has to do. Limit, in other words, is the primary idea in all quantity ; and the idea of limit arises because of some resistance met in the exercise of our activity. Economy of Energy. — Because we have to put forth effort, because we are confronted by obstacles, our enei-gy is limited. It therefore l)ecomes necessary to economize our energy — that is to say, to dispose of it or disti'ibute it in such ways as will accomplish the best j)ossible re- sults. This economy does not mean a hoarding up or withholding of energy, but rather (jiving it ont in the most effective way^ husbanding " oiT-r means so well they shall go far." If we put forth more energy than is needed to effect a certain purpose, and ecpially if we ])ut forth less than is needed, there is waste ; we fail to make the most of the resources at our disposal. We carry out our plans most successfully, and perform the THE ORIGIN OF NUMBER. 37 V hardest tasks ^vitll the least waste of power when we accurately adjust our energies to the thing required. Because of the limitation of human energy all activity is a halancing of energy over against the thing to be done, and is most fruitful of results when the balancing is most accurate. If the arrow of the savage is too heavy for his bow, or if it is too light to pierce the skin of the deer, there is in both cases a waste of energy. If the bow is so thick and clumsy that all his strength is required to bend it, or so slight or uneven that too little momentum is given to the arrow, there is but a barren show of action, and the savage has his labour for his pains. Bow and arrow must be accurately adjusted to each other in size, form, and weight ; and both have to be equated (as the mathematician would say) or bal- anced to the end in view — the killing of the game. This involves the process of measurement, and its result is more or less definite numerical values. Means and End : Valuation. — The same ])rinciple may be otherwise stated in terms of the relation exist- ing between means and end. If all our aims were reached at the moment of forming them, without any delay, postponement, or countervening occurrences — if to realize an end we had only to conceive it — the neces- sity for measurement would not exist, and there would be no such thing as number in the strictly mathematical sense. But the check to our activity, the limitation of energy, defers the satisfaction of our needs. The end to be realized is remote and complex, and in using adecjuate means, distance in space, remoteness in time, . .■»;'«ij;iiJ?Ht,iivcr»fi-ri. ■-■*■■•' 60 THE PSYCHOLOGY OF NUMBER. f * value in the units is made. If we are to divide fifteen apples "equally" among five boys, giving each boy three apples, this "equal" distribution assumes the eqicality of the units (apples) of measure. Much and Many. — The whole falling within a cer- tain limit supplies the muchness j for example, the amount of money in a purse, the amount of land in a field, the amount of pressure it takes to move an obsta- cle, etc. This " much," or amount, is vague and unde- fined till measured ; it is measured by counting it off into so many units. We "lay off" distance into so many yards, and then we know it to be so much. We reckon up the pieces of money in the purse and know how much their value is. A man has a pile of lumber; liow " much " has he ? If the boards are of uniform size, he finds the number (how many) of feet in one board, ?iA counts the number (how many) of boards, and finds the -whole soma7\y feet — that is, the indefinite " how much " has become, through counting, the definite " so much." Then, again, if he wishes to find the money value of the lumber, how much it is worth, he must count off the total number of feet at so much (so many dollars) per thousand, and the resulting so many dollars represents the worth of the lumber. The many, the counting up of the particular units, measures the worth of the whole. The counting has no other meaning, and the measurement of value can occur in no other way. It is clear that these two sides of all number are rel- ative to each other, just as means and ends are relative. The so many measures the so much, just as the means balance the end. The end is the whole, all that comes m « THE ORIGIN OF NUMBER. 61 witliin a certain limit ; the means are tlie partial activi- ties, the units by which we reaUze this whole. To build a house of a certain kind and value, we must have just so many bricks, so many cubic feet of stone, so much lumber, so many days' work, etc. The house is the end, the goal to be reached ; these things are the means. The house has been erected at a certain cost ; the counting off and valuing of the units which enter into these different factors, is the only way to dis- cover that cost. i. CHAPTER lY. THE ORIGIN OF NUMBEK I SUAiMAKY AND AITLICATIONS. SuM^iAKY : Complete Activity and Subordinate Acts. — Through the forciroino: illustrations — wliicli are illiis- tratioiis of one and the same principle regarded from different points of view — we are now prepared for the statement which sums up this preliminary examination of quantity. 21(at whicJt fixes the magnitude or quan- tity u'hicJt, in any given case^ needs to he measured is sofne activity or movement^ internally co?itinuous, hut externally limited. Tliat which measures this whole is some ntinor or partial activity into which the orig- inal continuous activity may he hrolicn up (analysis), and which repeated a cert<(in numher of times gives the s((iiie result {synthesis) as the original continuous actirify. This formula, embodying the idea that number is to be traced to measurement, and measurement back to ad- justment of activity, is the key t the entire treatment of number as presented in tliese pages, and the reader should be sure lie understands its meaniui:^ before ffoinji: furtlier. In order to test his comprehension of it he may ask himself such questions as these : The year is some unified activity — what activity does it represent ? At first sight simply the apjiarent return of the sun to , 1 THE ORIGIN OF NUMBER. 63 the same point in the heavens — a:i external change ; yet the only reason for attaching so much importance to this rather than to any other cyclical change, as to make it the unit of time measurement, is that the movement of the sun controls the cycle of human ac- tivities — from seedtime to seedtime, from harvest to harvest. This is illustrated historically in the fact that until men reached the agricultural stage, or else a con- dition of nomadic life in which their movements were controlled by the movement of the sun, they did not take the sun's movement as a measure of time. So, again, the day represents not simply an external change, a recurrent movement in IS^ature, but a rhythmic cycle of human action. Again, what activity is represented by the pound, by the bushel, by the foot ? '-^ What is the connection between the decimal system and the ten fingers of the hands ? What activity does the dollar stand for ? If the dollar did not represent certain pos- sible activities which it places at our control, would it be a measure of value ? AVhy may a child value a bright penny higher than a dull dollar ? And so on. Illustrations: Stages of Measurement. — Suppose we w4sh to find the quantity of land in a cei'tain field. The eye runs down the length and along the breadth of the field ; there is the sense of a certain amount of movement. This activity, limited by the boundaries of the field, constitutes the original vague muchness — the quantity to be measured — and therefore determines all succeeding processes. Then analysis comes in, the breaking up of this original continuous * The historical origin of these measures will throw light upon the psychological point. V i rrr :'! ' i* <' Jt- ff " ! 54 THE PSYCnOLOGY OF NUMBER.- activity into a series of minor, discrete acts. The eye runs down the side of the field and fixes upon a point wliich appears to mark half the length ; this process is repeated with each half and with each quarter, and thus the length is divided roughly into eight parts, each roughly estimated at twenty paces. The breadth of the field is treated in the same way. The eye moves along till it has measured, as nearly as w^e can judge, just as much space as equals one of the smallest divisions on the other side. The process is repeated, and we estimate that the breadth contains six of these divisions. Through these interrupted or discrete movements of the eye we are able to form a crude idea of tbe length and breadth of the field, and thus make a rough estimate of its area. The separate eye movements constitute the analysis w^hicli gives the unit of measurement, and the counting of these separate movements (units) is the synthesis giving the total numerical value. But the breaking up of the original continuous movement into minor units of activity is obviously crude and defective, and hence the resulting syntliesis is imperfect and inadequate. The only thing we are certain of is the number of times the minor act has been performed ; it is pure assumption tliat the minor act measures an equal length every time, and a mere guess that each of the lengths is twenty yards. In order, therefore, to make a closer estimate of the content of the field, we may mark off the length and breadth by pacing, and find that it is a hundred and seventy paces in length and a hundred and thirty ])aces in breadth. This is probably a more correct estimate, because {a) wo THE ORIGIN OF NUMBER. 55 can be much more certain that the various paces are practically equivalent to one another than that the eye movements are equal, and (h) since the pace is a more detinite and controlled movement, we have a much clearer idea of how much the pace or unit of measure- ment really is. J3ut it is still an assumption that the various paces are equal to one another. In other words, this unit of measure is not itself a constant and measured thing, and the required measurement is therefore still imperfect. Hence the substitution for the pace of some measuring unit, say the chain, which is itself defined ; the chain is applied, laid down and taken up, a certain number of times to both the length and the breadth of the field. Now the minor act is uniform ; it is controlled by the measuring instrument, and hence marks off exactly the same sjMce every thne!^ The partial activity being de- fined, the resulting numerical value — say, eight chains by six chains — is equally definite. Besides, the chain itself may be measured off into a certain number of equal portions ; we may apply a minor unit of measure — e. g., the link — until we have determined how many links make up the chain. By means of this analysis into still smaller acts, the meaning of the unit is brought more definitely home to consciousness.f * Note how the two factors of space and time appear in all meas- nrenient, .s;>rtce representing concrete value, time, the abstract num- ber, and both, the measured maji^nitude. f If it be noted that all we have done here is to make the original activity of running the eye along length and breadth first continu- ously and then in an interrupted series of minor movements, more controlled and hence more precise, the meaning of the proposition (page 52) regarding the origin of measurement in the adjustment of n i .,1 56 THE PSYCHOLOGY OP NUiMBER. u But tliis mathematical measurement, this analysis- synthesis, is still insufficient for complete adjustment of activity. What, after all, is the value of this measured quality ? What is it good for ? Until this question is answered there can not be perfect adjustment of activi- ties. To answer this brings us to the third and final stage of number measurement. This field will produce, say, only so many bushels of corn at a given price per bushel ; it is, therefore, not worth so much as a smaller field which will produce as much wheat at a larger price per bushel. Or, in addition to the mere size of the field, it may be necessary to take into account not only the value of the crop it will raise, but also the cost of tilling it. Here there must be a much more complete adjustment of activities. The analysis concerns not only so many square rods ; it includes also the money value of the crop and the cost of its production. The synthesis compares the result of this complex measure- ment with the results of other possible distributions of energy. Analytically the conditions are completely de- fined ; synthetically there can be a complete and eco- nomical adjustment of the conditions to secure the best possil)le results. The measured quantity representing the unified (or continuous) activity is the whole or unity ; the measur- ing parts, representing the minor or partial activities, are the components or units, which make up the unified whole. In all measurement each of these measuring parts in itself is a w^hole act — as a pace, a day's journey, etc. But in its function of measuring unit it is at once minor acts to constitute a comprehensive activity will be apparent once more. THE ORIGIN OF NUMBER. 67 reduced to a mere means of constructing tlie more com- prehensive act. The end or whole is one^ and yet made up of nnany parts. Summary. — All numerical concepts and processes arise in the process of fitting together a number of minor acts in such a way as to constitute a complete and more comprehensive act. 1. This fitting together, or adjusting, or balancing, will be accurate and economical just in the degree in which the minor acts are the same in Icincl as the major. If, for example, one is going to build a stone wall, the use of the means — the minor activities — will not be ac- curate until one can find a common measure for both the means, the use of the particular stones, and the end, the wall. Size, or amount of space occupied, is this common element. Hence, to define the process in terms of just so many cubic feet recpiired is economical ; to describe it in terms of so many stones would be impos- sible unless one had first found the volumes of the stones. Hence, once more, the abstraction and the generalization involved in all numerical processes — the special qualities of the stone are neglected, and the only thing considered is the number of cu})ic feet in the stone — abstraction. But through this factor of so much size the stone is referred at once to its place in the whole wall and to the other stones — general- ization. 2. An end, or whole of a certain quality^ furnishes the limit within which the magnitude lies. Quantity is limited quality, and there is no quantity save where there is a certain qualitative 'whole or l/wiitation. --' 3. Niimher arises through the use of means, or I" III *i ill 'Ml a 1 \ 68 THE PSYCHOLOGY OF NIjMBER. H ( ! M minor units of activity, to construct an activity equal in value to the given magnitude.. This process of con- structing an equivalent value is 7nimheri}Hj — evaluation. Hence, there are no mtmerical dlstmctlons (psycho- logically) except in the process of measuring some qualitative whole.* 4. This measuring or valuation (defining the original vague qualitative whole) will transform the vague quan- tity into precise numerical value j it will accomplish this successfully in just the degree in which the minor activity or unity of construction is itself measured, or is also a numerical value. Uidess it is itself a numerical quantity, a unity measured by being counted out into so many parts, the minor and the comprehensive activity can not be made precisely of the same kind. (Prin- ciple 1.) 5. Hence the purely correlative character of much and many, of measured whole and measuring part, of value and number, of unity and units, of end and means. >1 Educational Applications. We have now to apply the principle concerning the psychological origin of quantity and number to educa- ^ tion. We have seen {a) the need in life, the demand in actual experience of the race and the individual, w^hich brings the numerical operations ; the process of meas- uring, into existence. We have seen {b) what forms number is required to assume in order to meet the need, fulfil the demand. We have now to inquire how far * The pedagogical consequences of neglecting this principle will be seen in discussing the Grube method, or use of iha fixed unit. ( • 1 THE ORIGIN OP NUMBER. 50 these ideas and principles have a practical application in educational processes. The school and its operations must be either a nat- ural or an artificial thing. Every one will admit that if it is artificial, if it abandons or distorts the normal processes of gaining and using experience, it is false to its aim and inefiicient in its method. The development of number in the schools should therefore follow the principle of its normal psychological development in life. If this normal origin and growth have been cor- rectly described, we have a means for determining the true place of number as a means of education. It will require further development of the idea of number to show the educational principles corresponding to the growth of numerical concepts and operations in them- selves, but we already have the principle for deciding how number is to be treated as regards other phases of experience. The Two Methods : Things ; Symbols. — The prin- ciple corresponding with the psychological law — the translation of the psychological theory into educational practice — may be most clearly brought out by contrast- ing it with two methods of teaching, opposed to each other, and yet both at variance with normal psycho- logical growth. These two methods consist, the one in teaching number merely as a set of syiiibols ; the other in treating it as a direct jproperty of objects. The former method, that of symbols, is illustrated in the old-fashioned ways — not yet quite obsolete — of teaching addition, subtraction, etc., as something to be done with " figures," and giving elaborate rules which might guide the doer to certain results called " answers." I! !!' I" XT if. GO THE PSYCHOLOGY OP NUMBER. il' I It is little more than a blind manipulation of num- ber symbols. The child siniply takes, for example, the figures 3 and 12, and performs certain "operations" with them, which are dignified by the names addition, subtraction, multiplication, etc. ; he know's very little of what the figures signify, and less of the meaning of the operations. The second method, the simple percep- tion or observation method, depends almost wholly upon physical operations wdth things. Objects of various kinds — beans, shoe-pegs, splints, chairs, blocks — are sepa- rated and combined in various ways, and true ideas of number and of numerical operations are supposed neces- sarily to arise. Both of these methods are vitiated by the same fun- damental psychological error ; they do not take account of the fact that number arises in and through the ac- tivity of mind in dealing with ohjccts. The first meth- od leaves out the objects entirely, or at least makes no reflective and systematic use of them ; it lays the em- phasis on symbols, never showing clearly wdiat they symbolize, but leaving it to the chances of future expe- rience to put some meaning into empty abstractions. The second method brings in the objects, but so far as it emphasizes the objects to the neglect of the mental activity which uses them, it also makes number mean- ingless ; it subordinates thought (i. e., mathcmfi.tical ab- straction) to things. Practically it may be considered an improvement on the first method, because it is not possible to suppress entirely the activity which uses the things for the realization of some end ; but whenever this activity is made incidental and not important, the method comes far short of the intelligence and skill THE ORIGIN OP NUMBER. 61 that should be had from instruction hascd on psycho- logical principles. While the inetliod of syrnholft is still far too widely used in practice, no educationist defends it ; all con- dejnii it. It is not, then, necessary to dwell upon it lunger than to point out in the light of the previous discussion wiry it should be condemned. It treats num- ber as an independent entity — as something apart from the mental activity which produces it ; the natural gen- esis and use of nund)er are ignored, and, as a result, the method is mechanical and artificial. It subordinates sense to symbol. The method of things — of observing objects and taking vague percepts for definite numei'ical concepts — treats number as if it were an inherent pro])erty of things in themselves, simply waiting for the mind to grasp it, to "abstract" it from the things. I>ut we have seen that number is in reality a mode of ?ne(isifr- ing vahte^ and that it does not belong to things in them- selves, but arises in the economical adaptation of things to some use or purpose. Numljer is not (psychologic- ally) got from things, it is put i7ito them. It is then almost equally absurd to attempt to teach numerical ideas and ])roeess vnthovt things, and to teach them simply hy things. Numerical ideas can be nor- mally acquired, and numerical operations fully mastered only by arrangements of things — that is, by certain acts of mental construction, which are aided, of course, by acts of physical construction ; it is not the mere percep- tion of the things which gives us the idea, but the em- '• ploying of the things i7i a coiistructim way. The method of symbols supposes that number arises 1 A i 62 THE rSYCTTOLOGY OF NUMBRR. * i: wholly as a matter of abstract reasoning ; tlie metliod of objects snpposes that it arises from mere observation by the senses — that it is a property of things, an ex- ternal energy just waiting for a chance to seize upon consciousness. In reality, it arises from constructive (psychical) activit^y, from the actual use of certain things in reaching a certain end. This method of constructive use unites in itself the principles of both abstract rea- soning and of definite sense observation. If, to help the mental proc^ess, small cubical blocks are used to build a large cube with, there is necessarily continual and close observation of the various things in their quantita^'Vv. aspects; if splints are used to inclose a surface with, the particular splints must be noted. Indeed, ihis observation is likely to be closer and more accurate than that in wdiich the mere observation is an end in itself. In the latter case there is no interest, no purpose, and attention is laboured and wandering; there is no aim to guide and direct the observation. The observation wdiich goes with constructive activity is a part of the activity ; it has all the intensity, the depth of excitation of the activity ; it shares in the interest of and is directed by the activity. In the case where the observation is made the whole thing, distinctions have to be separately noted and separately memorized. There is nothing intrinsic by which to carry the facts noted ; that the two blocks here and the two there make four is an iWten^al fact to be carried by itself in memory. ]hit when the two sets are so used as to construct a whole of a certain value, the fact is internal • it is part of the mind's w\ay of acting, of seeing a definite whole through seeing its definite parts. Repetition in one THE OUIGIN OF NUMBER. G3 case means sini})ly learning by rote ; in the other ease, it means repetition of activity and formation of an in- telh'gent liabit. The rational factor is found in the fact that the con- structive activity proceeds upon a principle ; the con- struction follows a certain regular or orderly method. The method of action, the wav of combinini:^ the means to reacli the end, tlie ])arts to make the whole, is rela- tion / acting according to this relation is rational, ana prepares for the definite recognition of reason, for con- sciously grasping the nature of the operations, Ka- tional action will pass over of itself when the time is ripe into abstract reasoning. The habit of abstracting and generalizing of analysis and synthesis grows into definite control of thinking. Thk Factors in Kational Method. — In more de^ tail, dealing with number by itself, as represented by symbols, introduces the child at an early stage to ab- stractions without showing how they arise, or what they stand for ; and makes clear no reason, no /'^cessity, for tlie various operations performed, which are all reduci- ble to {(() synthesis — addition, multi])lication, involu- tion ; and (/>) analysis — subtraction, division, evolution. The object or observation method shows the relation of number to things, but does not make evident why it has this rchition ; does not bring out its value or meas- uiing use, and leaves the o})erations performed pui'ely external manipulations of number, or rather with things which may be numbered, not internal developments of its measuring power. The method which develops nu- merical ideas in connection with ^he construction of eonio definite thing, I/rings out clearly {a) the natural Gi THE PSYCHOLOGY OF NUMBER. unity, the limit (the mnj^nitude) to which all number refers ; (h) the unit of measurement (the particular thing) which helps to construct the whole ; and (c) the process of measuring, by which the second of these factors is used to make up or define the first — thus de- terminins: its numerical value. (a) Only this method ]iresents naturally the idea of a magnitude from which to set out. The end to be reached, the object to be measured, supplies this idea of a given quantity, and thus gives a natural basis for the develo})ment and use of ideas of number. In num- bers simply as objects, or in things s'lDiply as observed things, there is no princi})le of unity, no basis for nat- ural generalization. Only using the various things for a certain end brings them together into one ; we count and measure some (piantitative irhole. (h) While every object is a whole in itself, a unity in so far as it represents one single act, no object sim- ply as an observed object is a vn/'f. Objects which 'ive recognise as three in number may l)e before the child's senses, and yet there may be no consciousness of them as three diUerent units, or of the sum three. Some writers tell us that each object is one, and so gives the natural ]){isis for the evolution of nund)er ; that the starting ])oint is one o])ject, to which another ob- ject is *'' addt J," then a third, etc. But this overlooks j the fact that each object is one, not a twit but one 'irhifle. differinii: from and exclusive of every other whole. That is, to take it as an (>h.^('rve(l ol>ject is to centre attention wholly u])on the thing itself ; attention would discriininate and unify the (pialities which make the thing what it is— a q>udltative whole ; but there THE ORIGIN OF NUMJ3ER. 65 would be little room for the abstracting and relating action involved in all nniiiber. A numerical unit is not merely a whole, a unity in itself, but is, as we have seen, a unity employed as a means for constructing or meas- uring some larger whole. Only this iise, theji, traiis- for)ns the ohject from a qiialitatwe %mity into a im- onei'ical imit. The sequence therefore is : iirst the vague unity or wdiole, then discriminated })arts, then the recognition of these parts as measuring the whole, wliicli is now a defined unity — a sum. Or, briefly, the undefined whole ; the parts ; the related parts i^iow units) ; the sum, (c) Beginning with the numbers in themselves, as represented by mere symbols, or with perceived objects in themselves, there is no intrinsic reason, 710 reason m t/ni nihid itself for performing the operations of put- ting together ])arts to make a whole (using the unit to measure the mngnitude), or of breaking uj) a whole into units — discovering the standard of reference for meas- uring a given unity. These operations,"^ from eitlier of these standpoints, are purely arhitraiw ; we mny, if v»e wish, do something with number, or rather with num- ber symbols : the operations arc not something that we onust do from the very nature of nund)er itself. From tlie point of view of the constructive (or psychical) use of objects, this is reversed. These ])rocesses are simply ])hases of the act of eon struct nni. IMoreover, the oper- ations of adcess. 66 THE PSYCHOLOGY OF NUMBER. i J ^ f H 2)hi/(^teal lieaping up, physical iucreusc, p/u/ncal par- tition ; while in that of number by itself they are purely mental and abstract. From the standpoint of the psychological use of the things, tliese processes are not performed npon physical things, but with ref- erence to establishing definite values ; ^ while each pro- cess is itself concrete and actual. It is not something to be grasped by abstract thought, it is something done. Finally, to teach symbols instead of number as the instrument of measurement is to cut across all the ex- isting activities, whether impulsiv^e or habitual. To teach number as a property of observed things is to cut it otT from all other activities. To teach it through the close adjustment of things to a given end is to re-enforce it by all the deepest activities. All the deepest instinctive and acrpiired tendencies are towards the constant use of means to realize ends ; this is the law of all action. All that the teaching of /lumber has to do, when based upon the principle of rationally u^ing things, is to make this tendency more detinite and accurate. It simply directs and adjusts this piocess, so that we notice its various factors and measure them in their relation to one another. ^AFore- * 'I ..I conipHi'utioiis iutroducod in schools — o. g., that you can not multiply l)y u fraction, nor increase a number by division, etc., because multiplication means increase, etc. — result from conceiving the operations as physical ai^f^repition or separation instead of syn- thesis and analysis of values — mental i)rocesses. To multiply |1() by one third is absurd if multiplication means a physical increase; if it means a measurement of value, takii'g a nuuK rical value of $\i) (a measured (plant it y) in a certain way to iind the resulting numerical value, it is perfectly rational. M THE ORIGIN OF NUMBER. 67 over, it relics constantly upon the principle of rhythm, the regular breaking up and putting together of minor activities into a whole ; a natural principle, and the basis of all easy, graceful, and satisfactory activity. i >i ' I CIIAPTEE y. THE DEFINITION, ASPECTS, ANJ) FACTORS OF NUMERICAL IDEAS. We may sum up the steps already taken as follows : (1) The limitation of an energy (or quality) transforms it into quantity, giving it a certain undelined muchness or magnitude, as illustrated by size, hulk, weight, etc. (2) This indefinite whole of quantity is transformed into definite 7uimerical value through the process of measurement. (3) This measuring takes place through the use of units of magnitude, by putting thetn together till they make '^p an equivalent value. (4) Only when this unit of magnitude has been itself measured (has itself a definite mnierical value) is the measurement of the whole mai^nitude or construction of the entire nu- merical value adequate. Forty feet denotes an ade- quately measured (piantity, ])ecause the unit is itself defined ; f^rty eggs denotes an inade(iuately measured (juantity, Ijecause the unit of measure is not d« finite. Were eggs to become worth, say, twenty t'nies as much as they are now worth, they woidd Ik- weighed out by the pound — that is, inexact measurement would give way to exact measurement. Having l)efore us, then, the psychological process whi^*h c« tistitutes measured quantity, w^e may define nu!ii[*-r. vV.5 DEFINITION OP NUMBER. C9 II Definition of Xumuer. — The simplest expression of quantity in numerical terms involves two components : 1. A Standard Unit / a Unit of lifference. — This is itself a magnitude necessarily of the same kind as tlie quantity to be measured. Or, as it may be otherwise expressed, the unity of (piantity to be measured and the unit of quantity which measures it are homogeiieous quantities. Thus, inch and foot (measuring unit and measured unity), pound and ton, minute and hour, dime and dollar are pairs of homogeneous (piantities. 2. Numerical Value. — This expresses hoic many of the standard units make up, or construct, the quantity needing measurement. Examples of numerical value are : the yard of cloth costs seventeen cents ; the box will hold thirty-six cubic inches; the purse contains eight ten-dollar pieces. The seventeen, thirty-six, eight represent just so many units of measurement, the cent, the cubic inch, the ten-dollar piece ; they express the numerical values of the quantities ; they are pure nmn- he/'s, the results of a purely mental process. The nu- merical value alone represents the relative value or ratio of the measured quantity to the unit of measurement. The numerical value and the unit of measurement taken together express the absolute value (or magnitude) of tlie measured (piantity. In the teaching of aritlnnetic much confusion arises from the mistake of identifying numerical value with ab^ohite magnitude — that is, nHtnhei\ the instrument of measurement witli nieasured quantity. Number is the product of the mere rej^etitinn of a unit of measure- ment ; it simply indicates hoiv many there are; it is ])urely abstract, denoting tin series of acts by which 'r 70 THE PSYCHOLOGY OF NUMBER. tlie mind constructs defined parts into a nniiied and definite wliole. Absolute value (quantity numerically defined) is represented by the application of this /ww manij to magnitude, to quantity — that is, to limited qual- ity. To take an example of the confusion referred to : we are told that division is dividing a (1) number into a (2) number of e(pial (3) numbers. This definition as it stands has absolutely no meaning ; there is confusion of number with measured (piantity. Doubtless the definition is intended to mean : division is dividing a certain definite (juantity into a number of definite (pian- tities equal to one another. Only in (2), in the definition as quoted, is the term number correctly used ; in both (1) and (8) it means a measured magnitude. A meas- nred or numbered (piantity may be divided into a num- ber of parts, or taken a number of times ; but no num- ber can be multiplied or divided into parts. Number simpUj as number always signifies how many times one "so much," the unit of measurement, i.s taken to make np another "so much," the magnitude to be measured. It is, as already said, due to the fundamental activities of mind, discrimination, and relation, working upon a qualitative whole ; and we might as well talk of multi- plying hardness and redness, or of dividing them into hard and red things, as to talk of multiply:?ig a number or of di voiding it into parts. It may be observed that the problems constantly used in our arithmetics, multiply 2 by 4, divide 8 by 4, are legitimate enough provided they are properly inter- preted, if not orally at least mentally, but taken literally are absurd. The first expression means, of course, that a quantity having a value of two units of a certain kind DEFINITION OF NUMBER. n is to be taken four times ; and similarly 8 -i- 4 means that a total quantity of a certain kind is measured by four units or by two units of the same kind. Of course, in all mathematical calculations we ultimately operate with ])ure symbols, and the operations do not affect the unit of measure ; but in the beginning we should make con- stant reference to measured quantity, and always should be })repared to interpret the syml)ols and the processes. 3. Is umber, then, as distinct from the magnitude which is the unit of reference, and from the nuigni- tude which is the unity or limited quality to be meas- ured, is : The repetition of a certain magnitude used as the unit of measurement to e(|ual or express the compara- tive value of a magnitude of the same kind. It always answers the question ''How many ?" This " how many " may assume tw o related aspects : either how many tunes one pai't as unit has to be taken or repeated to make up the whole quantity ; or how many parts as units, each taken once, compose the whole. In the first case, the times of repetition of the measuring unit is mentally the more prominent ; in the second, the actual number of measuring parts ; e. g., in thinking of forty yards, we may at one time dwell on the forty t tines the unit is repeated ; at another time, on the actual forty parts making the uniiied whole. As already said, the number and the measuring unit together give the absolute magnitude of the quantity. The number bv itself indicates its relative value. It always expresses ratio* — i. e., the relation which the * llcnct\ again, tlie absurdity of iniiUiplying pure iiutnhor or dividing it into parts. Wo may divide a ratio, but not into parts. s,* T2 THE PSYCHOLOGY OF NUiMRKR. ii < r fi If! i magnitude to bo measured bears to the unit of refer- ence. Seven, as i)ure number, expi'esses ecjually tlie ratio of 1 foot to 7 feet, of 1 inch to 7 inches, of 1 day to 1 week, of $1,000 to $7,000, and so on indelinitely. Simply as seven it has no meaning, no definite vahie at all ; it only states a possible measurement. Tliis definition arrived at from psychological analy- sis is that given by some of the greatest mathematicians on a strictly mathematical basis, as may be seen from comparison with the following definitions : Wewtoiis. — Number is the abstract ratio of one quantity to another quantity of the same kind. Uulers. — ]N"unil)er is the ratio of one quantity to another (pumtity taken as unit.'^ Phases of Xumuek. — The aspects of number follow directly from what has been said. Quantity, the unity measured, whether a ''collection of objects" or a l)hys- ical whole, is c())dinuous. an undefined how much j num- ber as measuring value is discrete, how many. The mag- nitude, muchness, hefore measurement is mere unity ; cfftiP njeasurement it is a sum taken as an integer — that is, an aggregation of parts (units) making up one whole; number as showing how many refers to the units, which put together make the sum. Quantity, measured mag- nitude, is always concrete ; it is a certain kind of mag- nitude, length, volume, weight, area, amount of cost. * J. C. Gliishan, one of the ticutcst of living mathenuiticians, de- fines thus : " A unit is any standard of reference employed in count- ing any collection of objects, or in measuring any magnitude. A number is that which is a])plied to a unit to exj)ress the comparative magnitude of a (juantity of the same kind as the unit." (See his Arith- metic far High Schools, etc.) DEFINITION OP NUMBER. etc. " Number," as sim])ly detiniiig the Low many units of measurement, is always abstract. The conception of measuring ^>a/'^6' and of times of repetition is inseparable from number as expressing the numerical value of a (quantity ; as discrete, it is so many ■ parts taken one time — constituting the unity ; as ab- stract, it is one part taken so many times. In the one ease, as before suggested, attention is more upon the numbered ^y'tt/'^*', in the other, more upon the nunther of the parts. They are absolutely correlative conceptions of the same measured magnitude. That is, a value of $50 may be regarded as determined by taking ^IJif'tf/ times, or by taking $50 — that is, a w/io/e of Jiftf/ jjarts — 07ie time. The numerical process and the resulting numerical value are the same, however we arrive at the numhcy- — i. e., the ratio of measured quantity to measur- ing unit. As this conception of the relation between parts and times in the measurement of quantity is essen- tial to the interpretation of numerical operations, we may give it a little further consideration. AVe wish to know the amount of money in a roll of dollar bills. We take five dollars, say, as a convenient measuring unit ; we separate our undefined whole into grou])s of live dollars each ; we count these groups and find that there are ten of them — i. e., the numerical value is ten ; we have now a definite idea both of the measuring unit and of the times it is repeated, and so have reached a definite idea of the amount of money in the roll of bills. "We began with a vague whole, an un- defined unity ; we broke it up into parts (analysis), and by relating (counting) the parts we arrived at our unity again ; the same unity, yet not the same as regards the T ^r^x. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I lis 1^ 1^ |2j2 2.0 m IL25 iU 1.6 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 T A iiT^^^ '^ ^^" V.x \ % ^ u THE PSYCHOLOGY OF NUMBER. I I attitude of the mind towards it. It is now a definite unity constituted by a known number of definite jmrts ; it is a sifht of vnits. On the analytic side ot this defining process the emphasis is on the parts, the units ; on the syntlietic side the emphasis is on tlie defined unity, the sum. The j)arts are means to an end ; they exist only for the sake of the end, the sum. The ten units — that is, the unit repeated ten times — make up, are, the one sum — i. e., the sum taken one time. Further, since the unit of measure is itself measured by a smaller unit, the dollar, the same psychological explanation applies to the measurement of the quantity by means of this smaller unit. The five-dollar unit taken ten times is identical with ten of these units taken once. We are conscious, also, that any part of this iive-dollar unit taken ten tiines is identical with ten such parts taken once. That is, $1, taken ten times, is a whole of $10 taken once ; and since this is true of every dollar in the five, our measurement gives a whole of $10 taken once, a whole of $10 taken twice, and so on ; that is, altogether, a whole of $10 taken Jive times. In other words, the measurement, ten groups (or units) of five dollars each necessarily implies the correlative measurement. Jive groups of ten dollars each. This rhythmic process of parting and wholing which leads to all definite quantitative ideas, and involves the correlation of times and parts, may be illustrated by simple intuitions. In measuring a certain length we find it, let us suppose, to contain four parts of three feet each ; then the relation between parts (measuring units) and numerical value (times of repetition) may be TIMES AND PARTS. 75 perceived in the following, where the dots symbolize both times and units of quantity : • • • a b Measuring l)y the 3-feet unit we count it off four times — that is, the (juantity is expressed l)y 3 feet taken four times. This is represented by the four vertical columns of three minor units each. J]ut this measuring process necessarily involves the correlated process which is ex- pressed by 4 feet taken three times. For, in measuring by three feet, and finding that it is repeated four times, we perceive that each of its tln-ee parts is repeated /"our times, giving the three horizontal rows a, h, c — that is to say, a is one whole of 4 feet, h a second whole of 4 feet, and c a third whole of 4 feet ; or, in all, 4 feet taken three times. Briefly, 1 foot four times is one wliole of 4 feet ; this is true of every foot of the origi- nal measure, 3 feet; and therefore 3 iiiGt four times is 4 feet three times. It is clear that the two questions, {a) in 12 feet how many counts of 4 feet each, and {h) how many feet in each of 4 counts making 12 feet, are solved in e^ractJij the same XLHiy • neitlier the three counts (times) in the first case nor the three feet in the second case can be found irlthout eounthuj the twelve feet off in (jroujKs (f • four feet each. This necessary correlation, in the measurement of quantity, between "parts" and "times" — numerical value of the measurincr unit and numerical value of the measured quantity — gives the psychology of the fun- .■ M ; I n u V'5l 76 THE PSYCHOLOGY OF NUMBER. fl damenttal principle in multiplication known as the law of eonmuitation : the product of factors is the same in whatever order they may be taken — i. e., in the case of two factors, for example, either may be multiplicand or nndtiplier ; a times h is identical with h times a. It is asserted i)y some w^riters that this connnntative law does not hold when the mukiplicand is concrete ; ''for,"' we are told, ''though there is meaning in re- quiring ^i to be taken three times, there is no sense in proposing that the number 3 be taken four-doHars times" — which is perfectly true. Nevertheless, the ob- jection seems to be founded on a misconception of the psychical nature of mnnber and the i^sychological basis of the law of commutation. Psychologically speaking, can the multiplicand ever ha a pure number? If the foreffoinir account of the nature of number is correct, the multiplicand, however written, nuist always be nn- derstood to express measured quantity ; it is always concrete. As already said, 4x8 must mean 4 units of measurement taken three times. If number in itself is purely mental, a result of the mind's fundamental process of analysis-synthesis — what is the meaning of 8x4 where both symbols represent pure numbers, and where, it is said, the law of commutation does hold ? There is no sense, indeed, in proposing to multiply three by four dollars; but equally meaningless is the propo- sition to multiply one pure numl)er by another — to take an abstraction a number of timcp. Thus, if the commutative law " does not hold when the multiplicand is concrete'' — indicating a measured quantity — it does not hold at all ; there is no such law. Lut if the psychological explanation of number as aris- TIMES AND PARTS. 77 ing from measurement is true, there is a law of com- mutation. We measure, for example, a quantity of 20 pounds weight by a 4:-pound weight, and the result is expressed by 4 pounds x 5, but the psychological cor^ relate is 5 pounds x 4. Here we have true commuta- tion of the factors, inasmuch as there is an interchange of both character and function : the symbol which de- notes measured quantity in the one expression denotes pure number in the other, and vice verm. If the 4 pounds in the one expression remained 4 pounds in the commuted expression, would there be commutation ? We have referred to the fallacy of identifying actual measuring parts with numerical value ; it may now be said that, on the other hand, failure to note their neces- sary connection — their law of commutation — is often a source of perplexity. To say nothing at present of the mystery of "Division," witness the discussions upon the rules for the reduction of compound quantities and of mixed numbers to fractions. To reduce 41 yards to feet w^e are, according to some of the rules, to multiply 41 by 3. According to others, this is wrong, giving 123 yards for product ; and we ought to multiply 3 feet by 41, thus getting the true result, 123 feet. Some rule- makers tell us that though the former rule is wrong it may be followed, because it always brings the same nu- mcrical result as the correct rule, and in ])ractice is gen- erally more convenient. It seems curious that the rule should be always wronc: vet always brinii: the riffht re- suits. AVitli the relation between parts and times before us the difficulty yanishes. The expression 41 yards de- notes a measured quan'iity ; 41 expresses the numerical value of it, and one vard the measuring unit; our con- ' 1 1 I 'da I' i! i i 1 I 78 THE PSYCHOLOGY OF NUMBER. I. .{ i 11- f. »i ception of the quantity is therefore, primarily, 41 parts of 3 feet each, and we multiply 3 feet by 41 ; but this conception involves its correlate, 3 parts of 41 feet each ; and so, if it is more convenient, we may multi ply 41 feet by 3. A similar explanation is applicable to the reduction, e. g., of '^3 J to an improper fraction. The denominator of the fraction indicates what is, in this case, the direct unit of measure, one of the four equal parts of the dollar; and so we conceive the $3 as denoting 3 parts of 4 units (quarter dollars) each, and multiply 4 by 3; or, as denoting 4 parts of 3 units each, and multiply 3 by 4. / Educational Applications. 1. Every numerical operation involves three factors, and can be naturally and completely apprehended only when those three factors are introduced. This does not mean that they must be always formulated. On the contrary, the formulation, at the outset, would be con- fusing ; it would be too great a tax on attention. But the three factors must be there and must be used. Every problem and operation should (1) proceed upon the hasis of a total magnitude — a unity having a certain numerical value, should (2) have a certain unit which measures this whole, and should (3) have num- ber — the ratio of one of these to the other. Suppose it is a simple case of addition. John has $^2, James $3, Alfred $4:. IIow much have they altogether ? (1) The total magnitude, the amount (muchness) altogether, is here the thing sought for. There will be meaning to the problem, then, just in so far as the child feels this amount altogether as the whole of the various parts. 1 , v> EDUCATIONAL APPLICATIONS. 79 (2) The unit of measurement is the one dollar. (3) The number is tlie measuring of how many of these units there are in all, namely, nine. When discovered it de- lines or measures the how much of the magnitude which at lirst is but vaguely conceived. In other words, it must be borne in mind that the thought of some inclu- sive magnitude must, psychologically, precede the oper- ation, if its real meaning is to be apprehended. The conclusion simply defines or states exactly how much is that magnitude which, at the outset, is grasped only vaguely as 7nere magnitude. Are we never, then, to introduce problems dealing with simple numbers, with numbers not attached to magnitude, not measuring values of some kind ; are we not to add 4, 5, 7, 8, etc. \ Must it always be 4 apples, or dollars, or feet, or some other concrete magnitude ? iV/>, not necessarily as matter of practice in getting fa- cility in handling numhers, Number is the tool of measurement, and it requires considerable practice with the tool, as a tool, to handle it with ease and accuracy. But this drill or practice-work in " number" should never be introduced until after work based upon definite magnitudes ; it should be introduced only as there is formed the mental habit of continually referring number to the magnitude which it measures. Even in the case of practice, it would be safer for the teacher to call attention to his reference of number to concrete values in every case than to go to the other extreme, and neg- lect to call attention to its use in defining quantity. I'or example, when adding "numbers," the teacher might say, " Now, this time we have piles of apples, or we have inches, etc., and we want to see how much I :; 'It' 80 THE PSYCHOLOGY OF NUMBER. i.:! I iiii we have in all " ; or the teacher might ask, at the end of every problem, "What were we counting up or measur- ing that time?" letting each one interpret as he pleased. Just how far this is carried is a matter of detail ; what is not a matter of detail is that the habit of interpreta- tion be formed by continually referring the numbers to some quantity. 2. The unit is never to be taught as a fixed thing (e. g., as in the Grube method), but always as a unit of measurement. One is never one thing simply, but al- ways that one thing Ui 1 I'M .; i i 11 : ) i r I memory will take care of itself. Tlie facts do not need to be seized and carried by sheer effort of memory, but are reproduced, whenever needed, out of tlie mind's own power. The learning of facts, the preservation and re- teution of information, is an outcome of the formation of habit, of the attainment of power. The method which neglects the measuring function of number can not pos- sibly lead to a definite habit ; it can result only in the ability to remember. There is no question here about the need of drill, of discipline, in all instruction. But there is every ques- tion about the true nature of drill, of discipline. The sole conception of drill and of discipline which can be afforded by the rigid unit method is that of ability to hold the mind fixed upon something external, and of ability to carry facts by sheer force of memory. By the psychological method of treating the unit as means to an end, a basis of measurement, the discipline con- sists in the orderly and effective direction of power already struggling for eocpression or utterance. One is the drill of a slave to fit him for a task which he himself does not understand, and which he docs not care for in itself. The other is the discipline of the free man in fitting him to be an efiicient agent in the realization of his own aims. {c) Finally, the fixed unit method deadens interest and mechanizes the mind in not allowing free play to its tendencies to variety, to continual new development. As already said, according to the Grube method, the fact that 2 -h 2 = 4 always remains precisely the same, no matter how much its monotony is disguised by per- mutations with blocks, slioe pegs, pictures of birds, etc. EDUCATIONAL APPLICATIONS. 89 According to the measuring method, the habit or gen- eral direction of action remains the same, but is con- stantly differentiated through application to new facts. According to the Grube method unity is one thing, and that is the end of it. According to the measuring method unity may be 12 (the dozen oranges as meas- ured by the particular orange, the day as measured by tlie hour, the foot as measured by the inch, the year as measured by the month, etc.), or it may be 100 — e. g., the dollar as measured by the cent. Instead of relying upon a minute and exhaustive drill in numbers from 1 to 5, allowing next to no spon- taneity, severing nearly all connection with the child's actual experience, ruling out all variety as diametrically opposed to its method, it can lay hold of and give free play to any and every interest in a whole which comes up in the child's life. Unity as 12, as a dozen, is likely to be indefinitely more familiar and interesting to a child than 7 ; the desire to be able to tell thne comes to be an internal demand, etc. But the Grube method must rule out 12. Twenty -five as a imity (of money, the quarter- dollar), 50 (as the half-dollar), 100 (as the dollar), are continual and lively interests in the child's own activi- ties. Each of these is just as much one as is one eye or one block, and is arithmetically a very much better type of unit than the block by itself, because it is capa- ble of definite measurement or rhythmic analysis into sub-units, thus involving division, multiplication, frac- tions, etc. — operations which are entirely external and irrelevant to the fixed unit. Some will probably say, "But 100, or even 12, is altogether too complex and difiicult a number for a f[ : : ill Ji i: m i: ■'I 90 THE PSYCHOLOGY OF NUMBER. t 1 1^ I i cliild to grasp." Yes, if it is treated simply as an ac- cumulation or aggregation of individual separate fixed units. Very few adults can definitely grasp 100 in that sense. The Grube method, proceeding on the basis of the separate individual thing as unit, is quite logical in insisting upon exhausting all the combinations of all the lower numbers. No, if 100 is treated as a natural whole of value, needing to be definitely valued by being meas- ured out into sub-units of value. One dollar is one^ we repeat, as much as one block or one pebble, but it is also (which the block and pebble as fixed things are not) two 50's, four 25's, ten lO's, and so on. It may be well to remind the reader that while we are dealing here only with the theory of the matter, yet the successful dealing with such magnitude as the dozen and the dollar is not a matter of theory alone. Actual results in the schoolroom more than justify all that is here said on grounds of psychology. During the six months in which a child is kept monotonously drilling upon 1 to 5 in their various combinations, he may, as proved by experience, become expert in the combina- tion of higher numbers, as, for example, 1 ten to 5 tens, 1 hundred to 5 hundred, etc. If the action of the mind is judiciously aided by use of objects in the measuring process which gives rise to number, he knows that 4 tens and 2 tens are 6 tens, 4 hundred and 2 hundred aie 6 hundred, etc., just as surely as he knows that 4 cents and 2 cents are 6 cents ; because he knows that 4 units c asurement of any kind and 2 units of the same kind are 6 units of the same kind. Moreover, this introduction of larger quantities and larger units of measurement saves the child from the chilling effects EDUCATIONAL APPLICATIONS. 91 of monotony, maintains and even increases liis interest in numerical operations through variety and novelty, and through constant appeals to his actual experience. Many a child who has never seen " four birds sitting on a tree and two more birds come to join them, mak- ing in all six birds sitting on the tree," has heard of one of his father's cows being sold for $40 (4 tens), and an- other for $20 (2 tens), making in all 6 tens or $60 ; or of one team of horses being sold for 4 hundred dollars, and another for 2 hundred dollars, in all 6 hundred dollars. AVhen it is urged that these higher numbers are be- yond the child's grasp, what is really meant ? If the meaning is that the child can not picture the hundred, cannot visualis*^ it, this is perfectly true ; but it is about equally trr the case of the adult. No one can have a perfect mental picture of a hundred units of quantity of any kind. Yet we all have a conception of a hun- dred such units, and can work with this conception to perfectly certain and valid results. St a child, getting from the rational use of concrete objects as symbols of measuring units the fact that 4 such units and 2 sucli units are 6 such units, gets a clear enough working con- ception for any units whatever. The opposite assump- tion proceeds from the fallacy of tlie fixed unit method, and from the kindred fallacy that to know a quantity numerically we must mentally image its numerical value ; grasp in one act of attention all the measuring parts con- tained in the quantity. Neither adult nor child, we re- peat, can do this. We can not visualise a figure of a thousand sides — perhaps but few of us can " picture " one of even ten sides — but we nevertheless know the i m 1:1 1; II 'i ^» :, '1 , I ^ ^! i I 92 THE PSYCHOLOGY OP NUMBER. « i ■' t' I! i figure, have a definite conception of it, and with certain given conditions can determine accurately the proper- ties of the figure. The objection, in short, proceeds from the fallacy that we know only what we see ; that only what is presented to the senses or to the sensuous imagination is known ; and that the ideal and universal, the product of the mind's own working upon the mate- rials of sense perception, is not knowledge in any true sense of the word. To sum up : One metlijod cramps the mind, shutting out spontaneity, variety, and growth, and holding the mind down to the repetition of a few facts. The other expands the mind, demanding the repetition of activi- ties, and taking advantage of dawning interest in every kind of value. One method relies upon sheer memo- rising, making the " memory " a mere fact-carrier ; the other relies upon the formation of habits of action or definite mental powers, and secures memory of facts as a product of spontaneous activity. One method either awakens ?io interest and therefore stimulates no de- veloping activity ; or else appeals to such extrinsic in- terest as the skilful teacher may be able to induce by continual change of stimulus, leading to a varying ac- tivity that produces no unified result either in organis- ing power or in retained knowk dge. The other method, in relying on the mind's own activity of parting and wlioling — its natural functions — secures a continual sup- port and re-enforcement from an internal interest which is at once the condition and the product of the mind's vigorous action. T h n <\ n f c 1 \)k '»s CHAPTER YI. the development of number ; or, the arithmetical operations. Numerical Operations as External and as Intrinsic to Number. Addition, Subtraction, Multiplication. — As we have already seen, nninher in the strict sense is the measure of quantity. It definitely measures a given quantity by denoting how many units of measurement make up the quantity. All immerical operations, there- fore, are phases of this process of measurement ; these operations are bound together by the idea of measure- ment, and they differ from one another in the extent and accuracy with which they carry out the measuring idea. As ordinarily treated, the fundamental operations — addition, subtraction, etc. — are arithmetically connected but psychologically separated. Addition seems to be one operation which we perform with numbers, sub- traction another, and so on. This follows from a mis- conception of the nature of number as a psychical process. Wherever one is regarded as one thing^ two as two things^ three as three things^ and so on, this thought of numerical operations as something exter- nally performed upon or done with existing or ready- 93 h \ f« (■ ■ Mi 94 THE PSYCHOLOGY OF NUMBER. ill made numbers is inevitable. Number is a fixed exter- nal something upon which we can operate in various ways : it simply happens that these various ways are addition, subtraction, etc. ; they are not intrinsic in the idea of number itself. But if number is the mode of measuring magnitude — transforming a vague idea of quantity into a definite one — all these operations are internal and intrinsic de- velopments of number; they are the growth, in accu- racy and definiteness, of its measuring power. Our present purpose, then, is to show how these operations represent the development of number as the mode of measurement, and to point out the educational bearing of this fact. The Stages of Measurement. — We have already ?een that there are three stages of measurement, differing from one another in accuracy and definiteness. We may measure a quantity (1) by means of a unit which is not itself measured, (2) with a unit which is itself measured in terms of a unit homogeneous with the quantity to be measured, (3) with a unit wliich is not only defined as in (2), but has also a definite relation to some quantity of a different kind. If, for example, we count out the number of apples in a peck measure, we are using the first type of measurement; there is no minor unit homogeneous with the peck measure by which to define the apple. If we measure the number of pounds of apples, we are using the second type of measurement ; each apple may itself be measured and defined as so many ounces, and as therefore capable of exact comparison with the total number of pounds. In this case we have a continuous scale of homogeneous THE ARITHMETICAL OPERATIONS. 95 measuring units — drachms, ounces, etc. ; in the first type of measuring we have not such a scale. Finally, the pound itself may be defined not only as 16 ounces, but also as bearing a relation to some other standard ; as, e. g., a cubic foot of distilled water at the temperature of 39'83 weighs 62J pounds, the linear foot itself being defined as a definite part of a pendulum which, under given conditions, vibrates seconds in a given latitude (see page 46). The Specific Nximerical Operations. — The funda- mental operations, as already said, are phases in the de- velopment of the measuring process. 1. We have seen that the comparison of two quan- tities in order to select the one fittest for a given end not only gives rise to quantitative ideas, but also tends to make them more clear and definite. Each of the quantities is at first a vague whole ; but one is longer or shorter, heavier or lighter, in a word, more or less than the other. Here we have the germinal idea of addition and subtraction. The difference between the quantities will be a vague muchness, just as the quan- tities themselves are vague, and will become better de- fined just as these become better defined. This better definition arises with the first stage of measurement — that of the undefined unit. We begin with measuring a collection of objects by counting them off, and this suggests the measuring of a continuous quantity in a similar way — that is, by counting it off in so many paces, hand-breadtlis, etc. Now, in the use of the inexact unit there is given a more definite idea of the quanti- ties and of the more or less which distinguishes them, but no explicit thought of the ratio of one to the other ; 8 / : A\'^ i^'^h M i il m 96 THE PSYCHOLOGY OF NUMBER. lllj :i t:i 1. * »■:' I i ■> :i 1 I ii! f i :) ' there is a counting of lH'e tilings but not of ^qval things. In other words, the process of counting witli an unmeas- ured unit gives us aritlnneticully AdiUtio7i and /Suhtrac- tlon. The result is definite sini})ly as to mare or leas of magnitude. It shows how many more coins there are in one heap than in another, liow many more pa/;es in one distance than another, in, etc. It gives an idea of the relative value in this one imint of moreness or aggre- gation^ but it does not bring into consciousness what mul- tiple, or part, one of the quantities is of the other, or of their difference. This is a more complex conception, and so a later mental product. 2. With the development of the idea of quantity in fulness and accuracy the second stage of measurement is reached, in which the measuring unit is uniform and defined in terms homogeneous with the measured quantity. This principle of measuring with an exact unit — i. e., a unit which is itself made up of minor units in the same scale — gives rise to Midtiplication and Divi- sion, and is in reality the principle of ratio. In the addition or subtraction of two quantities we are not conscious of their ratio ; we do not even use the idea of their ratio. In multiplication and division we are con- stantly dealing with ratio. We do not discover merely that one quantity is more or less than another, but that one is a certain part or multiple of another. When, for example, we multiply $4 by 5 we are using ratio ; we have a sum of money measured by 5 units of $4 each, where the number 5 is the ratio of the quantity measured to the measuring unit. In division, the in- verse of multiplication, ratio is still more prominent. ri TT THE ARITHMETICAL OPERATIONS. ©T The idea of ratio involved in multiplication and division is a much more practical one than that of mere aggregation (more or less) involved in addition and subtraction, because it helps to a more accurate adjustment of means to end. Suppose a man in re- ceipt of a certain salary knew that the rent of one of two houses is $100 a year more than that of the other, but could not tell the ratio of the $100 to his salary, it is obvious that he would have but little to guide him to a decision. But if he knows that $100 is one fifth or one fiftieth of his entire income, he has clear and posi- tive knowledge for his guidance. "With ratio — nmltiplication and division — go the simpler forms and processes of fractions.* 3. The principle of measuring one scale in terms of another gives us arithmetically ^/•^o/'^ton, and the operations involving it, such as percentage and multi- plication and division of fiactions, and brings out the idea of the equation. The Order of Arithmetical Instruction. — We have already seen one fundamental objection to the ordinary method of teaching number, whether as car- ried on in a haphazard way or by what is known as " the Grube " method ; it takes number to be a fixed * In external form, but not in internal meaning, other fractions belong here also. For example, the ratio of 14 to 3 may be written 14 -^ 3 or "i/ ; in any case, the idea is to discover how many units of the value of 3 measure the value of 14 units, but the very fact that 3 is taken as the unit shows the meaning to be the discovery of the ratio of 14 to 3 as unity. Whether the result can actually be writ- ten in integral form or not is of no consequence in principle, so long as the process is the attempt to discover the ratio to unity ; the pro- cess is ^ of 14. 1, I , ii ii' i jj '* l\ ^ ' 'i '^1 1 1 lyjj \ 1 " m \ lyi ;! M 11.; l!:i| ! 'i^ .1 i ]\i « > i 98 THE rSYCIIOLOGY OP NUMBER. quantity, instead of a mental operation concerned in measuring quantity. We can now appreciate another fundamental objection : it attempts to teach all the operations simultaneously^ and thus neglects the fact of growth in psychological complexity corresponding to the development of the stages of measurement. It takes each number as an entity in itself, and exhausts all the operations (except formal proportion *) that can be performed within the range of that number. It as- sumes that the logical order is the order of growth in psychological difficulty. All operations are implied even in counting^ hut are not therefore identical. Logically^ or as processes, all operations are implied^ even in counting. To count up a total of four apples involves multiplication and division, and thus ratio and fractions. When we have counted 3 of the 4 apples, we have taken a first 1, a second 1, and a third 1 — that is, a total of three I's — out of the 4 which com- pose the original quantity. We \\2iYQ divided the origi- nal quantity of apples into partial values as units, and have taken one of those nnits so many times ; this is multiplication. But it does not follow that, because the operations are logically implied in this process, they are therefore the same in their complete development and all equal in point of psychological difficulty ; much less that they should be definitely evolved in conscious- id all taught together. The acorn implies the it the oak is not the acorn. Multiplication is im- ness .1 i k * Why not proportion, or even logarithms, on the principle that everything that is logically correlative should be taught at once f The logarithm is just as much involved in say 8, as are all the mul- tiplications and additions which can be deduced from it. TUE ARITHMETICAL OPEUATIONS. 99 plied in tlie simple act of counting:, and lios its genesis in addition ; but multiplication is not merely counting, nor is it identical with addition. The operation indi- cated in $2 + 82 + $12 + $2 = $S may be performed, and in the initial stages of mental growth is performed, without the conscious recognition that eight is four times two. The latter is implied in the former, and in due tinie is evolved from it ; but for this very reason it is a later and more complex conception, and therefore makes a severer demand upon conscious attention. The summing process is made comparatively easy through the use of objects ; it is little more than the perception of related things. The nuiltiplication process is more complex, because it demands the actual use and more or less conscious grasp of ratio, or times, the abstract ele- ment of all numbers ; it is the conception of the rela- tion of things. We might go on adding twos, or threes, or fours, instantly merging each successive addend in the growing aggregate, and, never returning to the ad- dends, correctly obtain the respective sums without the more abstract conception of times ever arising — that is, without ever being conscious that the " sum • ' is a prod- uct of which the times of repetition of the addend is one of the factors. Certainly this more abstract notion does not arise at first in the development of numerical ideas in either the child or the race. If anyone still maintains that addition (of equal ad- dends) and multiplication are identical processes, let him prove by mere summing (or counting) that the square root of two, multiplied by the square root of three, is equal to the square root of six ; or find by loga- rithms the sum of a given number of equal addends. .' fill ti i^V,:: III "I'M 100 THE PSYCHOLOGY OF NUMBER. \ ill- ii i •!)-^ '*!- ;ii Simultaneoiis Method not Psychological. — It seems clear, therefore, that the fundamental operations as for- mal processes should not be all taught together ; on the other hand, rational use should be made of their logical and psychological correlation. It is one thing to per- form arithematical operations in such a way as to in- volve the use of correlative operations, and it is another thing to force these operations into consciousness, or to make them the express object of attention. The natural psychological law in all cases is first the use of the process in a rational way, and then, after it has be- come familial', abstract recognition of it. The method usually followed violates both sides of the true psychological principle. Because it treats num- ber as so many independent things or unities, it can not mentally or by interpretation bring out how the opera- tions are correlative with one another. It is only when the unit is treated not as one thing, but as a standard of measuring numerical values, that addition and multi- plication, division and fractions, are rationally correla- tive. And it is because this correlation is not brought out and rationally used that — in spite of the teaching of all the operations contemporaneously — division is still a mystery and fractions a dark enigma. Then, the common method errs in the opposite ex- treme by attempting to force the recognition of ratio, and fractions, into consciousness before the mind is sufficiently mature, or sufficiently exercised in the use of ratio, to grasp its meaning. The result of tliis un- natural method is that mechanical drill and memoiiz- ing, with the sure effect of waning interest and feeble thought, is forced upon the pupil. To master all the 1 THE ARITHMETICAL OPERATIONS. 101 numerical operations contained in 0, 7, 8, and 9 is a slow and tedious process, and so the method is com- pelled in self-consistency to limit the range of numbers which are to be mastered in a given time. In reality it is easy for the mind to grasp the fact that $1 is a hun- dred ones, or fifty twos, or ten tens, or five twenties, long before it has exhausted all possible operations with such numbers as 7 or 11 or 18. It might, indeed, be maintained that a return to the old-fashioned ways of our boyhood, by which we soon became expert in the mechanical processes of addition and subtraction, would be preferable to this monotonous drill on " all that can be done with the numbers" from 1 to 10 and from 10 to 20 in the second year ; for this new method is just about as mechanical as the old, and, while leaving the child little if any better prepared for the " analysis " of the higher numbers, leaves him also without the expert- ness in the operations which is essential to progress in arithmetic. Dimsion and Siihtraction not to precede MuUvpli- catlon and Addition. — On the ground that the " first ])rocedure of the mind is always analytic," * some main- tain that division and subtraction (the "analytic pro- cesses ") should be taught before multiplication and ad- dition. But just as multiplication, definitely using the idea of ratio, is a more complex process than addition, so division, the inverse of multiplication, is more complex * It might be asserted with some truth tiiat the first procedure of the mind is synthetic : there must be a "whole" — a synthesis — however vague, for analysis to work upon. Ctrtainly the la&t pro- cedure of the mind is " synthetic." •i ' I I I u ■W ! , . 1.1 102 THE PSYCHOLOGY OF NUMBER. >'. * i!i i than subtraction, the inverse of addition. We may, as we liave seen, add a number of threes, for example — giving eacli addend a momentary attention, and then dropping it utterly from consciousness — without grasp- ing the /actor, which, with three as the other factor, will give a p?'odu€t equal to the smn of the addends. So in division, the inverse operation, this factor does not come merely from the successive subtractions of three from the sum until there is no remainder ; here, as in addition, a further mental operation is necessary before the factors are discovered — that of counting the times of repetition ; i. e., of finding the ratio of the sum (dividend) to the repeated subtrahend. We are told, too, that when we separate 8 cubes into 4 equal parts it is instantly seen that 8 contains 2 four times, that 2 is one fourth of 8, that 2 may be taken four times from 8, and that these results being obtained independently of addition and multiplication, division and subtraction may be taught first. There seems to be a fallacy lurking liere. "VYe may, indeed, separate 8 cubes into two parts, or four parts, or eight parts ; but that is mere physical separation. Granting recognition of the concrete (spatial) element • — the measuring units — how does the abstract element — the idea of times — arise? How do we know that there is four times 2 or eight times 1 ? Only by count- ing, by relating, by an act of synthesis — the last pro- cedure of the mind in a complete process of thought. Thus, the fallacy referred to ignores one of the two necessary factors (relation) in the psychical process of number. It must presuppose that counting does not imply addition and multiplication. What is counting / THE ARITHMETICAL OPERATIONS. 103 / but addition by ones ? What is five, if not one more than four ; and four, if not one less than five ? How is four, e. g., defined except as that number which, ap- pUed to a unit of measure, denotes a quantity consist- ing of three such units and one unit more ? This count- ing, which begins with discrete quantity (collection of objects) in the first stage of measurement, is addition (with subtraction implied) by ones, and the idea of mul- tiplication and division involved in it becomes evolved (in counting with an exact unit of measure) with the growth of numerical abstraction and the consequent development of the measuring power of number. It seems plain, then, that in the development of num- ber as the instrument of measurement there is first the rational use, leading to conscious recognition, of the aggregation idea — that is, addition and subtraction ; then the definite use, leading to conscious recognition, of the factor (times) idea — that is, multiplication and division. In other words, the psychological order as determined by the demand on conscious attention is the old-time arrangement — Addition and Subtraction, Multiplication and Division. It is the order in which numerical ideas and pro- cesses appear in the evolution of number as the instru- ment of measurement ; the order in which they appear in the reflective consciousness of the child ; the order of increasing growth in psychological complexity. This order may be said to reverse the order of logical de- pendence, but the psychological order rather than that of logical dependence is to be the guide in teaching. N'ot Exclusive Attention to One Hide or Process. — But the true method, as based on this psychological m "I I. s 104 THE PSYCHOLOGY OF NUMBER. ', i I; I m . ■' ■n order of instruction, by no means implies that addition and subtraction are to be completely mastered before the introduction of any multiplication or division or fractions. Quite the contrary. On account of their greater complexity the higher processes are not to be taught analytically — made, that is, an object of con- scious attention from the first ; but they may and should be freely used, and thus relieve the monotony of too much addition and subtraction, and at the same time prepare the way for their conscious (analytic) use. Because of the rhythmic character of multiplica- tion such forms of it as can be objectively presented in simple constructions — the putting together of tri- angles, squares, cubes, etc., to make larger or more complex figures, of dimes to make dollars — are much more easily learned than many of the addition and sub- traction combinations. The ideas of ratio should be in- cidentally introduced in connection with certain values (e. g., 9, 12, 6, 16, 100, etc.) practically from the begin- ning ; and consequently the process of fractions in sim- ple forms, and its symbolic statement. Kothing but the demands of a preconceived theory could so nullify ordi- nary common sense as to suppose that there is no alter- native between either exhausting all operations with every number before going on to the next higher, or else mastering all additions and sul)tractions before go- ing on to ratio — multiplication and division. Practical common sense and sound psychology agree in recom- mending first the e?nphasis on addition and subtraction, with incidental introduction of the more rhythmical and obvious forms of ratio, and gradual change of emphasis to the processes of multiplication and division. If the \ THE ARITHMETICAL OPERATIONS. 105 idea of number as a mode of measurement is followed, it will be practically impossible to w^ork in any other way. Even while working explicitly w^th addition and subtraction — inches, feet, ounces, pounds, dollars, cents, etc. — the process of ratio is constantly being introduced. The child can not help feeling that 1 inch is one third of 8 inches, 10 cents (1 dime) one tenth of a dollar, etc. ; and this natural growth towards the definite conception of ratio is only checked, not forwarded, by compelling a premature conscious recognition of the nature of the process. Addition and Subtraction. — The general nature of these operations as concerned with measurenient through the process of aggregating minor units or parts has al- ready been dealt with. Two or three points may, how- ever, be considered in more detail. 1. Work from and within a Whole. — Here, as every- where, the idea of a magnitude — a whole of quantity — corresponding to some one unified activity should be present from the first. Some vague quantity or whole, which is to be measured by the putting together of a number of parts, alone gives any reason for performing the operation and sets any limit to it. The process of breaking up the whole into parts and then putting to- gether these parts into a whole, measures or defines what was originally a vague magnitude and gives it precise numerical value. In dealing, say, with 6, we may begin with a figure like this ^."^ This is a unity or whole — it is one. But its value is indefinite. The * This may, of course, be constructed out of splints, or whatever is convenient. ii 106 THE PSYCHOLOGY OF NUMBER. Ill k: counting off of the various sticks changes the vague unity into a measured unity, but these parts always fall within the orujinal unity. Thei*e is always a sense of the whole connecting them together. If the square lias already been mastered, the figure will be recognised as one 4 -\- one 2. Or, if one of the diagonals is changed thus \/y^, it will be recognised as two triangles — that is, as one 3 -|- one 3. Or, of course, it may be taken all to pieces and put together again and recognised as 6 parts of the value of 1 each. Or, the pupil may be told to make "pickets" or "tents" of the figure, and, arranging them as follows, A A A , see that there are three groups of the value of 2 each. The principle kept in mind in this instance is that of the equation and its rhythmic construction, {a) Accord- ing to the prevalent method, six, when reached, would be simply six ones, six separate unities, that is — not, as in the foregoing illustration, six parts of unit value each. No matter how much the teacher is urged to have the pupils recognise six at a glance, and not count up the various unities in it separately, still the fact remains that it can not, by that method, be grasped as a whole ; while by the psychological method it can not be grasped in any other way. (?>) It is also, upon the psychologic- al method, regarded as having a value equal to (meas- ured by) its constituent minor wholes. We are alwavs ^>i .-■,,<; -;^- vjiihin a value, simply making it more clear .ij 1 detinite, not blindly or vaguely from fixed unities to their accidental sum — accidental, that is, so far as tiie action of mind is concerned. As a result, the psy- chological method appeals directly to the power of break- ing up a larger whole into minor wholes, and putting THE ARITHMETICAL OPERATIONS. 107 these together to make a larger whole. It appeals to the constructive rhythmic interest, never to mere memoriz- ing. It gives the maximum opportunity for the exer- cise of power ; it leaves the minimum for mere mechan- ical drill. Because, dealing with wholes, intuition may be used ; the rationality of the principle — the construe- tlon of a complete whole hy means of jpartial wholes — may be objectively seen and clearly appreciated. It may be laid down, then, in the most emphatic terms, tliat the value of any device for teaching addi- tion depends upon whether or not it begins with a whole which may he intuitively presented^ and whether or not it proceeds by the rhythmic partition of this original whole into minor wholes, and their recombination. 2. Use of Subtraction as Inverse Operation. — Upon this basis the process of subtraction is always iised simulta- neously with addition. In beginning with a fixed unity, or an aggregate of such unities, the '' method " may tell us to teach addition and subtraction together (or, what is really meant, one immediately after the other), but they can not be employed at the same time. If 1 is one thing, 2 two things, 3 three things, and so on, it requires one mental act to unite two or more such things, and notice the resulting sum ; and another act to remove one or more, and note the resulting diiierence. But in beginning with ^ and noting that it is made up of Q, or 4, and X, or 2, the synthesis (recognition of the whole of parts) and analysis (recognising the parts in the w^iole) are absolutely simultaneous. It is one and the same act (6 = 4-f- 2), which becomes in outward statement addi- tion or subtraction, according as the emphasis is directed upon both of the parts equally, or upon the whole and ■ 1 (' I 1. ■if m '"i u «:." S' I i\ -ii 108 THE PSYCHOLOGY OF NUMBER. '!>' It ■ |1 " Mi ^iii ;.-. t one of the parts. If, for example, in the above in- stance the Q and the X are both equally familiar, then the construction would probably appeal to the child as addition, putting together the more familiar to make the more unfamiliar. But if the []] alone is very familiar, he might rather notice that the dijfcrenee between the square and the original whole, namely, X, or 2 units. 3. The Conscious Process of Subtraction slightly more Complex. — The conscious recognition of subtrac- tion, however, is a slightly more complex process — makes more demand upon attention — than the con- scious interpretation of addition. In addition, the whole emphasis is upon the result; it is not necessary to keep the parts separate at all. The sum of 5 and 4, e. g., is first of all supplied by intuition, and where the association is complete the mind merely touches, as it were, the symbols, and the sum appears in conscious- ness. If, for example, we know that James and John and Peter have a certain amount of money — the unde- fined whole — of which James has 6 and John 8 and Peter 12 cents, we instantly merge or absorb each pre- ceding quantity in the next greater — 6, 14, 26. As soon as the two parts are added they are dropped as separate parts, the resulting whole is alone kept in mind. But in subtraction it is necessary to note both the whole and the given part, and the relation between them. If we say that of the total amount * James has 6 cents, John * While it is not necessary always to introduce the idea of the total first in words, it should be done even verballv until we are sure that the child's mind always supplies the idea of a whole from and within which he is constantly working. W i = THE ARITHMETICAL OPERATIONS. 109 I -1 2 more than James, and Peter 4 more than John, then the addition problem re(|uires the same attention to the two terms separately and to the result as is required in subtraction. There is the idea of definiteness or relative moreness, and not merely the idea of an aggregate more- ness. Here, as in subtraction, we are approaching nearer to ratio. Multiplication : Genesis of The Factor Idea. We have seen that, though multiplication is not iden- tical with addition (even with the special case of addi- tion where the addends are all equal), it has its genesis in addition, taking its rise in counting^ which is the fundamental numerical operation. Counting is the re- lating process in the mental activity which transforms an indefinite whole of quantity into a definite whole. It begins with discrete quantity, and is first of all largely mechanical — an operation with things. The child in his first countings does not consciously relate the things ; his act is not one of rational counting. He is apt to think that the number-names are the names of things ; that three^ e. g., is not the third of three related things, l)ut the nam^e of the third thing ; and on being asked to take up three he will fix upon the single thing which in counting was called three. But starting with groups of objects and repeating the operations of parting and wholing, he soon begins to feel that the objects are related to one another and to the whole. This is a growth towards the true idea of number, but the idea is not yet developed. There is a relating, but not the relating which constitutes num- ber. In the process of counting one, two, etc., getting »i \ w I i i I 1 1 ! f1 i'- -i^ ■ ^ ■■If I no THE rSYCriOLOGY OF NUMBER. 4 t, t [ ml' i ijt ; 11 fel lu as far as five, e. g., lie is conscious that five is connected with what goes l)efore. This perception is one of inore- ness or lessness, of aggregation ; live is more than four, and at last, detinitelj, it is one more than four. With the continuance of tlie physical acts there is further growth towards the higher conception. He separates a whole into parts and remakes the whole : he combines (using intuitions) unequal groups of measuring units (e.g., 3 feet and 4 feet) to express them as ories ^' he counts by o?ies, (/roups of two things, of three things, etc., and at last the idea of times, of pure number, is definitely grasped. The "five" is no longer mereli/ one more tlian four, it is five times one, whatever that one may be. In other words, he has passed from the lower idea to the higher ; from the idea of mere aggre- gation to that of times of repetition ; from addition to multiplication. It is plain that there must be time for the develop- ment of this abstracting and generalizing power. In fact, the complete development of the " times " idea, this factor relation, corresponds with the stages of the measuring power of number. The higher power of numerical abstraction is the higher power of the tool of measurement. This normal growth in the power of abstracting and relating can not be forced by any — the most minute and ingenious — analyses on the part of the teacher. The learner may indeed be drilled in such analyses, and may glibly repeat as well as " reason out " the processes ; just as he can be drilled to the repeti- tion of the words of an unknown tongue, or any other product of mere sensuous association. But it does not follow that he knows number, that be has grasped the I ''i \ • i fl 1 THE ARITHMETICAL OPERATIONS. Ill idea of times. The difficulty is not in the word times, as some appear to think ; it is in the idea itself, and would not disappear even if the word were (as some propose) exorcised from our arithmetics. It has not yet been proposed to eliminate the idea itself — i. e., the idea of number — from the science of number. Summary. — (1) Counting is fundamental in the de- velopment of numerical ideas ; as an act or operation with objects it is at first largely a mechanical process, but with the increase of the child's power of abstrac- tion it gradually becomes a rational process. (2) From this (partly) physical or mechanical stage there is evolved the relation of more or less, and addition and subtrac- tion arise — that is, e. g., five is one more than four. (3) The addition, through intuitions, of unequal (meas- ured) quantities, which are thus conceived and expressed as a defined unity of so many ones, is an aid to the de- velopment of the times idea. (4) Continuance of such operations — appealing to both eye and ear — brings out this idea more definitely — e. g., five is not now simply one more than four, it is five times one. (5) Counting (by ones) groups of twos, threes, etc., brings out still more clearly the idea of times. (6) Through repeated intuitions, sums (the results of uniting equal addends) become associated with times, the factor idea (times of repetition) displaces the part idea (aggregation), and multiplication as distinct from addition arises explicitly in consciousness. The Process of Multiplication. — The expression of measured quantity has, we have seen, two components, one denoting the unit of measure, and the other de- I h 1 1 \- h v'--\ ^ h-l ^ ^% urn I ill'.-;;,*! w 112 TIIK PSYCIIOLCKJY OF NUMBER. if' ;M 'li'i noting the number of tliese units constituting the ({uan- tity. But since the unit of nieasure is itself composed of a definite num])er of parts — is definitely measured by some other unit — it is clear that we actually con- ceive of the quantity as made up of so many given units (direct units of measure), each measured by so many minor units. For convenience we may call these minor units " primary," as making up the direct unit of meas- ure, and this direct unit, as being made up of primary units, may be called the '' derived " unit. We shall thus have in the complete expression of any measured quan- tity, (1) the derived unit of measure, (2) the number of such units, and (3) the number of primary units in the derived unit of measure. For example, take the following expressions of quan- tity : In a certain sum of money there are seven counts of five dollars each ; here the derived unit of measure is Jive dollars, the number of them is seve?i, and the pri- mary unit is one dollar. The cost of a farm of sixty acres at fifty dollars an acre is sixty fifties ; hei'e the derived unit of measure is fifty dollars, the number of them sixty, and the primary unit one dollar. The length of a field is fifteen chains — that is (in yards), fifteen twenty-twos ; here the derived unit is twenty-two yards, the number of them fifteen, and the primary unit one yard. In the quantity $fxf the primary unit is $1, the derived unit ${, and f is the nwnher expressing the quantity in terms of the derived unit. Now, when a quantity is expressed in terms of the derived unit, it is often necessary or convenient to ex- press it in terms of a primary unit. Thus, in the fore- going examples, the sum of money expressed as seven ■ \ m TIIK AIUTIIMETICAL OPERATIONS. 113 fives may be expressed as thirty-fve ones; the cost of tlie farm, expressed as sixty fifties, may he expressed as three thousand ones ; and tlie length of the field, ex- pressed as jifteoi twenty-twos, may he expressed as three hundred and thirty ones (yards); and ^fxf is measured hy ig^^ in terms of the primary unit, in each of these cases the second expression of the measured quantity merely states explicitly the number of minor (or primary) units which is implied in the first ex- ])ression. The operation by which we find the number of primary units in a quantity expressed by a given number of derived units is Multiplication. It is plain that the idea of times (pure number, ratio) is prominent in this operation ; we have the times the i^rimary unit is taken to make up the derived unit, and the times the derived unit is taken to make up the quantity. The multiplicand always rei)resents a number of (primary) units of quantity ; the multiplier is always pure num- ber, representing simply the times of repetition of the derived unit. But from the nature of the measuring process the two factors of the product may be inter- changed, the times of repetition of the primary unit may be commuted with the times of repetition of the derived unit; in other words, the 7zw??^j6^/' which is ap- plied to the primary unit may be commuted with the numher whicli is applied to the derived unit. Correlation of Factors. — In our conception of meas- ured quantity these two ideas are, as has been shown, absolutely correlative. Measuring a line of tioelve units by a line of tvm units, the numerical value is six / if we consciously attend to the process, the related conception instantly arises ; we can not think six times two units [!! "■I ' ij \{ [i 114 THE PSyCHOLOGY OP NUMBER. 'li jl •^1 ;lt (i ) '■ il without tliinking two times six units, because we can not think one unit six times without thinking one whole of six units. So, in measuring a rectangle 8 inches long by 10 inches wide, we can not analytically attend to the process which gives the result of 8 square inches taken ten times w^ithout being conscious of the inevitable correlate, 10 square inches taken eight times. In gen- eral : To think the measurement of any quantity as h units taken a times, is to think its correlate a units taken h times; for h units is h times one unit, and every one in h is repeated a times, giving a units once, a units twice, etc. — that is, a units b times. Educational Applications. 1. Just as, in addition, we must always begin with a vague sense of some aggregate, and then go on to make that definite by putting together the constituent units, while in subtraction we begin with a defined aggregate and a given part of it, and go on to determine the other parts ; so, in multiplication, we begin with a compara- tively vague sense of some whole which is to be more exactly determined by the " product," w^hile in division we begin w^th an exactly measured whole, and go on to determine exactly its measuring parts. In multiplica- tion the order is as follows : (1) The vague or imper- fectly defined magnitude ; (2) the definite unit of value (primary unit), which has to be repeated to make the derived (direct) unit of measure — the multiplicand ; (3) the number of times this derived unit is to be repeated — the multiplier ; and (4) the product — the vague mag- nitude now definitely measured. The operation of multiplication, therefore, already THE ARITHMETICAL OPERATIONS. 115 t s implies division / the definite unit of measurement which constitutes the multiplicand is plvvajs a certain exact (equal) portion of some whole. Hence multipli- cation always implies ratio ; the whole magnitude bears to the unit of measure a ratio which is expressed in the number of times (represented by the multiplier) the unit has to be taken to measure that magnitude — to give it accurate numerical value. In fact, the process ' is simply one of changing the numher which measures a magnitude by changing the unit of measure — i. e., by substituting for the given unit of measure the primary unit from which it was derived.* 2. In multiplication, then, as in addition, we are not performing a purposeless operation, or one with unre- lated parts and isolated units ; rather, we begin and end with some magnitude requiring measurement, keeping in mind that what distinguishes multiplication is the kind of measurement it uses — that, namely, in which a unit itself measured off by other units is taken a certain number of times. 3. The psychology of number, therefore, impera- tively demands that the quantity which is to be finally expressed by the " product " should first be suggested,, just as in addition the quantity given by " sum," within which and towards which we are working, is kept in : t J 1 [jS r i Hi| i I ;jr * In such instances as multiply 7 apples by 4, the idea of exact division or ratio is not so evident, but the 7 apples must be taken as one of four equal portions — i. e., as having the ratio i to the whole quantity. The fact, however, that the idea of an exact unit of measurement is not so clearly present, is a strong reason for using fewer examples of this sort, and more of those involving standard units of measure. vu 116 THE PSYCHOLOGY OP NUMBER. P'l ni .', mind from the first. If the child sees, e. g., tliat there is a certain field of given dimensions whose area is to be ascertained, or a piece of cloth of given lengtli and price per yard, of which the cost is to be determined, the mind has something to rest upon, a clearly defined purpose to accomplish. Beginning with a more or less definite image of the thing to be reached, the subsequent steps have a meaning, and the entire process is rational and consequently interesting. But when he is asked how much is 4 times 8 feet, or 9 times 32 cents, there is no intrinsic reason for performing the operation ; psycho- logically it is senseless, because there is no motive, no demand for its performance. The sole interest which attaches to it is external, as arising from the mere ma- nipulation of figures. Under an interested teacher, in- deed, even the pure" figuring" work may be interest- ing ; but this interest is re-enforced, transfoi'med, when the mechanical work is felt to be the means by which the mind spontaneously moves by definite steps towards a definite end. This does not mean, we may once more remark, that examples like 8 feet x 4, or even 8x4, are to be excluded, but only that the haJnt of regard- ing number as measuring quantity should be perma- nently formed. The ])upil should be so trained that all addends, sums, minuends, products, multiplicands, dividends, quotients, could be instantly interpreted in tlieir nature and function as connected with the process of measurement. For example : A farmer has 8 bush- els of potatoes to sell, and the market price is 55 cents a bushel : how much can he get for them ? This and similar examples are often presented in such a way that when the pupil gets the product, $4.40, THE ARITHMETICAL OPERATIONS. 117 r^ t his mind stops short with the mere idea of the product as a series of figures. This is irrational ; $4.40 in itself is not a product ; no quantity or value is ever in itself a product ; but as a product it measures more definitely the value of some quantity. In other words, the prod- uct must always be interpreted / it must be recognised as the accomplished measurement of a measured quan- tity in terms of more familiar or convenient units of measure. 4. The multiplicand must always be seen to be a unit in itself, no matter how" lai-ge it is as expressed in minor units. It signifies the known value of the unit with which one sets out to measure ; it is the meas- uring rod, as it were, which is none the less (rather the more) a unit because it is defined by a scale of parts. A foot is none the less one because it may be written as 12 inches or as 192 sixteenths ; nor is a mile any the less a unit because it is written as 320 rods or as 5280 feet. The ineradicable defect of the Grube method, or any method which conceives of a unit as one thing instead of as a standard of measuring, is that it can never give the idea of a multiplicand as just one unit — a part used to measure a whole. 5. It is important so to teach from the beginning that a clear and definite conception of the relation between parts and times may be developed. Of course, nothing is said till the time is ripe about tlie law of "commu- tation " ; but the idea should be present, and should be freely used. If a quantity of 12 units is meas- ured by 3 units repeated four times, the child can be led to see — will probably discover for himself — that this measurement is identical with the measurement 1 i J :!: -i Il ' > Vv m s " !i'i Mii i • t' > Ji In multiplication, as already snggested, we may look at a product of two factors in two ways : For example, 20 feet = 4 feet x 5 = also 5 feet x 4, or five times four times 1 foot = four times five times 1 foot — that is, we may nse the primary unit of measure " 1 foot " with either tlip f; . • times or the five times. Or, stated in general terms, 1/ times a times the primary unit of measure is identical with a times h times this primary unit — that is, we may interchange at pleasure the nu- merical value of '^'.c rr,p,o,suring imit (the derived unit as made up of pr.ir^;. ;- uTiits) with the numerical value of the whole quaiuity as made up of these de- rived units. This is in^nort' " as interpreting the pro- cess and result in division. J r 'ave 20 feet and the factor 4 feet given to find the other factor, we use the measurement 4 feet x 5 = 20 feet. If, on the other hand, we have the 20 feet and the numljer 5 given to find the other factor, we may use eitJier measurement; we may divide directly by the number 5, or we may " concrete " the 5 (consider it as denoting 5 feet), and get the other factor 4 (times) ; for we know that 4 times 5 feet is identical with 5 times 4 feet, and the conditions of the question require tlie latter interpre- tation. In other w^ords, we first of all determine what the problem dt.^iands, times or parts, then operate w^ith the pure number symbols, and interpret the result ac- cording to the conditions of the problem. Illustrations of Division. — Let us take a few^ illus- trations of these inverse operation^ : (1) We count out fifteen oranges, by groups of five, and the mimher of groups is three. We count them out in five groups, and the number of oranges in each group is three. These NATURE OF DIVISION AND FRACTIONS. 121 are said to be two totally different operations ; for, it is alleged, in one case we are searching for the size (the numerical value) of a group — the unit of measurement ; in the other for the nnmher of groups. But a little reflection will show that they are not " radically differ- ent" operations; they are psychological correlates, if not identities. In counting out fifteen oranges in groups of five there is a count of ^'w, then another count of five, then another count of five, and finally a counting of the number of groups. Psychologically, in counting out five there is a mental sequence of five acts (a partial synthesis) ; this is repeated three times, and finally the number of these sequences is counted (complete synthesis), and found to be three. In the second case, where the number {Jive) of groups is given, we begin by putting one orange in each of five places, making, as before, a *' count " of Jive oranges ; this opera- tion is repeated till all are counted out; and finally we count the number in each of the five groups. Tliat is, there is a mental sequence of five acts, which is repeated three times, and finally the number of such sequences is counted in counting the number of oranges in a group. It would be hardly too much to say that these two men- tal processes are so closely correlated as to be identical. Neither the three times in the one question nor the three oranges in the other can be found without count- ing out the whole quantity in groups of five oranges each (see page 75). There is hardly a difference even in the rhythm of the mental movement. This division by counting is the actual process w4th things ; it is the way of the child and of the savage or the illiterate man ; it is exactly symbolized in the " two kinds of division" I 1 m Mi I I I ill: I iM : I i: "I lit i sli 1, I ^hl 11 122 THE PSYCHOLOGY OF NUMBER. — that by a concrete divisor when we are searching for the number of the parts as actual units of measure ; and that by an abstract divisor when we are searching for the size of the parts — i. e., for the number of minor units in the actual unit of measure. With this actual process of counting out the objects the arithmetical operation exactly corresponds. Work- ing by long division as more typical of the general arithmetical operation, we have : I. Division : 15 oranges -r- 5 oranges ; i. e., 15 oranges are to be counted out in groups of 5 oranges ; how many groups ? 5 oranges 15 oranges 5 10 5 5 5 1° — 1st partial multiplier l°_2d r_3d « " = 3 times. II. Partition : 15 oranges in 5 groups ; how many in each group ? 5 times 15 oranges 5 10 5 5 5 1 orange — 1st partial multiplicand " 1 " —2d " " —3d « « = 3 oranges. That is, once more, both problems are solved by count- ing out the whole quantity in groups of five. (2) Solve the following problems : {a) Find the cost of a town lot of 36 feet frontage at $54 a foot. (J) At the rate of $54 a foot, a town lot was sold for $1944, NATURE OF DIVISION AND FRACTIONS. 123 find the number of feet fronta<^e. (c) Find the price per foot frontage when 36 feet cost $1944. (a) i) $54 36 1620 = 30 times $54 324= 6 " $1944 = 36 Or, by the correlate, $36 x 54 : (u) $36 54 1800 = 50 times $36 _144= 4 " $1944 = 54 " $54)$1944 1620 = 30 times $54 324= 6 " 324 ; 36 times $54 (c) Partition. 36)$1944 1800 = $50 36 times 144 = $4 36 '' 144 ; $54 36 times On comparing the successive steps in {b) with those in {a) tliey will be seen to correspond exactly — that is, (b) is the exact inverse of {ci). But the steps in {c) do not correspond with those in (a), the operation is not the exact inverse of {a) ; it is seen to be the exact inverse of the correlative {ii) of {ct). This indicates the con- nection of the operations through the law^ of commuta- tion ; and shows, once more, that either of the corre- lated measurements (i) and ((it) may be used in the solu- tion of {c). It should be noted, further, that {c) is a case of so-called partition^ yet involves a series of subtrac- tions that is a series of partial dividends (why not par- tlendsf) and partial quotients. N'ot Two Kinds of Division. — From the foregoing we see that just as a product of two factors may be in- terpreted in two ways, so there may be two interpre- tations of the result of the inverse operation, division. The factor sought may be either the numerical value of the dividing part (" derived unit ") in terms of the pri- I h' e i< I '1 II If 124 THE PSYCHOLOGY OF NUMBER. ir't IT I i :' i '■\l I!; 1.' V i ..!, f mary luiit which measures it, or the numerical vahie of the quantity in terms of the derived unit. But these numerical values may be interchanged at convenience, provided the results are rightly interpreted. There are not two kinds of division ; there is one operation lead- ing to one numerical result having two related mean- ings. It seems therefore unnecessary, either on psy- chological or practical grounds, to institute two kinds of division — viz., division (why not quot'dhmf) in the ordinary sense of the word, and " partition " — when the search is for the numerical value of the measuring quan- tity. When the search is for the mimerical vahie of the measuring unit, is not the pupil likely to become per- plexed by a series of parallel definitions — of divisor, dividend, quotient — for the two divisions when he finds that the operations in both cases are exactly alike ? If there is confusion in using the term division in two senses, is there not more confusion in using the two terms, divisor and quotient, each \\\t\\ two different meanings ? Without doubt, the meaning of the result should be grasped ; but this can not be done by simjfiy giving two names to exactly the same arithmetical opera- tion. Better give one name to one operation resulting in two correlated meanings than to have two names for one and the same operation. The new name does not help the pupil either in the numerical work or in the interpretation of the result. How is the child to know whether a given problem is a case of division or of " partition " ? He can not know without an intellectual operation, analysis, by which he grasps what is given and what is wanted in the problem. In other words, he must know the meaning of the problem, must know NATURE OF DIVISION AND FRACTIONS. 125 whether it is times or measuring parts he is to search for, before he heglns the operation ^ to this knowledge the different names afford him no aid wliatever.''^ Partition^ like Division^ depends on Suhtractions. — It is said, indeed, that in " jmrtition '' we are search- ing for the numerical value of one of a given number of equal parts which measure a quantity, and as a num- ber can not be subtracted from a measured quantity, the problem can not be solved by division. To this the answer is easy : In the first place, the divisor in the arithmetical operation can be a number, and the sub- tractions rationally explained (see page 122). And, be- sides, we can by the law of commutation concrete the number, find the related factor, and properly interpret the result. But, in the second place, if the divisor can not be an abstract number, what magic is there in a strange name to bring the impossible within the easy reach of childhood ? It seems, according to the parti- tionists^ that 20 feet -i- 5 feet represents a possible and intelligible operation ; but that 20 feet -r- 5 becomes possible and intelligible only by calling the implied operation a case of " partition " ; it is then simply one fifth of 20 feet — that is, 4 feet. Certainly, if we know the multiplication table, we know that one fifth of 20 feet is 4 feet, but we know" equally well that 5 feet is one fourth of 20 feet. These are not typical cases for the ai-gu- * Owing to the fixed unit fallacy, the theory of the '• two divi- sions" makes an unwarranted distinction between the actually meas- uring part and its times of repetition. The measuring part, as well as the whole, involves both the spatial element (unit of quantity) and the abstract (time) element ; it is itself a quantity that is meas- ured by a minor unit taken a nrmber of times. ' I fli •t ;■ HMli P Ml r' 126 TIIK PSVCIIOLOUV OF NUMIJKR. I n m ■h inent ; though attention to the processes even in these cases (see page 122) will show that if 20 feet -;- 5 is impos- sible because " division " is a ])rocess of subtraction, so also is the process one fifth of 20 feet, because " pai'ti- tion " is equally a process of sithtraction (page 1 22). For example, the operation indicated in $ll-809f)28 -^ §4081 it is admitted involves subtraction — i. e., the separation of the dividend into parts, and the obtaining of partial quotients. But it is clear that l-4()81tli of this dividend (partition) is obtained by exactly the same 2>vocess — i. e., in both cases we have a first sul)traction of 3000 times the divisor, a second of 100 times, a third of 80 times, and a fourth of 3 times, getting the same numerical quotient of 3183 in both operations ; but 3183 is inter- jpreted as pure mi7nher in the first case, and as measured guantity in the second — the so-called partition. In fine, when it comes to pass that there can be a clear conception of a foot as measured by inches without the thought of hoth the factors, one inch and tivelve times^ then, but not till then, it may be rationally aftirmed that the "two divisions" are radically different and totally unrelated processes. Fractions. The process of fractions as distinguished from that of " integers " simply makes explicit — especially in its notation — hoth the fundamental processes^ division and midti plication {analysis-synthesis)^ which are involved in all numher. In the fundamental psychical process which con- stitutes number, a vague whole of quantity is made definite by dividing it into parts and counting the [i: r; NATURE OF DIVISION AND FRACTIONS. 127 parts. Tills is essentially tlie process of fractions. Tlic "fraction," therefore, involves no new idea ; it helps to bring more clearly into consciousness the nature of the measuring process, and to express it in more definite form. The idea of ratio — the essence of numher — is implied in simple counting ; it is more definitely used in multiplication and division, and still more completely present in fractions, which use both these operations. Fractions are not to be regarded as something different from number — or as at least a different kind of number — arising from a different psychical process ; they are, in fact, as just said, the more complete development of the ideas implied in all stages of measurement. So far as the psychical origin of number is concerned, it would be more correct to say that " integers " come from frac- tions than that fractions come from integers. Without the " breaking " into parts and the " counting " of the parts there is no definitely measured Avhole, and no ex- act nnmerical ideas ; the definite measurement is sim- ply (a) the number of the parts taken distributively (the analysis), and {h) the number of them taken col- lectively (the synthesis). The process of forming the integer, or whole, is a process of taking a part so many times to get a complete idea of the quantity to be meas- nred ; and at any given stage of this operation what is reached is both an integer and a fraction — an integer in reference to the units counted, a fraction in reference to the measured unity. Even in the imperfect measurement of counting with an unmeasured unit, the ideas of multiplication and division (and therefore of ratio and fractions) are implied in the operation. We measure a whole of : ii I V. fi I I I! ' 'i I m ■hi n :■ I m ■.l( '' m If 128 THE PSYCHOLOGY OF NUMBER. a I fifteen apples by threes ; we count the parts — i. e., re- late or order them to one another, and to the \\'hole from which and within which we are working. This counting has a double reference — i. e., to the unit of measure, and to the whole which all the units make up. When, for example, we have counted two, three, . . . we have taken one unit of measure two, three, . . . times, and each count is expressed or measured by the numbers two, three, . . . — i.e., by ''integers" ; but also in reaching any of these counts we have — in reference to tlie whole — taken one of the five, two of the five, three of the five, etc. ; that is, one fifth of the whole, two fifths, three fifths, etc. JVo 3feasure7nent without Fractions. — AYhen we pass to measurement with exact units of measure, this idea of fractions — of equal parts making up a given whole — becomes more clearly the object of attention. The conception, 3 aj^ples out of 5 apples (three fifths of the whole) has not the same degree of clearness and exactness as that of 3 inches out of a measured whole of 5 inches. "VYhy % Because in the former case we do not know the exact value., the how much of the measuring unit ; in the latter case \\\q, unit is exactly defined in terms of otlier unities larger or smaller; in 3 apples the units are alike ^ in 3 inches the units are equal. So in measuring a length of 12 feet we may divide it into 2 parts, or 3 parts, or 4 parts, or 6 parts, or 12 parts ; then we can not really think of the 6 parts as making the whole without think- ing that 1 is one sixth ; 2, two sixths ; ' . three sixths, etc. In the process of inexact measurement the idea of fractions is involved ; in that of exact measurement, •! i NATURE OF DIVISION AND FRACTIONS. I09 I f:^ I tliis idea is more clearly defined in consciousness. In short, wherever there is exact measurement there is the conception of fractions, because there is the exact idea of number as the instrument of measurement. The process of fractions, as already suggested, simply makes more definite the idea of number, and the notation em- ployed gives a more complete statement of the analysis- synthesis, by which number is constituted. The num- ber Y, for example, denotes a possible measurement ; the number j^ states more definitely the actual pro- cess. It not only giv^es the absolute number of units of measure, but also points to the definition of the unit of measure itself — that is, the 7 shows the absolute nmnber of units in the quantity, while the 10 shows a relation of the unit of measure to some other stand- ard quantity, a primary unit of reference by which the actual measuring unit is defined. If a quantity is di- vided into 2 equal parts, or 3 equal parts, or 4 equal parts, or 71 equal parts, the 2, 3, 4, . . . n shows the entire number of parts in each measurement, and cor- responds with the *' denominator" of the fraction which expresses the measured quantity as unity y nnd in count- ing up (^-numerating) the parts (units) we are constantly making "nnmerators" — e. g., 1 out of n, 2 out of n^ 11 3 out of /?, etc. : or 1-nth, 2-wths, . . . ?i-nths, or - , ■n which is the measured unity. Or, if attention is given to tlie measuring units — the ones — the parts are ex- pressed by 1, 2, 8, etc., and the measured quantity itself n is expressed by y. Again, measuring the side of a certain room, we find it to contain ^ yards. This is a ( 1::^= I 'I 1 1 1 1 t' i ■■ iii ll \ - : W it u J ::;i m I 130 TOE PSYCHOLOGY OF NUMBER. full statement of the process of measurement ; it means (1) that the primary unit of measurement (the standard of reference) is one yard, (2) that the derived unit of measurement is one third of this, and (3) that this de- rived unit is taken nineteen times to measure the quan- tity. This is seen to agree with the mental process of the exact stage of measurement in which the unit of measure is itself defined or measured (see page 94). There must be, as we have seen, (1) a standard unity of reference (the primary unit), (2) a derived unit (the unit of direct measurement), and (3) the number of these in the quantity. The fraction gives complete expression to this process : In $f , for example, (1) the dollar is the unit of reference ; (2) it is divided into four parts to get the derived unit — the actual unit of meas- ure ; (3) the " numerator " 3 shows how many of these units make up the given quantity, and expresses the ratio qf this quantity to the standard unity. So, again, the measurement — 19 feet — of the side of a room can be stated in tei-ms of other units of the scale. It is 12 X 19 inches, or 19 ~- 3 yards, and the first of these expressions, as well as the second, is one of fractions ; it is -^-f-S- — that is, not 228 ones merely, but 228 of a definite \mit of measure — namely, one twelfth of a foot ; just as the second is ^ — i. e., 19 times a unit of measure defined by its relation to the yard. In the former case we do not generally state the measurement in fractional form, but the interj>retation of it demands an explicit reference to a dtmominator. Note what this brings us to : 19 (feet) — 228 (inches) = ^^- (yards) — t¥" (I'ods) — that is, four entirely different numbers equal to one another; a result which must appear i s NATURE OF DIVISION AND FRACTIONS. 131 I utterly meaningless to a child who has been trained by the fixed unit method. Any method which treats num- ber as a name for physical objects can not but reach just such absurdities. Only the method which recognises that number is a psychical process of valuation (analy- sis-synthesis) is free from such difficulties. The unit does not designate a fixed thing ; it designates simply the unit of valuation, the how much of anything which is taken as one in measuring the value (or how much) of a group or unity. It defines how many units each of so much value make up the so much of the whole. The complete process is one of fractions, and the full statement of it is a fraction, whether written out in full or necessarily understood in the interpretation. The 228 inches is -S^, signifying that the number of the derived units of measure in one inch is 1 ; 19 feet is ^^- yards, signifying that the number of the derived units of measurement in one yard is 3 ; the ^^ rods show that the number of units of measurement in one rod is 11 ; in other words, the unit of measure in ^- is one of the three equal parts of one yard, etc. It appears, therefore, that every numerical operation which makes a vague quantity definite, when fully stated, involves the " terms " of a fraction — that is, a fraction may be considered as a convenient language (notation) for expressing quantity in terms of the process which measures or defines it — which makes it " number." A fraction, then, completely defines the unity of ref- erence, and thus determines the unit of measure for the quantity that is to be measured. Thus the inch may be defined from |f foot, the foot from } yard, the ounce from -J-J pound, the cent from -J J^ dollar, etc. In each ' [M is fV 132 THE PSYCHOLOGY OP NUMBER. I ; in ' .1 I ; 1 i } "V lii;: li: I'l'' ; case the denominator shows the analysis of a standard unity into units of measurement — i. e., the unity in terms of the units taken collectively. Thus the meas- urements of the quantities Y inches, 5 ounces, 35 cents are more explicitly stated by the respective fractional forms -^ foot, -^ pound, y^/^j- dollar, because the unit of measure in each case is consciously defined by its re- lation to a standard unity in the same scale. It is clear that the definition of number (page 71) includes the fraction, for in both fraction and integer the fundamental conception is that of a quantity meas- ured by a number of defined parts — the conception of the ratio of the quantity to the measuring unit. The fraction differs from the integral number — in so far as it differs at all — in defining the measuring unit, and thus giving more completely the psychical operation in the exact stage of measurement. If the fraction, as being a number, is a mode of measurement, tliere appears to be no need of a special definition of it as the foundation of a new or different class of numerical operations. The definitions which ignore fractions as a mode of measurement are in gen- eral vague and inaccurate, and lead to much perplexity in the treatment of fractions. It is hardly accurate to gay tliat a '' fraction is a number of the equal parts of a unit," or that " it originates in the division of the unit into equal parts." Here the important distinction be- tween unity and a unit is overlooked. Measuring a piece of cloth we find it contains four yards : before meas- urement it was mere unity, after measurement a defined unity ; but in neither case is it a iintt. It is, after meas- urement, a unity of U7i{ts — a sum. Is or is it entirely u NATURE OF DIVISION AND FRACTIONS. 133 consistent with the measuring idea to say that a frac- tion is one or more of the equal parts of a unity. Of course, in counting the equal parts of a measured whole — a unity — we take a number of parts in making the synthesis of all the parts. But since a fraction is a number, and therefore denotes measured quantity, it denotes a whole quantity, a unity — e. g., -J of a yard is as much a quantity — a measured xinity — as 4 yards or 40 yards ; it is a fraction in its relation to a larger unity, the yard taken as a standard of reference. The Improper Fraction. — From the same misappre- hension of the nature of number endless discussions arise regarding the classes of fractions " proper," " im- proper," etc. With a right conception of the meas- uring function of a fraction there is no mystery about the " improper " fraction. From the definition of a fraction as a " number of the equal parts of a unit," it is inferred, e. g., that -f of a yard can not be a fraction, because it represents not parts of a unit, but the whole unit and something more. Since 3 thirds make up the yard (the unit), whence come the 4 thirds ? In this objection we have the fallacy of the fixed unit as well as the misapprehension of the nature of number. The fraction in the expression |- of a yard is a number. It means the repetition of a unit of meas- ure to equal a certain quantity. This unit of measure is not the yard ; it is a unit defined by its relation to the yard ; it is one of the three equal parts into which the unity yard is divided to get the direct unit of meas- ure ; and there is absolutely nothing to make the yard the limit of quantity to which this unit can be applied. The yard is the primary unit of reference from which ■■%] " '.II m 134 THE PSYCHOLOGY OF NUMBER. i'[ m ill' I!.:., ^1 ;:<^ IM!' • r the actual measuring unit is derived, and there is no more mystery in the application of this unit to measure a quantity greater than the primary unit than in the measured quantity, 3 feet x 4, because it is greater than one foot^ the primary unit from which the meas- uring unit (3 feet) is derived. The expression 4 thirds of a yard indicates an exactly measured quantity ; exactly measured, because the unit of measure is itself measured in its relation to another quantity of the same scale. This properly defined unit (1 third of a yard) can be applied to any homogeneous quantity whatever, and may be contained in such quantity one, two, three, four, , , . n times ; in fact, 4 thirds yard, 5 thirds yard, . , . n thirds yards are only different and more exact ways of stating the measurements — 4 feet, 5 feet, . . . n feet. The Compound Fraction. — Nor is there any dif- ficulty in interpreting a " compound " fraction. The value of 8 yards of cloth at %\ a yard is expressed by $|- X 8, a measurement which ought to occasion no more perplexity than $3 x 8, when it is understood that the denominator merely defines the unit of meas- ure with reference to the primary unit. So the value of f yard of cloth at $8 a yard is expressed by $8 x J, a measured quantity where, once more, the denominator shows how the unit of measure is to be obtained — i. e., it shows which of the myriad ways of parting and wholing $8 — the unity of reference — will give the direct or absolute unit of measure. This explanation applies to %-^-i X J, and to any compound fraction whatever. The Complex Fraction. — It is said that the complex fraction is an impossibility, because a quantity can not NATURE OF DIVISION AND FRACTIONS. 135 be divided into a fractional number of equal parts — e. g., if tlie denominator of such a fraction is J, it implies the division of some unity into 3 fourths equal parts, which is absurd. This is to restrict the term fraction by the imperfect definition already quoted, which ignores num- ber as measurement and fractions as an explicit state- ment of the measuring process. Division and multi- plication are fundamental in the psychical process of defining quantity ; the fraction simply brings the pro- cess articulately into consciousness, and by its notation gives it complete expression. The statement that the fraction process and the division process are totally dis- tinct is so far from being true, that there is no division without the fraction idea, and no fraction without the division idea. Both are identified by the law of com- mutation — a law which is the expression of a necessary and universal action of the mind in the measurement of quantity. The symbol f foot is an exact expression for a measured quantity ; like every other such expression, it defines the unit of measure and denotes the times this unit is repeated ; and, like every such expression, it has two interpretations corresponding to the related concep- tions of the measured quantity : it is i foot x 3, or 3 feet X J. We shall be justified in treating these two things (the fractic-n and division) as entirely distinct when we are able to conceive that 3 feet x 4, and 4 feet X 3 are unrelated measurements of totally different quantities. It may be noted, then, that in the " complex " frac- tion just as in division there may be two interpretations. In $12 -s- $4 the measuring is not hyfour parts — it is a parting hy fours ; while in $12 -j- 4 there is a measur- ^hM I ! : «' 'i %\ m )''■ I ir ■■♦ i I-' , m ■ !1' III, i<^ ■ m !;i Rii m li.lf m U' : 1' h \t " |ii; 1 136 THK rSYCHOLOGY OF NUMBER. ing by four parts — it is a parting by threes. But in every case, as has often been shown, either the size of tlie parts is given to iind their number, or the number of the parts to find their size. The same thing holds in so-called complex fractions. In ^ — as, e. g., find how much cloth at $J a yard can be bought for $9 — it is not proposed to divide $9 into $J equal parts, but to find the times the measuring unit ^J is taken to make $9. N'or in ;o — as, e. g., find cost per yard ^vhen $9 was paid for J yard — is there any attempt to divide $9 into 3 fourths equal parts. The purpose is to find the quantity which, with J as multiplier (i. e., taken J times), will give $9 ; and it is a matter of indifference whether the expression is called a "complex" fraction or di- vision of fractions, for fractions are necessarily corre- lated with multiplication a7id division by the uniform action of the mind in dealing w^ith quantity. Summary and Applications. 1. All numerical operations are intrinsically con- nected with number as measurement, and distinguished from one another through the development of the measuring idea in psychological complexity. Addi- tion and subtraction have their origin in the oper- ation of counting with an unmeasured unit — they do not explicitly use the idea of ratio, but merely that of more or less — the idea of aggregation. Multiplication and division have their origin in the use of an exact unit of measure — a unit w'hich is itself defined — and, NATURE OF DIVISION AND FRACTIONS. 137 ■I \i\ besides tlie idea of aggregation, use the idea of ratio. It follows, accordingly, that addition and subtraction should precede in order of formal instruction, inulti})li- cation, and division. Addition and subtraction, being inverse operations, should go together, with the em- 2))jasis at first slightly upon addition. 2. Multiplication and division, being inverse oper- ations, should go together, with the emphasis first upon multiplication. Multiplication should not be taught as a case of addition, nor division as a case of subtraction. But the factor idea (ratio or number) should in each case displace the idea of aggregation. While this is the order of analytical instruction, the processes involved in multiplication should be used — that is, in primary teaching there should be frequent excursions into these processes in accordance with the fundamental psycho- logical law : " First the rational use of the process, and ultimately conscious recognition of it." 3. In multiplication the multiplicand, strictly speak- ing, always represents a measured quantity, and is com- monly said to be "concrete"; the multiplier always represents pure number — the ratio, in fact, of the prod- uct to the multiplicand. But, as the multiplicand always involves the idea of number (it expresses the mimher of primary measuring units), the two factors of the product may be interchanged — that is, the multiplier may be made the concrete quantity, and the multiplicand the pure num- ber denoting times of repetition. 4. Division is the inverse of multiplication. We have the product given and one of the two factors which produce it to find the other factor. And since there are two interpretations of the process of multipli- '4 i il! '^ m If^ .8 if .' I 'I ■ i h ill 138 THE PSYCHOLOGY OF NUMBER. •I I 11 « 'ft I 1 1 n II: i r !H* ;^i cation there may be two interpretations of its inverse process, division. But there are not two kinds of mul- tiplication nor two kinds of division. In each case there is one process and one result with two interpreta- tions. No assistance to this interpretation can be af- forded by giving the name " partition " to the process by which we find the size, or numerical value, of the measuring part. The student knows what he is look- ing for before he begins the operation — whether for the value of the parts in terms of the primary unit, or the number of them in the whole quantity. It is not neces- sary for him to give a new name to an old operation. Besides, if he does not know what he is searching for before he begins the numerical work, the new name throws no light upon the subject. From the relation existing between nmltiplication and division, it is seen that in division the dividend — or multiplicand, as being the product of two factors — always represents a measured quantity — i. e., it is concrete ; the divisor may denote either a concrete quantity or a pure number ; and the quotient is of course numerical in the one case and interpreted as concrete in the other. 5. In fractions there are no mental processes differ- ent from what are involved in number as a mode of measuring quantity. The psychical process by which number is formed is from first to last essentially a pro- cess of " fractioning " — making a whole into equal parts and remaking the whole from the parts. In the pro- cess of number we start with a whole ; we have a unit of measurement; we repeat the unit of measurement to make up the whole. In a measured quantity repre- sented by a fraction we do exactly the same thing. We 1*1 1 NATURE OF DIVISION AND FRACTIONS. 130 begin with a whole of quantity ; we use a unit of meas- ure of the same kind as the quantity; we repeat the unit of measure to make up the whole. The fraction by its notation brings out more explicitly the actual pro- cess of measurement — that is, it not only gives the num- ber of units of measure, but actually defines the unit itself in terms of some other unit in the same scale. In other words, a fraction sums ujp in one statement the mental pr^ocess of analysis-synthesis Jjy which a vague whole is made definite. 1. As fraction at all it expresses a portion of some group or whole with which the quantity represented by the fraction is compared, and wdiich defines the measuring unit. Thus, |- yard of cloth is itself a whole, a definitely measured quantity ; but it is a frac- tion as regards the standard of reference, yard., which defines the direct measuring unit, one eighth of a yard. 2. A fraction, therefore, always denotes {a) the ab- solute number of units in a measured quantity ; {Jj) the number of such units in some standard quantity which defines the measuring unit in {a) ; and {c) the ratio of the given quantity (represented by the fraction) to this standard of reference. The numerator of the fraction gives {a) and {c\ and the denominator gives (b). Of course, any part (or multiple) of the standard of refer- ence may be taken as the unit of measure for a given quantity ; a given length may be measured by 1 foot, or by 1000 feet, or by -r^Vir ^^ ^ f^*^*- ^^^ ^^ begin- ning the explicit treatment of fractions it is better to use certain standard measures, their subdivisions, and their relations to one another. Thus, as a process of analysis-synthesis, the foot is defined by -J-f, the yard , :. M i J ^; where 12 refers to inches, 3 to feet, KJ to ounces, 10 to dimes, and 100 to cents. Familiarity with fractions thus defined by and connected with the ordinary scales of measurement means easy mastery of all forms of fractions as a mode of definite measurement. 3. As to the teaching of fractions, it will be enough, for the present, to note the following points : 1. In the formal treatment of fractions nothing new is involved ; there is simply a eonscums direction of at- tention to ideas and processes which, under right teach- ing, have been used from the first in the formation of numerical ideas, and which have been further developed in the fundamental arithmetical operations. 2. As in "integers" so in teaching fractions, the idea and process of measurement should be ever present. To begin the teaching of fractions with vague and un- defined "units" obtained by breaking up equally unde- fined wholes— the apple, the orange, the piece of paper, the pie — may be justly termed an irrational procedure. Half a pie, e. g., is not a numeral expression at all, un- less the pie is defined by weight or volume ; the con- stituent factors of a fraction are not present ; the unity of arithmetic is ignored ; the process of fractions is assumed to be something different from that of num- ber as measurement ; it becomes a question — it ac ally has been questioned — whether a fraction is really ,i number; and all this in spite of the fact that from the beginning fractions are implicit in all operations; that from first to last the process of number as a psychical act is a process of fractions. 3. The primary step in the explicit teaching of frac- NATURE OP DIVISION AND FRACTIONS. 141 tioiis — that is, in making tlie liabit of fractioning already formed an object of analytical attention — is to make perfectly definite the child's acquaintance with certain standard measures, their subdivisions and relations. In all fractions — because in all exact measurement — there must be a definite imit of measure. This implies two things : {a) The definition of a standard of reference (the " primary " unit) in terms of its own unit of meas- ure ; ih) the measurement of the given quantity by means of this " derived " unit. If the foot is unit of measure, it is unmeaning in itself ; it must be mastered, must be given significance by relating it to other units in the scale of length ; it is 1 (yard) -^- 3 in one direc- tion ; or (taking the usual divisions of the scale) it is If (i. e., -^ X 1^) in the other direction, i. e., as meas- ured in inches. The teaching of fractions, then, should be based on the ordinary standard scales of measure- ment ; on the fundamental process of parting and whol- ing in measurement, and not upon the qualitative parts of an undefined unity. 4. Under proper teaching of number as measure- ment the pupil soon learns to identify instantly 4 inches, -^ foot, -J foot as expressions for the same measured quantity. He is led easily to the conscious recognition of the true meaning of fractions as a means of indicating the exact measurement of a quantity in terms of a meas- uring unit which is itself exactly measured. 5. Addition and subtraction of fractions involve the ] 'nciple of ratio, multiplication and division the prin- 'le of proportion. In all cases the meaning of frac- v)ns as denoting definitely measured quantity should i/e made clear. For example, not i x J, but J foot x j ; >("■:■ I 1j" I 1^ ( ^1 \i^ m ■ li'i I . ':!l' :■!< m 142 TIIH PSYCHOLOGY OF NUMBER. not f X I", but $f X J, as indicating, e. g., the cost of -J yard of cloth at $£ a yard. Since a fraction expresses a quantity in a form for comparison with other quantities of the same kind, the fundamental operations as applied in fractions carry out these comparisons. Addition is always (as in " whole numbers ") of homogeneous quantities — i. e., thr.jje meas- ured in terms of some unit of length, surface, volume, time, etc. ; so wdth subtraction. All the first examples should deal only with definite measures ; after the prin- ciple is quite familiar, and only then, fractions having denominators not corresponding to any existing scale of measurement — e. g., IT, 49, 131 — may be introduced for the sake of securing mechanical facility. The same remark applies to multiplication and divi- sion of fractions — opcations wdiich involve no princi- ples different from the, corresponding operations wath " whole numbers." Multiplication of fractions is 7nul- tipUcation, and division is divisio7i / they are not new processes under old names. They make explicit use of ratio (the comparison of quantities), which is implied in the operations with " integers," by defining the measur- ing unit whicl^ defines a measured quantity. They put in shorthand, as it \vere, the complete psychical process of measurement, and thus make a severer demand on conscious attention. But if number has been from the first taught upon the psychological method, the pupil will be quite prepared to meet this demand. Tliere will be nothing strange in reducing ^^ractions to a com- mon denominator, nor any mystery in a product less than the multiplicand, or in a quotient greater than the dividend ; so far as the nature of the processes is con- I NATURE OF DIVISION AND FRACTIONS. I43 : ined, $-|- + $J will be just as intelligible as $3 + §i. If, too, the nature and relation of twies and nieasurir.g jparts have become fannliar, there will be no more mystery in 18 feet x j- = 9 feet than in measuring half- way across a room 18 feet wide ; the peculiar thing would be if taking a quantity only a part of a time did not give a smaller quantity. So in division, when the mutual relati« i between times and parts ig understood, the operation %^ -r- ?^yV) or $f -r- xV> ^^ j^'^t ^^s intelligible as $80 -r- ^10, or us $80 -f- 10. To say that the quotient e'ujld^ the result of $-f -^ $ii)-) is greater than the dividend ($f ) is to talk nonsense ; is to compare incomparable tilings — is to confuse parts with times, quantity with number, matter wdth a psychical process. *i^ lis 'i h ■1 ii i %\ nil 11 ifi v.i 'i'V >]i ■ I 5 .i . ;•* n c II u ll hi In « Ifi '■ fir"" ill if CHAPTER YIII. ON PRIirARY NUMBER TEACHING. The Ntimler Instinct. — AYe liave seen that number is not something impressed upon the mind by external energies, or given in the inere perception of things, but is a product of the mind's action in the measurement of quantity — that is, in making a vague whole definite. Since this action is the fundamental psychical activity directed upon quantitative relations, the process of num- bering should be attended with interest; that is, con- trary to the connnoidy received opinion, the study of arithmetic should be as interestino; to the learner as that of any other subject in the curriculum. The training of observation and perception in dealing with nature studies is said to be universally interesting. This is no doubt true, as there is a hunger of the senses — of sight, hearing, touch — which, when gratified by the j^resenta- tion of sense materials, affords satisfaction to the self. But we may surely say with equal ti'uth that the exer- cise of the higher energy which works upon these raw materials is attended ^nth at least e(jual pleasure. The natural action of attention and judgment working upon the sense-facts must be accompanied with as deep and vivid an interest as the normal action of the observing powers through which the sense-facts are acquired. 144 Kt'l !l number external ngs, but uremcnt definite, activity i of niun- ; is, con- study of as that traininoj li nature lis is no of sight, I'csenta- thc self, he exer- lese raw •e. The ng upon Jeep and hserving red. ON PRIMARY NUMBER TEACHING. 145 For numerical ideas involve the simplest forms of this higher process of mental elaboration ; they enter into all human activity ; they are essential to the proper intei'pretation of the pliysical world ; they are a neces- sary condition of man's emancipation from the merely sensuous ; they are a powerful instrument in his reac- tion against his environment ; in a word, number and numerical ideas are an indispensable condition of the development of the individual and the progress of the race. It would therefore seem to be contrary to the "beautiful economy of Nature" if the mind had to be forced to the acquisition of that knowledge and power which are essential to individual and racial develop- ment ; in other words, if the conditions of progress involved other conditions which tended to retard })rogress. The position here taken on theoretical grounds, that the normal activitv of the mind in constructinc: number is full of interest, is confirmed by actual experience and observation of the facts in child life. There are but few children who do not at first deliijjht in number. Counting (the fundamental process of arithmetic) is a thing of joy to them. It is the promise and potency of liigher things. The one, two, three of the " six-years' darling of a pygmy size " is the expression of a higher energy struggling for complete utterance. It is a proof of his gradn'^l emergence from a merely sensuous state to that hiffher staffe in which he bcinns to assert his mastery over the physical world. We have seen a first- year class — the whole class — just out of the kindergar- ten, become so thoroughly interested in arithmetic under a sympathetic and competent teacher, as to i)refev an I • \ii 'I r^A < vm 146 THE PSYCnOLOGY OP NUMBER. .<:! ' * . r ' I ^< ! t ,1 ,ij.i I :r I, ..r- exercise in aritliiiietic to a kindergarten song or a romp in the playground. Arrested Development. — Since, then, the natural ac- tion of tlie child's mind in gaining his first ideas of niunber is attended with interest, it seems clear that Avhen Tinder the formal teaching of number that inter- est, instead of being quickened and strengthened, actu- ally dies out, the method of teaching must be seriously at fault. The method must lack the essentials of true method. It does not stimulate and co-operate witli the rhythmic movement of the mind, but rather im- pedes and probably distorts it. The natural instinct of number, which is present in every one, is not guided by proper methods till effective development is reached. The native aptitude for number is continually baffled, and an artificial activity, opposed to all rational devel- opment of numerical ideas, is forced upon the mind. From this irrational process an arrested development of the number function ensues. An actual distaste for num- ber is created ; the child is adjudged to have no interest in number and no taste for mathematics ; and to nature is ascribed an incapacity which is solely due to irrational instruction. It is perhaps not too much to say that nine tenths of those who dislike arithmetic, or who at least feel that they have no a'^titude for mathematics, owe this misfortune to wrong teaching at first ; to a method which, instead of working in harmony with the number instinct and so making every stage of develop- ment a preparation for the next, actually thwarts the natural mov^ement of the mind, and substitutes for its spontaneous and free activity a forced and mechanical action accompanied with no vital interest, and lead- '. •( i-1 ON PRIMARY NUMBER TEACHING. U7 ing neither to acquired knowledge nor developed power. Characteristics of this defective method have heen frequently pointed out in the preceding pages, and it is unnecessary to notice them here further than to caution the teacher against a few of them, which it is especially necessary to avoid. Avoid what has been called the "fixed-unit" meth- od. No greater mistake can be made than to begin wdth a single thing and to proceed by aggregating such independent wholes. The method works by fixed and isolated unities towards an undefined limit ; that is, it attempts to develop accurate ideas of quantity without the presence of that which is the essence of quantity — namely, the idea of limit. It does not promote, but actually warps, the natural action of the mind in its con- struction of number ; it leaves the fundamental numer- ical operations meaningless, and fractions a frowaiing hill of difliculty. No amount of questioning upon one thing in the vain attempt to develop the idea of " one," no amount of drill on two such things or three such things, no amount of artificial analysis on the numbers from one to five, can make good the ineradicable defects of a beginning which actually obstructs the primary men- tal functions, and all but stifles the number instinct. Avoid, then, excessive analysis, the necessary conse- quence of this "rigid unit" method. This analysis, making appeals to an undeveloped power of numerical abstraction, becomes as dull and mechanical and quite as mischievous in its effects as the "fissure svstem," which is considered but little better than a mere jug- glery with number symbols. n , ■!'. mm m ■I . 148 THE PSYCHOLOGY OF NUMBER. ■ ^ri. A ' 1 1 1 h: '1 f,J f' H .•1 ' 1 ! i. ;, ri. in |Uii W'U iiiii "1 Hi 1 ;/ ^^ ■ i i IL .1 Avoid the error of assuming tliat there are exact numerical ideas in the mind as the result of a number of things before the senses. Tliis ignores the fact tliat number is not a thing, not a property nor a perception of things, but the result of the mind's action in dealing with quantity. Avoid treating numbers as a series of separate and independent entities, each of which is to be thoroughly mastered before the next is taken up. Too much thoroughness in primary number work is as harmful as too little thorouglmess in advanced work. Avoid on the one hand the simultaneous teaching of tlie fundamental operations, and on the other hand the teaching which fails to recognise their logical and psychological connection. Avoid the error which makes the "how many" alone constitute number, and leaves out of account the other co-ordinate factor, " how much." The nieasuping idea must always be prominent in developing number and numerical operations. Without this idea of meas- urement no clear conception of number can be devel- oped, and the real meaning of the various operations as simply phases in the development of the measuring idea will never be grasped. Avoid the fallacy of assuming that the child, to know a nwnher, nmst be able to picture all the num- bered units that make up a given quantity. Avoid the interest-killing monotony of the Grube grind on the three hundred and odd combinations of half a dozen numbers, which thus substitutes sheer me- chanical action for the spontaneous activity that simul- taneously develops numerical ideas and the power to retain them. I ON PRIMARY NUMBER TEACHING. U9 national Method. — The defects which have been enuiTicrated as marking the "fixed-unit" method sug- gest the chief features of the psycliological or rational method. This method pursues a diametrically opposite course. It does not introduce one object, theu another " closely observed " object, and so on, multiplying in- teresting questions in the attempt to develop the num- ber one from an accurate observation of a single object. It does as Kature prompts the child to do : it begins with a quantity — a group of things which may be meas- ured — and makes school instruction a continuation of the process by which the child has already acquired vague numerical ideas. Under Nature's teaching the child does not attempt to develop the number one by close observation of a single thing, for this observation, however close, will not yield the number one. He de- velops the idea of one, and all other numerical ideas, through the measuring activity ; he counts, and thus measures, apples, oranges, bananas, marbles, and any other things in which he feels some interest. ^NTature does not set him upon an impossible task — i. e., the getting of an idea under conditions which preclude its ac(piisition. She does not demand numerical abstrac- tion and generalization wlien there is nothing before him for this activity to work upon. Let the actual work of the schoolroom, tlierefore, be consistent with tlie method under which by Nature's teaching the child has already secured some development of the number activity. In all psychical activity every stage in the develop- ment of an instinct prepares the way for the next stage. The child's number instinct bei^ins to show itself in its I (i i m f ifc v^f \ i m\ in J ■* I It ;■ 1 • M 150 THE PSYCHOLOGY OF NUMBER. working upon continuous quantity — that is, a whole requiring measurement. Every successive step in the entire course of development should harmonize with this initial stage. To get exact ideas of quantity the mind must follow Nature's estahlished law; must meas- ure quantity ; must break it into parts and unify the parts, till it recognises the one as many and the many as one. There can be no possible numerical abstraction and generalization without a quantity to be measured. AVhere, then, does the " single closely observed object " come in as material for this parting and wholing ? Beginning with a group is in harmony with Nature's method ; promotes the normal action of the mind ; gives the craving numerical instinct something to work upon, and wisely guides it to its richest development. This psychological method promotes the natural exercise of mental function ; leads gradually but with ease and cer- tainty to true ideas of number ; secures recognition of the unity of the arithmetical operations ; gives clear conceptions of the nature of these operations as succes- sive steps in the process of measurement ; minimizes the difficulty with which multiplication and division have hitherto been attended ; and helps the child to recognise in the dreaded terra incognita of fractions a pleasant and familiar land. Forming the Habit of Parting and Wholing. — The teacher should from the first keep in view the im- portance of forming the hahit of parting and wholing. This is the fundamental psychical activity ; its goal is to grasp clearly and definitely by one act of mind a whole of many and defined parts. This primary activity work- ing upon quantity in the process of measurement gives ON PRIMARY NUMBER TEACHING. 151 rise to numerical relations ; the incoherent whole is made definite and unified — becomes the conception of a unity composed of units. Every right exercise of this activity gives new knowledge and an increase of analytic power. At last the hahit of numerical analysis is formed, and when it is found requisite to deal with quantity and quantitative relations, the mind always conceives of quantity as made up of parts — measuring units; not invariable units, but units chosen at pleasure or convenience ; parts, given by the necessary activity of analysis, a whole from the parts by the necessary ac- tivity of synthesis. This means that always and inevi- tably from first to last the process of fractioning is present. A Consti'uctwG Process. — This wholing and parting, as far as possible, should be a constructive act. The physical acts of separating a whole into parts and re- uniting the parts into a whole lead gradually to the corresponding mental process of number : division of a whole into exact parts, and the reconstruction of the parts to form a whole. It can not be said that even the physical acts are wholly mindless, for even in these acts there must be at least a vague mental awareness of the relation of the ])arts to one another and to the whole. These physical acts of wholing and parting under wise direction lead quickly, and with the least expenditure of energy, to clear and definite percepts of related things, and finally to definite conceptions of number. The child should be required to exercise his activity, ^o do as much as possible in the process, and to notice and state what he is really doing. lie should actually apply, for instance, the measuring unit to the measured quan- I \\ tm m ; " . ! f ) it-';-' iJK ' t , I'* i' ,;( ^i ii, 1 li'- ; i'l Hlr* ^5? 152 THE PSYCHOLOGY OF NUMBER. ''I A .^1 \ ■ ■ f m -J w tity. If the foot is measured by two G-incli or three 4- inch or four 3-iiicli units, lot liiiii lirst apply the miniber of actual units — two G-inch, three 4-ineh, and four 3-incli units — to make up the foot, and so on. By using the actual number of parts required he will have a more definite idea of the construction of a whole than if he simply applies one of the measuring units the necessary number of times. This operation with the actual units should precede the operation by which the whole is mentally constructed by applying or repeating the sin- gle unit of measurement the required number of times. It is the more concrete process, and is an effective exer- cise for the gradual growth of the more abstract times or ratio idea. When the child actually uses the 1-incli or the 3-inch unit to measure the foot, his ideas of these units as well as of the measured whole are enlarged and defined. He applies the inch to measure the foot, and this to meas- ure the yard, and the yard to measure the length of the room and other quantities. Let him freely practise this constructive activity, thus practically applying the psy- chological law, " Know by doing, and do by knowing." The 2-inch square is separated into four inch squares, or sixteen half-inch squares, and these measuring units are put together again to form the whole. Similarly a rec- tangle 2 inches by 3 inches, for example, is divided into its constituent inch squares or half-inch squares, and again reconstructed from the ])arts. A square is divided into four right-angled isosceles triangles, into eight smaller triangles, and the parts rhythmically put to- gether again. Value of Kinde^'garten Constructions. — In this con- ON PltlMARY NUMBER TEACHING. 153 nection it may be noted that most of the exercises of the kindergarten can be etl'ectively used for training in number. The constrnctive exercises which are so ])rominent a featnre in tlie kindergarten are admira- bly adapted to lead gradually to mathematical abstrac- tion and generalization, ^o doubt much has been done in this direction, but much more could be done were the teacher versed in the psychological method of deal- ing with number. No one questions the general value of kindergarten i '-.lining, which on the whole is founded on sound psycliological principles ; but, on the other hand, no educati >nal psychologist doubts that its phi- losophy as commonly understood needs revision, and that its methods are capable of improvement. If its aim is, as it should be, an effective preparation of the child for his subsequent educational course, it is thought that its practical results are far from what they ought to be. It is ofti n maintained with considerable force that kinder'^arte I methods should be introduced into the primary and even higher schools. On the other \\pj\L ooiiie'hing might be said with a good show of reasoii ia favour of introducing primary and grammar school methods into the kindergarten. What is radi- cally sound in the kindergarten methods will harmonize with what is radically sound in the methods of the pub- lic school. On the other hand, what is psychologically sound in the methods of the public school should at least influence the aims and methods of the kindei'gar- ten. Is the p'^esent kindergarten training, speaking generally, really the best preparation for the training given in a thoroughly good public school ? The func- . tion of such a school is to give the best possible prepa- ^ 5^1 :|! I \ M \ M\ ■.I ■ . .•y i" M 'i' h fi 1, ; i ' 1 , . 1 I'.'l :■ ri'ii li 154 THE PSYCHOLOGY OP NUMBER. ration for life by means of studies and discipline, wliich, as far as inevitable limitations permit, seciiro at the same time the best possible development of character. Amon£«: these studies the three IV s must always hold a prominent place, in spite of theories which seem to assume that language, the complement of man's reason, and number, the instrument of man's interpretation and mastery of the physical world, are not essential to hu- man advancement, and may therefore be degraded from the central position which they have long occupied to one in which they are the subjects of merely haphazard and disconnected teaching. The practical methods founded on these theories seem to treat the world of Nature as one whole, which even the child may grasp in its infinite diversity and total unity. The " flower in the crannied wall " is made the central point around which all that is know- able is to be collected. But as the human mind is lim- ited, and must move obedient to the law of its consti- tution, the theories and metliods wdiich overlook these facts are not likely permanently to prevail ; and the old subjects that have stood the test of time will no doul)t stand the test of the most searching psychological inves- tigation, and regain their full recognition as the "core" subjects of the school curriculum. Does the kindergarten, then, accomplish all tliat may be done as a preparation for such a curriculum ? It is to be feared that with regard to many of them the an- swer must be in the negative ; and this is perhaps es- pecially true concerning the subject of arithmetic. We have known the seven-year-old " head boy " of a kin- dergarten, conducted by a rioted kindergarten teacher, ON I'RLMARY NUMBER TExVCIIING. 155 m who could not recognise a (jiiaiitity of tliroe tliinj:;s with- out countiiiii^ thorn hv ones. Jjuin<«^ asked to heirin tlic construction of a certain form at a distance of three inches from tlie edge of the table, he invariably had to count off carefully the inch spaces — a clear proof, it is thought, that his training had not l)een the best pos- sible preparation for the arithmetical instruction of the higher schools. It is certain that arithmetical instruction in the higher grade of school may be greatly improved ; it is alike certain that as a preparation for this better in- struction the ti'aining of the kindei-garten also may be greatly improved ; and there is every reason to believe that with this improvement its rational training in other things, its ethical aim, its educative interest, and its character-forming spirit, would be materially en- hanced. In some kindergartens, at least, the monotony of continuous play would be pleasurably relieved by a little recreation at work. It is certain that the interest associated with many exercises necessarily connected with the number activity, especially the constructive and analytic processes of the kindergarten, can be made under right teaching the menus by which numerical ideas may be gradually and pleasantly worked out. The little builder of many forms of beauty and utility would in due time find, when the inevitable and harder tasks began, that he had been building better than he knew. There is surely something lacking either in the kindergarten as a preparation for the primary school, or in the primary school as a continuation of the kin- dergarten, when a child after full training in the kinder- garten, together with two years' work in the primary ll'ili I 150 THE PSYCriOLOGY OF NUMBER. 1^1 WW school, is considered able to undertake nothing beyond the "number 20." It might reasonably be maintained that, under rational and therefore pleasurable training of the number instinct in the kindergarten, the child ou<>;ht to be aiithmeticallv strong enouii-li to make ini- mediate accjuaintance with the number 20, and rapidly accjuire — if he has not already ac(|uired — a working con- ce])tioii of much larger numbers. Iin2)ortant Points of Bational Method. — In apply- ing!: the rational method of teaching arithmetic there are important things that the teacher must keep in view if he is to phI the child's mind to v;ork fioely and nat- urally in tit; evolution of mmiber. The child's niind must be guided aloui:^ the lines of least resistance to the true idea of n%iiiJjer. This movement, in the very nature of things, must be slow as compared with the gathering of sense facts ; but under the psychological method it may be sure and pleasurable. The result aimed at can not be reached b^' banishinj>: the word times from arithmetic ; nor [)y working continually with indefinite units of measure ; nor by exclusive at- tention to manual occu])ations under the vague idea that physical separation of things is analysis of thought ; nor by making counting — emphasizing the vague how many — the single purpose, and unmeasured units the sole matter of the exercises, to the exclusion uf the how 7nvch, and the measuring idea which is the essence of number ; nor by substituting for the rhythmic and bpontaneous action of the child's mind in dealing with wholes, both qualitative and (juantitative, a minute and formal analysis which properly finds ])lace only in a riper stage of mental growth ; nor by any amount of i ON PRIMARY NUMBER TEACIIINO. 15' drill, liowever industrious and deviceful the drill-mas- ter, which substitutes mechanical action and factitious interest for spontaneous action and intrinsic interest, the very life of the self-developing soul. Kumber is the measurement of quantity, and there- fore the only solid basis of method and sure guide for the teacher is the measurvmj idea. 1. The Memured Whole. — The factors in number are, as before shown, the unity (the whole of quantity) to be measured, the unit of measurement, and t^e times of its repetition — the number in the strictly mathemat- ical and psychological sense of the word. The teacher must bear in mind the distinction between unity and unit as fundamental. The entire dilference between a good method and a bad method lies here, because the essential principle of mnnber lies here. Vague unity, units, defined unity, is the secjuence as determined by psychological law. In the child's lirst deahng with number there must be the grouj) of th: 2:s, tin whole of quantity to start from ; and in every step of the ini- tial stage the idea of a whole to be measured is to be kept prominent. In addition, there is a whole (the sum) to be made more definite by putting together its component parts (addends) — not equal measKrlna units, but each part defined l)y a common unit — so as to com- ph^tcly define the quantity in terms of this specifically defined unit. In subtraction we have a given quantity (minuend) and a component part (subtrahend) of it to find the other conq)oneMt (remainder) — a process which lielps to a more definite idea of the given whole, and especially makes explicit the vague idea of the "remain- der" with ;vhich we began. In nmlti])lication there is M m ,:'■■' 5; I r-M 158 THE PSYCHOLOGY OP NUMBER. 1 1 ^ • 1 '. ■ 1 1 :f 1 » i given a quantity (mnltiplicand) defined by a measuring unit and the times (multiplier) of its repetition ; and the process makes the quantity articulately defined (in the product) by substituting a more familiar unit for the derived unit of measurement ; in other words, by expressing the quantity in terms of the j^rimary unit by which the derived unit itself is measured. In divi- sion we have a whole quantity given (dividend), and one of two related measuring parts (the divisor) to find the other part (quotient), and the operation makes clearer the whole magnitude, and at the same time makes the first vague idea of the other measuring part (quo- tient) perfectly definite. Briefly, in all numerical oper- ations there is some magnitude to be definitely deter- mined in numerical terms, and the arithmetical opera- tions are simply related steps expressing tlie correspond- ing stages of the mental movement by which the vague whole is made definite. Keep clearly in mind, there- fore, the inclusive magnitude from which and within which the mental movement takes place — which justi- fies and gives meaning to both the psychical process and the arithmetical operation. 2. The U?nt of Measure — Its True Functw\ — From the vague unity, through the vnll.^, to the deii- nite unity, the snm^ is the law of mental movement. The second point of essential importance is to make clear the idea of the unit of measure. More than half the difiiculty of the teacher in teaching, and the learner in learning, is due to misconce])tion of what the ''unit'' really is. It is not a single unmeasured object ; it is not even a single defined or measured thing ; it is any measuring jjart by which a quantity is numerically de- :!l ON PRIMARY NUMBER TEACHING. 159 fined ; it is (in the crude stage of measurement) one of the like things used to measure a collection of the things ; it is (in the second or exact stage of measure- ment) one of the equal parts used to measure an ex- actly measured quantity. It is one of a necessarily related many constituting a whole. It is, therefore, an utterly false method to begin with an isolated object — false to the fact of measurement, false to the free activity of the mind in the measurin §• i ' 1 f ' : I f » t .( : I' « ■1! hi; I, , • I >l t rill' 162 THE PSYCHOLOGY OF NUMBER. as the quantity 12 inches measured by the unit 4 inches, by the unit 3 inches, etc. ; or the quantity 20 cents measured by the unit 10 cents, by the unit 5 cents, etc. The movement towards the real number ;idea began in operations with undefined units, and is strengthened by these supplementary exercises with ox- Jactly measured quantity ; thei-e is a more rapid growth towards the numerical discriminating and unifying power, {h) Count by o}ic% but not necessarily by sin- gle things ; in fact, to avoid the fixed unit error, do not begin with counting single things. The 12 things in the group have been measured off, for example, into four groups, or into three groups ; these are units, are ones, and in counting there is a first one, a second oi^e, a third 07ie — that is, in all ''three times^' one; and so with the four ones when the quantity is divided into four equal parts. Proceed similarly with exactly measured quanti- ties : the four f3-inch ones or the six 2-incli ones making up the linoai . )ot, or other exactly measured quantity. As before said, the child first of all sees related things, and with the repetition of the exercises — parting and wholing — begins to feel the relations of things, and in due time consciou>lv recoiniises these relations, and the goal is at last reached — a definite idea of number. {c) Use the ActiKilTn (ti<. — In these constructive proc- esses let the child at first use — as before suggested — the actual concrete units to miike up or etpial the measured ([uantity ; then apply the shxjle r'oncrete unit the requi- site number of " times." It first case, in measuring, for example, a length of l:i feet, four actual units of measure (3 feet) are put toi/'^'^'er to equal the 12 feet; in the second case, one unit ^ applied, laid down, and 1 I ON PIlIMArtY NUMBER TEACHING. 103 taken up three times. This application of tlie single unit so many times is an important step in the process of numerical abstraction and generalization ; it is from .the less abstract and more concrete to the more abstract and less concrete. It may be noted, also, that the other senses, especially the sense of hearing, may be made to co-operate with sight in the evolution of the thnes idea. Appeal by a variety of examples to the trusty eye, but appeal also to the trusty ear — strokes on a bell, taps on the desk, uttered syllables, etc. Here, as in all other cases, we do not confine ourselves to single bell- strokes or svllables ; we count the number of double strokes, triple strokes ; of double and triple syllables, as, for example, oh^ oh ^ oli^ oh; oh, oh — i. e., 3 counts of two sounds each, etc. Counting and Measuring. — In the separating and combining processes referred to, counting goes on. This is at first chiefly mechanical, and care must be taken in the interest of the number idea to make it become rational. Through practice in parting and wholing the idea of the function of the unit is grad- ually formed ; it is the concrete, spatial thing used to measure quantity. The point is, not to neglect either the sy)atial element or the other essential factor in num- ber, the counting, the actual relating process. In the method of number teachint!: usuallv followed, counting is the prominent thing, to the almost total exclusion of the measuring idea ; the emphasis is upon the how manv, with but little attention to the how much. But the counting is largely mechanical. There is a repe- tition of names without definite meaning. The child is groping his way toward the light. He can not help I'll ■■pp w M I r .1: M l< ' I I' I 164 THE PSYCHOLOGY OF NUMBER. feeling, as lie counts liis units, tliat one, two, three is not so much — because it is not so many — as one, two, three, four. These first vague ideas must he made clear and definite ; the natural movement of the mind is aided by the proper presentation of right material ; the initial mechanical operation of naming the units in order gives place to an intelligent relating of the units to one another, and finally to a conscious grasp of the relation of each to the unified whole ; the counting — one, two, three, etc. — is now a rational process. /So much; so many. — In tlie development of this rational process there must be no divo rr;e between the how much and the how many, between the measuring process and the results of measurement. The so nmch is determined only by the so many, and the so many has significance only from its relation to the so much. These are co-ordinate factors of the idea of number as measurement. Now, the development of counting — determining the how many that defines the how much — is aided by symmetrical arrangements of the units of measure (see page 34). The child at first counts the units one, two, . . . six, with only the faintest idea of the relations of the units in the numbers named. Both the analytic and relating activities are greatly aided by the rhythmic grouping of tlie units of measure, or of the counters used to represent them ; the mastery of the number relations (of both addition and multiplica- tion) as so many units making up a quantity, becomes much easier and more complete. Thus, when exercises in parting and wholing (accompanied witli counting) a quantity, say a length of 12 inches, have given rise to even imperfect ideas of unit of measurement and times :ri; ■■^ ON PULMaRY number TEACHING. 1G5 of repetition, the symmetric forms may be used with great advantage; indeed, they may be used in the exer- cises from the first. We lave counted, according to the unit of measure used, one oart, two parts ; one part, two parts, three parts, etc. Both the times and the unit values are more easily grasped through the number forms ; for example, six, one of the two measuring units, may be shown as a whole of related units (threes, twos, ones) as in the arrangement, 1,1; and so on with the whole quantity and all its minor parts (addition) and repeated units. Heal meaning is given to the operation of count- ing when, instead of using unarranged units, we have the rhythmic arrangement : e o • o • e o o • • e o • • Six, five, four, three, two, one. The actual values of the measuring units, and the mean- ing of counting — necessarily related processes — are fully brought out. Six is at last perceived as six without the necessity of counting. :il It i I ' 1 . • 1. 1 I 1 ■ J (' t lfi:H CHAPTER IX. ON PRIMARY NUMJ}ER TEACHING. lielation lettceen Times and Parts. — AYitli the growth of tlie idea of the unit as a measure itself measured by minor units, and of number as indicating times of repetition of a unit of measure, there is grad- ually developed a clear idea of the relation between the value of the actual measuring unit (as made up of minor units) and the number of them in tlie given quantity ; in other words, of the relation between the number of de- rived units in the quantity and the number of primary units in the derived unit. A common error, as has been often pointed out, is that of making too broad a distinc- tion between these related factors in the measuring proc- ess. They are said to be totally different conceptions. It has been shown that they are absolutely inseparable. They are, in any and every case, two aspects of the same measurement. The direct unit in a given measurement is not wholly concrete ; it is a quantity measured by a number of other units ; and so it involves, as every meas- ured quantity involves, the space element in the single concrete (minor) unit, and the abstract element in the number applied to the unit. AVhen we speak of the " size" of the numbered parts (derived units) composing a given quantity, we mean the number of minor units 166 ON PRIMARY NUMBER TEACHING. ir.7 of wliicli one part is composed. Wlicii, for instance, we conceive of $15 as measured by the nnit J^3, we get the number live ; when we are required to divide lj>15 into live equal parts, we are searching for the " size " of the measuring unit — i. e., for the numerical value of the unit in terms of the minor unit ($1) by wliich it is measured. The relation, then, between "times and parts" is the relation between the number of derived units in the measured quantity and the number of primary units in the derived unit. It is clear that the rational processes of parting and wholing that ultimately give clear ideas of tmit and number, must also bring out clearly the re- lation between these two factors in measured quantity : the smaller the unit the larger the number ; or, the num- ber of the measuring units in the quantity varies in- versely as the number of primary units in the derived unit. Measuring a lengtli of one foot by a 6-incli unit, by a 3-inch unit, and by a 1-inch unit, the numbers are respectively 2, 4, and 12 ; measuring a lengtli of one decimetre by 10, 20, 30, 40, 50 centimetres, the num- bers are respectively 10, 5, 3J, 2^, 2 ; measuring $20 by the $1 unit, $2 unit, $4 unit, the numbers are 20, 10, 5, etc. In the constructive exercises already described, attention to measuring unit and its times of repetition must lead to the conscious recognition of this principle, which is fundamental in number as measurement. It has already been given in the complete statement in " fractional " form of the process of measurement : Any 'measured quantity may be expressed in the form 12 3 4 n 75 o' oi 7^ • • • -• This principle is, of course, the basal 12 3 4 n r JT 7 7 m ■>. IMAGE EVALUATION TEST TARGET (MT-3) m /. * *'-V^^. v.. %° i/.. 1.0 US 1^ 1.1 iriis L25 i 1.4 I 1.6 V] <^ n :> ^5 ^>. ^ W ^'^ V ^ %i^/ ^ Hiotographic Sdences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. USSO (716) 872-4503 "^ ^ ^; f/. ■^ s \> I' III .» lit' J' ;'!: ! ,' * w. m '.*!) I ■ 1G8 THE PSYCHOLOGY OF NUMBER. principle in the treatment of fractions (because it is tlic primary principle of number, and " fractions " are num- bers) — namely, both terms of a fraction may be multi- plied or divided by the same number without altering the value of the fraction. The Laiv of Coniinutation. — The development of this principle through the rational use of its idea — that is through the use of the facts supplied by sense- perception in the rational use of things — is the develop- ment of the psychological law of commutation which is primary and essential in all mathematics. The method that ignores this necessary relation between times and parts, or regards them as totally different things, never leads to a clear conception of this important principle. It is, as a consequence, always iinding difficulties where, for rational method, none really exist. The principle is difficult for the child only when the method is wrong. With right presentation of nuxterial he can have no dif- ficulty in seeing that the larger the units the smaller their number in a given (pumtity. When he counts out a collection of 24 objects into piles of 3 each, and into piles of 6 each, can he fail to see that the re- spective numbers differ ? And with rightly directed attention to the concrete processes, may he not be led slowly, perhaps, but surely, to a clear thought of how and why they differ ? The learner can not help seeing, for example, the difference between the 2-inch unit and the 6-inch unit, and the corres})onding difference between six times and three times. To see clearly is to think clearly ; there is a rationality in rationally presented facts, and this rationality leads with certainty to a com- plete recognition of the meaning — the law — of the facts. ' ON PRIMARY NUMBER TEACHING. 109 Should he Used from the First. — Thus the idea of the law of coninuitation can be used from the first ; rather, must be used if clear and adequate ideas of number are to be gained, and numerical operations are to be a thing of meaning and of interest. As already seen, it is impossible to know 12 objects, as measured by fours, without at the same time know- ing it as measured by threes. So it is in the actual operation with things : we count out a quantity of 12 things into groups of four things each, and find the number of the groups to be three y and we count out the 12 things into four groups, and find in each group three things. In both cases the operations are alike ; in neither is it possible to get the result without using counts of four things each. The child counts out 12 things into groups of 4 things each ; how many groups ? lie counts them out into 4 groups ; how many things in each group ? In both cases he sees that he counts by fours. lie counts 10 things in groups of 5 things each ; how many groups ? He counts them out into 5 groups; how many in each group ? In both operations he sees that he counts out into groups of five things each. Twenty cents are to be equally divided among 5 boys; how many cents will each boy get ? Twenty cents are divided equally among a number of boys, giving each boy 5 cents ; how many boys are there ? The child per- forms the operation with counters, and finds in both cases /6>z«/", meaning 4 boys in the one case, 4 cents in the other ; he sees that in both examples the operation was a counting hyfves, and he will soon be in posses- sion of the important truth — which many teachers, and even teachers of teachers, seem not to know — that one m ii 'M m Jill iro THE PSYCHOLOGY OF NUMBER. Pi!,* i\ :,) f ^i'U I i' ' 51 "I J I I I I ■fi• ::! •I ^y^/i of 20 units of any kind is four units, becavse Jive units must be repeated four times to measure 20 units. There will be but little difficulty in further illus- trating the principle by applying it to exact measure- ment. Three 4-inch measures make up the linear foot ; the child sees that one inch counted out of each of the 3 4-inch units gives a whole of 3 inches ; another inch thus counted out gives another whole of 3 inches, etc. ; so that in all there are 4 units of 3 inches each. Since, however, the principle is more distinctly illustrated by symmetrical groupings of the measuring units (see page 34), the separate primary units niay be used with advan- tage — for example, for the 4-inch measure use the four single units which compose it. With the proper use of such parting and wholing exercises the child can not fail to comprehend in due time this fundamental prin- ciple in number and numerical operntions. Seeing clearly that the unit is composed of minor units, that it is repeated to make up or measure some quantity, he can not fail to see that each and all of its minor units are repeated the same number of times. What has been joined together by a psychical law can not be divorced without checking or distorting mental growth. It is known from experience that when the constructive exercises already referred to are carried on under rational direction, the use of the re- lated ideas grows natui'ally and surely into a conscious recognition of the relations : how much and how many, quantity and its instrument of measurement, parts and times, measuring unit and measured whole, measuring minor unit and its measured minor whole, correlated m ON PRLMARY NUMBER TEACHING. ITl ' M factors of product, correlated processes of division, are all seen in their true logical and psvcholonical inter- relation. These things are organically connected by necessary laws of thought ; the method which is ra- tionalized by this idea makes arithmetic a delight to the pupil and a powerful educating instrument. A method which violates this necessarv law of mind in dealing with quantity — constantly obstructing the origi- nal action of the mind — makes arithmetic a thine: of rule and routine, uninteresting to the teacher, and prob- ably detested by the learner. Ifake Ilasie slowli/. — As ali-eady suggested, time is necessary for the completion of the idea of nund)er. Under sound instruction a working conce])tion suitable for a primary stage of development may be readily ac- quired, and may be used for higher development. But a perfectly clear and delinite conception of number is a ])roduct of growth l)y slow degrees. The power of numerical abstraction and generalization can not be im- parted at will even by the most painstaking teacher. Ilence the absurdity of making minute mechanical analysis a substitute for nature's sure but patient way. It seems to be thought that mechanical drill upon a few numbers — a drill which, if rational, would really use ratio and proportion — will in some unexplained way " impart " the idea of number to the child apart from the self-activity of which alone it is the product. And this fallacious idea is strenc^thened by the fluent chatter of the child — the apt repeater of mere sense facts — about the " equal numbers in a number," the " equal numbers that make a number," and all the rest of it. This routine analysis and parrotlike expression of it are f f ^ ^ ■■ 1 I < 1 ■ li m H ;ii II . 11 172 THE PSYCHOLOGY OF NUMBER. a direct violation of the psychology of number. The idea of number can not be got from this forcing process. The conscious grasp of the idea, we repeat, must come from rational use of the idea, and is all but impossible by monotonous analyses with a few "simple" numbers ; it is absolutely impossible in the immature state of the child-mind in which it is attempted. The child must, once more, freely and rationally use the ideas ; must operate with many things, using many numbers, before the idea of number can possibly be developed. We are omitting the things the child can do — rationally iise numerical ideas — and forcing upon him things that he can not do — form at once a complete conception of number and numerical relations. It is high time to change all this : to omit the things he can not do, and interest him in the things he can do. In the compara- tively formal and mechanical stage tliere must be a cer- tain amount of mechanical drill — mechanical, yet in no small degree disciplinary, because it works with ideas which, though imperfect, are adequate to the stage of development attained, and through rational use become in due time accurate scientific conceptions. Besides valuable discipline, the child gets possession of facts and principles — of elementary knowlc^^e, it may be said — which are essential in his progress towards sci- entific concepts and organized knowledge. It seems absurd, or worse than absurd, to insist on thoroughness, on perfect number concepts, at a time when perfection is impossible, and to ignore the conditions under which alone perfect concepts, can arise — the wise working with imperfect ideas till in good time, under the law con- necting idea and action, facile doing may result in per- ON PRIMARY NUMBER TEACHING. 173 feet knowing. Following the nonpsycliological method liinders the luitural action of the mind, and fails to pre- pare the child for subsequent and higher work in arith- metic. The rational method, promoting the natural ac- tion of the mind by constructive processes which use number, leads surely and economically to clear and definite ideas of number, and thoroughly prepares for real and rapid progress in the higher work. The Stat'timj Point. — It is commonly assumed that the child is familiar with a few of the smaller numbers — with at least the number three. He has undoubtedly acquired some vague ideas of number, because he has been acting under the number instinct ; he has been counting and measuring. But he does not, because he can not, knoio the number, lie knows 3 things, and 5 or more things, when he sees them ; he knows that 5 apples are more than 3 apples, and 3 apples less than 5 apples. But he does not know three in the mathemat- ical or psychological sense as denoting measurement of quantity — the repetition of a unit of measure to equal or make up a magnitude — the ratio of the magnitude to the unit of measure. If he does know the number 3, in the strict sense, it is positively cruel to keep him drilling for months and months upon the number five and " all that can be done with it," and years upon the mimber twentv. The Nxmiljer Two. — There can be little doubt that among the early and imperfect ideas of number the idea of two is first to appear. From the first vague feeling of a this and a that through all stages of growth to the complete mathematical idea of two, his sense ex- periences are rich in twos : two eyes, two ears, two .liiik'i >l f * i ■ ■ > HI i : hill m 1. I ■: il,^ i i 1 1 M i , i I'Vi (!ii m 5: :i ' Li': i : ■ ^ is 1 i,.^ 174 THE PSYCHOLOGY OF NUMBER. hands, this side and that, up and down, riglit and left, etc. The whole structure of things, so to speak, seems to abound in twos. But it is not to be supposed that this common experience has given him the number two as expressing order or relation of measuring nnits. The two things which he knows are qualitative ones-, not units. Two is not recognised as expressing the same relation, however the units may vary in quality or mag- nitude ; it is not yet one + one, or one taken two times — two apples, tivo 5-apples, two 10-apples, two 100-apples ; or tivo 1-inch, two 5-inch, two 10-incli, two 100-inch units ; in short, o?ie unit of measure of any quantity and any value taken tivo times. But his large experience with pairs of things, and the imperfect idea of two that neces- sarily comes iirst, prepare him for the ready use of the idea, and the comparatively easy development of it. There must be a test of how far, or to what extent, he knows the number two. This is supplied by construct- ive exercises with things in which the idea of two is prominent. The child separates a lot of beans (say 8) into two equal parts, and names the number of the parts two / separates each part into two equal parts, and names the number of the parts tivo / separates each of these parts into two equal parts, and names the number of the single things tivo. Or, arranging in perceptive forms, how many ones in ? How many twos in ? How many pairs of twos in • ? Similar exercises and questions may be given with splints formed into two squares, and into two groups of two pickets each ; with 12 splints formed into two squares with diagonals (see page 106) ; then each square (group) formed into ON PRLMARY NUMBER TEACHING. 175 two triangles ; how many squares ? how many ti'iangles (unit groups)? how many pairs of triangles? Exact measurements are to accompany such exercises : the 12-inch length measured by the 6-inch, this again by the 3-inch ; how many 6-inch units in the whole ? 3-inch units in the 6-inch units ? pairs of 3-inch units in the whole ? Put 1-inch units together to make the 2-inch unit, the 2 inch units to make the 4-inch unit, two of these to make 8 inches, etc. It is not meant that these exercises are to be continued till the number two is thoroughly mastered ; they carry with them notions of higher numbers, without which a conception of two can not be reached. Beware of " thoroughness " at a stage when thoroughness (in the sense of complete mastery) is impossible. The Number Three. — The number three is a much more difficult idea for the child. As in the case of two, he knows three objects as more than two and less than four ; three units in exact measurement as more than two, etc. But he does not know three in the strictly numerical sense. He may know tw^o fairly well — as a working notion — without having a clear idea of the ordered or related ones making a whole. In three, the ordering or relating idea must be consciously present. It is not enough to see the three discriminated ones ; they must at the same time be related^ unified — a first one, a second one, a third one, three 07ies^ a one of three. Three must be three units — measuring parts of a quali- tative whole — units made up, it may be, of 2, or 3, or 4 ... or 71 minor units. If 12 objects are counted off in unit-groups of 4 each, 15 objects into unit-groups of 5 each, 18 objects into unit-groups of 6 each, 30 ob- 13 ; 1' ; .1 , 1 1 1 ■• m i ii •'I !( h u i: 176 THE PSYCHOLOGY OF NUMBER. jects into uiiit-groiips of 10 each, the number in each and every case ninst be recognised as three. The num- ber three, in fact, may be taken as the test of progress towards the true idea of number, and of the child's abil- ity to proceed rapidly to higher numbers and nunierical relations. If he knows three, if he has even an intelli- gent working conception of three, he can proceed in a few lessons to the number ten, and will thus have all higher numbers within comparatively easy reach. The child is not to be kept drilling on the number thi-ee until it is fully mastered. "What was said of two applies equally to three : it can not be mastered without the implicit use of higher numbers. But, while dealing with higher numbers, three may be kept in view as the crucial point in the development of exact ideas of num- ber. There are to be constructive exercises with vari- ous measuring units in which the threes are prominent. In getting the number two, there has been practically a use of three ; for where two is pretty clearly in mind, four is not far behind it in definiteness. When, for example, it is clearly seen that two 2-inch units make up the 4-inch unit, and two 4-inch units make up 8 inches, the two 2-inch units are perceived as four — i. e., and are seen as . But for the complete con- ception of J'our it nmst be related not only to two ( \ but also to three ; in other words, there must be rational counting — we must pass througli the mim- her three to complete recognition of the number four. But, as already suggested, there are to be special exercises in threes. The dozen splints are to be used in constructing squares and triangles. How many . f ON PRIMARY NUMBER TEACHING. 177 squares? IIow many splints needed to make one tri- angle ? Each of the two 6-splints is to be made into as many triangles as possible. How many triangles in each group ? How many in all ? " Two twos, or three triangles and one more." Each of the twf> 6-splints is to be made into as many pickets (A) as possible. How many pickets in each group ? Hc>w many in all ? " Two 3-pickets." In the two 3-pickets how many 2-pickets ? Thus, also, with exact measurements. The 4-inch units in the foot, the 3-inch units in the foot and in the 9- inch measure, the 2-incli units in the 6-inch measure ; the number of 3 square-inch units in the 3-inch square, the number of 4-inch units in the 3-inch by 4-inch rec- tangle, the number of 3-centimetre units in the 9 cen- timetres, etc. Other Numhers to Ten, — Just as when the child has a good idea of two he implicitly knows four, so when he has a good idea of three he has a fair idea of six as two threes. In (symbolizing any units whatever) the two 3-units are jperceived as six units. This perception is connected with 1, 2, 3, 4, of which the child has already good working ideas, and has only to be related with the number five in order to fix its true place in the sequence of related actf3, one, two . . . six, which completes the measurement. In this we pass through five ; five is the connecting link between four and six in the completed sequence. Attention to the perception of six units in discriminates the five • • • • • • units • • • from both the four and the six, and con- 1 1 I M ceives their proper relation to the four units whicli are f .1" * 1.1), 1 ji 1 M ■ '' ,!•> ' ^ ' i: i ^:;U ^i!= }: ; :\.\- &. ■ ^:i'^'' 178 THE PSYCHOLOGY OF NUMBER. • • a part of tliein, and to the six units of which they are a part. When there is a fair idea of four it is an easy step to a fair idea of eight ; 4 + 4 = 8. is not much more difficult than 2 + 2 = 4 — From the examination of eight comes the perception of seven, and a conception of its relation as one more than seven and one less than eiorht. From five, • ? it is by no means a difficult step to ten, as two lives, • • ; and a comparison of this with eight read- ily leads to the perception of nine, • , and to a conception of its relation to both eight and ten. Is Ten twice Five? — From the error of consider- ing the unit as a fixed thing, and number as arising from aggregating one by one other isolated things, arises apparently the fallacious idea that to master ten, for instance, is twice as difficult a task as to mas- ter five. Hence the prevailing practice of devoting six months of precious school life in wearying and repul- sive " analyses of the number five," and finding " a year all too short " for similar analyses of the number tenT But it must be clear that by proper application of the measuring idea, wisely directed exercises in parting and wholing which promote the original activity of the mind in dealing with quantity, it is easy and pleasant to get elementary conceptions of number in general in the time now given to barren grinds on the number ten ; not scientific conceptions, indeed, but sufficiently clear working conceptions, capable of large and free appli- cations in the measurement of quantity — applications ■'I ON PRIMARY NUMBER TEACHING. 179 ' which alone can make the vague definite, and at last evolve a perfect conception of number from its first and necessarily crude beginnings in sense-percej)tion. The /Sequence m Prhnart/ Number Teachlmj. — The measuring exercises suggested, operations of sepa- rating a whole into equal parts and remaking the whole from the parts, correspond to the processes of multipli- cation and division. But they are not these processes, though they are presentations of sense facts which help the unconscious growth towards the conscious recogni- tion of these processes as phases in the development of the measuring idea. It has been shown that addition and subtraction, multiplication and division, differ in psy- chological complexity ; that eacli operation in the order named makes a severer demand on attention ; and that, therefore, while operations corresponding to the higher processes may — indeed, must — with discretion be used from the first, they should not be made the object of conscious or analytic attention. We separate a quan- tity into parts — the presentative element in division ; we put the parts together again — the presentative ele- ment in multiplication. But the ideas of subtraction are involved in these operations — because division and multiplication are involved — and as demanding less power of discrimination and relation are the first to be analytically taught. The separation into parts is not enough ; the mere putting together of the parts is not enough. There must be a counting of the parts making up the whole: one, two, three . . .; and this means addition by ones. We may separate any 12-unit quantity into equal 2-unit parts ; we do not know that there are six parts till we have counted the parts, one, I I - ! J ( J ■ ■ •!l! : :: ^f'( « ' » i. i h ■ 1 ■ .' "-■''111 ' X^*\ 1 •!,'K«; i! ^H!: ;!!i^ ■! , II: i I II liii ( / 180 THE PSYCHOLOGY OF NUMBER. two . . . six — tliat is, a Jlrst one, a second one . . ., six ones. Nor will the most ingenious presentations of sense material free us from this fundamental process. Such presentations help to make the process rational.; they do not supersede it. If a good working idea of four has been got, it carries with it the idea of three. But it does not follow that seven (4 H- 3) is known without includes the presenta- does counting. The presentation tion ; but the mere perception of • not give the number seven. We perceive the four and the three, and we know these numbers because we have previously analyzed each of them, and put its units in ordered relation to one another — that is, counted them. We have a perception of the group which represents the union of these two numbers, but we do not really know seven till we put the three additional units in ordered relation to the four; or, in other words, till (starting with four) we count five, six, seven, thus fix- ing the places of the new units in the sequence of acts by which the whole is measured. We are not to rest satisfied, on the one hand, with mechanical counting — mere naming of numbers — nor, on the other liand, with mere perceptions of units unrelated by counting. This consciously relating process gives the ordinal element in number ; counting, for example, the units in a seven- unit quantity, when we reach four we must recognize that four is the fourth in the sequence of acts by which the whole is constructed. The following points in the development of number seem, therefore, perfectly clear : A A lis ON PRIMARY NUMBER TEACHING. 181 1. The measuring idea should be made prominent by constructive exercises such as have been suggested. 2. There must be rational eountinir — relating — of the units of measurement. This is addition by ones. It is impossible to know, for example, ten times, with- out having added ten ones. 3. While use may — rather should — be made of the ratio idea (division and multiplication), the mastery of the combinations to ten should be kept chiefly in view ; that is, addition and subtraction, with the emphasis on addition, should be first in attention, but with exer- cises in the higher processes. This 7mist be the course if the ideas of unit and number are to be rationally evolved. In counting up, for example, the four 3-inch units in the foot measure, the child first feels and at last sees that one unit is 1 out of four ; two, 2 out of four ; three, 3 out of four ; or 1-fourth, 2-fourths, 3-fourths of the whole. 4. To help in this relating as well as in the discrim- inating process, rhythmic arrangements of the actual units, and of points or other symbols of them, may be used with remarkable effect. The real meaning of five, as denoting related units of measure, w411 clearly and quickly be seen when the units (or their symbols) are arranged in the form and so with other num- bers. The results of the entire mental operation of analysis-synthesis, by which the vague whole has been made definite, are given in these perceptive forms. 5. Hence, during the first year at school we need not confine our instruction to 5, or 50, or 500, if we follow rational methods. The first thing, then, is to m m I i WT 182 THE PSYCHOLOGY OF NUjMBER. M'i {' uiii "« :: I i|' it; .1 ' see that the child gains not a thorough mastery but a good working idea of the number ten.* After exercises in parting and wholing, some notions of unit and number have been gained, and a formal start is made for ten^ the instrument of instruments in the development of number. Many a 12-unit quan- tity has been divided into equal parts. Working ideas of the numbers from one to six have been ob- tained ; the child works with them to make the num- bers one to six more definite. If the preliminary con- structive operations have been w^isely directed the task is now an easy one. Not long-continued and monoto- nous drill on all that can be done upon the number ten is needed, but a little systematic work, with the ideas which from free and spontaneous use are ready to flash, as it were, into conscious recognition. Using the number forms for six, , it will need but few ex- • • • ercises to make perfectly clear the real meaning of six, and then the remaining numbers Y ... 10 are, as means for further progress, within easy reach. In using these number percepts the " picturing " power should be cul- tivated. Five, six, seven as five and two, eight as two * For the constructive exercises referred to, various objects and measured things may be used as counters; but since exact ideas of numbers can arise only from exact measurements, and since ten is the base of our system of notation, the metric system can be used with great advantage. The cubic centimetre (block of wood) may be taken as a primary unit of measure ; a rectangular prism (a deci- metre in length), equal to ten of these units, will be the 10-unit, and ten of these the 100-unit. The units may be of different colours, and the units of the decimetre alternatelv black and white. A foot- rule, with one edge graduated according to the English scale, and the other according to the metric scale, is a most useful help. If ON PRIMARY NUMBER TEACHING. 183 fours, ten as two fives, and five twos, etc., should be instantly recognised. In exercises upon the combina- tions of six we have the whole and all the related parts distinctly imaged. The analysis of the visual forms has been made 5 + 1 = 6, 4 + 2^6, etc. Now cover the 5 ; how many are hidden ? how many seen ? Cover the four ; how many are hidden ? liow many seen ? And so with all the combinations, taking care that the related pairs of number are seen as related — for example, 6 = 5 -f- 1 = 1 -f- 5. This insures a repetition of the number activity in each case, and the ability to recognise, on the instant, any number ; to see not only a whole made lip of parts, but also the definite numher of parts in the whole. Comlnnations of the 10 - Units. — It is hardly neces- sary to say that, with a fair degree of expertness (not a perfect mastery) in handling these twenty -five primary combinations, rapid progress may be made in the use of the higher numbers. As soon as a good idea of ten is gained, the pupil handles the tens, using as the ten- unit the decimetre already described (or other convenient measures), and going on to 10-tens, even more easily and pleasantly than he proceeded to 10 "ones." Is there any known law, save the law of an utterly irra- tional method, that confines the child for six months to the number five, for twelve months to the number ten, for another year to the number twenty, actually exact- ing two whole years of the child's school life — to isay nothing of the kindergarten period — before letting him attempt anything with the mysterious thirty ? If a child has really learned that five and three are eight, does he not know that 5 inches and 3 inches are 8 inches, m s\m m '\m' W mr III r - i'i J' r Sfci i!Hr ■I - I ' ; ! ■ .'i: 1! ■'I H Ihl: Nl^t:n ■ . r 184: THE PSYCHOLOGY OF NUMBER. 5 feet and 3 feet are 8 feet, 5 yards and 3 yards are 8 yards, 5 miles and 3 miles are 8 miles ? Do live and three cease to be eight, or ten repudiate 5 times two when the unit of measure is changed ? As a matter of fact, when the child, under rational instruction, gets a good grip of three he quickly seizes ten, the master key to number. As soon as he comes to ten let him for- mally practise with the tens the combinations he has learned with ones : the ten is a unit, because it is to be repeated a number of times to make up another quan- tity, just as much as " one " is a unit because it is used a number of times to make up a quantity. When it is known that live units and three units are eight units, it is known for all units of measure whatever. There is absolutely no limitation save this, that the child should have a reasonably good working idea of the unit of measure. If he knows, for instance, that 5 feet and 3 feet are 8 feet, he not only knows that 5 miles and 3 miles are 8 miles, but also has a good idea of the dis- tance 8 miles, provided he has a good idea of the unit mile. So, when he has a working idea of a 10-unit quantity as compared with the 1-unit quantity which measures it, he passes with the greatest ease to a good idea of a magnitude measured by ten such 10-unit quantities. In using the mr' 'C units, previously re- ferred to, he has analyzed the c^oimetre prism, has com- pared it with the minor unit — the cubic centimetre — has found that it takes ten of these minor units, or one of 1 taken ten times, to equal it, etc. He goes through J..Ki same constructive process with ten of these 10-unit measures ; he puts ten of them together to make a square centimetre ; he analyzes and compares ; he uses ON PRIMARY NUMBER TEACHING. 185 li^ this new unit measure just as he used the minor-unit measure, the single cube ; he finds that 5 units and 3 units are 8 units, 6 units and 2 units are 8 units, . . . 7 units and 3 units are 10 units, 8 units and 2 units are 10 units ; and all the rest of it. He finds also, just as certainly as lie found in operating with the 10 cubes, that to make up the whole he is measuring, one 10-unit has to be repeated ten times, 2 ten-units five times, and 5 ten-units two times. In fine, he knows as much about the whole, the ten 10-units, the 100 minor units, as he knows about the 1 decimetre, the 10 minor units ; knows it just as well, is just as clear as to the re- lations of the several parts ; for he has analyzed the undefined — the unknown — down to its known constit- uf" i, and built it up again l)y known relating pro- poses. He does not perceive it just so well, does not image it as a quantity of a hundred parts ; but, never- theless, he has a fairly definite conception of the quan- tity as a measared whole. Naming the Numhers. — in counting the tens the names of the new numbers may be given ; or, rather, a name or two may be given, and the child will discover the others for himself : 1 ten, 2 tens, 3 tens, etc. ; for two tens we have the somewhat special name twen-^y (twain-ten), and for three tens ihiv-ty i what name for four tens ? iovjive tens ? Let the pupils experience as often as possible the joy of discovering something for themselves. While the work with tens is going on, practice may be had in the analysis of two tens, so as to lead to count- ing and naming the numbers from ten to twenty, twenty to thirty, etc. Here, again, the children will do some- il ■a] i w ';.« i 1 > ' V,:. |.|; 194 THE PSYCHOLOGY OF NUMBER. iinil — i. e., the unit of reference. Thus the number ex- pressed by the ones previously given will be expressed by lll'lll. The iirst figure to the left of the unit-figure Avhose position is thus marked, for example, in 453»453 metres, denotes tens/ the first figure to the right, tenths I the second figure to the left, himdreds j the second figure to the right, hundredths ^ the third figure to the left, thousands I the third figure to the right, thousandths, etc. It will be readily observed, too, that the figures to the right of the decimal point are read in groujis of tliree, just as those to the left are. As denoting a mim- her, 453 is always four hundred and fifty-three. In this example it is on the left side of the decimal point, 453 inetres (primary units) ; on the other it is 453 milli- tnetres, etc. Here, as everywhere, there must be a good deal of drill, in order that the pupil may acquire perfect facility in reading and writing numbers ; this means, ability to read automatically any number and its unit of measure, and similarly to express any quantity that may be named. For example : naming each period ac- cording to its unit of measure, name the first period (group of three figures) to the left — the units period ; the second — the thousands (thousand-unit) period ; the third to the left — the millions period ; the first period to the right — the thousandths period ; the second to the right — the millionths period. Make the figure 7 ex- press billionths, hundred thousands, tens, tenths, bill- ionths ; make 45 express tens, hundreds, thousandths, millionths, etc. Care is to be taken to name correctly the measuring units in the periods to the right — for example, '00573 is five hundred and seventy-three hun- NOTATION, ADDITION, SUBTRACTION. 195 dred-thousandths I •0006734 is six thousand seven hun- dred and thirty-four ten-millionths. ADDITION AND SUBTRACTION. Addition. — In addition, as we have seen, we work from and within a vague whole of quantity for the pur- pose of making it definite. If a quantity is measured by the parts — 2 feet, 3 feet, 4 feet, 5 feet — we do not arrive at the definite measurement by simply counting the nuirJjer of the parts ; we have to count the number of the common unit of measure in all the parts, and so find the whole quantity as so many times this common unit. In learning addition, the countings are associated with intuitions of groups of measuring units, and the results stored up for practical use. The pupil who has been properly trained does not, in the foregoing ex- ample, start with 2, count in the 3 by ones, then the 4 by ones, etc., though this counting was part of the ini- tial stage even when aided by the best arrangements of objects, by which he at last perceives that 5 + 4 = 9, without now counting by ones. Addition may there- fore be considered as the operation of finding the quan- tity which, as a whole, is made up of two or more given quantities as its parts. The parts are the addends (quan- tities to be added), and the result which explicitly de- fines the quantity is the smn. It follows that in every addition, integral or fractional, all the addends and the sum must be quantities of the same kind — i. e., each and all must have the same measuring unit. Kot only is it impossible to add 5 feet to 4 minutes ; it is impossible to add 5 feet to 4 rods — i. e., to express the whole quantity by a mimher (denoting so many units of measurement) — \G'f^'' li « ! • / ■ ''•■,' :,:-! m r-. -Hi. i ■!"(('■ « »t V 7 1 i j! : •'■ r ■J* ii Si rlir' 1^ ;• ;j*" ij ! 'hi I' i 196 THE PSYCHOLOGY OF NUMBER. 'I' I without first expressing the addends in the same unit of measurement. Thorough mastery of the addition tables must be acquired, and rapidity and accuracy in both mental and written work. Exercises on combinations, not with the single units only, but with the 10-units, the 100-units, the 1000-units — any units of which the pupil has ac- quired a fair working idea — will greatly aid (are a ne- cessity) in attaining this knowledge of sums and diifer- ences as well as skill in its application. Practical facility in handling numbers mtist be acquired, at first with partial meaning, afterwards with full meaning of the operations. Some additional points may be noticed : 1. Use must be made of the knowledge of the tens as acquired in the way referred to ; for example, if 8 + 8 = 16, then 18 -f 8 = 26, 28 -f- 8 = 36, etc., should follow instantly as a logical consequence. The pupil may at first " make np " to the next ten by separating, for example, 8 into 2 + 6, giving, that is, 18 + 2 -F 6 = 20 + 6 = 26, just as at first lie may get the sum 8 -f 8 through the steps 8 + 2-h6 = 10-|-6 = 16.. But in all cases the intermediate step should be dispensed with as soon as possible, and the perception of the addends — for example, 28 + 8 — should instantly suggest the sum 36, no matter what the kind or magnitude of the unit that may be used. 2. In this connection it may be noticed that expert- ness in two-column addition, summing such numbers as 75, 68, no matter, again, what the unit may be, can be easily acquired — both acquired and used with the greatest interest. There is hardly a more interesting exercise in that "mental" practice which is essential u NOTATION, ADDITION, SUBTRACTION. 197 f^mri the heginnmg to the end of the entire course in arithmetic, if knowledge, power, and skill are to be really and tlioronghly gained. Tims, in finding the sum of 78 and 89, the mental movement would he the sum of the tens, the sum of the units, the tens and the units in the latter, the total in tens and units. Very soon the two "sums" are obtained simultaneously, and, with a little practice, the total (15 tens, 1 ten, 7 units) is named on the instant. With a degree of facility in adding by single columns, it is not far to equal facility in adding by double columns. 3. There should be also plenty of mental practice in addition (and subtraction) by equal increments. Count by 2's from 2 to 24 ; by Vs from 1 to 31 ; by 3's from 3 to 36, from 1 to 37, etc. 4. It affords excellent practice in written work to set down separately the sum of each colwrnn, the right- hand figure of each column- sum being placed under the column from which it is de- rived, and the other figures in their order diagonally down- ward to the left. These par- tial sums are then added to- gether to obtain the total ; thus : In this example the sum of the first column is 42 ; the 2 is placed under the first col- umn, and the 4 under the second column in the line 42 -|- 51 -f 45 + 57 = 195 1 ill $9874 28 8768 29 3425 14 8267 23 1 2482 16 9341 17 ■{ •, 2345 14 8273 20 2834 17 6443 17 1 7512 45 1 5454 15 195 1 $62052 tv 1 1 AV-. 1 K.^ r 1 Af it rn ill ii^ * 1/ J' I i: '^r' 198 THE PSYCHOLOGY OF NUMBER. below tliat of the 2. The sum of the second cohimn is 61 ; the 1 is placed under the second column on the left of the 2, and the 5 is placed on the left of the 4. The sum of the third column is 45 ; the 5 is placed under the third column just to the left of the 1, and the 4 diag- onally below to the left of 5. The sum of the fourth cohinm is 57 ; the 7 is placed under the fourth column from which it was obtained, and the 5 next to the 4 in the line below. These partial sums are now added to get the total, $62052. Some advantages of this method may be noted : (1) It helps the pupil to a clearer idea of the carry- ing process. (2) In case of a mistake in the additions, it enables the pupil to detect the error more easily. What pupil has not felt the drudgery of having to go over the whole work in order to find where an error had crept in ? By this arrangement the addition of any column can be tested independently of the addition of the pre- ceding column, no knowledge of the " carried " num- ber being required. If it is known, for example, that an error has occurred in the addition of the thou- sands, the error can be discovered and corrected with- out adding the hundreds to ascertain the number carried. (3) Because the columns are added independently the I'esult may be tested by adding the digits in each row, then adding these sums and comparing the total with the total obtained from adding the colunm-sums treated as separate numbers. These two totals ought to be equal. In the example, the sum of the digits in the first row is 28, in the second 29, etc., and the total 1 J- 1 \ NOTATION, ADDITION, SUBTRACTION. 199 of these sums is 195, wliicli is the same as the total of the cohimn-siims, 42, 51, 45, 57. (4) This method is especially useful in additions of tabulated numbers which are to be added both verti- cally and horizontally. 5. Another excellent practice, for the more ad- vanced student, is in the addition of two numbers, be- ginning on the left. AVhen the common plan of adding two numbers by beginning with the right-hand digits is becoming monotonous, the new method may be prac- tised with an awakened interest because of its novelty, and at the same time a broader view of the arithmetical operation is obtained. The only point to be attended to is whether the sum of any pair of digits we are work- ing with has to be increased by a one from some lower rank. In adding a pair of digits of any order, the stu- dent at the same time glances at the lower orders to see if a one is coming up from below to be added. In 628 adding ^rc\y while adding 6 and 3, we see at a glance that their sum is not to be increased, and write down 9 at once ; in adding the next pair, 5 and 2, we instantly see that their sum is to be increased by 07)C from the sum of the next pair (9 + 8), and we instantly write down 8. The student should practise this method till he can use it with ease. He may exercise himself to any extent by writing down two numbers and finding their sum, then adding this sum to the last of the two num- bers, then this sum to the preceding sum, etc. As ad- dition is a further development of the fundamental process of counting, and is itself "the master light of all our seeing " in numerical operations, perfect facility m III im I- !| 200 THE PSYCHOLOGY OF NUMBER. ^•1-^' m \ y should be acquired, though not, as before said, by ex- cluding all other ideas and operations till this perfec- tion is attained. Get complete possession of addition, with full knowledge of numbers^ if possible; without it, if necessary. SMraction. — Addition and subtraction are inverse operations. The one implies the other, and in primary operations the two should go together, with the em- phasis on addition. Subtraction in actual operations with objects would seem logically to precede addition. If we wish to get a definite idea of a 14-unit quantity, and separate it into two known parts of 8 units and 6 units each, it seems that logically the 6 unit-quantity is taken away from the whole, and both the minor quan- tities are recognised as parts of the whole before the final process of constructing the whole from the parts is completed. There is no need, therefore, of making a complete separation between these two operations. On the contrary, they should be taught as correlative operations, wnth addition slightly prominent first for reasons already set forth. From what has been shown as to the logical and psychological relation between addition and subtrac- tion, it appears that subtraction is the operation of finding the part of a given (juantity which remains after a given part of the quantity has been taken away. As in addition, so in subtraction, all the quantities with which we are working — minuend, subtrahend, remain- der — must have the same unit of measurement. Further, as in addition we are working from and within a vague whole by means of its given parts, so in subtraction we are working from a defined whole, through a defined m NOTATION, ADDITION, SUBTRACTION. 201 it remain- part, in order to make the vaguely conceived der " perfectly definite. Reinainder or Difference. — From the nature of sub- traction as related to addition, there seems to be no strong reason for the "important distinction" that should be noted between "taking" one number out of another and finding the difference between two num- bers. We can not take away a given portion of a given quantity (to find the remainder) without conceiving this given portion as part of the whole ; we can not get a definite idea of the "difi^erence" between two measured quantities without conceiving the less as a part of the greater. If $5 is given as a part that has been taken from $9, w^e primarily count from 5 to 9 to find the remainder. If $5 and $9 are given as two quantities, we have to count from 5 to 9 to determine the differ- ence. We have to conceive the $5 as a part of the $9. If the preliminary work of parting and wholing to develop good ideas of number and numerical processes has been rationally done, there will be but little difli- culty in the actual operation in formal subtraction. The following points with respect to the long-time mystic operations of "borrowing and carrying" may be no- ticed : 1. The operation involved in, e. g., 75 — 38, may and should be made perfectly clear by counters. The ten-unit in its relation to the unit has been made clear through many constructive acts. The mental process here, then, is indicated simply by 60 -f (10 -f 5) — 30 - 8 = 30 -f 2 -H 5 := 37. ?SM ^r i 202 THE PSYCHOLOGY OF NUMBER. If the pupil has acquired facility in the addition com- binations, the operation of adding 10 and 5 and taking 8 from the sum (getting 7) is probably as easy — may become as easily automatic — as taking 8 from the 10 and adding 5 to the difference (getting 7). But the meaning and identity of both processes can be made perfectly clear. The pupil may lind it at lirst a little easier to take 8 from the " borrowed " 10 and add 5 to tlie remainder (2), than to add 5 to the borrowed 10 and take 8 from the sum 15. But, in any case, these analytic acts are to lead to the clear comprehension of the process, and especially to its automatic use. There should be, of course, large practice in finding the differ- ences of pairs of tens, as well as in finding their sums. 2. The second method of explaining the " borrow- ing and carrying " in subtraction — that of adding equal quantities to minuend and subtrahend — may be made equally clear. That the difference between two quan- tities remains the same when each has received equal in- crements, the pupil will discover for himself by "doing" such operations. In 75 — 38 we add one ten-unit — i. e., ten ones — to the 5 ones, and subtract 8, as in the first case considered ; i. e., 15 — 8, or 10 — 8 -f 5 ; we then increase by 1-ten the 3 tens in the subtrahend, getting 4 tens, which we take from the 7 tens. This process is not a direct solution of the problem, but it is one that can be made quite intelligible. There appears to be but little difference in psychological complexity between the two methods. In both methods 8 is to be taken from 15 — i. e., we have 10 -f 5 — 8. In the method of borrowing from the tens, we have to bear in mind, when we come to the subtraction of the tens, that the actual number NOTATION, ADDITION, SUBTRACTION. 203 ,5> of tens to be dealt with is one less tlian the number of written tens. In the case of equal additions, we have to bear in mind that the actual number of tens to be dealt with is one more than the number of written tens. 3. Probably the best way to treat subtraction is the method based on the fact that the sum of the remainder and subtrahend is equal to the minuend. If we wish, for example, to find the difference between $15 and $8, we make up the 8 to 15, i. e., count from 8 up to 15, noting the new count of 7, which is the " difference " between 8 and 15. To find the difi:'erence between 45 and 38 is to find what number added to 38 will 45 make 45 : 38. The 8 units of the subtrahend can not be made up to the 5 units of the minuend ; we make it up, therefore, to 15 by adding 7 units, and put down 7 as a supposed part of the remainder. As this addition of 7 to 8 makes 15, we have 1 ten to carry to the 3 tens of the subtrahend, making it 4 tens, which requires no tens to make it up to the 4 tens of the minuend ; the remain- der is therefore 7. Proceed similarly with 75 — 38, etc. Take an example with larger numbers. From 873478 take 564693— that is, find what number added to the latter will give a result equal to the for- ^^^ mer. Write the subtrahend under the minu- — '— — end, as in the margin, so that the figures of o/io^ok the same decimal order shall be in the same column. To 3, the right-hand figure of the subtrahend, 5 must be added to make up 8, the right-hand figure of the minuend ; this is the right-hand figure of the remainder. AVe add 8 to 9, making the 9 up to 17 m fk 1. w n b II It 1 1"( ll f ,w I . I '. 1 ^I <, 1'' H ' < I M ■*;. 204 THE PSYCnOLOGY OF NUMBER. (ten-nnit), and putting down 8 as the second figure of tlie remainder. We carry tlie 1 (hundred) from the 17 (ten) to the 6 hundred in the subtrahend, making it 7 (hundred) ; this 7 (hundred) is made up to 14 (hun- dred) by adding 7 (hundred), which is set down in the tliird place of the remainder ; carrying 1 from the 14 to the 4 (thousand) in the subtrahend, we have 5 (thou- sand), which is made up to 13 (thousand) in the minu- end by adding 8 (thousand), and 8 is set down in the thousands' place in the remainder. Similarly, carry- ing 1 from the made-up 13 to the next figure, 6, of the minuend, we have 7, which requires 7iothing to make it up to 7, and a zero is therefore set down in the 10- thou- sands' place in the remainder; finally, 5 requires 3 to make it up to 8, and so 3 is set down as the last figure of the remainder. Using italics to denote the numbers to be set down in figures as the remainder^ the state- ment of the mental process will be : 3 and Jlve^ eight ; 9 and eighty seventeen ; 7 and seven, fourteen ; 5 and eight, thirteen ; 7 and naught, seven ; 5 and three, eight. After some practice the minuend-sums need not be pronounced, and we shall have simply 3 and Jwe, 9 and eight, etc. This method is usually adopted in making change, and may be used with great facility in making calculations involving both additions and sub- ^^^^^^ tractions. Thus, suppose a merchant, having 2714 $19128 in bank, cheques out the sums $2714, ^^^^ $996, $3952, $166, $7516, how much has he remaining in bank ? The several subtrahends are arranged in columns under the minuend, just as in addition. Add the subtrahends and 166 7516 $3784 NOTATION, ADDITION, SUBTRACTION. 205 make up to the minuend in the way described, setting down the making-up number. The process is — Ist column: 12, 14, 20, 2-1 and four, 28 — carry 2; 2d " 3, 9, 14, 28, 24 and eifj/it, 32— carry 3 ; 3d " 8, 9, 18, 27, 34 and seve^i, 41— carry 4 ; 4th " 11, U, 16 and three, Id ; this makes up the 19 (thousand) of the minuend, and the whole " making - up " number, or remainder, is $3784, the amount of money the merchant has left in bank. The principle of " carrying " is exactly that of addition. We are making up, by successive partial addends, a smaller number to a greater. When we have come to 24 (tens) — for instance, in the second column in the example — we add 8 (tens) to make it up to 32 (tens), and so have 1 ten more — i. e., three in all — to carry to the next " making-up" column. There seems to be no good ground for the assertion sometimes made that this method is illogical, and wastes a year or more of the pupil's time. The first statement is refuted by the psychology of number ; the second, by actual experience in the schoolroom. If to think from 15 down to 7 is logical, it would be no easy task to show that to think from 8 up to 15 is illogical. We can nei- ther think down in the one case nor up in the other without thinking of a measured whole of 15 units as made up of two parts, one of 7 units, the other of 8 units. As a conscious process, 8 + 7 = 15 carries with it the inevitable correlates 15 — 8 = 7, 15 — 7= 8. From what has been shown as to the relations of the fundamental operations, it might even be inferred that if there is any difference in difiiculty between the making-up method and the taking-away method, the m .* f ; ( J" • t .< :t'' ' 1 i -ftr. • ' ,1 ';; j .''' ■ If ;'«■ ( 1,1 206 THE PSYCHOLOGY OP NUMBER. difference is in favour of the making-up method, as in- volving less demand upon conscious attention. How- ever this may be, it is certainly known from actual knowledge of school practice that pupils who have been instructed under psychological methods have had but little diiticulty in comprehending the making-up method, and have quickly ac(juired skill in the application of it. Fmiilainental Principles of Addition and Sub- traction. — When a quantity is expressed by means of several terms connected by the signs -f a,nd — , the ex- pression is called an aggregate ', and when the several operations are performed the result is the total or sum of the aggregate. Some of the fundamental principles connecting the operations of addition and subtraction are : (1) If equals be added to equals, the wholes are equal. (2) If equals be subtracted from equals, the remain- ders are equal. (3) Adding or subtracting zero from any quantity leaves the quantity unchanged. (4) Changing the order of performing the additions and subtractions in any aggregate does not change the total or sum of the aggregate. The pupil can use these principles, and abstract rec- ognition of them will come in good time. I! •! CHAPTER XI. t|'1 MULTIPLICATION AND DIVISION. Multiplication. — From the pieceding discussion (see especially page 109 et seq.) of multiplication as a stage in the development of number, it is clear that certain points are to be kept steadily in view, if the process is to be made really intelligible to the pupil. 1. It is not simply addition of a special kind. It means development and conscious use of the idea of number — that is, of the ratio of the measured quantity to the unit of measure, whatever the magnitude of the unit may be in terms of minor units. In counting with a 1-unit measure, one, two, three, . . . nine, the number is known when the unit it names is recognised as the ninth in a series of nine units constituting a whole — when, that is, the defined quantity is grasped as nine times the unit of measure. 2. In the development of the measuring process (as in the exact stage of measurement) there is the explicit recognition that the measuring unit is itself measured olf into a definite number of minor units. This gives rise to the process of multiplication, and of course to a more definite and adequate idea of numher as denoting times of repetition of the unit to make up or equal the magnitude. Nine times one is nine is understood in it& full significance. 15 207 ii;^: M t 'I w 208 THE PSYCHOLOGY OF NUMBER. i ('Si i \iVir «> n^ ■Ml ■■! li; : ''I 3. A quantity expressed in terms of a given unit of measure is, by multiplication, expressed in terms of the minor units in the given unit of measure ; in other words, for the number of derived units in the quantity is substituted the mmiher of primary units in the quan- tity. If we buy 7 barrels of flour at $5 a barrel, the measured cost is $5x7; seven units of $5 each. By multiplication this is changed to $35 — i. e., $1 x 35. This product, as it is called, this new measurement, is not seve?i Jives. It denotes the same quantity under a difierent though related measurement ; it is thirty-five ones. In one of these measurements the nicmhe?' is seven, in the other it is thirty -five. 4. The multiplicand must always be regarded as a unit of measure — a measure made up of primary units ; and the operation looked upon as simply making the quantity more definite by expressing it in a better known or more convenient unit of measure. 5. While the multiplicand as multiplicand must al- ways be interpreted to mean measured quantity, we can take either factor as multiplier or multiplicand. This idea must be used from the first, even in the primary stage. In finding the number of primary units (dollars) in 12 yards of velvet at $5 a yard, there is no known law that decrees 12 as unchangeably the multiplier, and $5 as the only multiplicand. On the contrary, by a necessary law of mind, every measuring process has two phases, and so the measurement $5 x 12 carries with it the measurement $12 x 5. Only a total mis- conception of number and the measuring process could prompt the question. How can 12 yards become $12 ? The proposition $5 x 12 = $12 x 5, is not a proposition ii yi MULTIPLICATION AND DIVISION. 209 IS ion about things j it is a proposition concerning a psychical process — the mind's mode of defining and interpreting a certain quantity. This principle of measurement — of interchange of times and parts — is essential to the proper understanding of numerical operations, and can from the beginning be intelligently used. Intelligent use leads to perfect mastery. The problem of multi- plication then is : Given the numher of unit-groups in a measured quantity, and the numher of minor units in each unit-group, to determine, from these related fac- tors, the number of minor units in the quantity. The Formal Process of Multiplication. — It may be well to consider the logical steps in learning the process :- (1) The multiplication of a quantity by powers of ten. Beginning with some ultimate or primary unit of measure, we conceive a measured quantity as mak- ing up ten such units — that is, we multiply the unit by ten ; we may further conceive this 10 unit quantity used as a unit of measure, and repeated ten times to make up a larger quantity — that is, the 10-unit quantity is multiplied by ten to express this larger quantity iri terms of the minor unit, it is 100 of them, etc. It has already been shown how the notation corresponds with this process. The 1-unit multiplied by 10 becomes 10, the 10-unit multiplied by 10 becomes 100 ; in oilier words, the 1 increases 10 times with every removal to the left of the decimal point. So the product of 5 ones is 10 fives or 5 tens — i. e., 50 ; the product of 5 tens by 10 is 50 tens or 500 — i. e., 6 multiplied by 100, etc. (2) We may find the total product which measures •,j i ' ' $ '■Hi: 210 THE PSYCHOLOGY OF NUMBER. Ivi ::i:i Is -U ■ * ! I: 'ii' ■1 '■ .;)'^ r."'! m a quantity by finding the sum of partial products. If a given quantity is measured by 4 feet x 28, we may multiply the 4 feet by 20 and by 8, and the sum of the partial products will be the total product — in all 28 times the multiplicand. This is the basis of the work in long multiplication. (3) We may multiply by the factors of the multi- plier. This is using the relation between parts and times. If we have, e. g., a quantity expressed by $2 X 20, it is expressed equally by $2 x 5 x 4 — i. e., by $10 X 4 ; in other words, we have made the meas- uring unit 5 times as large, and the number of them 5 times as small. In the following, e. g., we take the multiplicand 5 times, getting the first partial product ; in multiplying by 4, we have in fact taken the multiplicand 10 times, and this product 4 times, obtaining the second partial product 11,120. 278 45 1390 1112 12510 . . 5 times multiplicand. . . 40 (i. e., 4 times 10 times multiplicand). = 45 times multiplicand. Special Processes. — Special processes may be used in many cases. These afford good practice for mental work, and give better ideas of number and numerical operations, as well as preparation for subsequent work. A few of these processes may be noticed. 1. When the multiplier is any of the numbers 11 to 19, the product can be obtained in one line, thus : MULTIPLICATION AND DIVISION. 211 8765 19 166535 (( (( li Nine 5's, 45 — carry 4 ; 6's, 58 and Jive, 63 — carry 6; T's, 69 and six, 75 — carry 7 ; 8's, 79 and sevefi, 86 — carry 8 ; 8 and eight, 16. The number in italics is in each case the number in the multiplicand just to the right of the one multiplied. To multiply by 31, 41, 91, it is best to write the multi- plier over the multiplicand, and use the multiplicand itself as the partial product from the digit 1 in the multiplier. For example : the multiplier. product by 1. product by 8 (tens). product. 81 96478567 771829536 7814773927 . . The product can be obtained in one line, as in multiply- ing by 19, but there is greater risk of error in the mental working. Such examples as 84 X 76 afford interesting and useful mental practice. Multiplying crosswise and summing the products, 76 tens ; multiplying the units, 2 tens 4 units ; multiplying the two tens, 56 hundreds ; hence 63 hundreds, 8 tens, and 4 units — i. e., 6384. 2. Practice in finding the squares of numbers is very useful. The rule for finding the square of the sum of two numbers and the difference between the squares of two numbers may be readily arrived at. For example, multi- ply 85 by 85 : 80-1-5 80+5 80' -j- 5 X 80 + 5X80-f5' 80» + 10 X 80 + 5' = 7225 I wW^ 212 THE PSYCHOLOGY OF NUMBER. ^<» * [.'I' '. . : i jt This may be illustrated by intuitions, symbolising units by dots. Let the following indicate the square of 7 (5 ~|- 2). We see at once that to make up the whole square there is (i) the square of 5, {it) 5 taken twice, and (iii) the square of 2 — that is, the square of the first number, twice the product of the first by the sec- ond, and the square of the second number. It will be readily seen that the difference of the two squares {V — 5') is twelve times two j but twelve is the sum of the numbers and two their difference. Does this hold for other num- bers? The pupil will be greatly interested in discov- ering for himself the general principle : the difference of the squares of two numbers is equal to the sum of the numbers multiplied by their difference. If, in the figure, he compares the square of 3 with the square of 4, of 4 with that of 5, he will see that the square of any of these numbers is got from the square of the next lower by simply adding the sum, of the numbers to the square of the lower. The square of 3 is 9 ; the square of 4 is 9 (= 3") -f (3 -f 4) ; the square of 5 is 16 + (4 + 5) ; and the square of 7 is 36 + (^ + ^)- The pupil will de- duce for himself that, given the square of any number, the square of the next consecutive number is obtained by adding the sum of the numbers to the given square. All these principles, and many others, may be made the basis of exercises equally interesting and useful in mental arithmetic : Square of 95 ; multiply 95 by 105, (100 - 5) (100 -f 5) ; 295 by 305, (300 + 5) (300 - 5) ; the square of 250 ; the square of 251, etc. MULTIPLICATION AND DIVISION. 213 3. The making-iip metliod in subtraction may be conveniently used when the product of one number by another has to be taken from a given quantity. From 89713 take 8 times 8793. Tlie work is done as follows : '^lii 89713 8793 8 Eight 3's, 24 and nine = 33 — carry 3 ; 9's, 75 and six = 81 — carry 8 ; 7's, 64: and three = 07 — carry 6 8's, 70 and nineteen =^ 89. a a 19369 The numbers in italics indicate the remainder, 19369. 4. Advantage may often be taken of the fact that Bome of the numbers (tens, etc.) of the multiplier — and, once more, either factor may be made the multiplier — represent a multiple of some of the others. If, for in- stance, we want to find the cost of 2053 bags of flour, at $3,287 a bag, w^e may use the latter for multiplier, and write only three partial products : 2053 3287 14371 57484 615 9 ^ 6 7 4 8.2 1 1 7 times. 2 8 times. 3 times. In tliis example we multiply by 7, and, observing that 28 is 4 times 7, we multiply the first line of the product by 4, getting the second line ; then the multi- plicand by 3, taking care, of course, to put the product in the thousands' place. We may often take advantage of this method by breaking the order of finding the partial products. 1 I ii . J I' 1 ■li » >J! ,!;■ »i Iv.'- ]^^' ^ HH- ■i!! M.* r^ ^% 214 THE PSYCHOLOGY OF NUMBER. Thus, if the product of 567392 by 218126 is required, we may use the former as multiplier, and work thus : 218126 567392 15 2 6 8 8 2 215056 85505 3 92 7 times. 5 6 0,0 times. 3 9 2 times. 1 2 3,7 6 2,9 4 7,3 9 2 "We nutict- iiiat ^Q is 8 times 7, and that 392 is 7 times 56. Le;^;i i, therefore, with 7. Multiplying this pro^n) the second line of partial prod- ucts; and, jinFu^y, \n '-Jplying this second line by 7, we get the third line of partirii products. Or we might have used 218126 for multiplier, ob- serving that 9 times 2 are 18, and 7 times 18 is 126 ; thus; 567392 21812 6 1134784 10213056 71 491392 1 2 3,7 6 2,9 4 7,3 9 2 Multipiying first by 2 (hundred thousand) ; multiply- ing this product by 9 ; multiplying this second partial product by 7, taking care as to the proper placing of the products, we have the complete product. 5. Another plan that affords a good exercise in mental additions, and subsequently proves useful, is the method of finding a product of two factors in a single line. To multiply, e. g., 487 by 563, write the MULTIPLICATION AND DIVISION. 215 multiplier, with the digits in inverted order, on the lower edge of a slip of paper, thus, 3 6 5 1 . Place the paper over the multiplicand so that the units (3) shall be just over the units of the multiplicand. The artifice con- sists in moving the slip of paper along the multiplicand, figure by figure, till the last digit (5) of the inverted multiplier is over the last digit of the multiplicand, and taking the product of any pair, or the sum of products of any pairs, of numbers that may be in column. Thus : Three T's, 21 — one and carry 2 ; Three 8's, 26 ; six T's, ^^— eight and carry 6 ; Three 4's, 18 ; six 8's, seven 5's ; 101 — one and carry 10 ; Six 4's, 34 ; five 8's ; U—four and carry 7 ; Five 4's and 7 carried — twenty-seven. The numbers in italics, taken in order, are the product, 274181. 3 65 487 365 487 365 487 365 487 3 65 487 Proofs of Multiplication.— {V) By repeating the operation with the factors interchanged. (2) The prod- uct divided by either factor should give the other factor. (3) By casting the nines out of the multiplier and the multiplicand, then multiplying these remainders together and casting the nines out of their product ; the remainder thus obtained should equal the remainder from casting the nines out of the product of multiplier and multi- plicand. For example, test the following by casting out the nines ; "'■■n iKk'h 216 THE PSYCHOLOGY OF NUMBER. ri, • ■ i 987761 X 56789 = 56,093,959,429. 7 . . out of product of 2 and 8. Out of multiplicand . . 2 X ^ • • out of multiplier. 7 . . out of product. This proof is not a perfectly sure test of accuracy. It does not point out an error of 9, or of a multiple of 9, in the product. Thus, if has been written for 9 or 9 for 0, or if a partial product has been set down in the wrong place, or if one or more noughts have been inserted or omitted in any of the products, or if two figures have been inter- changed, or if one figure set down is as much too great as another is too small, casting out the nines will fail to detect the error, for the remainder from dividing by 9 will not be affected. Still the proof is interesting, as throwing light upon the decimal system of notation. The Ifultiplication Table. — The sure groundwork for this table is, of course, facile mastery of the addition and subtraction tables. Though scraps of it given from time to time — as the 2's and 3's in 6 — are perhaps of no great value as contributing to the making and mas- tering of the entire table, yet some complete parts of the table — as, for example, two times, three times, ten times^ — may be kept in view, and may be expertly han- dled quite early in the course. It has been said that the table is a grand effort of the special memory for sym- bols and their combinations, and that the labour can not be extenuated in any way. The labour is, indeed, heavy enough, but it is believed that it may be somewhat light- ened. The table, as the key to arithmetic, must be learned, and it must be learned perfectly — i. e., so that any pair of factors instantly suggests the product ; there must be no halting memory summoning attention and .^1 MULTIPLICATION AND DIVISION. 217 judgment to its aid. It is tlieiefore worth while to ''extenuate" the lahour of learning it, if this can pos- sibly be done. To this end some suggestions are made which are believed to be rational, while they have cer- tainly stood the test of experience. 1. The Meaning of the Tahle. — Pnpils rightly taught know how to construct the table ; they know what it means. The symbol memory, like every other kind of memory, is always aided where intelligence is at work. In former times, not so long past, the table used to be said or sung — rattled off in some familiar tune — with- out a glimmer of what it all meant ; but under rational instruction the children know several important things about it, and the teacher should use these things in less- ening the labour of complete mastery. 2. Memory aided hy Intelligence. — (1) The pupils have learned how to construct any part of the table, two times, three times, etc. (2) They know exactly what such construction means, for they have acquired a fair idea of times — of number as denoting repetition of a measuring unit. They know, therefore, the meaning of every product : 6 oranges at 5 cents apiece, 6 yards of calico at 9 cents a yard, etc. (3) They can derive the product of any pair of fac- tors from the product of the immediately preceding pair. Knowing that 6 yards of cloth at 8 cents a yard cost 48 cents, they know that 7 yards cost 8 cents more. Similarly they quickly learn that if 10 oranges cost 50 cents, 9 oranges will cost five cents less, and 8 oranges one ten less, etc. Thus they will have various ways of constructing, and recovering when momentarily forgot- ten, the product of any pair of digits. n V'"-' i' 1 I 1 ^ v1 hi ■ 4 u * 1,1 1 m ■'r- ■■A\ ■ ::'::i'j J* Si 1 M '■i'lr' f, 1»' ! m i ' !■ 218 THE PSYCHOLOGY OF NUMBER. 3. The Commvtatwn of Factors. — In learning tlie table the relation of the factors must be kept in view. This greatly reduces the labour. There ought to be little difficulty in this if a fair idea of the relation be- tween parts and times has been brought out. At 3 cents apiece, 5 oranges cost 3 cents X 5 ; this is seen to be iden- tical with 5 cents X 3. Each of these expresses meas- ured quantity, a sum of money ; the thought " oranges " disappears from this conception. The table is often taught without reference to this principle, and so the labour of learning it is at least one half greater than it ought to be. In our boyhood we learned 9 X 6 = 54, without a suspicion that 6 X = 54. Let us see to it that the present things be made better than the former. 4. Memory fiirther aided. — Associations. In this connection the following suggestions are worthy of at- tention : (1) The thing to be kept in view is that, so far as possi- ble, associations are to he formed directly Jjetween a jpro- duct and one or hoth of the factors ivhich produce it. (2) Ten times is already learned in addition — in the counting of the tens. The pupil know^s how to "mul- tiply" any number by ten by simply affixing a zero to the number. The association of product and factors is direct ; the product is the multiplicand with the zero of the 10 affixed. Ten times, then, is well in hand. (3) Eleven times is almost equally easy. The prod- uct in each case, for the first nine digits, is directly associated with the digit j the digit is simply repeated — 11, 22, 33, etc. Eleven tens is known from ten elevens, and the other two products (11 X 11, 12 X 11) must be built up from this. MULTIPLICATION AND DIVISION. 219 ens, it be (4) Nine times can be formed and remembered in a similar way. The pupil will note : («) That a product is made up of tens and units. {])) That in 9 times (up to 10 X 9) the number of tens is always one less than the number multiplied, [c] That in every product the sum of the digits is 9 ; and thus, having written down the tens directly from the multiplicand, he can at once write the units. He should be led to notice also that (i) holds good as to the law of tens up to 10 X 9, after which the number of tens is two less than the multi- plicand up to and including 20 X 9, after which the number of tens is three less, etc. He should note, too, in his formed table, how the t^ns increase by one and the units decrease by one. This may seem somewhat complex, but it works well. We have known a boy of six years to construct and learn 9 times up to 9 times 10 in fifteen minutes. (5) Probably two times has been completely learned before a formal attack is made upon the table as a whole. There has been much practice in counting by 2's — backward and forw^ard — and by 3's, etc. There seems to be no way of making a mnemonic association between a product and its factors ; but addition by two is an easy operation, and two times is quickly learned. (6) In twelve times (assuming two times) the memory can be aided by association. The product of any multi- plicand may be obtained by taking it, the multiplicand, as so many tens^ and doubling it for the units ; tw^elve times 3 = three tens and six (twice 3) units. For 5 and up to 9, doubling the unit gives more than 10, but the additions are easy. 12 times 5 =^five tens and ten units (twice five) - 60. Or, consider the products I' *l Ml ^p^ Wi 220 THE PSYCHOLOGY OF NUMBER. «i \ '■hi \W m* *A tliiis : tlio products of 1 — 4 are 12, 24, 36, 48 ; those of 5 — 9 are each 07ie ten more tliari the multiplicand, and the units increase by 2's — i. e., 0, 2, 4, 6, 8 ; the products of 5, 6, 7, 8, 9 are therefore 60, 72, 84, 96, 108. (7) Some assistance from association may be had in learning 5 times by observing : (a) that the units are alternately 5 and 0, 5 for the odd multiplicands, for the even ones ; {h) if the multiplicand is even, the tens are half of it ; if odd, the tens are half the next lower number : 8X5 = 4 tens and units ; 9X5 = 4 tens and 5 units, etc. More advanced students will take pleasure in extending the multiplication table according to these laws, as well as in accounting for the laws. For example : in 9 times, why are the tens one less than th multiplicand up to 10 X 9, then two tens less up 20 X 9 ? etc. In 8 times why are the tens one less than the multiplicand up to 5 X 8, two less from 6X8 to 10 X 8 ? etc. Division. — Division is, w^e have seen, the operation of finding either of two factors when their product and the other factor are given. After what has been said in Chapter Y upon the nature of division and its relation to multiplication and fractions, little further need be added, especially as most of the text-books ex- plain clearly enough the actual arithmetical work. A few points, however, may be briefly noticed : (1) If, in the method of teaching, the idea of number as meas- urement has been kept steadily in view, the nature of division as the inverse of multiplication will be fully understood. (2) Knowing the relation of the factors in multiplication, the pupil wall, with but little difficulty, ^ less MULTIPLICATION AND DIVISION. 221 cornpreliend the o[)enition and be al)le to interpret tlie results in every case. Practised from tlie lirst in usini^ the idea of correlation — of number detininc: the measnr- ing unit and numher defining the measured whole — in both multiplication and division, he can tell on the in- stant which of these factors is demanded in any problem. (3) There does not seem to be any necessity for begin- ning formal division by the "long division" process. The pupil knows that 2 x 5 = 10, and that 10 -r- r> = 2, whatever 7nay he the unit of measure. He knows that ten ones divided by 5 is two ones, tliat ten tens divided by 5 is two tens, ten hundred-units divided by 5 is two hundred-units, etc. He has learned that 12 units of any order in the decimal system when divided by 5 gives 2 units of that order, with 2 units of that order, or 20 units of the next lower order, remaining ; which 20 units on division by 5 gi\ 's 4 units of that order, mak- ing the total quotient 24. In short, if the pupil has been taught to divide a number of any two digits by any of the single digits, he can divide any number by a single digit. Thus, suppose 497C is to be divided by 8 : , ^ here eight will not divide 4 giving a quotient ~— - of the same order — i. e., in the thousand units ; the 4 is changred to 40 units of the next lower order, making, with the 9 of that order, 49. This di- vided by 8 gives 6, with 1 over. Similarly this 1 is 10 of the next order, which, with the Y of that order, makes 17 ; this divided by 8 gives 2, with 1 over ; this 1 is 10 of the next order, and with the 6 makes 16 of that order, which, divided by 8, gives 2, the last figure of the quotient. ISTo matter what the series of figures, the process is the same, and the pupil should experience ii^y M\ TTT 222 THE PSYCHOLOGY OP NUMBER. liii HHv* VA d' ' ''.I ! ! no real diflSculty if rational method and practice have been followed. A few practical points may be noted : (1) The division by any power of 10 is as easy as multiplication by any power of 10 — is, in fact, derived directly from it. (2) So with division by factors of the divisor, which is directly connected with mnltiplication by factors of the multiplier. To the pupil it will prove an interest- ing exercise to discover the " true remainder." Take, for example, 5795 — 48. 8) 5795 6) 72-1 ... 3 rem. in ones, the quotient being 724 eights. 120 ... 4 rem. in 8-unit groups ; hence remainder in 07ies is 8 X 4 + 3 = 35. This is the old rule : Multiply the first divisor by the second remainder and add the product to the first remainder. The same method is applicable to the case of three or more factorial divisors ; apply the rule to the last twc divisions, and use the result with the first divisor and first remainder. Or, reduce each remainder to units as it occurs ; for example, divide 2231 by 90 (= 3 X 5 X 6). 3 )2231 5 )743 unit-groups of 3 with rem. 2 units ; 6)148 unit-groups of 15 with 7'e7n. 3 groups of 3=9 units ; 24 groups of 90 with rem. 4 groups of 15 = 60 units. The remainder is therefore 60 + 9 + 2 = 71. Other- wise, applying the rule with the last two divisions : 5 X 4 -|- 3 = 23 ; use this as th^ "second remainder" from the " first divisor," and remainder 23 X 3 + 2 = 71. (3) In long division the multiplications and subtrac- MULTIPLICATION AND DIVISION. 223 tions may he combined, as described under multiplica- tion and subtraction— e. g., 635040 -;- 864. 864)635040(735 3024 4320 (1) Seven 4'8, 28 and two = 30— carry 3. (2) Seven 6's, 45 and zero = 45— carry 4. (3) Seven 8's, 60 and three = 63. Tliis gives 302, which, with 4 brouglit down, makes the first remainder. Proceed similarly with 3 and 5, the other figures in the quotient. The student may note the application of the method in a longer operation ; Divide 217,449,898,579 by 56437. The following is the work : 3852967 56437)217449898579 481388 298929 167448 545745 378127 395059 Three 7's, 21 and eight = 29— carry 2. Three 3's, 11 and three - 14— carry 1. Three 4'8, 13 and one = 14 —carry 1. Three 6's, 19 and eight = 27— carry 2. Tliree 5's, 17 and four = 21. This gives 48138, which, with the 8 (heavy-faced type) brought down, makes the complete first remainder. With this pro- ceed exactly as before, and so on with the other re- mainders. (4) Casting out the Mnes.-^-lt is seen that 9 (and 16 i nil Hi w 224 THE PSYCHOLOGY OF NUMBER. i '■!'( ■i -'if '■i. it' . -I I® of course 3) is a measure of 9, 99, 999, 9999, etc.— that is, of 10 — 1, 100 - 1, 1000 — 1, etc. Hence, if from any number there be taken all the ones, and 1 from every 10, 1 from every 100, etc., the remainders from the tens, the hundreds, the thousands, etc., constitute a number which is a multiple of 9. The original number will therefore be a multiple of 9, if the total of the de- ductions made is a multiple of 9 ; this total is the num- ber of ones -\- the number of tens + the riiimher of hundreds, etc. — that is, this total is the sum of the digits of the given number. For example, is 39273 divisible by 9 ? 30000 = 3 times 10000 = 3 times 9999 and 3 9000 = 9 " 1000 = 9 " 999 " 9 200 = 2 " 100 = 2 " 99 " 2 70 = 7 " 10 = 7 " 9 " 7 1=3" 1=3" " 3 Adding 39273 = some multiple of 9 and 24 Hence the given number is exactly divisible by 3, but leaves a remainder of 6 when divided bv 9, because 24 -T- 9 leaves 6 remainder. The principle is : any number divided by 9 leaves the same remainder as the sum of its digits divided by 9. To cast the nines out of any number, therefore, is to find the remainder in dividing the number by 9. In casting out the nines from the sum of the digits we may conveniently omit the nines from the partial sums as fast as they rise above 8. Proofs of Division. — (1) By repeating the calcula- tion with the integral part of the quotient for divisor. (2) By multiplying the divisor by the complete quo- MULTIPLICATION AND DIVISION. 225 de- tient. (3) By casting out the nines, as in multiplica- tion. For example : 3,893,865,223 -^ 179 = 21,753,437. 4 . . out of product 8x5. 9's out of divisor . . 8 X ^ • • out of quotient. 4 . . out of dividend. If there is a remainder the method can still be applied. Test the accuracy of 3,893,865,378 -f- 179 = 21,753,437 j|| where the remainder is 155. divisor j" • • ^X^? 2 . . out of quotient and remainder, 6 . . out of dividend. The disadvantages of this proof are similar to those in the proof of multiplication by casting out the nines. Fundamental Principles connecting Multiplication and Division. — From the theory of number as meas- urement and numerical operations as a development of the measuring idea, there are certain fundamental principles— fundamental also in fractions— connecting the operations of multiplication and division. The principal of these are the following : (1) If equals be multiplied by equals, the products are equal. (2) If equals be divided by equals, the quotients are equal. (3) If an expression contains a series of multipliers and divisors, changing the order of the multipliers and divisors does not change the value of the expression. ! './ mr^ VU: f Hi'' » , 'Hi i^j»' ' i 1 1 ■ "'r(; ti ; ! ■ 1 it, /I ^ 226 THE PSYCHOLOGY OP NUMBER. The last principle includes several principles of use- ful application, either implied or stated explicitly in the discussions upon number and its development. (a) The order of numerical factors may be changed. (h) Multiplying a factor by any number multiplies the product by the same number, (c) Dividing a factor of any number divides the product by the same number. {d) Multiplying the dividend by any number multiplies the quotient by that number, (e) Dividing the divi- dend by any number divides the quotient by the same number. (/*) Multiplying the divisor by any number divides the quotient by the same number. ((/) Dividing the divisor by any number multiplies the quotient by the same number, (h) Multiplying or dividing both divisor and dividend by the same number leaves the quotient unaltered. (^) All these principles are neces- sarily involved in the principles of number as already unfolded. The following is worthy of attention : In an aggregate whose terms contain multipliers and divisors, the inidtiplicatums and the divisions are to he ^ler- formed befoke the additions and the subtractions are made. i! CHAPTER XII. MEASURES AND MULTIPLES. Greatest Common Measure. — The pupil who has been led to have a clear idea of number — who has been taught to look upon the unit as the m£asurer — will find no difficulty in mastering greatest common measure. With all the preliminary notions he is familiar, and it will be an easy matter to pass to the formal process. While in the illustrations given in this chapter we generally use the pure number symbols, it must be borne in mind that here, as everywhere in number and nu- merical processes, the idea of measurement is to be kept prominent, especially in the introductory lessons. A common factor is a common measure — a unit of meas- ure that is contained in two or more quantities an exact number of times. A common multiple is a definitely measured quantity, which can be measured by two or more quantities, themselves measured by units of the same kind and value as those of the given quantity. The teacher should see to it, then, that all his illustra- tions and examples deal with the concrete ; that the measuring idea be kept prominent from first to last. Easy Resolution into Factors. — Taking the num- ber 15, the learner sees that it can be considered 3 fives, or 5 threes ; the five or the three is a measurer 227 n X |i'! m if I'M? * ■if .■if I ' -i- . i ^ 1 I \\{;. I ,( 228 THE PSYCHOLOGY OP NUMBER. or measure of 15, and the equation 15 = 5 X 3 puts in evidence the fact that 5 and 3 are measures or factors of 15. Taking 35, he sees the significance of the equa- tion 35 = 5 X 7. He further notes that 5 is a meas- ure of each of the numbers 15 and 35, and is therefore a common measure. If, next, the numbers 12 and 18 are taken, he will see that all the measures of 12 are — 1, 2, 3, 4, 6, 12 ; and that all the measures of 18 are — 1, 2, 3, 6, 9, 18. Then it will be seen that 1, 2, 3, 6 are common measures of 12 and 18, and that while there are several such meas- ures, there is one that is the greatest — the one that will be called the greatest common measure. Before any pro- cess is taught the class should be exercised in the work- ing of easy examples, both mental and written ; being asked to find common measures, and the greatest com- mon measure of 16 and 2-1, of 24, 36, 48, etc. An ad- ditional interest will be secured by proposing some sim- ple practical problems. It will be better, before beginning the ordinary for- mal treatment, to have exercises in finding the greatest common measure, by resolving the numbers given into their simple factors. It would be necessary, then, to re- call or develop a certain fundamental principle. The division 2)60 3)30 10 is to be interpreted, first, that 60 is 30 twos, and, next, that the 30 twos are 10 three-twos or 10 sixes ; and thus that if a number contains the factor i ! MEASURES AND MULTIPLES. 220 30 2, and if the quotient contains the factor 3, the number itself contains tlie factor or measnre 6. Then, since 108 = 2 X 2 X 3 X 3 X 3, and 72 = 2 X 2 X 2 X 3 X 3, we may see that all tlie single common factors are 2, 2, 3, 3 ; and that, therefore, 2 X 2 X 3 X 3, or 36, is the greatest common measure. Practice on this method will find a place : the pupil has a new interest, and the teacher can take advantage of it to secure further train- ing in number and in the elementary processes. Hie General Method. — But soon it will be found that this method is limited, as its successful application depends on the pupiPs ability to discover a factor. An example, such as. Find the greatest common measure of 851 and 1073, we may suppose to have been given the class, and found beyond their present power of factoring. The reason for the failure will be manifest to them — their inability to find any factor of either number. The need for some new, or, it may be, ex- tended method, is felt ; and this need is the teacher's opportunity for introducing the more powerful method, and for the development of it he has his class in a state of healthy, natural, unforced interest. The Fundamental Principles. — To develop the method, it would be well to turn aside from the ex- ample attempted and give attention to certain facts upon which the method is based. Taking for illustra- tion the numbers 21 and 35, we see, as before, that 21 is 3 sevens and 35 is five sevens. Thus, if 21 is added to 35 we shall have 3 sevens, and 5 sevens or 8 sevens ; the seven being the unit of measure, or meas- urer. Similarly, if 21 is subtracted from 35 the result 'ifi w w. Ill -h • i m h«! I t'! la -^5' I;- *H ■ 1 230 THE PSYCHOLOGY OF NUMBER. will be 2 sevens. Further, if to 21 is added 3 times 35, we liave 3 sevens and 3 times 5 sevens — that is, a cer- tain number of sevens. This is seen to be true for any number of times seven, any number of times eiglit, or nine, . . . etc. Actual measurements will make the principle still clearer. Thus, if A B and C D have a common measure, it must measure A B exactly, and C D exactly : B C E D and measuring off on C D a part C E = to A B, the common measure must measure C E exactly, and there- fore E D exactly, because it measures the whole of C D ; but E D is the difference of the quantities, etc. In the same way E D may be measured off on A B, and the same reasoning will apply. Thus the pupils are led to see certain general principles, and to see them in their generality. 1. From the fact that if we take the sum or the dif- ference of 21 and 35 — that is, of 3 sevens and 5 sevens — or the sum or the difference of any number of times 21 and any number of times 35, we are sure to have a number of sevens (seven representing any measured quantity whatever), it is plain that any number which measures two numbers will measure their sum or their difference, or the sum and also the difference of any of their multiples. The pupils can be got to develop the general form of this principle. If c is a common meas- ure of a and h, so that a = 7nCj and h — nc, then a-\-h = mc -\- no, etc. 2. Because the common measure of two numbers measures their sum, and because the minuend, in a sub- MEASURES AND MULTIPLES. 231 traction operation, is the sum of the remainder and the subtraliend, it is plain that every common factor of the remainder and the subtrahend is a factor of the minuend. The Application of the Method. — We pass now to the application, and shall take the numbers 851 and 1073. The difficulty has been that these numbers are large, and in reply to the question, What smaller num- ber will have in it any common factor that 851 and 1073 may have ? there might be expected the answer, 1073 — 851. But there must be an examination of this statement. 851 1073 851 222 If 851 and 1073 have a common factor, this factor will also measure 222 ; and if 222 and 851 have a common factor, this factor will measure 1073. Thus the greatest common measure of 851 and 1073 is a factor of 222, and the greatest common measure of 851 and 222 is a factor of 1073. Therefore the greatest common measure of 851 and 222 is the greatest common measure of 851 and 1073. It will now be easy to show that if 222, or 2 times 222, or 3 times 222, be taken from 851, 222 and this remainder will have for greatest common factor the greatest common factor of 851 and 222, and the advantage in taking from 851, 3 times 222 is ap- parent. 851 222 185 m. 'A \\ w^ M 'i J"- f iiiii: 111 '•J .If M . 232 THE PSYCHOLOGY OP NUMBER. It will be easy to follow this out through the suc- cessive steps : 185 222 185 185 37 37 37 divides 185 exactly, and is thus the greatest common measure of 185 and 37 ; so that 37 is the greatest com- mon measure of 851 and 1073. The class will now see that 851 = 23 X 37 1073 = 29 X 37 and a conviction will be added to the proof. Then the identity of the work with the following may be shown : 851)1073(1 851 222j851(3 185)222(1 185 37)185(5 185 We see now that a definite method has been evolved, and when the class has been exercised in applying it, it may be well to explain certain artifices by means of which the work may be shortened, or exhibited in a neater form. For example, the work of finding the MEASURES AND MULTIPLES. 233 SUC- greatest common measure of 851 and 1073, as given above, may be presented as follows : 1 3 1 185 185 1073 851 851 CyC)Q 222 185 222 185 37 37 = G. C. M. Or the work might be conveniently arranged as in the following example : Find the greatest common measure of 158938 and 531206. 158938 108784 3 2 1 11 1 5 27 531206 476814 54392 50154 46618 3536 50154 4238 3536 3510 702 26 702 The quotients appear in the middle column, and the work explains itself. It is to be observed that if any common factor is easily discoverable in the two given quantities, it is bet- ter first to divide both quantities by the common factor. If, also, a prime factor is found in only one of the quan- tities which are in operation for the greatest common measure, it may be struck out. In the last example, for instance, tlie first remainder is divisible by 8, while the corresponding number on the other side (the first divisor) is divisible by 2. We may therefore divide this number by 2 and the other by 8, reserving 2 as I! ^^ ■Hi li'.^TT^: ii\ :' 111''' I ^! I 1 ■If 23i THE PSYCHOLOGY OF NUMBEK. part of the required common measure. These factors being removed, we operate with the quotients, 79469 and 6799. The latter divides the former with remainder 4680 ; this, it is obvious, lias the factors 40, 13, 9. llence, if the two original quantities have a common factor, it is 13 X 2 — a result obtained by the actual work. This study of the measures of numbers suggests classilications of numbers. Numbers may be (1) even or odd, according as they do or do not contain 2 as a factor ; (2) composite or prime, according as they are or are not resolvable into simpler factors. Two numbers may have no factors in common, though each of them may be composite ; they are then said to hQ prime to each other. It will be supposed that the class is familiar with these classifications and defini- tions before proceeding to a study of least common multiple. ;.?: lifi LEAST COMMON MULTIPLE. In the presentation of least common multiple, it is necessary — as indeed it always is in the introduction of a new process — first to bring out clearly the essenti-' facts and ideas upon which the process rests. Tlrn pupil must first get a clear idea of the term .iHj common multiple, least common multiple. .. factor (or measure) of 15 is 3 ; 15 is called a mndtiple of l ; it represents the quantity that 3 exactly measures. The pupil will now be asked to name different multiples of 3, say, and will see that he may name or write down as many as he chooses. Then, if a series of multi- MEASURES AND MULTIPLES. 235 pies of 2 be written down, so that we have the two series : 3, 6, 9, 12, 15, 18, 21, 21, . . . 2, 4, (>, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . . he will see that there are numbers which are at the same time multiples of 2 and 3, and are therefore cnm- ni07i multiples of 2 and 3. These common multiples are, here, 6, 12, 18, . . . Then, because we have started with the smallest multiples of the numbers, 6 is the small- est or least common multiple of 2 and 3. At this point the pupil can hardly fail to see that the second common multiple is Q-\-(S^ the third is 6 + 6 + 6, etc. ; in other words, that all the common multiples of two numbers are formed by repeating as an addend the least common multiple. He can then be led to see the reason for this, viz., that (referring to the foregoing exam{)lc), in order to get a common multiple of 2 and 3 larger tlian 6, it Avill be necessary to add to 6 a common multiple of 2 and 3, so that 6 is the least number that can be used. Numhers Prime to Each Other. — A necessary step preUminary to teaching the formal process is the bring- ing out of the fact that the least common multiple of two numbers prime to each other is their product. For example, take the numbers 5 and T : a common multiple must have 5 as a factor and 7 as a factor ; it is, there- fon^ 5 niulti})lied by another factor. But since the mul- tiple contains 7, and since 7 is prime to 5, the other fac- tor of the multiple must contain 7. Hence, since tlie smallest multiple of 7 is 7, the least common multiple of 5 and 7 is 5 X 7. Similarly, a common multiple of 4 and 9 is a multiple of 4, and is therefore 4 multiplied f)>fM '^'W b quan- P a and sary to to this lon the y meas- having mclude \g of a CIIAPTEE XIII. FRACTIONS. After the previous discussion on the nature of frac- tions (see especially Chapter VII) and their psychological relation with the fundamental operations, a brief ref- erence to some of the points brought out is all that is needed as an introduction to the formal teaching of fractions. dumber depends upon measurement of quantity. This measurement bemns with the use of inexact units — the counting of like things — and gives rise to addition and subtraction. From this first crude measurement is evolved the higher stage in wdiich exactly defined nnits of measure are used, and in which multiplication, di- vision, and fractions arise. Multiplication and division brins: out more clearly the idea of number as measure- ment of quantity — as denoting, that is, (?') a unit of measure and (/«) times of its repetition. The fraction carries the development of the measuring idea a step further. As a mental process it constitutes a more definite measurement by consciously using a defined unit of measure ; and as a flotation, it gives complete ex- pression to this more definite mental process. Frac- tions therefore employ more explicitly both the con- ceptions involved in multiplication and division — name- 241 n im^ II 4 V •' ). I li;: ^ 115 %. ■'I I. u -'! 1 i 1 .1 ■1 liki tJi 2i2 THE PSYCHOLOGY OF NUMBER. ly, analysis of a whole into exact units, and synthesis of these into a defined whole. The idea of fractions is present from the first, because division and multiplica- tion are implied from the first. There is no number without measurement, nor measurement without frac- tions. Even in whole numbers, as has been pointed out, both " terms " of a fraction are implied in the accurate interpretation of the measured quantities. Since there is nothing new in the process of frac- tions, so in the teaching of fractions there is nothing essentially different from the familiar operations with whole numbers. If the idea of number as measurement has been made the basis of method in primary work and in the fundamental operations, the fraction idea must have been constantly used, and there is absolutely no break when the pupil comes to the formal study of fractions. There is only before him the easy task of examining somewhat more attentively the nature of the processes he has long been using. The suggestions made in reference to primary teaching and formal in- struction in the fundamental operations apply with equal force to the teaching of fractions. The meas- uring idea is to be kept prominent : avoidance of the fixed unit fallacy and its logical outgrowth, the use of the undefined qualitative unit — the pie and apple method — as the basis for developing " fractional " units of measure, and the " properties of fractions " ; the es- sential property of the imit in measurement — the 7neas- ured part of a measured whole ; the logical and psycho- logical relation between the niimher that defines the measuring unit and the number that defines the meas- ured quantity j or as it is sometimes expressed, the FRACTIONS. 213 Lidy of " relation between the size of tlie parts " (the measur- ing^ units) and the numher of tlie parts composing or e(|ualling the measured (juantity ; these and all kindred })uints that have been brought out in discussing number as measurement, and numerical operations as simply })hases in the development of the measuring idea, can not be ignored in the teaching of fractions, because they can not be ignored in the teaching of whole numbers. Exact number demands definition of the unit of meas- ure ; the fraction completely satisfies this demand by stating or defining expressly the unit of measure. In all number as representing measured quantity the ques- tions are : What is the unit of measure, and how is it defined or measured ? How many units equal or con- stitute the qaantity % These questions only number in its fractional form completely answers. It is the com- pletion of the josychical process of number as measure- ment of quantity ; the idea of the quantity is made definite, and it is definitely expressed. While the following treatment of fractions is in strict line with the principles of number set forth in these pages, and has stood the test of actual experience, it is given only by way of suggestion. The principles are universal and necessary ; devices for their effective application are within certain limits individual and con- tingent. Principles are determined by philosophy, de- vices by rational experience. The teacher must be loyal to princi})les, but the slave of no man's devices. 1. The Function of the Fraction. 1. In its primary conception a fraction may be con- sidered as a number in which the unit of measure is -i I i 1 1 r II I ' } '<}' '• Ih i.! iiU, 1: 244 THE PSYCHOLOGY OF NUMBER. expressly defined. In the quantities 4 dimes, 5 inches, 9 ounces, the units of measure are not explicitly defined ; their value is, however, implied, or else there is not a definite conception of the quantity. In $-3^, -f-^ foot, -j^ pound, the units of measure are explicitly defined ; and each of these expressions denotes four things : 1. The unity (or standard) of reference from which the actual unit of measure is derived. 2. IIow this unit is de- rived from the unity of reference. 3. The absolute ■y number of these derived units in the quantity. 4. This number is the ratio of the given quantity to the unity of reference. For example, in -^-q pound, the unity of reference is one pound ; it is divided into sixteen equal parts, to give the direct measuring unit ; the number of these units in the given quantity is nine; the ratio of the given quantity to the unity of reference is nine. 2. If properly taught, the pu])il knows — if not, he must be made to know — that any quantity can be divided into 2, 3, 4, 5 . . . n ecpial parts, and can be expressed in the forms f, f, f, |- • • • ^- Familiar with the ideas of division and nnilti plication which be- come explicit in fractions, he learns in a few niinutes (has already learned, if he has been rationally instructed) that any quantity may be measured by 2 halves, or 3 thirds, or 4 fourths ... or ?2 7iths ; that to take a half, a third, a fourth ... an 7?^th of any quantity, it is only necessary to divide by 2, or 3, or 4 . . . or ?i ; that if, for example, 16 cents, or 16 feet, or 16 pounds has been divided into four parts, the counts of the units in each case are one, two, three, four, or one fourth, two fourths, three fourths, four fourths ; that each of these units — L fourths — is measured by otlier units, and can be ex- pressed as integers, namely, 4 cents, 4 feet, 4 pounds, and so on, with kindred ideas and operations. 3. The Primary Practical Prhiciple in Fractions. — It is clear that this complete exj^ression for the num- ber process is the fundamental principle employed in the treatment of fractions : if both terms of a fraction be multiplied or divided by the same nund)er, the numer- ical value of the fraction will not be changed. This principle is usually " demonstrated " ; it is, however, involved in the very conception of number, and seems as difficult to demonstrate as the delinition of a triangle ; but intuitions and illustrations to any extent may be given. Any 12-unit quantity, for example, is measured by f , f , -}-|, or by f , f , \\ ; the identity of the quan- tity remains unchanged in the changing measurements. Moreover, if half the quantity be measured, the identity of J, f, f, 3^ is seen at once. The principle is, of course, that in a given measured quantity the " size " of the units varies inversely witli their number. This principle is said to be beyond the comprebension of the pupil. On the contrary, if constructive exercises, such as have been described, have been practised, there comes in good time a complete recognition of the principle. When, for instance, the child measures off any 24-unit quantity by twos, threes, fours, sixes, eights, lie can not help feeling the relation between the magnitude and the number of the measuring parts. This is, in fact, the process of number. Proof of the Principle. — If the first vague aware- ness of the relation does not grow into a clear compre- hension of it, clearly the method is at fault. In any ■ii ,''f, -^ Hi !!■ jj^ » ir^«" ' 'ii' ' i; *: ■ • 1 ( ( I •1 .' i A 240 THE PSYCHOLOGY OF NUMBER. ease, if the pupil does not understand the principle after rationally using it, any formal " demonstration '* is a mere delusion ; for anv so-called demonstration is grounded on the principle — in general is the principle — merely illustrated or used in a disguised form. For example : Prove l^^f = ^l-J-. Since $|- = ,^ J x 3 : mul- tii)ly by 4, and we have 8f X 4 = s^ X 4 X 3 = ^3 ; multiply these equals by 5 : . • . si^f X 4 X 5 = §3 X 5 = $15 ; but also $fg- X 20 = $15; . • . $f X 20 = $^1^ X 20; dividing equals by equals; . • . $1 = $JJ^ Or, generally : ^ X h = a', multi- ply both sides by n. . • . ^ X nh — na ; but ^X nh = na ; . • . |- X nh = '^X nh; dividing both sides by nh ; n T nn nb' These and similar proofs are in essence the idea already considered : that if a quantity is divided into a certain number of equal parts, each part has a certain value ; if into twice the number of parts, each part has half the value of the former part ; if into three times as many parts, each has a third of the value; if into n times as many parts, each has 1 nth of the value. If formal proof is wanted of this important princi- ple (which is, onco more, the principle of number), the following is perhaps as intelligible as any other. To prove, for example, that $| = $|^- = $jV^ —^ etc., we tCl — 09 — __2 5_ — pf p . C>H — C:3 7 — C< 7 5, — pfp <1\ I ^1 I FRACTIONS. 24: ix 4. The Fraction as Division. — AVliile in its primai-v concoption tlie fraction if not simj^ly a formal division, it nevertlieless involves the idea of division, and can not be fully treated without identifvin<]^ it with the formal process. The quantity -j^^ foot, first re ■ t m > Ml' » 1,1 : '^ ? !•■ - !i ' m ill I'Hi 1: 248 THE PSYCHOLOGY OF NUMBER. = 7 feet -T- 12. The unit of reference, 1 foot, may bo thought of and expressed as 12 twelfths foot : . • . 7 feet = 84 twelfths foot ; . • . 7 feet X iV = "^ twelfths foot = J^- foot X 7. (3) To prove that SJ X 3 is equal to $3 X i : 4 times $J == ^1 ; multiply these equals by 3 ; . • . 4 times ^J X 3 = S3 ; . ' . 8i X 3 = $3 -r- 4. Or, using q for any quantity, 4 times ^q = q\ . • . 4 times ^^^ X 3 = 3^' ; hence, J*/ X 3 = 3^/ -^ 4. (4) Or, generally, ~qXrri = iiiq -r- n. For n times ^ ^ z= (^ ; ,' . n times ~q X fn = mq ; j^q X m = mq -i- n. Such formal proofs are useful and even necessary, but are likely to be misleading unless the pupil has evolved, from rational use of the principle, a clear idea of the relation between times and parts, the importance of which has been emphasized in this book ; he is apt to become a mere spectator in the manipulation of sym- bols, rather than a conscious actor in the mental move- ment which leads to complete possession of the thought. II. Change of Form in Fractions. 1. From what has been already said, it appears that any quantity may be expressed in the form of a frac- tion having any required denominator. Express 9 yards in eighths of a yard. Since the unit of measure is f , 9 FRACTIONS. 2-19 Piich nnits is ■^,^^-. Similarly, >«^7 expressed «as liundredtlis is -J^^, etc. In gciieml, any qiuintity of q units of meas- ure exiiressed as ?/tlis is -"-. 2. In the same way, any quantity expressed in frac- tional form may be changed to an e([uivalent fraction having any denominator. Transform Jr'| into an e([niva- lent fraction having denominator 2(J. We can follow either of two plans : (1) 20 is a multij)le of 5 by 4 ; we therefore mulri- ])ly both terms of the given fraction by 4, getting s|5-. This is best in prac^tice. (2) Since the new denominator is to be 20, m'C regard the unit of measure as 'tlJ-; ^| is ^i X -i ; but ^ = one fifth of Sf J- = 8/o ; S4 "0 — K 1 (5 It may be remarked that in such transformations the new denominator is generally a multi])le of the original denominator. If it is not, the new equivalent fraction will be complex, it will have a fractional numerator. Thus, if it is re([uired to transform ^ yard to an equiva- lent fraction with denominator 12, we niultiply both • ' a terms by 1^, with the result .^*^. 3. It is often necessary or convenient to reduce a fraction to its lowest terms — that is, to express it in terms of the largest unit of measure as defined by the unity of reference. This is done by dividing both terms of the fraction by their greatest common measure ; thus, ^tV ^^'^ e(|uivalent to %^, in which the quantity is ex- pressed in the largest unit of measure, as delined by the unity of reference, the dollar. The principle in- m .1 i ^*"^^ 'If i 250 THE PSYCHOLOGY OF NUMBER. volved is that stated in I, 3 — viz., the numerical value of the units is increased a number of times, tlie number of them is diminished the same number of times. In practice, the greatest common measure can gener- ally be foiMid by inspection, as described in Chapter XII. 612 ^ 2 X 2 X 3 X >3 X 17 _ 17 ~ 19* Thus, In some cases ' ()84 2 X 2 X 3 X 3 X ID the greatest common measure must be found by the general method described in the same chapter. Thus, 79409 if the proposed fraction is ^rrr^r—-, , we should discover ^ z()obOo the greatesi: common divisor to be 18 ; en dividing both P1 1 ^ terms of the given fraction there results ^.-7777 , which is the simplest of all equivalent fractions. 4. In changing a mixed number to an improper frac- tion, and vice versa, the primary principle of fractions applies at once : (1) Reduce 75f yards to an improper fraction. The expression = 75 yards -|- f yard ; express 75 yards in form of a fraction with denominator 3 : 1 yard is J yard ; 75 yards is |- yard X 75 = ^^ yards ; . • . 75 yards + f yard = ^l^^- yards = ^^ yards. (2) In the converse operation either consider the problem as a case of formal division giving 75f , or con- sider the expression as denoting so many thirds of a. yard ; then 3-thirds — one yard ; how many 3-thirds in 227 thirds? Evidently, as before, a case of division, pving 75 ones and two-thirds remainder — that is, 75J yards. It may be observed that in (1), while tht^ ])ri- mary measurement of the qnantity is 3 units X 75 -|- 2 FRACTIONS. 251 ds. the con- of a Is in sioii, pri- + 2 I units, and we multiply 3 by 75, it is equally lop;ic'al to use the correlate 75 units X 3. (See pa10-V (2) How much was left of 16f yards of cloth after 6| yards :<.M;re cut from it? deducing the fractional parts to common denomi- nator : m FRACTIONS. after 255 16| 10 ,2 (\8 — f^64 15 A6 4 72 + 27 72 ^tI y^^ds remainder. Here the fractional part of the subtrahend is greater than the fractional part of the minuend ; the minuend is therefore changed to the form 15+1 + f| = i^r2 + 2j _ 72 1 n99 10 J J In actual work we may take 64 from 72, the denominator of the minuend fraction, and add to the remainder the numerator of the minuend fraction. Thus, we can not subtract 04 from 27 ; we subtract it from 72, and add the remainder, 8, to 27, getting -fl-. This is equivalent to taking 1 (= ^) from the minuend, and uniting it with the minuend fraction, as has been done in the example. 3. Multiplicatioii of Fractions. — (1) When the multiplicand is a fraction and the multiplier a whole number, the operation is exactly like multiplication of integers. To find the cost of 12 yards of cloth at $J a yard, we multiply 3 by 12 and define the product by the proper unit of measure. In finding the cost of 12 yards at $3 a yard, the complete ]irocess is $1 X 3 X 12 ; we operate Avith the pure numbers 3 and 12, getting 36, and define the product by naming the proper unit of measure one dollar y the cost is then 36 dollars. In the proposed case we do exactly the same thing : $J X 3 X 12 — that is, 36 times the p!'oper unit of measure ($J) — and the product is 36 qxicrter dollars. Neither the process nor the product changes, because the nnit^ or the manner of writing it, happens to change. (2) AVhen the multiplier is a fraction, exactly the 18 IH I /i.^^'l M^K" 256 THE PSYCHOLOGY OF NUMBER. II I' \ n. same principles hold ; in fact, the measured qnantity, $i X 12, is identical with $12 X f . In this conception of quantity (money value) we have nothing to do with yards, and either form of the measurement may be taken. In fact, $i X 12 is f of $1 + | of $1 + J of $1, and so on 12 times — that is, f of $12. The multi- plicand is always a unit of measure ; the multiplier always shows how this unit is treated to make up the measured whole. It is purely an operation. In this example the denominator shows how the unit $12 is to be dealt with in order to yield the derived unit of meas- ure : it is to be divided into four parts, and the derived unit thus found is to be taken three times. As already shown, from the nature of the fraction it denotes three times one fourth of the multiplicand, or one fourth of three times the multiplicand — that is, $J^X 3, or $ ^ 4— • (3) The explanation usually given of the process is in harmony with this. This explanation considers the multipT and as a case of pure division ; that is, J is one fourth of 3, and to multiply a (juantity by f is to take one fourtii of 3 times the quantity. In fact, in all operations with fractions the idea of division, as well as of multiplication, is present; a factor and a divisor are always elements in the problem. (4) The method to be followed when both factors are of fractional form involves nothing different from the other two cases. Tlie price of f yard of cloth at $f a yard is to be found. The result is indicated by $f X 4 ; that is, as before, 4 times a certain quantity io to be divided by 9, or \ of the quantity is to be multiplied by 4. In the first case, $| X 4 is fher ^J- X 3 X 4 = $i X 12 = Hi 1 ' i FRACTIONS. 257 to be IS, as o $3 ; or 8i X 4 X 3 = $1 X 3 = $3 ; and -J of this is $f , or ^. It may be ob?ervcd tliat we may change the multi- pheand into an equivalent fraction with a miit of meas- ure determined by a multiple of the denominators. In $1 X i, for example, we have ^} Xi = $^X4: = ^ = $-|-. Tiie complete process is seen to be J X i = Jf . But since numerators are always factors of a dividend, and denominators factors of a divisor, common factors may be divided out. In $g\ X 4, for instance, the value of the quantity is the same whether we take 4 times the number of units (— $g-|-), or make the units -I times as large ($-|). Tliis is nothing but the application of fun- damental principles (see page 22G) of multiplication and division. If we have to divide 210 by 21, we may pro- ceed thus : -^V- - ^"3 1 y- = I X -?- X 5 = 5. Again, 32310 -r- 385 == ^J±l^m^^^ = ,1 X H X I X r X 2 X 6 = 1 X 84: =:. 84. 4. Divimm of Fractions. — (1) When the divisor is an integer and the dividend a fraction. Paid %-^ for 5 yards of calico, what was the price per yard ? One yard will cost one fifth of %-^, or $/^. At $5 a yard, how much lace can be bouglit for ^-^-^ ? The answer is indicated in $^ --- §5 ; the quantities must have the same unit of measure, and the expression is equivalent to $-i^ -^ 8t¥ ~ "^ -^^*^ = A 5 ^^ence, -^ yard of lace can be bought. (2) When the divisor is a fraction, and the dividend an intesjer. At 8f a yard, how many yards of dress goods can It for $6 % number of \ards is iriven in ^6 -~ tl. where. be bou given •'4' i ■'■ mmf ii < !■ ''i'*> « t r , '. ; ■^■'^'' '« f i: 1 1) ^ :'!; 258 THE PSYCHOLOGY OF NUMBER. again, tlie quantities must be reduced to the same unit of measure : $6 -r- Jj^f =: ^-^^ -r- 1 = 21 -f- 3 = 8 ; hence, 8 yards can be bought. Paid $6 for f yard of velvet, what was the price per yard ? The cost is given in $6 -i- f , wliich means that J of 3 times the quantity sought is $6, and therefore it is $6 X 4 -7- 3 = $8. Or, by the law of commutation, $8 X J iz. $f X 8 = SO ; and §6 ~ 8f = 88, as before. (3) When both divisor and dividend are fractions. What quantity of cloth at 8/75- a yard can be bought for 8-5-? The quantity is given in 8|- -i- 8A, where again the quantities must be expressed in terms of a common unit of measure : there results 8J|- -v- 8-^ =16-7-3 = 5^, which is the number of yards. If 5^ yards of calico cost 8y? what is the price per yard ? AYe have 8f -r- J/ — that is, one third of 16 times some quantity = 8f ; 1^> times tlie quantity = 8f X 3 ; the quantity is 8i X 3 X j\ = 8^^. The Inverted Dimsor. — It is obvious in all these cases that practically the divisor has been inverted and then treated as a factor with the dividend to get the quotient. It must be clear, too, that this is simply re- ducing the quantifies to he compared to the same imit of measure. When 8^2 is to be divided by Sf — i- e., when their ratio is to be found — they must be expressed in the same unit of measure. The divisor is measured olf in ffths of a dollar ; the dividend, then, must be expressed in ffths of a dollar — that is, it becomes 5 X 12, or 60. The question is now changed to one of common division : $12 -^ i =: 4 = 15. Similarly, in StI ~~ $4 — |o — ^1^ ^\^Q divisor is expressed in fifths of a "5" f = 60 FRACTIONS. 251) dollar; the dividend $|| iiiiist be expressed in fifths; how is this done ? By multiplying 5^} f l)y 5, which gives the number of jifih.< in ^} |, namely, %\% ; f<_»r if $12 is GO fifths, ^V of $12 must be \% fifths. The unit of measure is now the same, and we have f | (fifths) -r- 4 (fifths) = 1 J. " Inverting the divisor," then, makes the problem one of ordinary division by expressiiuj the quantities in the same nmnher measure. Though formal proofs of rules are in general too abstract to begin with, yet after the pupil has freely used and learned the nature of the processes involved in concrete examples, he will quite readily comprehend the more abstract proof, and even the general demon- stration. Take a few instances : 1. To prove that the product of two fractions has for its numerator the product of the numerators of the given fractions, and for its denominator the product of their denominators : (1) Trove | X | - ^^ 73 I X 5 = ^f- ; but this product is 9 times too great, and therefore the required product is \ of 16 15 (2) f X 8 rr: 3 ; for f X S = 3 X i X 8 ; and ^ X 9 = 5 ; multiply these equals ; .-. |X8X|X0 = 3X5; divide by 8 X 9 ; . • . I X -| = I Jg ; i. e., the product of numera- tors, etc. (3) Generally, let ^ and ~ be any two fractions. -^ X ^ = « ; (because, from the nature of number, -1- X ?> = 1) ; similarly, ■J X ^ " <2 ; multiply these equals ; i I: III «•!* L i |< rlT' • i U ' nl f ill '■JM ( 260 THE PSYCHOLOGY OF NUMBER. .". J X ^ Xlcl = a X c; divide these equals hyhd; • * • T ^ ^ ~ tv^ ' ^^^^^ ^®' ^^^^ product of tlie numer- ators of the given fractions is the numerator of the re- quired fraction, and the product of their denominators its denominator. 2. To prove the rule for division of fractions, " in- vert the divisor and proceed as in multiplication." (1) f -i- 4 5 divide f by 5 and there results -^-^ ; but it is required to divide not by 5 but by ^ of 5 ; the re- quired quotient must therefore be 9 times -^^ — that is, a, which is m . (2) |- X 8 — 3 ; multiply these equals by 9 ; ..-.1X8X9 = 3X9. (1) Similarly, | X 9 = 5 ; multiply these equals by 8 ; . • . I X 9 X 8 = 5 X 8. (2) Divide (1) by (2) : • 3. _i_ 5 — 3 x9 • • 8 • "? ~ 5 X 8 • Similarly, a general proof may be given, as in mul- tiplication. h i' by JcZ; lumcr- tlie re- inators It' is, " in- 5? the re- tliat is, (1) by 8; (2) D mul- CIIAPTER XIY. DECIMALS. As already indicated in Chapter X, decimals may be regarded as a natural and legitimate extension of the notation with which the pnpils are already familiar. Takino; this view of decimals as a basis for teachinir the subject, we shall see how easily and naturally all the ordinary processes are established, and, further, liow this mode of treatment recalls and confirms all that was said in building up the simple rules. Notation and Numeration. — Consider the number 111 : the first 1, starting at the right, denotes one unit ; the second, one ten, or ten units ; the third, one hun- dred, or ten tens, or one hundred units. The third 1 is equivalent to one hundred times the first 1, and to ten times the second 1 ; the second 1 is equivalent to ten times the first 1, and to one tenth of the third 1 ; the first 1 is equivalent to one tenth of the second 1, and to one hundredth of the third 1. Let us now rewrite tlie number already taken, ])lace a point after the first 1 to indicate that that 1 is to be reii^arded as the unit, and then place after the point turee I's, so that we have 111-111. AYe may ask what each of these I's should mean, if the same relation is to hold among successive digits that we 2G1 IMAGE EVALUATION TEST TARGET (MT-S) fe / ^ ^ < V^% ,^% r/.^ ^ 1.0 I.I 11.25 tii lis 2.5 2.2 i -^ IIIIIM iiiiim 1.4 vl ^ /a m '/ ^. Hiotographic Sciences Corporation m \ :\ \ r ^ ». c^ «J '^^ 4' 33 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 "^ Cv W(rV' • 202 THE PSYCIIULOGY OF NUMBER. have supposed hitherto to hold. The 1 after the point, standing next to the 1 which the point tells us is to be looked upon as a unit, would naturally mean one tenth of that 1 — that is, one tenth of a unit, or, as we shall say, one tenth. The next 1, passing to the right, stand- ing two places to the right of the unit, is one hundredth of the unit, or one hundredth ; it is one tenth of the pre- ceding 1 — that is, one tenth of one tenth. Similarly, the next 1 would signify one thousandth, and would equal one hundredth of the one tenth or one tenth of the one hundredth. Thus the number above written may be read as follows : One hundred, one ten, one unit, one tenth, one hundredth, and one thousandth. But just as in ordinary numbers it is convenient, for the purpose of reading, to combine the elements into groups, here also it will be well to adopt a similar method. The 1 to the extreme right is 1 thousandth ; the next 1 is, from its position, equivalent to 10 thousandths ; and the next 1 is 100 thousandths ; so that to the right of the point we have 111 thousandths. The whole number may now be read, one lumdred and eleven, and one hundred and eleven thousandths. Very little practice will suffice to acquaint the pupil wdth the extended notation and nu- meration. A few questions, such as the following, will prove useful : (1) Read 539*7423, and show that the reading prop- erly expresses the number. (2) Explain how it is that the insertion of a zero between the point and the 6 in the decimal '5 changes the value of the decimal, but that the addi- tion of zeros to the right of the 5 does not change the value. DECIMALS. 263 ri (3) N^ame the decimal consisting of three digits which lies nearest in value to '573245. These will serve to bring out in a new relation some of the essential features of the decimal system, and throw light on some facts that at an earlier stage in the pupil's progress were necessarily somewhat dimly seen. I. Simple Rules. Multiplication. — When once the notation is under- stood, addition and subtraction of decimals can oifer no difficulties, and we pass them by to consider multiplica- tion. In this connection the most striking application is the multiplication by 10, 100, etc. The pupil will be asked to compare 7 and '7, '3 and '03, '009 and '0009, and he will see at once that the first number in each case is, in virtue of the position of the point, 10 times the second number. Next, when asked to compare the numbers 37 and 3*7, he will see that the 3 in the first number is 10 times as great as the 3 in the second, that the 7 in the first number is 10 times as great as the 7 in the second, and that therefore the first number is 10 times as great as the second number. lie has thus been led to discover that by moving the points one place to the right we get a immber 10 times as great as the origi- nal number. Similarly, a corresponding conclusion may be reached for multiplication by 100, 1000, etc., and the conclusions in each case should be arrived at and stated by the pupil. It will at once follow that to divide a number by 10, 100, 1000, etc., we have only to move the point one place, two places, three places, etc., to the left. "VVe pass next to the multiplication by any integral number. ■I!!] iSB [' ■ i[ .1 Hi '■ii ■'Hi, "V ! it ^'1: 2G4 THE PSYCHOLOGY OF NUMBER. 5-37 3 16-11 4-42 57 30-94 221-0 251-94 The multiplication in eacli of the foregoing cases is based on the same considerations as the multiplication of integers by integers. Thus, in the second case, 7 times 2 hundredths are 14 hundredths — that is, 1 tenth and 4 hundredths, and the 4 must be in the hundredtlis place ; 7 times 4 tenths are 28 tenths, which with the former 1 tenth make up 29 tenths or 2 units and 9 tenths, and the 9 must be in the tenths place ; thus, the 4 and the 9 will be properly placed if the point is in- troduced before the 9, etc. ; next, multiplying by 5, we must write the results one place to the left, for reasons explained in an earlier chapter. The pupil will now understand multiplication by an integer, and is ready to proceed with multiplication by a decimal. 31 2 3-1 2 23 2-3 9-3 6 6 2-4 71-7 6 7-17 6 He will be asked to multiply some number, say 3*12, by some number, say 23 ; the result is 71*76. If, then, we propose to multiply 3'12 by 2-3, it will be seen that this differs from the former only in that the multiplier is 10 times as small ; the product then will be 10 times as small, and may at once be written down 7-176. A further example or two, in which a different number of DECIMALS. 2G5 decimal places are taken, will suffice to show that to multii^ly two decimals we proceed as in the multiplica- tion of integers, and mark off in the resulting product as many places as there are in both multiplier and mul- tiplicand. Division. — To teach division, it is well to begin with the division by an integer, as this will connect the pro- cess with what is already known. Consider the follow- ing examples : (1) (2) T)2 1(3 Y)2-l(-3 2 1 21 (3) 5)-0 1 5(-0 3 •0015 (4) 2 3)1 4 5-8 1(5-4 7 135 10-8 92 1-61 1-6 1 The pupil who can explain the first division can at once explain the second, the third, and the fourth ; and he will see how to divide whenever the divisor is a whole number. Then he may be asked to explain why, in the following divisions, we have the same quotient : 3115 12)60 5 5 He will be led to recognise a principle that he already knows, namely, that the multiplication of divisor and dividend by the same number does not change the quo- tient. He may then be asked to state a quotient equi va- il w~ lU' 20C THE PSYCHOLOGY OF NUMBER. I '■ .1 ::??^ \lp..- ViUi W ( .,,1 1 i ' 'it •' ! (.,;■ lent to 35 — "7, but having for divisor a whole number. An answer to be expected is, 350 -r- 7 ; at any rate, he can be led to this result, and, as this quotient is seen to be 50, he can conclude that 35 -i- '7 = 50. An exami- nation of a few more examples will show how always to proceed. It would be well to have the student, in his earlier practice, write out a full statement of what he does. Suppose he is required to lind the quotient 1'375'i -r- "23 ; his solution should stand somewhat as follows : The quotient of 1"3754 by "23 is the same as (multi- plying each number by 100) that of 137*5^1: by 23. 2 3)1 3 7-5 •l(5-98 11 5 2 2-5 2 0-7 1-8 4 1-8 4 .-. 1-3754 -^ -23 = 5-98. The delation of Decimals to Vulgar Fractions. — The simple rules being understood, we may now con- sider the conversion of decimals to the ordinary frac- tional form, and the conversion of ordinary fractions into the decimal form. From the definition, '273 = -f^^ -\- -j-J^ + ttott ~ iVo^o , and the student sees at once how to write a deci- mal in the form of a fraction. A\^e may next ask the pupil to divide 1 by 2, as an exercise in the division of decimals. 2)l-0(-5 1-0 DECIMALS. 2G7 Similarly, 4)3-00(-75 2-8 20 8)7-00(-8r5 6-4 60 56 '« ; 40 But the quotients l-f-2, 3-^4, 7-^8 have up to this point been taken as equivalent to J, j, J. ,'. i = %i = -75, 1 = -875. As an exercise these results might be verified thus : •^K — 75 — 3 A method has now been found for converting an ordi- nary fraction into a decimal ; at the same time another method has been suggested in the verification above made ; for we see that 4- = — ^- = — 3^5-^ 6_ _ 75^ _ ' 4 2^2 2x5x2x5 — 1^0^ — •75. The latter method is very valuable from the point of view of theory, and the pupil should work several examples in this way. We shall next consider the example f . 3)2-00(-66 1-8 20 18 2 . • . f = '6666 . . ., the 6's being repeated without end. This fact is expressed thus : | = '6, and f is said to give rise to a recurring decimal. Let us now seek to convert | to a decimal by the other method. Accord- ing to it, we multiply the denominator by some number which will change it into 10, 100, 1000, etc.— that is, into n « i p'™ 11 ; Sin: li J ! t '/t 'l^'T ,4 1 ^'1 J' ''.1. i ;i ■; ■: J; i '1 ! : 2GS THE PSYCHOLOGY OF NUMBER. some power of 10. !Now, any sucli power is made by multiplying 10 by itself some number of times ; but 10 itself is made by multiplying 2 and 5 ; therefore every power of 10 is made up wholly of the factors 2 and 5, and in equal number. We can not, then, multiply 3 by any number that will make it into a power of ten — that is, we can not convert f into an ordinary decimal with a finite number of digits. "We have thus a com- plete view of the case. The following are examples of recurring decimals : •33^ = -09. 1^ - -11:2857 i--l Take next ^ : 6)l-000(-166 6 40 36 40 36 4 1 the 6's recurring, and this is •16666 . . written |^ = -16. Here the first figure of the decimal does not recur, and -J- is said to give rise to a mixed recurring decimal, those formerly met with being called pure recurring decimals. Similarly, ■^ = -583 ^ = -05. If we try to apply the second method to these examples, we get — 1 — 1 — 8 5 f 8 Q 2x3 2x5x3 iF'-'^Tf Is 2x2x3 "i» 2 X 3 ^~3 2x5x3 7x5x5 5x2x5x2x3 5 rh of -4^ 6x2x3x3 = tV oi I DECIMALS. 2G9 The examples given lead np to tlie following proposi- tions, for the truth of which it will be easy to state the general argument : Proposition I.— A fraction whose denominator con- tains only the factors 2 or 5 leads to a decimal, the num- ber of w^hose digits is the same as the number of times the factor 2 or the factor 5 is contained in the denomi- nator, according as the former factor or the latter occurs the greater number of times. Proposition U.—A fraction whose denominator con- tains neither the factor 2 nor the factor 5 leads to a pure recurring decimal. Proposition III.— A fraction whose denominator con- tains in addition to the factors 2 and 5 a factor prime to these factors, leads to a mixed recurring decimal the number of digits that are before the period being' the same as the number of times the factor 2 or the fJictor 5 is contained in the denominator, according as the for- mer factor or the latter occurs the greater number of times. Questions similar to the following afford a valuable exercise on this i^art of the work : ^ (1) In the case of fractions, such as I, ^V, etc., lead- ing to pure recurring decimals, what limit is there to the number of iigures in the period ? (2) I = -142857 : explain why any other fraction with denominator 7 will lead to a recurring decimal with a period consisting of the same digits following one another in the same circular order. It will now be in place to consider the converse pro- cess of changing recurring decimals into their equiva- lent vulgar fractions. A difference of opinion exists as m m W ' 1 . -1 ^ \ IJ ;i:i CHAPTER XV. 1 '« ill PERCENTAGE AND ITS APPLICATIONS. Percentage.— In some text-books on arithmetic per- centage is treated as if it were a special process involv- ing certain distinctive principles and therefore entitled to rank as a separate department. In these books, ac- cordincrlj, percentage has its definitions, its " cases," and it? s and formulas. This elaborate treatment seems to be a mistake on both the theoretical and the practical side : on the theoretical side, because it asserts or assumes a new phase in the development of number ; on the prac- tical side, because it substitutes a system of mechanical rules for the intelhgent application of a few simple prin- ciples with which the student is perfectly familiar. In the growth of number as measurement percentage pre- sents nothing new. It has to do with the ideas and processes of ratio with which fractions are more or less explicitly concerned, and its problems afford excellent practice for enlarging and defining these ideas, and securing greater facility in using them. But the mere fact that, in this new topic with its cases and its rules, a quantity is measured oif into a hundred parts instead of into any other possible number of parts, appears to be no valid reason for constituting percentage a new process marking a new phase in the evolution of num- 279 'I:ill ii Sim Tf'ra- 280 THE PSYCHOLOGY OF NUMBEU. I' » I ■ n* ' ! ') m ,! t •I '"i ber. It is no doubt correct enough to say that " per- centage is a process of computing by hundredths"; but is such a process to be broadly distinguished as a mental operation, from a process of computing by eighths, or tenths, or twentieths, or fiftieths? If the dilference between fractions and percentage is not a dif- ference in logical or psychological process, but chiefly a difference in handlincj number symbols, is it worth while to invest the subject with an air of mystery, and invent, for the edification of the pu})il, from six to nine " cases " with their corresponding rules and f ornmlas ? The real facts regarding percentage indicate clearly enough that, to say tiie least, there is no need for this formal treatment and the complexity to which it gives rise. (1) The phrase ^^^y cent^ a shortened form of the Latin per centum, is equivalent to the English word hundredths ; and a rate per cent is, then, simply a number expressing so many hundredths of a quantity. Thus, 1 per cent, 2 per cent, 3 per cent, 4 per cent . . . n per cent means 1, 2, 3, 4 ... ?i of the hundred equal parts into which a given quantity may be divided, ji]st ^s tV) "bV? t^ ' ' ' Tq J^cans 1, 2, 3 ... ^i of the fifty parts into which a quantity may be divided ; or, in gen- eral, as ^i, "I, "I, -^^ . . . represents 1, 2, 3, 4 ... of the 71 parts into which a quantity may be measured off. (2) All problems in percentage involve, then, simply the principles discussed in fractions, and may be solved by direct application of these principles. Indeed, for the mental w^ork with which every arithmetical "7'z) one in percentage ; in the one case a certain quantity is ex- pressed as 3-fiftlis of another; in the other it is ex- pressed as CO-hundredths of it. 2. To find lohatper cent one quantity is of another, {a) What part (fraction) of $325 is $195? (h) What per cent of $325 is $195? In a certain sense question (a) may be said to be indefinite — i. e., any one of an unlimited number of equivalent fractions may be taken as a correct answer. Thus, the answer is i|| = f| = f(=: ^%o^ = ^%o_ etc.). But if the question were, how many 325tlis of $325 in $195 ? How many 65ths of it ? How many fifths of it? — the respective answers to each of these are the first three of these fractions, and they are all found by exactly tlie same reasoning. For example : 1. -^^j of $325 is $1 ; this is contained 195 times in $195; tliere- (; li ^:i . m m I " '< li'' ■ t I il '4.'' ' '( ' ! .r( li h ' \J i 282 THE PSVCIIOLOGY OF NUMBER. fore, $195 is 5|| of $325. 2. ^V o^ ^'"^25 is $5 ; this is contained 39 times in $195; therefore, $195 is §» of $325. 3. ^ of $325 is $65 ; this is contained 3 times in $195 ; therefore, $195 is f of $325. Simihxrly for other equivalent fractions which answer corresponding questions. In question (h) we are, strictly speaking, limited to one answer, but it is found in exactly the same way ; may, in fact, be obtained from any of the unlimited series of fractions that answer question (a). The question really is, how many hundredths of $325 are there in $195 ? We reason as before : j-J^ of $325 is $3i (or $3.25) ; this is contained 60 times in $195 ; therefore, $195 is j%\ of $325. The solutions of these questions might, of course, have been varied by Jirst multiplying $195 by 1, 65, 5, and 100 respectively ; thus, in question {h) the comparison is to be made be- tween $195 and the hundredth of $325 — i. e., how often is $f^|- contained in $195, where (see Division of Fractions) the quantities must be expressed in the same unit of measure, and the division is 19500 (hundredths of $1) -r- 325 (hundredths of $1). In general, the most direct way is to find any convenient fraction expressing the ratio of the quantities, and then change this to an equivalent fraction having 100 for denominator. 3. To find the nuinber of xoliicli a certain per cent is given. (a) A dealer bought goods for $195, which was ^ of cost ; find the cost. (h) A dealer bought goods for $195, which was 60 per cent of cost ; find the cost. In (a) the cost is measured off in 5 equal parts, and PERCENTAGE AND ITS APPLICATIONS. 283 1 TOTT 3 of them are given : 3 of them = $105, 1 of them = $05, 5 of them (the whole) = $05 X 5 = $325. In {h) the cost is conceived of as measured olf in 100 equal parts, and GO of them are given: 00 of them = $11)5, 1 of them = $195 -^ 00, 100 of them (the whole) = $195 ~ 00 X 100 = $325. Here, as in the last case, in accordance with the principle connecting factors and divisors, we might have nmltiplied hy the respective factors before dividing by the respective divisors — e. g., 5 times J of a quantity = -J of 5 times the quantity — that is, $05 ^3X5 ="$05 X 5 -r- 3. Introductory Lesson. — Different teachers will use different devices in applying in percentage the simple principles of fractions. The following points are merely suggested : 1. It will hardly be necessai'y, at this stage of the pupil's development, to use concrete illustrations. It will certainly not be necessary if the pupil has been taught arithmetic according to the psychology of the subject. Begin the teaching of arithmetic with the use of things, but do not continue and end with things. So long as pupils have to use objects, they are apt to attend to the mere practical processes at the expense of the higher mental processes through which alone num- ber concept can arise. The infantile stage of depend- ence on objects is only a stage ; it is not to be a perma- nent resting place ; the method of crawling on all-fours may seriously arrest development. 2. The first aim will be to get the pupil to identify per cents with fractions. He already knows how and why a fraction may be changed to an equivalent frac- tion having any given numerator or denominator. (1) i' n 1 ryrp I! J li I' * ' . \ I ' I . I ' ,1 1; • ! . :!i'l I I :M :ir 2S4 THE PSYCHOLOGY OF NUMIJER. Give, then, exercises exprcssiiiir certain simple fractions in (exact number of) Imndredths: J = 1% ; i — i^/a 5 1 — _2 0_ . JL -- 10 . 1 6 — 10 0? 10 — T tnr ' ITU 5 . 3 TTJIJ 9 ¥ XS 60 . ^\ = ^ijQg., etc. It will readily be seen that a lar«jje number of fractions can be changed into equivalent sliHjle fractions liaving 100 for a denominator ; in other M'ords, into fractions expressing an exact number of hundredths. (2) Then some exercises to show tliat any fraction may be expressed in hundredths : 1 — qqi 100 > 8 Ir* . t = ^i . 1 .. = ^^ = 'l^'ii etc loo 100 2100 100 The pupils already know that multiplication and divi- sion by ten and by a hundred are very easily performed ; in other words, that a number of tenths or hundredths of a quantity is more easily found than any other frac- tion of that quantity ; they will also see that the num- ber of fractions that can be expressed as a whole num- ber of hundredths is much larger than the number that can be expressed as a whole number of tenths; they will probably infer why the ])ractice of measuring off a quantity in hundredths has been so generally adopted. 3. The different ways of writing hundredths will be recalled, and the S3'^mbol for the phrase 2>^^ (^^nt will be given ; for example, 5 per cent has the symbol 5^, and is expressed by y-§-^, 5 hundredths, and '05. 4. Easy mental problems (followed by written work) connecting fractions with percentage, and illustrating the different "cases" of percentage. What fractions are equivalent to the following : 1 per cent, 10 per cent, 25 per cent, 30 per cent, GO per cent, 80 per cent, 90 per cent, etc. ? What per cent of a quantity is -J of it, J of rEJlCKNTACil^] AND ITS APPLICATIONS. 2S5 etc. off a and it, -J of it, -J- of it, J of it, I of it, J of it? Questions like these, together with practical i)rol)lems in the same line, will serve to show the identity in principle he- tween fractions and percentage. Percentage is but another name for fractions. 5. The pupils will be then prepared for more formal problems illustrating the general cases. These are not to be presented as speiyial cases demanding special rules, definitions, and formulas. The thing is to avoid mul- tiplying rules and hair-splitting definitions, and to give the pupil facility in the application of a few simple principles. It has been proved by actual experience that students who never heard of the nine cases of per- centage, and the nine rules or formnlas, have readily acquired the power to handle any problem in percent- age except, perhaps, such as, on account of their com- ])lexity, are more properly exercises in algebraic analy- sis. The pupil should not be confined to any one mode of solution in working problems in percentage. lie will sometimes use the purely fractional form, at others the so-called percentage form, and in still other cases a com- bination of both forms. He should be so instructed in the real nature of the principles and practised in their application as to be able to use all forms with equal facility, and almost instantly determine, in any given problem, which of the forms will lead to the most con- cise and elegant solution. It may be well to utter a caution against the vague use of the phrase jf?!?/* cent^ which too generally prevails. It is often used as if it possessed in itself a clear and definite meaning. It denotes simply a possible mode of measurement. Ten per cent, or one hundred per i ,ii- ■•',.' i;,t ( 280 THE PSYCHOLOGY OF NUMTiKR. cent, lias no more meaning tlian ten or one ; all num- bers sigmiy j}Ofi,Hihle measurements; tliey are empty of meaning till applied to measured qua/ititi/. It is not U!ieonnnon to tind in published solutions of percentage ])roblems (Itff('re)it quantities used as defined by the same unit of measure because they are expressed in per cents. AVe have before us, for example, a solution in which the author takes it for granted that the difference be- tween 110 per cent of one quantity and DO per cent of a different quantity is 20 per cent. *' Let 100 per cent equal the recpiired quantity " is a very conunon pre- supposition in the solution of a percentage problem, and equally common to it to lind the same 100 per cent '' doing " duty for some other quantity which demands recognition in the same solution. So, in a recent Eng- lish work of great pretensions, we have it posted, in all the emphasis of black letter — as a fundamental work- ing principle — that " 100 per cent is 1." One hundred per cent of any quantity — like 2-halves of it, or 8-thirds of it, or 4-fourths of it ... , or n-n\\\% of it — is indeed the quantity taken 07ice, or one time. But this loose way of making " 100 per cent equal to 1," or to any quantity, is due to a total miscon(;eption of the nature of number as measurement of quantity, and of the func- tion of the fraction as stating explicitly the process of measurement. It seems as if both teachers and pupils were often hypnotized by this subtle one hun- dred per cent. Some ArpLicATiONS of Percentage. 1. Profit and Loss. — We do not need either formal cases or formal rules, as " given the buying price and i :■ PERCENTAGE AND ITS APITJCATIONS. 287 the selling price to tiiid the guiii or loss per cent." A few examples will serve to illustrate the different '' cases." (1) Bought sugar for C cents per pound and sold it for 8 cents per pound ; lind the gain 2)er cent. The question simply stated is : 2 cents gain on 6 cents cost means how much gain on 100 cents of cost ; that is, I = how many hundredths ? Multiply hoth terms by lOf (= -Loh)^ ' q,.^ cents outlay ij^ains 2 cents ; 1 cent " " i cent ; . • . 100 cents '' " i X 100 = 3?4 cents. (2) Cloth was bought at GO cents a yard, and sold to gain 25 })er cent ; lind the selling price. Take the cost price as unit of comparison : Selling price =125 per cent of cost = f cost = | of 60 cents = 75 cents. Or, On 100 cents of cost gain is 25 cents. On 1 cent of cost gain is -^-^-^ cent. . • . On GO cents of cost gain is \ cent X GO = 15 cents. Hence 60 cents -|- 15 cents = 75 cents, the selling price. (3) By selling cotton at 12 cents a yard there is a gain of 20 per cent ; what was the cost price ? Take the cost price as unit of comparison : 20 per cent of cost is \ of cost ; therefore, 1^ cost = J cost = 12 cents. Therefore, cost = 10 cents. (•1) By selling coffee at 30 cents a pound a grocer lost 25 per cent ; what price would bring him a proUt of 10 per cent? Selling price = f of cost = 30 cents ; therefore, cost = 40 cents. New price = \^ of cost = ^ of 40 20 I 'hi' Ml h. .'I I! : III''!* 1- 288 THE PSYCHOLOGY OF NUMBER. cents = 44 cents. Otherwise, the losing price, f (Jf) of cost, must be increased to -J-J^ (|-|) of cost — that is, must be increased in the ratio ff ; therefore, f|^ of 30 cents = 44 cents, the price required. (5) A merchant gains 30 per cent by selling goods at 39 cents a yard ; at what selling price would he lose 40 per cent ? Gaining price is |f of cost. Losing price is j\ of cost ; therefore the latter is y\ of the former = -^^ of 39 = 18 (cents). 2. Stocl's, Commission, etc. — A few examples will show that there is no new principle in these rules. (1) How much cash will be realized by selling out $4,000 stock. Government 5's, at 95^ ? $100 stock brings $95^ cash ; $4,000 stock brings $95J X 40 = $3,810 cash. (2) What amount will be realized by selling out $4,400 six-per-cent stock at $106f , allowing brokerage -J- ? Every 100 of stock brings $(106f - \) = SlOOf ; therefore, $4,400 of stock brings $106J X 44 = $4,675. (3) What semi-annual income w'ill be derived from investing $9,000 in bank stock selling at $120 and pay- ing 4 per cent half yearly dividends? $120 will buy $100 stock, which brings $4 income — that is, the income is 4 -j- 120 = -^ of the investment = ^V of $9,000 = $300. (4) Which is the better investment, a stock paying 12 per cent at $140, or one paying 9 per cent at $120 ? What income from investing $1,400 in each ? In the first investment $140 brings $12 income ; therefore, $1 brings %^^-^ = $-^g-. In the second investment $120 brings $9 ; there- ;[|. i 3 /15\ hat is, of 30 ' fjjoods lie lose s j\ of - -6- of es will s. ng out brings ng out rage i ? $1061 ; $•1,675. )d irom nd pay- come — inent = paying t 8120^? Income ; : tliere- PERCENTAGE AND ITS APPLICATIONS. 289 fore, II brings $^^ = $^; $^3^ is greater than $^3_. therefore, the first is the better investment. Income from the first 3T of $1,1:00 = $120 ; Income from the second = ^\ of $1,400 = $105. (5) A commission merchant is instructed to invest $945 in certain goods, deducting his commission of 2i per cent on the price paid for the goods; find the agent's commission. Since the agent receives $2^ for every $100 he in- vests, $102i must be sent for every $100 that is to be invested in goods ; that is, for every $102^ sent, the agent receives $2J ; therefore, he receives 2^ -j- 102^ = jV of the whole amount sent; therefore, amount of commission = $945 -r- 41. (6) For how much must a house worth $3,900 be in- sured at 2J per cent, so that the owner, in case of loss, may recover both the value of the house and the premium paid ? Since the premium is 2^ per cent of the amount in- sured, the property must be 100 per cent — 2^ per cent = 971- per cent = f » of the amount insured ; therefore. If of this amount = $3,900, and the amount is $4,000. (7) What amount nmst a town be assessed so that after allowing the collector 2 per cent the net amount realized may be $24,500 ? The collector gets 2 per cent = j\ of total levy ; there- fore, town gets 11 of total levy ; therefore, || of total hvy = $24,500, and, therefore, total levy =. $25,000. Interest. The pupil, having learned the meaning and the use of the term per cent, should find very little difticulty in the subject of interest. However in the problems of I 'j 'is! rfr'rrfi'* ! D = !:•(: I » H I ■ ) ■ I : ' .t 1 !•;:;' J 'I i i i i'- 200 THE PSYCHOLOGY OF NUMBER. interest and kindred commercial work pupils frequently fail ; but if the cause of the failure is examined into, it will nearly always be found to be, not so much an in- ability to meet the mathematics of the i)roblems, as a want of accurate knowled<^e of the terms used, and of acquaintance with the business forms and operations involved. On this account, in taking up the appli- cations of arithmetic to commercial work, the teacher should be at great pains to ensure that every pupil understands well, and sees clearly through, all such forms and operations. Shnple Interest. — In accordance with what has been said, it is necessary first to explain to the class how men, when loaning money, require a certain payment for the use of the money, and how the amount to be paid for this use — -that is to say, the interest — depends on the time, twice, thrice, etc., the time (implying, as it does, twice, thrice, etc., the use), I'equiring twice, thrice, etc., the interest. The unit of time is generally taken as one year, and the 7'ate for the year is given as a per cent. Accordingly, if we say that a man loans money for a year at 5 per cent per annum, we mean that at the end of one year lie would receive as interest -j-lir ^^ ^^^^ money loaned ; if the money were loaned for half a year the interest would l)e J of y^y- of the sum loaned, and if for fifty-three days, it would be -^^ of ^^ of the sum loaned. The pupil is now prepared to do any problem of calculation of simj)le interest, and after be- ing trained in the formal working and stating of such problems — that is, after realizing the problem to the full — should be trained in making rapid calculations after the methods of men in business. PERCENTAGE AND ITS APPLICATIONS. 291 lently ito, it an iii- i, as a Lud of ations appli- gaclier pupil [ such s been ,v men, for the aid for on the it does, as one er cent. y for a the end of the • half a loaned, T^of » do any ifter be- of such L to the julations He should next be led to see the relations among the interest, the sum loaned (the principal), and the sum called the amount. Suppose the sum loaned to be $100, the time to be six months, and the rate 6 per cent per annum. Take the line A B to represent six months, A the beginning of the time, and B the end of the time. $3 interest. Principal : $100 $100 principal. A B At the end of the time the sum $103 has to be paid to the loaner— that is, the $100 has to be restored, and $3 paid as interest. The sum, $103, is called the amount. It is plain then that (1) The interest = -^ of the principal ; — T§3 ^^ t^^G amount. (2) The principal = ^p- of the interest ; — iSt of ^^^6 amount. (3) The amount = if2. of tlie interest ; — TW of the principal. The use of a line to represent time will assist the pupil greatly, and after examining a few examples similar to the foregoing, he will knoiv all the relations among prin- cipal, interest, and amount, and will see how to write them down when the rate and the time are given. When these relations are understood, tlie whole subject of interest is understood, the only care required on the part of the teacher consisting in making a careful gradation of problems. One of the most striking applications of interest is to problems relating to the so-called true discount — a term which should fall into disuse. There is but one dis- l.ir 1,1 ft III m ^VfvT^ 292 THE PSYCnOLOGY OF NUMBER. i LI W: "i- 1 ,r -' count, the discount of actual business life ; it is an ap- plication of percentage, and on account of its being calculated in the same way as interest it is erroneously spoken of as interest, and a confusion arises in the mind of the pupil. Accordingly, the problem, Find what sum would pay now a debt of $150 due at the end of six months, the rate of interest being 6 per cent per annum, is a definite problem in interest. To solve it we have re- course to the line ilhistration given above. It is plain that if one had $100 now, and put it out at ijiterest at the rate given, it would come back at the end of the time as $103. Thus, $100 now is the equivalent of $103 at the end of six months — that is, the sum now, equivalent to a certain sum due at the end of six months, is \^^ of that sum. Therefore, in the case in question, the sum is UJ of $150. It is true that there is here an allowance off, a discount, so to speak, but until the pupil understands the whole question of inter- est and discount the term should not be used in this connection. We shall suppose, then, that the student has mastered simple interest, and shall turn to com- pound interest. Com/pound Interest. — The teacher should explain that the value of money — as the pupil has seen — de- pends, in some measure, on where it is placed in time ; men in business always suppose interest to be paid when it is due, or if an agreement is made that its payment be deferred, they regard this interest in its turn a source of interest. An example worked out in detail will help the pupil to see just what is done. Suppose a sum of ,000 loaned for three years, at 5 per cent per annum, PERCENTAGE AND ITS APPLICATIONS. 293 interest to be paid at the end of the three years, and the interest at the end of each year to become a source of interest for the ensuing year or years : $10 Principal. 5 $5 0.0 First year's interest. $10 Sum bearing interest for the 2d year. $105 5 $5 2.5 $10 5 0. $110 2.5 5 $5 5.1 2 5 1 1 2.5 $1 1 5 7.6 2 5 1 0.0 $1 5 7.6 2 5 Second year's interest. Sum bearing interest for the 3d year. Third year's interest. Amount to be paid at end of time. Original principal. Amount of interest. The pupil will work several such examplefi, and will find not a little pleasure in determining just how much interest has been paid as interest on interest. He is then ready to make a more general study of compound interest. Suppose a sum loaned at compound interest for three years at 5 per cent per annum. What is the interest on any sum for one year at 5 per cent ? Plainly yj^ of the sum. What is the amount ? -fH- of the sum. What, then, is the amount of any sum for one year? -ffj of that sum. What sum bears interest for the second year ? fJJ of the original sum. What will the amount I ! ,, %m ■ ^^ II .it ^H'fTff'^ 294 THE PSYCHOLOGY OP NUMBER. ii'i lit r?- Mi' i III , ll. ■f. 'If of this be? |J| of itself, and therefore {^ of fg J of the original sum. Accordingly, the amount of the sum for two years is (xg-f)' of the original sum. What for three years? Plainly -}-§|- of (y§§-)' of original sum, and therefore ({-§■§/ of original sum. This is found to be "HfJ^I-J- of original sum. How much more have we than the original sum ? tVoVWjt ^^ original sum ; therefore, interest — iVVoVo^o ^^ sum — xVVtuVs" of amount, etc. The pupil should be told that in all transactions in- volving a time longer than one year (or it may be by agreement six months or three months) compound in- terest is alone employed where tlie interest is thought of as all being paid at the end of the time. From what has been said he will know at once how to solve the following problem of interest : Find what sum paid now will discharge a debt of $1,000, due at the end of three years, the 7'ate of interest being 6 per cent. He should acquire a facility in thus transferring money from one time to another. Annuities. — Afew words may be said on the sub- ject of annuities. If A gives B $100 to keep for all time, and the rate of interest be 6 per cent, B would be undertaking an equivalent if he would agree (for him- self and his heirs) to pay to B (and his heirs) $6 at the end of each year, for all time. This $0 supposed paid at the end of each year is called an annuity ; as it runs for all time, it is called a perpetual annuity, and is said to begin now, though the first payment is made at the end of the first year. The $100 is very properly called its cash value, and the relation of the $100 to the annuity of $6 is plainly that of principal to interest. Thus, it WW lly:; 3 sum at for sum, found more •iffinal )7635 ns in- be by nd in- lousjbt 1 wbat ^e tbe 1 paid end of t. He money le sub- for all Duld be )r him- ) at tlie paid at uns for said to tbe end died its annuity Thus, it PERCENTx\GE AND ITS APPLICATIONS. 295 will be easy to find tbe cash value of any given per- petual annuity, or to find the perpetual annuity that could be purchased with a given sum. To illustrate this we should need a line extending beyond all limits : $100 $() $6 $() $() $0 $0 $6 > (The divisions of the line represent each one year.) Kext we may su})pose an annuity to begin at the end of, say, three years, so that the first payment would be made at the end of the fourth year. Taking the annuity to be $6 and the rate 6 per cent per annum, we see that the value of this annuity at the beginning of the fourth year (represented by the point in the illus- tration below) is $100. $100 $0 $0 $6 I B C D E F J H > But that $100 is placed at the end of three years from now, and is therefore equivalent to (fg^)' ^^ ^^^^ "^^^'• We have thus the cash value of an annuity deferred three years. When the pupil knows how to deal with the two cases discussed he can easily be led to find the cash value of an annuity beginning now and running for a definite number of years. When asked to compare the two perpetual annuities represented below, he will see that the first exceeds the second by three payments — $6 at the end of the first year, $0 at the end of the sec- ond year, and $0 at the end of the third year, and these constitute an annuity for three years beginning now. \\ M ■f|Tr— w ! m :■ i.-,r' . I ; ; ' i I, : ; I , ■ 296 THE PSYCHOLOGY OF NUMBER. $6 $6 $6 $0 $0 $() 86 $6 1 1 1 1 1 1 1 1 $6 1 $6 $6 $6 $6 $6 1 1 1 1 1 1 1 1 $6 1 > > But the cash vahie of the first annuity is $100, and the cash value of the second is (|^J)' of $100. . • . The cash vahie of an annuity of $6 beginning now, running for three years, is $100 — (-}^§)' of $100 or U -(«!/} of $100. It will be easy to obtain a general formula, and also to find the value of a deferred annuity running for a definite number of years. m ii ■■if' i : f :'i ■ f '■ri t ! CHAPTER XYL EVOLUTION. Square Root. — The product of 3 and 3 is 9 ; of 5 and 5 is 25. The measures of squares whose sides measure 3 and 5 are 9 and 25. We say that 9 is the square of 3, and that 25 is the square of 5 ; 3 is the square root of 9, and 5 the square root of 25. The square of 3 is written 3", the square root of 3 is ex- pressed thus : V^d. The pupil can write at once the table of squares : 1'= 1 2»= 4 3'= 9 4" = 16 5' = 25 [10' = 100] 6' == 36 7^ = 49 8' = 64 9" = 81 He will note that the square of any number expressed by one digit is a number expressed by one digit or by two digits, while the lowest number expressed by two digits — viz., 10 — has for its square 100, a number ex- pressed by three digits. 297 !-i 1"^? w: f i 1 'i 1 1 1: 1 wm It .(• ' !:;■'•■ In: lit : ■ i K f 1 , ^j 4 - : ■J. t > ill . ^ m \ ^. ; I'll : i;jL ^ ■ ttiiiL 298 THE PSYCHOLOGY OF NUMBER. It is plain that the square of any number expressed by two digits has for its square a number expressed by three digits or by four digits. Also the square root of a number* expressed by three digits is a number ex- pressed by two digits, and the tens digit is known from the first digit on the left ; for example, 025 (if it has an exact square root), lying as it does between 400 and 900, will have for square root a mimber lying between 20 and 30 — that is, the tens figure of the root will be 2. Similarly, if a number is expressed b}^ four digits its square root is expressed by two digits, and the tens digit of the root can be determined from the first two digits (to the left) of the number ; thus the square root of 2709 — a number lying between 2500 and 3600 — will have 5 for a tens digit, and this is determined by the 27 of the number 2709. AVrite next the table of squares ; 10' 20" 30" 40' 50' 60' 70' 80' 90' 100 400 900 1600 2500 3600 4900 6400 8100 [100' = 10000] Now take 13 and square it : the result is 169. We wish to arrive at a method of recovering 13 from 169. * In general, when we speak of the square root of a number, wo suppose that it has an exact square root. EVOLUTION. 299 To do tliis we shall examine how the 1G9 is formed from the 13 : 13 13 • 9 30) 30 f 100 169 Thus 13, which is made up of two parts, 10 and 3, has for its square a number 169, which is seen to be made up of 100, the square of 10 ; 9, the square of 3 ; and twice the product of 10 and 3. This is familiar to the pupil who has worked algebra, and may be ilhistra- ted geometrically. The whole square ^^ is measured by 13", and its parts make up 10' + 2 X (10 X 3) + 3'. Now, to recov- er 13 from 169 : we see that its hundreds digit 1, showing that the number lies be- tween 100 and 400, gives' the tens digit of the root, so that we know one of the parts of the root, viz., 10. The square of this part is 100, and the rest of the given number, 69, must be 2 times 10, III 8 ipji' ?'! IT?: '(■ ' .•'If '*i 300 THE PSYCHOLOGY OP NUMBER. multiplied by the other part, together with the square of the other part. 1GO|10 lUO 69 Accordingly, if we multiply 10 by 2 and divide 09 by this product we get a clue to the other part. Dividing 60 by 2 X 10, or 20, we see that the quotient is a little greater than 3 ; if, then, after taking 3 times 20 from 09 there is left the square of 3, we have the root. Plainly this is the case : ' 100|10 + 3 100 20 01> CO 9 = 3' Now, this work might be written somewhat more neatly, thus: 109|10 + 3 100 20 ) + 3) 09 09 It may be further simplified by leaving out unneces sary zeros, thus : 10913 1 23 09 69 The pupil is now in a position to find the square root of all numbers expressed by three or four digits. It would be well, before considering the squaro root of li ! EVOLUTION. 301 larger mimbcrs, to exaniiiio for the square root of sueli numbers as IMJO, 27'09. The pupil will see at ouco that the scpuire root of the former number lies between 1 and 2, that of the latter between 5 and 0, and can easily be led to complete the process of extracting tlie roots, iinding as results 1*3 and 5*3. He will thus dis- cover for himself that the problem is not different from the one already solved. We are now ready to examine for the square root of larger numbers. Write first the following table : 100"= 10000 200'= 40000 300'= 90000 400' = 160000 500' = 250000 [1000' = 1000000] 600' = 360000 TOO' = 490000 800' = 640000 900' = 810000 A study of the table will lead to the conclusion that the square roots of numbers expressed by 5 or 6 digits are numbers expressed by 3 digits, and that, if expressed by 5 digits, the first (to the left) digit of the root is determined by the first (to the left) figure of the num- ber, and if expressed by 6 digits, by the first two digits of the number. Thus, the square root of 16900 will have 1 as the hundreds digit, while that of 270900 will have 5 as the hundreds digit. Next, by multiplication, we find that 230' = 52900 and 240' = 57600 'I iii 802 THE PSYCnOLOGY OF NUMBER. I' / I ; I ,1 1 Si I 1 . 1 c m ' i- !i 1 !i lii Conserjnciitly the square root of (say) 5475G must lie be- tween 230 and 240 — that is, mns't have 23 as its first two digits. The first tliree digits of the number 54756 are sufficient to show that this must be the case. Kow, sup- pose we seek the square root of 54750. Phiinlvj the first part of the root is 200 : 5475G|200 40000 "14750 Now, had we been seeking the square root of 52900, wliich is 230 — that is, consists of two parts, 200 and 30 — we should have worked thus : 529001200 + 30 40000 200 X 2 - 400 ) 30 S 12900 12900 for 57600 : 57600 200 + 40 40000 200 X 2 - 400 ) + 40 i 17600 17600 Then plainly we see how, in finding the square root of 54756, to determine the second figure : '?i I 200 X 2 = 400 I + 30 i 54756 1200 + 30 40000 14756 12900 1856 EVOLUTION. 303 ; lie be- rst two T56 are i\v, siip- 52900, >00 and 3 root of We have yet to find the units digit of the root. But at tins point we may say that the root consists of two parts, one 230, and the other to be foun^ and may pro- ceed as in the earlier case : 400 I 430 230 X 2 = 460 I 464 54756 1200 + 30 + 4 4000J) 1475^ 12900 1856 1856 The work may now be shortened : 54756|234 43 464 147 129 1856 1856 After the pupil has been exercised in extracting the roots of numbers expressed by 5 or 6 digits, he will find no difficulty in determining the square roots of such numbers as 547-56, 5-4756, '054756. The extension to numbers expressed by a higher number of digits will be easy, and the need for marking off into periods of two, starting from the decimal point, as well as its full significance, will have been realized by the ^""^Up to this point we have spoken of numbers whose 21 myi'r I « A.: m '> »! \• 4' - If, ) 436 1744 -/.'V' ll I'i .'Ik 1 ■ 308 THE PSYCHOLOGY OF NUMBER. I; ^i! «i B "II, I: 4;i ■|1. ,1, ''1 i i 1 1,1 ' ' ■ 1 1 1 1 m. It may be further sliorteiied, thus : 2744|14 1 1' X 300 = 300 1 X 4 X 30 = 120 • ^'^ 16 436 1Y44 1744 The further development of the method will follow lines similar to those followed in square root. We shall take spLr . Li .',y to indicate how, in the case of finding a cube root cohsi ling of several figures, a certain saving of work may be s^c^^red ; 814 780 504|934 729 OO^'XS 3 X 90 X 3 3' 24300 = 810^ = 9 3 X 90'+ 3 X (3 X 90)4-3'= 25119 ^ 9 2594700 11160 16 2605876 85 780 75 357 10 423 504 10 423 504 When ve reach the point where we wish to determine the third figure, we have to find three times the square of 93 — that is, 3(90*+2X90X 3 + 3') Kow, as indicated above, 810 is 3 X 90 X 3, 9 is 3', 25119 is 3 X 90' + 3 X 90 X 3 + 3', so that, if to the sum of 810, 9, 25119 we add 9, which is the square of EVOLUTION. 309 How ;hall nng 3, we shall have found three times the square of 93. If to the resulting number we affix two zeros, we shall have three hundred times the square of 93. This artifice may be employed when at each suc- cessive stage we need three hundred times the square of the part already found. We shall conclude this chapter with the remark that the fourth root of a number is to be found by extracting the square root of its square root, and the sixth root of a number by extracting the square root of its cube root. |934 line lare THE END. i3', the e of wr^ ■1 1 « ' ;i!' ',' fin [ ' \< m M \r\ D. APPLETON & CO.'S PUBLICATIONS. New Volumes in the International Education Series. "J^HE PSYCHOLOGY OF NUMBER, and its •^ Application to Methods of Teaching Arithmetic. By James A. McLellan, a. M., LI.. D., Principal of the Ontario School of Pedagogy, Toronto, and John Dewey, Ph. D., Head Pro- fessor of Philosophy in the University of Chicago. i2mo. Cloth, .$1.50. It is believed that this work will supply a special want. There is no subject taught in the elementary schools that taxes the teacher's resources as to methods and devices to a greater extent than arithmetic, and none that is more dangerous to the pupil in the way of deadening his mind and arresting its development, if bad methods are used. The authors of this book have presented in an admirable manner the psychological view of number, and shown its applications to the correct methods of teaching the several arithmetical processes. n^HE MOTTOES AND COMMENTARIES OF -^ FRIED RICH FROEBEVS MOTHER FLA V. " Mother Communings and Mottoes " rendered into English verse by Henrietta R. Eliot, and " Prose Commentaries " translated by Susan E. Blow. With 48 full-page Illustrations. i2mo. Cloth, $1.50. The increased interest in kindergarten work and the demand for a clearer exposition of Froebel's philosophy than has heretofore appeared have made a new version of the ' ' Mother Play " an imperative necessity. No one is better equipped for such a work than Miss Blow, as her late book, " Symbolic Education," has attested. It is an attractive volume of a convenient size, and a book of specific value to mothers as well as to teachers of every grade. It will be followed shortly by another volume contain- ing the songs and games. PRIEDRICH FROEBEVS PEDAGOGICS OF J- THE KINDERGARTEN ; or. His Ideas concerning the Play and Playthings of the Child. Translated by Josephine Jarvis. i2mo. Cloth, $1.50. This book holds the keynote of the " New Education," and will assist many in a correct comprehension of the true principles underlying the practical outcome of Froe- bel's thought. Although extant for nearly fifty years, his ideas are still in need of elucidation, and the avernge kindergartner and primary-school teacher giasps but a superficial meaning of the methods suggested. rHE EVOLUTION OF THE MASSACHU- SETTS PUBLIC-SCHOOL SYSTEM. A Historical Sketch in Six Lectures. By George H. Martin, A. 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