IMAGE EVALUATION TEST TARGET (MT-3) 1.0 l.i - Itt III 22 iMo III! 2.0 1.8 i 1.25 1.4 1.6 ^ 6" ► V] signifie "A SUIVRE", le symbole V signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent etre filmds 6 des taux de reduction diffdrents. Lorsque le document est trop grand pour etre reproduit en un seul ciichd, il est film6 d partir de Tangle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 2 3 1 2 3 4 (n 1 ■ 6 i THE GROUND- WOKK OF NUMBER ^.a THE GROUND-WORK OF NUMBER A MANUAL — FOR - THE USE OF PRIMARY TEACHERS. BV A. S. ROSE AND S. E. LANG, Inspkctors of Schools, Manitoba. TORONTO : THE COPP, CLARK COMPANY, LIMITED. 1898. Co«,.AXv, !.,„,„„, Torcto, i„ ,h,„„„e .,, the MiLter oXhcSu' re. PREFACE. Ill this manual an attempt has been made to explain the nature of number, and to set forth the conditions of clear thinking, with special application to the study of arithmetic. The main propositions are that num- ber is a relation based upon the idea of time and not that of space, and that all thought begins in analysis. From the former of these, certain inferences are drawn which have an important bearing on the teaching of the subject. It will be seen that in this view of the case all arithmetic which denls with measurement of space is applied arithmetic ; that in the scientific study of number the use of objects is not only unnecessary, but also inconvenient, and even destruc- tive of clear numerical ideas ; that no true idea of a fraction can be formed from a consideration of the spatial relations of objects. The second proposition furnishes us with a test for the validity of all methods of procedure in the acquisition of knowledge. CONTENTS CHAPTER I INTRODUCTION. 1. Arithmetic, the Type of a Deductive Science. 2. Num- ber, a Product of the Thouglit Power. 3. Number Distinguished from Form. 4. Number Distinguished from Products of Perception. 5. Number not a Generalization from Experience. 6. Number and Imagination. 7. Definition. 8. Conditions under which the Idea of Number Arises. 9. The Work of the Thought-Power. 10. Number and Language. 11. Tlie Real Value of Arithmetic— Degrees of Difficulty in Certain Judgments. CHAPTER IT. RETROSPECT. 1. The Old Methods — their Fundamental Weakness. 2. Grube's System — the Place of Sense-Perception, Physical "Operations," the Space Element. 3. The "Extensive" School. CHAPTER IH. THEORETICAL. 1. The Two-fold Movement of Mind. 2. The two Phases in Relation to Progress and Development. 3. The Dictum — Explanation and Illustration. 4. Every "Operation" a Tyt)e of the Analytic- Synthetic Action of Mind. 5. The Order of the Arithmetical Operations. 6. Tables — their Value. vii vni COXTEXTS. CHAPTKIl IV. PRACTICAL. 1. Adc(iUHto Knowledge «)f Number. 2. The Number Three. 3. Tlio Num1)er >S7x. 4. Tlio Lino of Least Resistance. 5. Age at which the Science of Arith- metic Should Begin. 6. The Use of Objects. 7. Pictorial hnages and Thought-Processes. 8, Oral and Written K.vercises Compared. {). Difficulty of Problems due to Lack (jf Data. 10. Notation. CHAPTKU V. EXEKCISKH. 1. The Number Six — Pure Arithmetic and Applierl. 2. The Number I'en. 3. Exercises on Ten as the Base for all Numerical Thinking. 4. Notatin the other hand, in the relation to which their distinctness is thus necessary, they are at the same time united. Bitt if it were not for the action of something which is not either of them or both together, there would be no alternative between their separateneas and their fusion." —Green : Prol. to Ethict, 28-99. 4 THE GROUND-WORK OF NUMBER. thinking ever takes place. Thought, as has often been pointed out, consists in the establislnnent of relations. In thinking we organize into syste- matic relations that which would otherwise be separate and discrete in our experience ; we unite ideas in thought because they appear alike, or keep them apart because they seem different ; the idea of causation is the bond or basis of rela- tion between certain ideas ; we relate objects and events in space and time — these are examples of the various ways in which thought introduces system into the facts of our experience. Thus we see that to think is to order and arrange the materials of thought into unities of various kinds. But thought does not stop with the mere act of relation. By a process of abstraction we may become conscious of the relations them- selves: abstracting from the ideas related, and fixing our attention upon the coiniection be- tween them, we reach such ideas of relation as resemblance, difference, causation, quantity, and number. All ideas of relation, then, are due to the activity of the thought powder, and number is a product of one of the phases of this mental activity. .1 INTRODUCTION. 3. Tlie science of number is to be distin- guished from that of form. Wo liave seen tluit the action of tlie mind in giving order and colie- rence to tlie impressions of sense, gives rise to various ideas of relation. The science of geom- etry is made possible by the existence of one of these products of mental action in relating phenomena in the form of space. Arithmetic deals with number, geometry with form. In numbering, the mind organizes the data of sense in a time relation. It matters not which of tlie sense organs supplies the data, the organization of the data is in every case performed in the same way. The same form or bond, — that of time, — unites the impressions of sense in the act which gives rise to this product; and into this product the element of space does not enter. No analysis of the idea of space can ever bring forth the idea of number. Objects, in so far as they are conditioned by the relation ■f space, are co-existent. But they can be numbered only in so far as they can be thought of as discrete, i.e., as successive. It will be seen that the possibility of numbering objects in 6 THE GROUND-WORK OF NUMBER. I!:; .• I II A space depends upon our considering them in the time relation.* * " If you wheel aboufc a burning coal with rapidity, it will present to the senses an image of a circle of fire ; nor will there be any interval of time betwixt two revolutions ; meerly because 'tis impossible for our perceptions to succeed each other with the same rapidity, that motion can be communi- cated to external objects. Wherever we have no successive perceptions, we have no notion of time, even though there be a real succession in the objects." — Hume: Treatise on Human Nature, Part 2, Sec.S. ' ' I have already observed that time, in a strict sense, im- plies succession, and that when we apply its idea to any unchangeable object, 'tis only by a fiction of tho imagination by which the unchangeable object is supposed to participate of the changes of the co-existent objects, and in particular of that of our perceptions. The fiction of the imagination almost universally takes place ; and 'tis by means of it that a single object placed before us, and survey 'd for anytime without our discovering in it any interruption or variation, is able to give us a notion of identity. For when we consider any two points of this time, we may place them in different lights : we may either survey them at the very same instant, in which case they give us the idea of number, both by themselves and by the object : or, on the other hand, we may trace the succession of time, by a like succession of ideas, and conceiving first one moment, along with the object then existent, imagine after- wards a change in the time without any variation or interrup- tion in the object ; in which case it gives us the idea of unity." — Hume: Treatise on Human Nature, Part U, See. 2. '* Kant concludes his theory of the a priori forms of sense by noticing that the two forms are not altogether independent of each other. For if we compare them we see that the space is a determination of objects as such, while time has to do with states, and principally with the states of the subject as repre- senting and perceiving objects. Hence, time stands between the conceptions of the understanding, and the perceptions of sense. — Caird : The Critical Philosophy of Kant, Ch, 5. INTRODUCTION. 4. The idea of number differs both in kind and origin from ideas of sound, touch, colour, and others which are due to the sense organs.* The senses furnish us with knowledge of the qualities of objects. Number is not a quality discernible in objects by means of any actual or possible sense organ. A very little reflection is sufficient to show that number is not due to the perceptive power. If it be held that the idea is in any way derived from sensation, we are entitled to ask from which department of sense, or from what combination of sensations, we get the idea. It is not due to the sense of sight, since colour is all that we get from that source ; it is not due to hearing, or it would be a sound ; and so of the others, touch, taste, and smell. The most delicate susceptibility to colour, sound, etc., is compatible with vague ideas of number. A person may be trained to appreciate slight differences in the various aspects, — inten- « « No one is more emphatic than Locke in opposing what is real to what we "make for ourselves," the work of nature to the work of the mind. Simple ideas or sensations we cer- tainly do not "make for ourselves." But relations are neither simple ideas nor their material archetypes. They therefore, as Locke explicitly holds, fall under the head of the work of the mind." —Green : Prol. to Ethict, § SO. 8 THE GROUND-WORK OF NUMBER. sity, extent, duration, — of a given sensation. This power of discrimination may be developed to a very high degree. But such exercises can- not give the observer any ideas of number. Nor, on the other hand, is the study of number suited to the development of the powers of observation. The power to form correct ideas of numerical relation is not in the smallest degree dependent upon the extent of one's training in distinguishing shades of colour, or exactly esti- mating distances, or noticing differences in pitch of tones. 5. Ideas of numerical relation are not general- izations from experience. The term generaliza- tion indicates that an idea has been formed as the result of a process of reflection, and is capa- ble ctf being applied equally to each member of a class of similar objects. When individuals are found to agree in the possession of certain com- mon attributes which distinguish them from other individuals, we collect them into a class, and assio-n to it a name. The name serves not only to denote each individual of a class, but also to suggest the abstract notion of the circum- stances in which these individuals were found to INTRODUCTION. 9 agree. It is obvious tliat we can establish a scale of general notions, in which the names stand for relatively larger and larger classes of individuals with con-espondingly fewer and fewer similarities. The terms employed to indi- cate higher and lower classes are genus and species, terms which are purely relative to each (jtlier. One of the chief features to be noticed in ' the nature of all ideas which have been formed in this way, is the fact that such ideas are con- stantly undergoing a process of modification in each individual's mind. The sciences of nature furnish numberless illustrations of tliis. Com- pare, for example, the scientist's idea of granite, or of a maple tree, with his idea of the same when a child. Compare your present idea o^ gravitation with your idea when you fust heard of it. All ideas reached as the result of a pro- cess of observation and classification of facts are subject to this tendency. A single newly dis- covered fact may modify an accepted scientific theory. The discovery of a new species may necessitate a change in the connotation of the genus. It is not, however, by an imaginary separation of qualities of objects into like and unlike, and the formation of a class upon this 10 THE GROUND-WORK OF NUMBER. basis, that we reach the idea six. There is no group of qualities known under the name six, which is possessed equally by all the members of a class. When objects are numbered, e.g., blocks, or books, or chairs, it is not in their v^haracter as possessing certain qualities, it is not as blocks, as books, or as chairs, that they are considered, but simply as discrete units, and hence as capable of being counted.* Further, we have no numerical ideas of which it can be said that they possess a higher degree of gener- ality than others. The objects to which the term six may be applied are not restricted in any way by the presence or absence of certain qualities. And again, ideas of numerical rela- tion differ from general notions in regard to the tendency to become modified. There is no such tendency to alter, there is no alteration possible in an idea of number, once correctly formed. After an analysis of what is involved in the number six, no new experience can add to, or in any way modify, our ideas concerning it. 6. Critics speak of the "style" of a literary production, meaning thereby an element which * See "Comte, Mill, and Spencer," (p. 82) by Dr. Watsfon. INTRODUCTION. 11 is due, not to anything in the nature of the subject itself, but to tlie way in wliich it was conceived and expressed, an element due to certain characteristics in the mental constitution of the writer. It may be safely affirmed that no two individuals conceive any given historical transaction in exactly the same way. If a statesman, a political economist, a military critic, a poet, a patriot, and an army contractor, were each to write an account of the battle of Water- loo, no two of the accounts would exhibit the same mental conception. Each account would be coloured by the personality of the writer. The facts would be regarded in different rela- tions in each case. The points of view would be different. Such is not the case with arithmetic. The idea of any numerical relation, if appre- hended rightly, is apprehended in the same way by all men. When we say that numerical ideas are not generalizatioiiS from experience, that no added experience can produce any change in an idea of numerical relation once correctly formed, that numerical ideas, if apprehended at all, are appre- hended in the same way by all men, that the 12 THE GROUND- WORK OF NUMBER. Ill i Hubject of arithmetic is not .suited to tlie de- velopment of the power of im{i<,nn.'ition, — all this is expressed shortly by saying that arith- metic is an exact science. 7. A number is an abstract idea, and involves these factors : a unit, taken discretely, so many times, to measure a collection or multifiuJe of things. The ratio between the unit and the multitude is tlie number. The expressions three, three pairs, three do/en, three score, all iniply repetition. That which repeats or is repeated is different in each case. The unit or measure of the group, collection, or multiplicity of things to be numbered may be the simple unit, or it may be some other unit, a couple, a dozen, a score, or what not. In any and all of these cases, there is a relation discerned between the unit employed and the multitude of things to be measured. The relation, e.g., between a group of things, — marbles, houses, or musical notes, — on the one hand, and the unit or measure on the other, may become the object of attention, and this relation is a number. We have the unmeasured multi- tude, the unit, and the (numerical) relation. It is clear that tlie same collection may be taken INTRODUCTION. 13 in relation to any ono of tlui possible units, and that with every change of unit we have a change in the relation, that is, in the number. The term ratio expresses the fundamental idea in numerical relation. A number is a ratio. If the foregoing is correct; if the idea of number does not originate in experience by the activity of sense-perception; if it is not an idea compounded of different sensations : it must follow that lunnber is a product of the thought power ; and the laws which govern thought activity in general will be found to operate in our apprehension of numerical relation. 8. The thought process may for convenience be considered under two heads, analysis and synthesis, although these are in reality but two phases or aspects of the one mental process. All tliouut what think you then, Glaucon, if a piison should ask them— you wonderful clever men, about what kiiul of nuud)er3 are you reasoning ; in which unity such as you deem it, is equal, each whole to the whole, without any difference whatever, and having no parts in itself? What think you they would reply? This, as far as I think, that they speak of such num- bers only as can be comprehended by tlie intellect alone, ])ut in no otlier way. You see, then, my friend,' I ol)served, 'that our real need of this branch of science, is ])robably liecause it seems to compel the soul to use pure intelligence in the siarch after pure truth.' " —Plato's liepublic, Eouk vii, Chap. S. (Davis Tr.) 22 THE GROUND-WORK OF NUMBER. nature of thinking, we may consider a few examples. A comparison may then be made between the various kinds, of which it is usual to distinguish those judgments which concern resemblance and difference, and from which result classification and definition of species; those which concern the relation of cause and effect ; and those which relate to quantity and number. The fusion of a number of concrete images into the so-called generic image, or type form including common features, is " largely passive,"* and probably accomplished without aid from language ; for example, when the child has formed a mental image of a dog. Abstraction and comparison are involved in making the transition from this pictorial image to the con- ception properly so-called. When in the course of later experience of animals the child observes marked difierence in size, colour, or temper, there will follow a seric3 of judgments in which pecu- liar features already incorporated in the generic image are set aside, and a new and more definite idea foniiad. It is by processes of this kind '*Sep I. i \ Psychology, Descriptive and Explanatory, I INTRODUCTION. 23 repeated many times in the experience of the individual, and fixed in later stages by the help of language, that the true general notion is finally reached ; and the term dog, metal, or animal, stands for this notion, which is found upon reflection to imply a certain definite set of qualities, and to apply to certain individuals. After having noted a few repetitions of a certain sequence of events, a child will begin to attribute an observed effect to personal agency. If the incidents are of practical interest, and particularly if he himself is the actor, the element of expectation comes in to deepen the idea of relation thus established. Later on, cases occur which do not agree with his previous experience, and these stimulate and guide him in his transition from the earlier stage to one in which genuine inference is made. It is probable that the establishment of numerical relations (see page 14) involves a greater idea of abstraction than judgments of resemblance and difference, those of causation or those of spatial qualities and relations ; that the idea of a class of things, the question, "Who broke the cup?" the idea of a circle or a straight 24 THE GROUND-WORK OF NUMBER. line, all of them involve mental processes capable of being performed by a child who is unable to form the abstract idea of six. It is true that young children may be taught to perform some interesting operations witli marbles, spools, pebbles, etc., and, like the Japanese tradesman, with his sorohan, get fairly correct results ; but the operations are physical operations, and may be performed with ecjual speed and greater accuracy by a calculating machine. What is referred to liere as involving a real effort of attention is the tnental process of establishing a numerical relation. The study of arithmetic, then, should be to the pupil an excellent means of mental discip- line. The name of arithmetic lias often been made a cover for tnany kinds of exercises from which thought is conspicuously absent. Its real function — that of giving employment to the fac- ulty or powder " properly denominated thought," tliough allowed in theory, has not generally been recognized in practice. CHAPTER 11. llETR()SPK(n\ 1. The use of the plirase, " the old methods," iniphes that there was an aim kept consciously in view by the school masters of a generation ago. This seems to liave been to make pu})ils accurate and rapid in performing certain "operations," and in "solving problems." The " method " by which these results were reached may be described briefly. It consisted in plenty of practice in doing such work according to rule. It would be unfair to say that this system prevented the possibility of exercising the rea- soning power. A bright boy often discovered for himself the reasons which und(>rlay certain of the rules, and " worked " the questions with a clear idea of the relation between process and result. There was nothing to prevent a kindly disposed or enterprising teacher from stepping out of the beaten course and makino- clear the reasonableness of a formula. It is enouoh to say. however, that while arithmetic was in the 2") 26 THE (J ROUND-WORK OF NUMBER. stage of rules and formulas, a boy might go through the course successfully, might reach a high degree of skill in doing sums and problems, without much exercise of the reasoning pow^er, i.e., by a steady and slavish adherence to the prescribed rules. The tendency, with such aims professedly in view, was to discourage reason- ing and encourage the mechanical performance of tasks consisting chiefly, if not wholly, in imitation. Great emphasis was laid upon a " knowledge " of the multiplication table, as an indispensable tool in the performance of these tasks. The great thing was to get ready and keep ready for use one hundred and forty-four statements of abstract truth. These statements were com- mitted to memory, parrot fashion. This pro- cedure effectually prevented the exercise of the thought power in the consideration of pure number. There was no attempt to relate one truth to another. There could not be, for the simple reason that it was the words that were learned and not the truths underlying the words. The mainstay of the general run of schools was not the teacher, but the text-book, and the •i RETROSPECT. 27 mainstay of the teacher of arithmetic was the key containing the answers to the problems. The chief defect in most text-books was that the subject was presented, not in the way in which it might best be apprehended by a young mind, but in the way in which a finished scholar might be supposed to review it. An example of this may be seen in the use of large numbers, long rows of figures representing something of which the child had, and could have only the faintest and vaguest notion. The only attempt made to clear up the meaning of these expressions was the early introduction of a chapter on notation and numeration. The result of this, however, was usually to increase, if possible, the confusion in the learner's mind between figures and num- bers. When this took place, and there were not many cases in which it did not take place, there was a complete absence of thought proper on the part of the child engaged in the manipulation of long rows of figures. Figures and not numbers being the object of attention, it is easy to see that there could be no idea of relation, and hence none of numerical relation. The difficulty generally and erroneously supposed to reside in fractions was due to the same cause. The habit 28 THE (UIOIJND-WORK OF NUMBER. of con.sidering only the fi<^air(;.s and i«^noring the numbers, added to the evils of wrong definitions and illof^ical sequence in teaching, rendered that branch jxiculiarly dark and difficult. In the working out of problems several errors are to be noted. The problems were based on matters foreign to the interest of the pupils (save only that attaching to the ever-recurring dollar or pound); the solutions consisted in the mechanical application of a rule or formula kept in mind ready to be applied to suitable cases, the cases being distinguished chiefly by the form in which the question was stated, and the student losing himself i.opelessly if there were any variation therefrom; and the teacher usually overlooked the fact tliat the chief difficulty in many problems is to understand thoroughly the terms used in stating them. The time at the disposal of the teacher was divided into two portions : one for mental arith- metic, the other for exercises with slate and pencil. Mental arithmetic consisted of the solu- tion of problems involving smaller quantities than those given in the written exercises, but similar in principle with them, the memory IlETUOSPECT. taking tlie place of the slate and pc^ncil. Save for the greater demands on the attention due to the fact that imaginary figures are less vivid than those on the slate, " mental " arithmetic, when it was merely a ivllection or shadow of the written, possessed about an equal value. 2. Obviously the chief defect in such a system lay in the confusion between figures and num- bers. It was to remedy this that the system with which Grube's name is associated was devised. Instead of teaehing inindjers in general, notation and numeration, and the formal oper- ations of addition, subtraction, multiplication, and division, with numbers lari»er than the pupils had any clear idea of, the new system proposed to deal with the smaller numbers ilrst, give easy practice in the employment of all forms of calculation from the start, and use objects — blocks, marbles, etc. — as an aid to the formation of clear ideas of number and numerical operations. In developing his system of teaching arithme- tic, Grube seems to have adopted as his guiding principle the dictum of his predecessor, Pestalozzi, that *' sense-perception is the basis of instruc- 30 THE GROUND-WORK OF NUMBER. tioii." In this view of tlie case, (1) a number is like an object in space, and possesses certain qualities or properties which may be discovered by a careful exercise of the senses ; it is only by means of the powers of observation that knowl- edge can be gained regarding the qualities of objects ; and hence, in order to learn the proper- ties of number, it is necessary to observe in the same v:ay a number of blocks or other objects. (2) As number is something external to the mind, the arithmetical operations consist in the combination and separation of things, a mere physical putting together and taking apart. (3) The primary units coiaposing any number possess spatial attributes, and the idea of frac- tion arises from the division of this primary spatial unit into parts. (1) The principle enunciated by Pestalozzi, — sense-perception the basis of instruction, — un- derwent considerable extension at the hands of Grube. When applied within its appropriate sphere, the principle is most useful. One of Pestalozzi's aims in education was " to popu- larize science," and in that sphere the principle has a very obvious application. It is indeed /■« RETROSPECT. 81 difficult to imagine how any progress in science could be made, if scientists were not trained to observe the facts of nature. It is easy to see that a student of any one of the sciences of nature will work at a disadvantage if his sense organs happen to be weak or untrained. A high degree of intellectual as well as perceptive power is necessary here ; but other things being equal, the scientist whose sense organs are in perfect condition has an immense advantage over one not so endowed. That an object is of a certain colour, gives forth a certain sound when struck, possesses a certain degree of hardness or rough- ness, has a sweet, bitter, or salt taste, gives off a pleasant or unpleasant smell, — these are facts to be observed by the scientist, and there is no way at present known to man by which such facts can be observed save by the exercise of the perceptive powers. There can be no doubt that in the domain of the sciences of nature, sense- perception is the basis of instruction. In adopting this as a guiding principle and unwarrantably extending its application to the science of number, Grube seems to have had before his mind two circumstances. In the first 32 THE GKOUND-WOlUv OF NUMP.ER. place li(3 noticed tluit oik; of tlio cliief faults in tlie tcachiiiii' of science — the niemorizins: of mere words — was also a gross defect in the teaching of elementary arithmetic. In the second place he saw the unwisdom of insisting upon the pupil's mastering the operations of additit md subtraction before going on to nuiltiplication and division; and casting about for an illustra- tion to show the defect of this procedure, he thought he discerned a j^'^^i^'^dlel in the way in which we acquire a knowledge of natural ob- jects. It is not by noticing the same quality in different objects, but b}^ observing one object thorouglily that a child comes to a clear intui- tion of tlie plant or other natural object. Mis- led by the fact of a common defect iu teaching, as well as by his own illustration, he came t(.) the conclusion that the dictum of Pestalozzi applied to arithmetic. He accordingly decided to study a number as he w^ould study a natural object. For him, a number is a thing in pretty much the same sense that a plant, or a stone, or a shell is a thing. A plant occupies a certain portion of space, and may be seen, touched or tasted. A number is a group of things and occTipies a certain portion of space. It may be examined, RETROSPECT. 33 not in exactly the same way as a plant but in a very similar way. The more thorough your observation of the plant, the more valuable will be your knowledge regarding it. The consider- ation of a number ought to be thorough for the same reason. Grube would not go so far as to say that any department of sense perception can give us an idea of number in the same way that the eye gives us ideas of colour, but he makes an equally fatal error. Having, as he thought, established a parallel between tlie genesis of our knowledge of a number and that of our knowledoe of an object, he becomes blind to everything in the process save the part played by sense perception. He recoo-nizes that without a series of sensations we can have no idea of number. He forgets that the idea of number could not arise but for the work of the mind in introducing order and relation into what is given to us by the senses. He must have known that relations are not sen- sations ; but he failed to provide for that very exercise of the thought power without which an idea of numerical relation can never rise into consciousness. His analysis of the number idea 8 34 THE GROUND-WORK OF NUMBER. : goes far enough to find out that its essential " property " consists in the " way " in which a group of things may be put together and taken apart* ; but it does not go far enough to show that this essential property of number is the element which is contributed by the thought power, and is not in any manner due to the senses. Number is a " way " of thinking about objects, not an inherent property which may be discovered by manipulating them. (2) A number, then, is not a thing, or a group of things. The process known as numbering is not a physical process ; it is a mental one. A pupil who can think of five as made up of two and three, may illustrate this truth by holding up his five fingers. On the other hand, he may learn to perform the operation of putting two objects with three objects without the mental process which the operation is supposed to embody. Grube, however, having decided to treat a number as a mere aggregation of spatial *J. S. Mill takes a similar view : — "What is then connoted by the name of a number ? Of course, some property belong- ing to the agglomeration of things which we call by the name; anil that property is the characteristic manner in which the agglomeration is made up of and maybe separated into parts." - -Logic: Book in, Chapter ^4, § 5. RETROSPECT. 35 units, as a mere something external to the mind and its processes, was naturally led to make physical operations do duty for psychical ones. Addition, the first arithmetical operation in his scheme, appears to consist in putting the two objects with the three objects. Numbers being thought of as simply groups of objects, the piling up or separating of such groups is sup- posed to be, not merely to represent, addition and subtraction. Grube has signally failed to understand that a number is a relation imposed upon things by the mind's action ; that the symbol six expresses, not an object or group of objects, but an idea. An idea is a mental process. When we are attending to any idea of relation, we have before us two constituents connected with each other in a certain way. That which is the especial object of thought is the connection between the constituents, not the constituents themselves. What are the constituents involved in an idea of number ? There is the multitude, on the one hand, and a unit on the other. That which occupies the attention is the connection between the multitude and the unit. 36 THE GllOUND-WORK OF NUMliER. Reflection upon what is involved in any idea of number will give prominence to one pluise or aspect, while tlie other phases or aspects retire to the background. The unit by which the whole is measured may be the simple unit, or it may be a complex unit. The kind of relation which is established or di.s^ cned between the two con- stituents of the idea may be the simple relation in w^hich ratio is implied, or it may be that in which the ratio is o,yr)V\o\t As the establishment or the discernmentf Cl i^' relation involves a judgment, this may be set iortr in a pro2X)sition. The idea exprci^jied by v^ie il ■■■i six may be taken to illustrate this. The x-oiciliua between the whole quantity and the simple unit is one phase of the idea. The proposition, 6 — 5 = 1, expresses one kind of relation; J of 6 = 1, expresses the other kind. What is wanted is logical control of these thought processes. All propositions set down to express relations between quantities and units may be classified as representing some one of the fundamental operations of arithmetic. There are operations wliich consist in piling heaps of objects together, and separating them into smaller heaps. Such RETROSPECT. 87 operations are to be distinguished from tlie psychical operations which have just been de- scribed. (3) The idea of a fraction as a portion of a primary spatial unit is quite consistent with that mistaken idea of a number wliich governs the theory of Grube. He would define a IVai'tiou as " a part of anything." The idea of fraction, in this view, arises in the mind when any object occupying space, and taken as a unit, is divided into an equal number of parts. Thus of a number of apples we may take one and divide it into three parts. One of these parts is properly known by the name of one-third, and two of the parts, two-thirds. It should be tolerably clear at this stage of our discussion that such a view of the nature of the fraction is erroneous. The errors involved are precisely those which have already been pointed out in the idea of number as conceived by the exponents of this system. The funda- mental defect is the assumption that there is a space element in number. Number is the repetition of a unit, i.e., of a 38 THE GROUND-WORK OF NUMBER. 'I: I 1 ;f J 31 ■ measure. The most careful search fails to reveal any space element in such a concept. A recent work on number states that measurement with "an undefined unit," as the unit ^ace, for example, is a "stage behind" that which employs the unit yard, because the yards are all equal to one another and the paces are not. " When we pass to measurement with exact units of measure, this idea of fractions — of parts making up a whole — becomes more clearly the object of attention. The conception, 3 apples out of 5 apples has not the same degree of clearness and exactness as that of 3 inches out of a measured whole of 5 inches. Why ? Because in the former case we do not know the exact value, the how omtch of the measuring unit ; in the latter case the unit is exactly defined in the terms of other unities larger or smaller; in 3 apples the units are alike ; in 3 inches the units are equal.* The error here is in supposing that the spatial equality of the actual things which are taken as units is brought into consciousness at all. As apples, or as inches, the spatial aspect may *McLellau and Dewey, Psychology of Number, p. 128. RETROSPECT. 39 receive attention ; as units, every other aspect is set aside, and that aspect alone is considered which they possess as units — an aspect which they would not possess but for the action of the mind in so constituting them. For various reasons it has been found convenient to agree upon certain units, as inches, ounces, etc. Very great efforts are made to secure uniformity in their use, and a high degree of exactness is reached. For all practical purposes these inches, etc., may be regarded as equal. They approach, of course, very much nearer to absolute equality than apples or paces. They are as nearly equal as human devices can make them ; but as abso- lute equality cannot be reached, it is necessary to take them as equal. In practical measure- ment there is a difference in degree of exactness between the pace and the yard ; in numbering, the pace receives its value as unit, from the mind, in precisely the same way as the yard, or other "measured quantity," receives its value. The wisdom of using the convenient yard rather than the pace, in actual measurement, is not called in question. The position taken is simply that the conception — 3 apples out of 5 apples — does possess the same degree of clearness and 40 THE GROUND-WORK OF NUMBER. definiteness as the conception, 8 inches out of a measured whole of 5 inches. The mental act which gives rise to the numerical product is the same in the cases given. The units receive their constitution as units from the mind; they are placed in precisely the same relative position ; they possess one and the same value ; they are thought of as identical. Unless taken " as " equal, the paces are not units at all. A unit is properly defined as a measure. The term simple, applied to the primary unit, empha- sizes the fact that it is indivisible, ultimate, simple. How does this unit come by its peculiar character of simplicity ? Granted that the strokes on a bell may be numbered by relating each sensation in its place in the series, and that the units so constituted possess this simple individual character acquired by the apple (or the inch) used as a unit, how does it lose its spatial attributes ? The answer is, that in num- bering objects in space the mind considers them in the time relation, as a series of discrete units, and must do so — owing to its own constitution — before they can be numerically related. The idea of fraction, then, cannot possibly RETROSPECT. 41 arise from the division of the primary unit, the distinctive feature of the primary unit being that of indivisibility. In this respect it recalls the Atom of tlie scientist, which is ultimate, and the point of the mathematician, which has no parts and no magnitude. The numerical unit is not an object in space whose spatial attributes are to be kept clearly in mind in the act of numbering, and from which the idea of fraction arises in the process of dividing it into parts. All fractions are numbers, and the idea of frac- tion arises in the same way as that of any other number. The idea of fraction is the idea of the relation of a unit to a multitude. As has already been pointed out, the complete idea of number, when all that it implies is fully set forth includes that of ratio. The ratio becomes promi- nent when the fractional relation is discerned and expressed ; and it may be expressed either as J or as 1 : 6. 3. It has been pointed out that the chief defect in the teaching of arithmetic prior to the Sid^ rent of Grube lay in the confusion betw een figures and numbers . It is not ] rieces- sary to do more than : refer to tliis fact . No 42 THE GROUND-WORK OF NUMBER. :i ; one to-day attempts any serious defence of the old system. To attack it is to waste energy upon an abandoned position. The recoil, how- ever, from the system of figures, though emin- ently desirable, was too violent. As often happens, undue emphasis was placed upon the other extreme. If we have hitherto neglected to take into account the part played by sense perception in the production of the idea of number, said the reformers, let us by all means proceed to remedy that defect. If the idea of number cannot arise in the mind without the intervention of objects, let us see to it that objects are provided. Accordingly, the handling of objects — blocks, marbles, stones, etc., was encouraged, but to such an extent, unfortu- nately, in many instances, as to obscure the psychical process itself. The really important consideration is the psychical process. At best the physical process of putting two blocks with three blocks could serve no purpose beyond that of illustrating the psychical one. Somehow or other it had got to be assumed that number was in the objects, and could be, and had to be, extricated from them by the mind, and that this was to be accomplished only by placing the Mi! RETROSPECT. 48 objects in certain positions relative to each other. The error was in " assuming there are exact numerical ideas in the mind as the result of a number of things before the senses"*; and this assumption was due to an incorrect theory as to the origin of the idea of number, namely, the assumption that it is due wholly to expe- rience. In the fulness of time, it was pointed out that the measuring idea in number had been neg- lected. By this measuring idea in number is meant that the function of the unit is to measure, that in numbering any collection of things, by any unit, the exact number is found when the whole group has b- en defined as so many of the unit, that in short the unit measures the collection or group so many times. There are a thousand men in ten companies of a hundred each. When the group is thought of as one thousand our unit is the simple unit; when thought of as so many companies, our unit is one hundred times what it was before. In thus numbering we have measured the group, though with different units in each case. •Paychology of Number, McLellan and Dewey, p. 62. 44 THE GROUND-WORK OF NUMBER. When the importance of this feature — the measuring function — was recognized, efforts were at once put forth to secure the psychical activity from which the idea of number, so con- ceived, might be expected to arise. " If, to help the mental process, small cubical blocks are used to build a large cube with, there is necessarily continual and close observation of the various things in their quantitative aspect ; if splints are used to enclose a surface with, the particular splints must be noted. Indeed this observation is likely to be closer and more accurate than that in which the mere observa- tion is an end in itself. In the latter case there ifi no interest, no purpose, and attention is labored and wandering ; there is no aim to guide and direct the observation." * It was necessary that the " method of sym- bols " should be abandoned for something better. Experience has proved the results to be as barren as might reasonably be expected from a method so artificial and mechanical. It was necessary, too, that the measuring idea in num- ber should receive due emphasis. But the course * McLellau and Dewey, I*s} diology of Number, p. 62. RETROSPECT. 45 proposed in the latter case, though right in purpose, was as wrong in method as that in the former. The handling of objects merely provided certain exercise for the senses, and the examination, comparison, and measurement of extensive magnitudes simply gave another direction to this sense activity. The building of a large cube with smaller ones is an interesting occupation. It requires "continual and close observation." Eye and hand alike are trained in the process. A very high degree of accuracy may shortly be reached by the observer engaged in the occupation of estimating what are technically called " the attri- butes of sensation " — that of " extent " among the rest. Exercises in measuring either with the eye or with a yard stick, as they demand close observation, tend unmistakably to the development of finer sense discrimination. Tlie more interested the observer becomes the more chance for development of the kind mentioned. Unfortunately, however, the more completely the child becomes immersed in the process of measurement and comparison, the farther lie is carried away from the idea of number. Just to 46 THE GROUND-WORK OF NUMBER. ill whatever degree the attempt to secure interest in the comparison of extensive magnitudes is successful, to that degree is the exercise de- structive of clear ideas of numerical relation. Delicacy of sense-discrimination is a desirable thing, but, as we have already pointed out, it is in no way connected with clear ideas of number. Here, again, it is only necessary to refer to the fact that any theory which assumes the existence of a space element in number will inevitably involve erroneous conclusions regarding the exercises which are necessary or suitable to the development of the idea. It has been shown that analysis fails to discover any such element. Whatever useful ends may be served by engaging in exercises of this kind — the examination and comparison of space forms — numerical accuracy is not one of them. m if CHAPTER III. THEORY. 1. In all psychical activity there is a two- fold movement, which it may be profitable to trace through the various stages of intellectual progress. In the case of sensation we observe that there is implied, in addition to the peripheral excita- tion, a certain responsive action necessary to the working up of the raw material into well- defined psychical states, and this central respon- sive action is called attention. We are said to attend when certain of our sensations, or psychical phenomena of any kind, are brought forward into clearer relief, while others are allowed to withdraw into the background of consciousness. It is an activity which has been well described as primarily a process of adjust- ment, of concentration, or narrowing, of the psychical area. It has a second function as well : that of combining a plurality of psychical elements. We may thus concentrate our minds 47 48 THE GROUND-WORK OF NUMBER. upon one of the constituents in a chord, or attend to the complex effect as a whole. Again, we find that the two functions just described — distinguishing sensations from each other, and combining them into a group — have a parallel in the phases or aspects of primary intellection. Primary intellection in- volves differentiation or the singling out of the constituents of any complex; and integration, or the conjoining into a whole, of the elements so distinguished. In the primary stage of the organization of our experience — the formation of the percept and its ideal representation, the image — intellectual development proceeds along this double track of separation and combination. Further, there is to be found in the formation of tliought products, what is simply a higher development of the same double process. Thought is one operation, having two sides or aspects: Analysis, the taking apart of what enters into a complex, and Synthesis, the put- ting together of the elements to form a related whole. In this highest phase of mental activity, the two-fold movement which we have been considering, thougli the same in nature with THEORY. 49 what has been described in the more elementary phases, is found to rec^uire a much greater effort of attention. The isolation of some particular feature in a presentation, when this feature is prominent, does not demand any special effort of attention. When, however, there are promin- ent features calling for notice, and the effort is to attend to some minor one which does not strongly assert its presence, we have abstraction, or abstract attention. And this is precisely what takes place in the thought process. It involves a high degree of abstraction, and results in the formation of those liighly elaborated products known as concepts and judgments. As has already been pointed out, the essence of the whole process of thinking is in that form known as judgment, and an examination of any judgment will show that the process consists in distinguishing two factors in order to relate them to each other. It is not considered necessary to enter into an elaborate defence of the position that analysis is the primary and preparatory^ and synthesis the complementary stage of all mental processes. The priority of the analytic phase in all psychi- 4 50 THE GROUND-WORK OF NUMBER. cal activity is assumed. No thinker of any repute has ever advanced or tried to maintain the contrary opinion.* This well-known law — •"Analysis and synthesis, though commonly treated as two different methods, are, if properly understood, only the two necessary parts of the same method. Each is the relative and correlative of the other. Analysis, without a.subseq^uent synthesis is incomplete ; it is a mean cut off from its end. Synthesis without a previous analysis is baseless ; for syn- thesis receives from analysis the elements which it recom- poses. And, as synthesis supposes analysis as the requisite of its possibility, so it is also dependent on analysis for the qualities of its existence. The value of every synthesis depends upon the value of the foregoing analysis. If the precedent analysis aiford false elements, the subsequent synthesis of these elements will necessarily afford a false result "The two procedures are thus equally necessary to each other. On the one hand, analysis without synthesis affords only a commenced, only an incomplete knowledge. On the other hand, synthesis without analysis is a false knowledge — that is, no knowledge at all. Both, therefore, are absolutely necessary to philosophy, and both are, in philosophy, as much parts of the same method, as in the animal body, inspiration and expiration are parts of the same vital function. But though these operations are each requisite to the other, yet were we to distinguish and compare what ought only to be considered as conjoined, it is to analysis that the preference nnjst be accorded. An analysis is always valuable ; for though now without a synthesis, this synthesis may at any time be added ; whereas a synthesis without a previous analysis, is raart be conceived without any reference to the whole, it becomes itself a whole — an independent unit ; and its rela- tions to existence in general are misapprehended. Further, the size of the part as compared with the size of the whole must be misapprehended unless the whole is not only recog- nized as including it, but is figured in its total extent. . . . Still more when part and whole, instead of being statically related oidy, are dynamically related, must there be a general understanding of the whole before the part can be understood. By a savage who has never before seen a vehicle, no idea can THEORY. 51 ^11 thought begins in analysis — governing as it does the whole range of intellectual activity, is surely one of first-rate importance. Pedagogi- cal science attempts to deduce from the facts of mental life a system of rules for the guidance of the teacher. This is perhaps the most signifi- be formed of the use and action of a wheel. . . . Most of all, however, where the whole is organic, does the complete comprehension of a part imply extensive comprehension of the whole. Suppose a being ignorant of the human botly to find a detached arm. If not C(»nceived by him as a supposed whole instead of being conceived as a part, still its relation to the other part, and its structure would be wholly inexplic- able. ... A theory of the structure of the arm implies a theory of the structure of the body at large. And this truth holds not of material aggregates only, but of immaterial aggregates — aggregated notions, thoughts, deeds, words." — Herbert Spencer: Data of Ethics, Chap. 1. " Successive states of consciousness may be represented as waves, of which one is forever taking the place of another, but such successive states cannot make a knowledge even of the most elementary sort. Knowled<:e is of related facts, and it is essential to every act of knowledge that the related facts should be present in consciousness. ... A known object is a related whole, of which, as of every such whole, the members are necessarily present together." —Green: Prol. to Ethics, Book I, Chap. i. *' Just as we saw that all intellectual elaboration is at once differentiation or separation, and integration or conil)ination of what is differentiated, so we shall find that thought itself is but a higher development of each phase. First of all, then, thought may be viewed as ji carrying further and to higher forms the process of differentiation of presentative elements by means of isolating acts of attention. ... In the second place, all thought is integrating or combining ; or as it is commonly expressed, it is a process of Synthesis. In think- ing, we never merely isolate or abstract. We analytically resolve the presentation complexes of our concrete experiences only in order to establish certain relations among them. " —Sully : The Unman Mind, Chap. 9, § S, 52 THE GROUND-WORK OF NUMBER. fil i cant psychological law with which the teacher has to reckon. For him it becomes a rule or guiding principle. The entire realm of peda- gogical science may be ransacked in vain to discover a principle of ecjual value. It is the golden rule of teaching. Having shown that this two-fold movement belongs to every intellectual act from the highest to the lowest, it will now be necessary to consider it in relation to mental progress and development. 2. The history of the mental experience of any individual, when viewed as a whole, exhibits an adjustment of means to an end, a stream of consciousness, self-directed in part, and in part impelled, to the accomplishment of a certain work. The flow of this stream is an unbroken one ; every wave is what it is by reason of its relation to every other ; and the currents and eddies tend more and more to become persistent. Continuity, relation, and tendency to solidarity care the obvious features of the intellectual life in its development.* Written out at length, the history of that development would be that of a *This is well worked out by Ladd, in his "Psychology, Descriptive and Explanatory," Chapter XXVII. THEORY. 5d continuous series of analytic-synthetic acts, re- lated to each other in a definite way, the whole course marked by the gradual establishment of habit. The law of habit is exemplified by every act of every human being in course of development, whether at work or at play, standing or walk- ing, feeling, desiring, perceiving, or thinking. As affected by the operation of this law, the analytic-synthetic principle seems to take on ad- ditional importance. A habit of thought marked by incorrect or inadequate analysis means simply the acquisition of a stock of vague and ill-arranged concepts; whereas habitual logical control of such processes will ensure clearness and orderly arrangement of one's intellectual stores. Turning now to the two features of con- tinuity and relation in mental development, we may include all that is implied in these prin- ciples in the statement that the whole of mental development consists in an unbroken series of interdei endent psychical processes. The point of interest for us is to discover the exact nature of the relation referred to. What exactly is the 54 THE GROUND-WORK OF NUMBER. Ill relation existing between one wave of progress and anotlier, in the stream of conscious life ? Is it possible to formulate a sta,teraent regarding the conditions of mental progress which shall sufficiently set forth the important features of the process, and at the same time avoid that vagueness which is a frequent accompaniment of general statements ? We must keep in mind the fact that the series of psychical acts which go to make up the mental life of an individual is a series of acta each of which includes a two-fold movement; and also that in this two-fold movement analysis is the prior, and synthesis the complementary pliase. We are to remember that an analytic process implies the presence to the mind of some complex, some whole, awaiting this process. Consciousness is always the consciousness of a complex. Our picture of mental life then would be that of a series of wholes, each of which undergoes a more or less complete analysis, ar. d a subsequent synthesis. Further, as we are con- stantly making additions to our mental stock, and assimilating and incorporating new and old togetlier to form new concepts, we recognize that THEORY. 55 all new ideas thus formed will receive their character and complexion very largely from the old, that adequate knowledge depends quite as much upon the character of wlmt we already possess as upon what we now accjuire. A total mental product, made up of a number of factors, will necessarily partake of the character of those factors. If the factors are clearly understood in the first place, and their relation to the new concept clearly apprehended, our new knowledge will be clear and distinct. If not, we must con- tent ourselves with inadequate knowledge, or else employ the natural remedy for such, i.e., closer analysis. 3. We are now in a position to gather up the results of this enquiry, and set forth the condi- tions of mental progress as follows : — All thought begin fi in analysis and ends in synthesis. This is true of the mental operations of the youngest child. It is equally true of the mental operations of the oldest and wisest philosopher. TJte progress of human knowledge, then, wxiy he considered as a series of points at which the mind considers wholes; and its power to understand these wholes depends upon the • M 56 THE OUOUND-WORK OF NUMBER. I extent to which it has previously mastered, as wholes, the imrts which now make up this more confiplex whole. A student is trying to understand tlie theory of winds. To compass this task he must perform a certain analysis. What is involved in that theory ? His power to understand the complex problem now presented to him depends precisely upon the extent of his knowledge of gravitation, and the expansive force of heat. No difficulty is experienced where these factors have been previously grasped. But an acquaintance with them is indispensable to any one who desires to understand the theory of winds. At some pre- vious stage in his career the student may have heard of both of these matters. But his ideas concerning them are indistinct. It may be, for example, that he has not fully realized the fact that the force of gravitation and that of heat are exerted upon air as well as upon solid bodies. Before he can make any progress with the theory of winds he must first devote his atten- tion to these factors, in turn dealing with each as a separate whole, and analyzing each into its separate constituents. Then, and not till then, 1 THEORY. 57 can he approach the greater wliolo witli any prospect of comprehending it. A student of British history unacquainted with the fundamental laws of the reahn, as vindicated by the barons at Runnymede in the thirteenth century, and by the Long Parliament in the seventeenth, can form no clear idea of the significance of the Revolution of 1G88. Why ? Simply because as already set forth, knowledge of any whole necessarily includes knowledge of the parts which go to make up that whole. An historical transaction can be comprehended in its entirety only when the factors have been previously comprehended. The disputes regarding the dispensing power, standing armies, the right of petition, arbitrary imprisonment, imposition of taxes, freedom of elections, possess no meaning to the student of history except in the light of the great charters. Examination of the sentence in grammar shows that there has been an act of judgment set forth in words, that two ideas have been distinguished and related. No effort to under- stand the expression of a judgm-ent can be successful without a previous clear understand- 58 THE GROUND-WORK OF NUMBER. ^li^ ing of the concept as expressed by a term. The question whether a given expression should be classed as a sentence or no cannot be answered by one who is unacquainted with the elements of the judgment. The position taken is that the examples given are illustrations of the application of a principle of universal scope. The application of this principle is not circumscribed, is not confined to any one science or group of sciences. It obtains in every department of human thought. The master mechanic in a great workshop who has been advanced from post to post, from simpler to more complex duties, occupies his present position by reason of the fact that he has mastered, as wholes, certain elementary tasks belonging to the previous stages of his experience ; and his fitness to perform his larger tasks rests fundamentally upon the use he is now able to make of adequate knowledge pre- viously acquired. A man who has failed in carrying on his business is usually an example of failure to pay due heed to the rule implied in this principle. Ignorance or inadequate knowledge of affairs of detail means inability THEORY. 69 to sufficiently understand matters of larger concern. If the principle under discussion demands recognition for the prosecution of studies in the inexact sciences, if close adherence to the rule it suggests is necessary in the departments of thought from which illustrations have been taken, what shall we say of the importance of its application to an exact science like arith- metic ? 4. It is necessary at this stage in our dis- cussion to notice some important features of educational theory regarding the arithmetical operations, addition, subtraction, etc. Thought processes involving numerical cal- culation conform to the general type commonly named analytic-synthetic. Each case of an arithmetical operation will be found on exam- ination to involve both the analytic and the synthetic phase of thought. The proposition "eight is seven and one" expresses this two- fold movement. Analysis of the concept "eight" results in the recojxnition and distinction of the two factors. Eight is thought of as separable into seven and one. The complement of the 60 THE GROUND- WORK OF NUMBER. idea tliat eight is so separable is that these parts together make the wliole; and the thought is not complete until both aspects have been con- sidered. Again, the proposition, " eight is four twos," obviously expresses the same two-fold movement, the initial idea being the separation into factors, the concluding one that of com- bination to form the whole. The proposition " one fourth of eight is two " similarly involves these features. 5. The arithmetical operations are analytic- synthetic operations, and of these the terms subtraction and division seem to apply to and emphasize the prior, or analytic, while addition and multiplication refer to the latter or synthetic phase of the process. It is often assumed and sometimes stated that in the acquisition of ideas of numerical relation, the mind proceeds by the addition of another unit to the last learned number ; that one who knows the number " two " gets to know the number " three " by adding " two " and " one " together, or that the idea " two " is reached by putting " one " and " one " together. If the idea of " two " is really reached in this way, that is to THEORY. 61 say, if a child gets to the idea of " two " by thinking of one and afterwards of one more to be added to the first one, then we have dis- covered an important exception to the operation of the supposed universal law which we have been considering, namely, that in the two-fold movement of mind analysis is the prior and synthesis the complementary stage. The procedure of the mind from " two " to " three," or from " three " to " four," is sometimes spoken of as a synthetic process. Let us con- sider how the mind really acts in taking this step. During the time that ' two " is being considered, it is not thought of in relation to any higher group. When the group " two " lias been grasped, the mind is ready to deal with the more complex problem which some external agency presents to it. The proposition, " two is made up of one and one," is the expression of a judgment. It has already been shown that judgment is an original and primary act of mind, that it consists in the distinction and relation of two factors, and that in every such act the elements are held before the mind as at once distinct and related. The III .> im 62 THE GROUND- WORK OF NUMBER. II elements spoken of may be parts of a complex of colour, form, etc., i.e., a datum of sense ; or they may be elements of an abstract idea. In either case the purpose and effect of an act of judgment is to make clear something which was obscure in the first place. Are we entitled to hold that the proposition " two is one and one " is the expression of just such a judgment as has been described ? The position taken is that w^e are entitled to do so, and that in reality there is no exception to tlie analytic-synthetic principle. What is present to the mind first is the idea cf " two." It is by a process of analysis that the constituent elements are made clear. We never find ourselves in possession of a perfectly simple mental experience: consciousness is always the consciousness of a complex. Any theory of number, then, which takes addition to be the first in logical or psychological order of the numerical " operations " must deny the priority of the analytic phase of thinking. It is true, certainly, that no ingenuity can succeed in inducing the mind to violate its own laws. This law of mind is as iu xorable as gravitation. But it is also true that the mental THEORY. 63 energy of the learner may be wasted through the effort of his teacher to induce him to violate it. Materials for thought may be presented in such a way as to encourage analysis and dis- courage synthesis. The emphasis which has been laid upon the priority of analysis should not obscure the fact that synthesis is its neces- sary complement. The mind in all stages of its progress disintegrates its materials for thought in order to establish relations among them. The first movement seems to stand to the second in the relation of means to end. The intimate and necessary connection between them shows the absurdity of employing, or attempting to employ, or emphasizing one phase for a considerable time to the exclusion of the other. It seems to be taken for granted by many that there is a great difference in point of diffi- culty between the operations of subtraction and addition, and the corresponding operations of division and multiplication. It has even been stated that the child has no actual experiences which suggest to him the necessity of the division operation, and we are solemnly warned of the danger of " forcing these operations into ; t 1 lit m li s G4 THE GROUND-WORK OF NUMBER. " * r consciousness. ^ As if a child who had been told by his mother to carry in two pails of water, and being unable to carry more than one at a time, would not, if asked, be able to state how many times he would have to bring in one pail of water. It has been noticed that students reach the High school quite unable to explain the differ- ence between addition and division. It will be found that the theory of a space element in number, and an undue emphasis laid on syn- thesis, will sufficiently explain the confusion which has grown up in the minds of such students. That analysis and synthesis are of the very essence of all intellection, that analysis is the first and synthesis the succeeding stage, and that this is the order in the case of every single thought process, — there is a greater degree of unanimity in the acceptance of these statements as theoretical truths, and in their violation as practical principles than in anything else of the kind with which we are acquainted. The study of number carried on in accordance with these * McLellan and Dewey, Psychology of Number, p. 100. THEORY. 65 • principles consists in the solution of a series of problems which present themselves naturally to the learner. The process which he will be called upon to perform will not be a set of exercises chosen arbitrarily and without refer- ence to the demands of the mind or of the subject matter. There are problems in connec- tion with the study of number which the child will, if properly led, if his thoughts are not misdirected, learn to discover for himself. 6. Not infrequently there are to be found individuals who, desiring to be considered prac- tical in the highest degree, put forth some such theory as the following : The elementary trutlis of number are not very numerous. They may be systematically collected in the form of a(]di- tion, subtraction and multiplication tables, which for all practical purposes will include the truths of number. The pupil should be made to learn these truths off by heart, and loarn them thoroughly. It is necessary that thoy should be learned in such a way as to be instantly avail- able. It will require considerable drill and constant review. Thus by a judicious use of >t: li! ii i m ''Vt Mi m GO THE GROUND-WORK OF NUMRER. the power of ineinoiy the desired result may in a very short time be attained. This proposal starts with the assumption that arithmetic is to be studied with an eye single to its so-called practical value, as enabling the student to get answers to questions that arise in daily life speedily and correctly; and it involves, moreover, a very wide-spread and pernicious error regarding the nature of the process called " review." It has been frequently remarked that arith- metic is a thought study. What is the character of the knowledge which the pupil possesses whore that knowledge is confined to " tables " learned in the usual way ? What is involved in the ready employment of those tables? The knowledge is simply a knowledge of words. A call has been made upon artificial memory. The pupil's entire numerical apparatus is built up of words. If he have a good memory for words, he will, to the casual observer, appear to possess real knowledge. But it is merely the semblance of knowledge after all. It is not a knowledge of number, but a memory of words. The thought power is not in the very smallest degree THEORY. 67 called into play. To revert to the figure of language as currency, the language of tables so learned is counterfeit coin. It pretends to be what it is not. It has the appearance, but not the reality, of numerical knowledge. Review does not consist in merely beating a path already traversed. It is rather the process involved in viewing a bit of country first from one standpoint and then from another, ascending to greater and greater altitudes in order to take larger and more comprehensive surveys, each of which will include all the preceding ones. The point to be noticed is that we are tlius constantly seeing old things in new relations. In order that progress r 'ly be made, old truths must constantly be employed in the discovery of new. To review, then, simply consists in making this use of truths already learned. The proper review of any subject is that review which is necessarily involved in the procedure marked out by the dictum (p. 55). The accjuisition of a new idea has been shown to depend upon our previous mastery of the elements which compose it. To grasp a new idea, then, we are forced to consider it as a whole made up of these )U .1 • ^ III I ■pi ,1 ; 5li 68 THE GROUND-WORK OF NUMBER. elements. No idea can bo acquired without this analysis. And what does this analysis involve but a reconsideration of previously learned fac- tors ? This is the true method of acquiring new knowledge, and it includes the true method of reviewing the old. A review of arithmetic which consists in the mere repetition of thought processes without any effort to incorporate new truths with the old is not a true review. To march back and forth in this way is about the same thing as marking time. h . CHAPTER IV. PRACTICAL. 1. We liave now to consider ii matter of the largest i)ractieal importance — the order of steps to be followed in the scientific study of number. In what order shall the numbers be considered, and to what extent shall each number be studied ^ A moment's consideration of what is implied in the principle laid down on p. 55 should make it clear that there is no course open to us but a consideration of the numbers in the order of their magnitude, and an exhaustive study of each number in turn. The first of these points requires no further remark. The second will need some elucidation. What is here meant by an exhaustive study of a number ? Briefly, such a study as will meet the conditions of clear thinking. These have been set forth in the form of a general principle. It will be necessary, however, to show somewhat in detail what is meant by clear and adequate knowledge of a number. 69 ■ '- '1 I i ! ii 70 THE r. HOUND- WORK OF NUMBER. •!«• . ''' The relation betwuiiii tho concept and the judgment lia.s been discussed sufficiently for tho purpose in hand. Some important consider- ations regarding tho functions of language have also been noticed. It has been pointed out that terms employed to represent mathematical (as well as other) ideas serve also as substitutes for thinking. Accompanying the use of any mathe- matical term is the vague consciousness that we are quite able to think out and state if we wish to do so, all that is involved in the thought pro- cess by which the idea was reached. Without the actual process of judgment, the so-called numerical concept can have no existence in the mind, and apart from the ability to recall this process of judgment the name is a mere counter- feit coin. In order to be ready for all possible emergencies, one should be able to recall every one of the series of judgments involved in the concept. Any given group may be thought into its factors. Several judgments of this kind may be performed in the case of the smaller numbers. Thus six may be thought of as five and one, as four and two, as three and three. In the PRACTICAL. 71 case of larger numbers, a ^rcat nunibor of sncli judgments may bo made. Wliat is lackiii;^ in a numerical concept which has been built n\) exclusively of judgments having relVircnci; to the parts into which a group may bo divided ? It is incomplete in that attention has not boon sufficiently directed to the number as a whole. It is necessary to consider six as made up of four and two, of throe twos, of two throes, and of six ones. Here, the emphasis is upon the units which measure the whole. It may bo added that a clear idea of number recjuiros complete analysis of the unit employed : and this is true of all numerical ideas whatever. A clear idea of ratio — which is of the very essence of number — requires that the idea of the whole shall be fully grasped. Not only must we consider the group six as made up of six ones, or as measured by one six times. We nnist now emphasize the fact of the equality of those units, and so reach the judgment " One is one- sixth of six." i'lli •^i Adequate knowledge of a number includes, then, those judgments which are reached as a result of the measurement of a group by all its i 72 THE GROUND-WORK OF NUMBER. li contained units ; and such judgments will fall into two classes: those in which what may be called integral relations only are involved, and those in v/hich the idea of ratio is explicitly set forth. 2. An illustration or two will serve to show the practical application of what has been laid down. The pupil is })eginning the study of three. He has already mastered two. What is the problem now presented to his mind ? The problem is the problem of measuring the new whole by its contained units. It has been point- ed out that thes*' measurements will result in the formation of two classes of judgments: judgments involving integral relations, and those involving fractional relations ; answers to the questions, how many ? and how much ? When this is accomplished the leanier will come nearer to perfect knowledge than is possible in the case of anything else we are acquainted with in the whole range of human thought. Following the line of least resistance, what is the simplest measurement of three that can be made ? Will the pupil naturally measure the group by ones, or will he employ as a unit the PRACTICAL. 73 group two which he has just analyzed ? The answer to the question as to which of these processes is the more complex will guide us into the path of least resistance. In the proposition, "Three equals two and one," there is involved the idea of the whole as made up of two factors, the original factor, which has already been thoroughly grasped, together with another unit. The proposition, " Three equals OTie, and one, and one," implies that each unit has been thought of singly, mani- festly a more complex operation. It involves the conscious analysis of two into its elements. If it should happen that the pupil at first begins to analyze the group into its simple elements, what inference could justly be drawn under such circumstances ? Is it not simply that he is not in a position to think of two as a whole ? Such procedure — thinking by ones — is conclusive evidence that he cannot think by twos, which is oidy another way of saying that he has not " mastered " two. Something has evidently been neglected in his previous consid- eration of "one of the parts which now make up this more complex whole." The parts have ■e taken from three ? " and reaches the judgment: "In three there is one two and one; o\'(?r.'* He is now in a j)()sition U) proceed to the PRACTICAL. 75 measurement of the whole hy tlie simple unit, and his judgment takers the form : "In three there are three ones," or " Three times one are three. " This involves the following stops : "Three is two and o\u) ; two is two ones ; there- fore, three is three ones." As already pointed out, our principle calls for the n\astery of each portion or element of knowledge as a whole, and since this cannot be accomplishe 1^ 6^ 4> %^ '^ v^^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 *^ «?, (/a I '<«* 84 THE GROUND-WORK OF NUMBER. in connection with his study of two and of four. At this point he would see at once that " Three is one half of six." As ratio is of the very essence of number, we may conclude that the pupil who has a thorough grasp of the ratios implied in a number is in possession of an adequate idea of that number. An exercise in the comparison of ratios is the most searching test that can be applied. The mere act of comparing two ratios, as one third and two sixths, does not of itself require a great effort of intellectual power. The discovery or discernment of resemblance is the initial step in this quasi-inductive process. The equality of the ratios 1 : 3 and 2 : 6, how over, can be observed only if there is a clear and definite idea of the ratios themselves. It is a commonplace to re- mark that the comparison of two ideas is ren- dered possible only by close acquaintance with the ideas themselves. On the other hand, the judgment, one third equals two sixths, or, more fully, 1 : 3 as 2 : 6, is a judgment easily reached by one who has a clear idea of the meaning of the judgment, *' Two is one third of six," and the judgment, " Two is two sixths of six." 1 PRACTICAL. 85 5. We have seen that the individual in giving order and system to what he encounters in the world of his experience employs among many other activities the power to number. It is to be borne in mind that the exercise o^ this power takes place without the interventioAi of the schoolmaster. It is not necessary to the healthy exercise of his spiritual powers that the child should call in the artificial and systematic aid of the teacher. The world of space and time is full of interest, and it is quite as natural that the objects which exist, and the events which happen therein, should be numbered, as that their size, colour, or duration should be observed and com- pared. We are not to suppose that the child on entering school is without any numerical ideas. Some teachers who are quite at home with the distinction between empirical and scientific in connection with knowledge of facts of botany or chemistry, are apt to overlook the fact that a foundation of numerical concepts has been laid in the mind of the child before he comes under the teacher's hands. Ifi m 'ti The child has become acquainted with the language of number in much the same way that ; 'V',1^%OBBBI6i 86 THu: GROUND-WORK OF NUMBER. is; ■v ll:;: •i ■ ■ i ' %r-m he has acquired the other parts of his vocabu- lary. Words like " some," " many," " few " are the first used. The want of something more definite is soon felt and supplied. It is probable, however, that prior to the employment of a unit for the more definite measurement of a vague multitude of things, the words one, two, three, etc., convey to the child the ordinal idea only. Numerical ideas he certainly has : though much of his knowledge is vague, much of his language inexact. It is for the teacher to increase his power to number by the judicious use of exer- cises and the correct use of conventional mathe- matical symbols. The systematic study of the truths of number — the study of arithmetic as a science — as con- templated in accordance with the ideas expressed in these pages, cannot be carried on in any other way than by the unremitting use of the thought powers : by the formation of clear and distinct notions, by a careful and constant use of judg- ment and inference. It is believed that much of what is unsatisfactory in connection with the teaching of primary arithmetic is due to the unfortunate custom of attempting the scientific U m PRACTICAL. 87 study of the subject at too early an age. As a rule, children are not ready for the prosecution of a thought study, an exact science like arith- metic, at the age of five : and it is probable that no time is lost by postponing it till the age of seven has been reached. 6. Meantime the use of objects, i.e., the em- ployment of numbers of stones, spools, dots, musical notes, and what not, will be carried on, not according to any system more orderly than that of nature, but in the ordinary course of the child's daily experience. It will not be necessary for the teacher to supplement this experience by the "use of objects," as that phrase is generally understood by teachers. It should also be kept clearly in mind that exer- cises in "measuring" — the employment of the eye or the foot-rule for the comparison of quan- tities — are chiefly useful in the development of sense discrimination, not for that of numerical knowledge. We are not to lose sight of the importance of exercises which look to the devel- opment of sense perception, but 't must be borne in mind that such exercises are in no way calculated to develop distinct ideas of number. f! II \, It f"'- 88 THE GROUND-WORK OF NUMBER. Long before the time has arrived for the scien- tific study of number, the child may and should acquire power to distinguish between the vari- ous kinds of space forms. This is the study of geometry. A child can discriminate between a triangle and a square without employing numer- ical ideas. It is true of course that he cannot express that difference in language without em- ploying the language of number. 7. The advocates of the "use of objects" in teaching primary arithmetic attempt to justify the practice not only by an appeal to the authority of Grube, but also by the claim that mental arithmetic consists in forming mental pictures of objects corresponding to former ac- tual experience in handling such objects. If this claim can be successfully established, the case of the advocates of the "objects" is a strong one. An illustration will show what, exactly, is meant hy this. A problem is to be solved : " How many threes in seven ? " The learner is asked to ar- range his objects thus, ••• ••• • : and when going through the ''mental" exercises he is expected to call up before his mind's eye a picture corresponding to this. The picture !■ PRACTICAL. 8d is which he thus calls up differs from what he actually saw with his bodily eye only in point of vividness. The judgment he reaches in both cases as the result of his examination of the objects, actual and pictorial, is expressed thus: " In seven there are two threes and one over." The mental processes, however, which are in- volved in performing arithmetical operations — whether mental arithmetic or any other kind of arithmetic — are thought processes. The pro- cesses which have just been described are not thought processes. The first is a perceptive process, and the second a representative one, and there is not any more thinking in the one case than in the other. We have noticed that a pictorial image is the intermediate step between perception and thinking in the formation of concepts (general notions): but " seven," as has been pointed out so often, is not a general notion. The true thought process by which the problem referred to is solved may be exhibited as follows : " Seven is six and one ; six is made up of two threes ; therefore in seven there are two threes and one over." Here the thought power is at work. We pass from one 1 ■ii i 90 THE GROUND-WORK OF NUMBER. in ^ijt judgment to another by a process of inference. So, in answering the question, " How many nines in sixty ? " " In ten there is one nine and one over ; in sixty there are six tens ; therefore in sixty there are six nines and six over." It is difficult to see how the pictorial power can be brought into effective use in connection with large numbers; whereas the thought power is adequate to the performance of such processes even though very large numbers are employed. If arithmetic is properly a thought study, the pupil should be trained to think out numer- ical relations. The mere exercise of the per- ceptive and representative powers in connection with groups of objects is not arithmetic. There is, after all, no reason why the pupil should handle blocks and spools in school. Before entering the school-room he has had a vast deal of experience in numbering objects and events, and is in possession of certain clear, though indistinct, ideas of number. He has not yet begun the study of the science of number, but he has made the necessary preparation for the study. And it may be remarked that the preparation referred to is quite as complete in PRACTICAL. 91 the case of a blind child as in the case of one who goes to the study of arithmetic fully equipped with all that is necessary for the formation of those pictorial images, which, in the theory we are combatting, are made to do duty for abstract thinking. If we could con- ceive a being deprived of the special senses of sight and touch — the senses by which we get our ideas of space and colour — such a being could certainly pursue the scientific study of arithmetic, the special sense of hearing pro- viding the material; but he would be quite unable to solve any arithmetical problem by the employment of pictorial images. 8. Written exercises in arithmetic have a value of their own apart from the training in thinking out numerical relations, and that is the value which attaches to any exercise in the written expression of one's thoughts. Written exercises in arithmetic are, or should be, simply the expression of thought. The important dif- ference between the two " kinds " of arithmetic is that in one case the thought is expressed orally, and in the other case in writing. It would seem at first sight as if for the purposes i T Til THE GROUND-WORK OF NUMBER. M « / of the teaclier one form of expression would be as useful as the other. Such is not the case, however. While the pupil is engaged in the oral expression of thought, the teacher is in a position to test by means of questions the accuracy of the pupil's thinking. Not so while written work is being done by the class. The great danger in seat work is that the pupil may be acquiring evil habits, such as mechanical counting, the employment of the perceptive instead of the thought power, and, almost inevit- ably, the use of set forms of expression which save the pupil the trouble of thinking. In connection with the teaching of arithmetic, careful attention is paid to the expression of the pupil's thought, not so much because the teacher wishes to cultivate the power of expression, but chiefly because he wishes to know exactly how and what the pupil is thinking. Expression here is scrutinized because it is the test of thought. Any expression which does not exhibit the process of thinking is of little value. As between the written and the oral expression of thought in the study of arithmetic, there is no hesitation in saying that the latter is far and PRACTICAL. 93 away the more valuable exercise. " Seat work " must be pronounced to be of very doubtful value. It is certainly liable to produce great evil to the thinking power of the children. The pupil gets into the way of setting down forms of expression which do not stand for real thought processes in his mind, but the use of which somehow enables him in a mysterious way to find the answer to the question. Such statements are not really the expression of the thought of the pupil. The figure of language as currency reminds us of the constant danger of counterfeit coin. Pupils who have been provided with a form in which they are to exhibit all solutions of problems do not necessarily acquire thereby either conciseness of expression or clearness of thought. They may, on the other hand, acquire the trick of appearing to perform mental pro- cesses which they do not really perform. The formal statement — 6 hats cost $15, .*. 1 hat will cost J of $15, hence 4 hats will cost 4 times J of $15 = $10, may be the expression of thinkmg by the pupils, or it may be a "blank cheque of intellectual I ■i il- i( 1 fi m 94 THE GROUND-WORK OF NUMBER. bankruptcy." A "right answer" may always and easily be found by rule if the pupil only knows how to do the trick. / The problem is evidently a problem in pro- portion. The pupil who really solves the prob- lem must know that the number required is the number which stands in the same relation to 15 that 4 does to 6. He applies his knowl- edge of pure number to a concrete case. He knows that the ratio 10 : 15 is always equal to the ratio 4:6. If there is any advantage to the pupil in writing down the entire chain of reasoning in a complicated problem that advan- tage will be made secure by insisting on plenty of oral expression before the pupil is asked to write. It is not too much to say that nine- tenths of the difficulty experienced by pupils in the solution of problems would disappear if the terms used in stating them were thorough- ly understood. Here again is the weakness of written work. Oral questioning would force the pupil to think out the solution, and the difficulty would disappear, because it would be impossible then to employ terms which were not thoroughly understood. PRACTICAL. 95 Much has been said about the difficulty of providing suitable employment at seats for chil- dren of the elementary grades. For a long time a principal source of supply for exercises at seat has been the " number work " ; and if this source is cut off the difficulty for some teachers will be greater than ever. The problem, however, is not to keep the children employed, but how to keep them profitably and happily employed. Arith- metic is an important subject, no doubt, and must always occupy a considerable place on our school programmes. But it should not be allow- ed to permeate every department of school study, and dominate every hour of the school day. It is not an overstatement of fact to affirm that there are many teachers who try to keep the children employed in the study of number dur- ing one-third of their time in the school -room, in class and at seats. There is a double blunder in these cases. Teachers who so arrange their time tables should first revise their ideas regarding education values; and then, when they have reduced arithmetic to its proper rank as a sch'^ol study, try to form a correct estimate of the .value of " written " arithmetic. i/' m. ^1^ m 96 THE GROUND-WORK OF NUMBER. 9. It may be added that much of the " useful " material which has been crowded into the text books, though called by the name of aiithmetic is not arithmetic at all. A knowledge of the usages of the shop and market-place, the num- ber of cents in a sovereign, how to write a non-negotiable promissory note, the buying and selling of bonds, etc., all these things are both interesting and useful, but they are not arith- metic, do not belong to the science. To solve an intricate problem in stocks is looked upon as a creditable feat for a school-boy. It is ; but the difficult part of the feat is not in the arithmeti- cal processes performed, but in understanding the ins and outs of the business of the broker. There are some teachers who would consider the following questions as quite difficult questions in arithmetic : How inany bushels in a ton of bran? How many pyramids the size of the great pyra- mid of Egypt would it take to cover a half-breed claim on the Red River? Find the weight of the gas in Andree's balloon. Reduce three York shillings to Egyptian currency. The difficulty, wherever it is, is not in the arithmetical pro- cesses necessary to answer these questions. PRACTICAL. 97 10. The system of notation which we use is called the decimal system, because ten is the radix or basis. If twelve were adopted as the base instead of ten, it is clear that there would be a great gain in one important respect, namely, in the fact that whereas the latter has only two divisors, the former has four. The foot of twelve inches, the pound of twelve ounces, the shilling of twelve pence, the year of twelve months, the day of twelve hours, the dozen, the gross, and great gross, have hitherto resisted the effort to secure complete conformity with the decimal system. The usual and obvious way of accounting for the universal adoption of ten as the basis of numerical calculation and nota- tion is by reference to the ten fingers of the human hand. When the number ten has been mastered, the learner has reached a very important point in his career as a student of the science of arith- metic: not, of course, on account of anything magical in connection with the number ten, but simply because ten has by agreement been fixed as the basis of our system of notation. The science of arithmetic should present but few 7 4 II 98 THE GROUND-WORK OF NUMBER. i; difficulties from this point onward. The learner is now in possession of knowledge which will enable him to solve any question that can pos- sibly arise in the domain of pure number. He is in possession of an exhaustive knowledge of ten, which has been well designated " the key /^ to number." If the procedure marked out by the dictum on page 55 has not been followed, if the pupil's knowledge is vague or incomplete through his failure to " master as wholes the parts which now make up this more complex whole," he will now more than ever be hampered in his pro- gress. The problem to which he has now to address himself is that of becoming acquainted with our system of notation. The numbers greater than ten present nothing that has not been practically mastered. Eleven means ten and one, fifteen means ten and five, twenty means two tens, forty-five means four tens and five. The pupil whose study of arithmetic has been a scientific study will now go forward by leaps and bounds. The manifest duty of the teacher at this stage is to show the pupil that he holds the key / PRACTICAL. 99 ,/ / ^ JJ^ \ which is to unlock all future problems in num- ber. Exercises are now to be provided which shall give the learner the consciousness of the power he possesses. He must form the habit of computing by tens. A few examples will suffice to show the method to be followed. The answers to the questions (a) 16-r5 ; (b) i of 20 ; (c) 30 -r 8 may be reached in either of two ways. I. (a) 5 + 5 = 10; 10 + 5 = 15; .'. in 16 there are 3 fives and 1 over. (b) iof 20 = 4; .-. t of 20 = 4 times 4= 16 (c) 8 + 8 = 16; 16 + 8 = 24; 30-24 = 6;.-. in 30 there are 3 eights and 6 over. II. (a) 16 = 10 + 6; in ten there are 2 fives; in 6 there is one 5 and 1 over ; . • . in 16 there are 3 fives and 1 over. (6) I of 10 is 8 ; there are 2 tens in 20 ; . • . I of 20 is twice 8 or 16. (c) In 10 there is one 8 and 2 over ; in 30, or 3 tens, there are 3 eights and 6 over. The latter of these modes is preferable in that it explicitly recognizes the decimal system ; it tends to establish the habit of analysis of large : .ir If Ii.\') 100 THE GROUND-WORK OF NUMBER. I ■• f i I Mi I I groups into tens; and it makes perfectly clear all that needs to be learned at this stage regard- ing the conventional method of notation. The former plan is faulty in that it does not do any of these things. It may further be remarked that in the second case there is a more system- atic use made of old truth in the discovery of new truth, than in the first. Sufficient practice in exercises of this kind must be given in order to make quite clear to the mind of the pupil the fundamental fact that numbers greater than ten are expressed in terms of ten, and must be so thought of and treated in numerical calculation. If this point is clearly recognized in all the work done at this stage, there will be no difficulty at a later stage in understanding that " every figure placed to the left of another represents ten times as much as if it were in the place of that other." 1 CHAPTER V. EXERCISES. 1. The Number Six.— (A) Pure Arithmetic. (a) 1. What number with five makes six ? 2. What number with one makes six ? 3. Six, take away five, leaves what ? \ 4. Six, take away one, leaves what ? 5. How many times can we take five away from six ? 6. One five and one more make how many ? (b) 1. What number with four makes six ? 2. What number with two makes six ? 3. Six, take away four, leaves how many ? 4. Six, take away two, leaves how many ? 5. How often can we take four away from six? 6. One four and two make how many ? (c) 1. What goes with three to make six ? 2. Six, take away three, leaves how many ? 3. How often can we take three away from six? 101 if^ w m^mmmmmm 102 THE GROUND-WORK OF NUMBER. •I. 1 liiit ^—-^ 4. Two threes (or twice three) are how many ? (d) 1. What goes with two to make six ? 2. How often can we take two away from six ? 3. Three times two make how many ? (e) 1. What goes with one to make six ? 2. How many times can you take one away from six ? 3. Six is how many times one ? (/) 1. When six is divided into ones, it is divided into how many equal parts ? (Give name, if necessary.) 2. Two is what part of six ? 3. Two is what other part of six ? What is the same part of six that one is of three ? 4. Three is what part of six ? 5. Tliree is what other part of six ? What number is the same part of six that one is of two ? or that two is of four ? 6. Four is what part of six ? EXERCISES. 103 1. Four is what other part of six ? What number is the same part of six that two is of three ? 8. Five is what part of six ? 9. One half of six and one third of six are how many ? 10. Two thirds of six, less one half of six, are how many ? etc. (B) Applied Arithmetic. (a) 1. An exercise book costs six cents. John has only five cents. How much more money must he get in order to buy it ? 2. A boy has six marbles. There is one in one pocket. How many in the other ? 3. A boy has to carry six armfuls of wood. He has already carried five. How many more has he to carry? 4. It is six o'clock. The clock has struck once. How many times has it yet to strike ? 5. There are six pupils in the school. How many classes of five can the teacher make ? 'it 104 THE GROUND-WORK OP NUMBER. |i!'. ' !5" ' 6. Pencils are sold in bundles of five. A boy wants six pencils.. How many- bundles must he get? (b) 1. A pole is six feet high. A spider has crept up four feet. How far has he yet to go ? 2. There are six calves in the stable. Two • of them are lying down. How many are standing ? 3. I have six silver pieces. There are four of them in a dollar. How many dollars have I ? 4. There are four farthings in a penny. How many farthings are there in one penny and two farthings ? (c) 1. You have been promised six plums, and have already received three. How many are left ? 2. There are six boys in a room. Three go out ? How many are left ? 3. Six men wash to cross a river. The boat will only carry three passengers. How many trips must be made ? J\ EXERCISES. 105 4. It takes three horses to draw the binder. There are two binders at work in the field. How many horses ? (d) 1. A man has six cows. Two of them are in the stable. How many are outside ? 2. There are six horses, and two can stand in each stall. How many stalls do you need ? s 8. Find the cost of three two-cent stamps. (e) 1. There are six freight cars on the track. Only one is loaded. How many are empty ? 2. I have six cartridges. How many chick- ens can I shoot with them if I shoot one with each cartridge ? 3. How many apples can you buy at one cent each if you have six cents ? (/) 1. Six oranges are to be divided among six boys. How many will one boy get. 2. What part of the oranges does one boy get? 3. There are six working days in a week. A boy works Monday and Tuesday. What part of the week has he worked ? t 1^1 1'^ imw 106 THE GROUND-WORK OF NUMBER. 4. If he is to get three dollars a week, what will he get for the two days' work ? 5. Six books cost two dollars. What will three books cost ? How many can I buy with one dollar ? 6. It is six miles from the school to the post office. A boy has walked four miles. What part of the distance has he walked ? 7. A man has worked five sixths of the week. How many days has he worked ? 2. The Number Ten. — (A) Pure Arithmetic. (a) 1. What number with nine makes ten ? 2. What number with one makes ten ? 3. Ten, take away nine, leaves what ? 4. Ten, take away one, leaves what ? 5. How many times can we take nine away from ten ? G. One nine and one make how many ? (Z>) 1. What number with eight makes ten ? 2. What number with two makes ten ? 3. Ten, take away eight, leaves how many ? 4. Ten, take away two, leaves how many ? EXEBCISES. 107 6. How many eights in ten ? 6. One eight and two make how many ? (c) 1. What number with seven makes ten ? 2. What number with three makes ten ? 3. Ten, take away seven, leaves what ? 4. Ten, take away three, leaves what ? 6. Ten is how many sevens ? 6. One seven and three make how many ? (d) 1. Six, with what other number makes ten ? 2. Four, with what number makes ten ? 3. Ten, less six, leaves what ? 4. Ten, less four, leaves what ? 5. How many sixes in ten ? 6. One six and four ones make what ? (e) 1. Five, with what other number makes ten? 2. Ten, take away five, leaves what ? 3. How many fives in ten ? 4. Two fives make what ? (/) 1. Four with what number makes ten ? 2. Ten, take away four, leaves what ? 3. How many fours in ten ? 4. Two fours and two ones make what ? 108 (a) THE GROUND-WORK OF NUMBER. 1. What number goes with three to make ten ? 2. Ten is how many threes ? 3. Three threes and one are liow many ? (h) 1. Two and what number makes ten ? 2. Ten is liow many twos ? 3. Five twos make how many ? (i) 1. One and how many makes ten ? 2. Ten is how many ones ? 3. Ten times one is what number ? (j) 1. Think ten into ones. How many equal parts are there ? 2. What shall we call one of these parts ? 3. What shall we call two of these parts, three of these parts, four of these parts ? etc. 4. Think ten into twos. How many equal parts are there? Therefore two is what part of ten ? Compare one fifth of ten with two tenths of ten. 6. Compare two fifths of ten with four tenths of ten, three fifths of ten with six tenths of ten, etc. EXERCISES. 109 bs, (B) Applied Arithmetic. (a) 1. There are ten pupils in a rural school. There are nine boys. How many girls are there ? 2. There are ten pupils in two classes. In one of the classes there is only one pupil. How many pupils are there in the other ? 3. A boy has ten cents. He spent nine cents. How many cents has he left ? 4. John has ten cents. How often can he give his brother nine cents ? 5. There are nine sheaves in a stook. One stook and one sheaf makes how many ? (b) 1. It is ten miles from Brandon to Harrow. A man has ridden eight miles. How far has he yet to go ? 2. Ten pigeons on the roof. Two have flown. How many are left ? 3. How many lead pencils at eight cents each can be bought with ten cents ? 4. Johnny can carry eight bricks in his little hod. Mary carries two in her hands. How many bricks have they ? ')| 110 THE GROUND-WORK OF NUMBER. KM (c) 1. A boy wishes to buy a ten cent scribbler. He has seven cents. How much more must he get ? 2. A hen has ten chickens. A hawk takes three. How many are left ? 3. John has ten sheep. He can put seven in each stable. How many stables are needed ? 4. How many days in one week and three days ? (d) 1. Henry has ten marbles in two pockets. He hat six in one pocket. How many in the other ? 2. Mary has ten cents. She spends four cents. How many cents has she left? 3. There are six days in a working week. A man has worked ten days. How many weeks has he worked ? 4. One working week with four days is how many days ? (e) 1. A boy has ten cents. He loses five. What is left ? 2. How many exercise books at five cents each can be bought for ten cents ? 1:1 K Ill IS EXERCISES. 3. Two lead pencils at five cents ea<;h would cost how much ? if) 1. There are four farthings in a penny. How many pence in ten farthings ? 2. A man has two four horse teams, and two drivers. How many horses has he? fo) 1. John has ten matches. How many triangles can he make ? 2. On a farm three hoi^s are required for each of three binders, and one as a carnage horse. How many horaes in all? (h) 1. How many double seats will be required for ten boys ? 2. Find the cost of live two-cent stamps. (i) 1. At one cent each, how many slate pencils can be bought for ten cents ? 2. What is the cost of ten alleys at one cent each ? 0') 1- A hen hafi ten chickens. She loses one What part of her flock has she lost? What part has she left ? P1 :i; f livV 112 THE GROUND- WORK OF NUMBER. 2. A boy starts to ride ten miles. At the end of the eighth mile he punctures his tire, and walks the remainder. What part of the distance does he walk ? What part has he ridden ? 3. Mary has ten cents. She spends five cents for an exercise book. What part of her money has she spent ? 8. Exercises Emphasizing "Ten" as the Base FOR ALL Arithmetical Operations. (a) 1. What number with ten makes twenty ? 2. Twenty, take away ten, leaves what ? 3. How many tens in twenty ? 4. Two tens make what ? 5. In twenty, how many nines ? (Solution : First step — Think twenty into tens. Second step — Think each ten into nines. Third step — Combine nines and ones.) 6. In twenty, how many eights ? 7. In twenty, how many sevens ? 8. In twenty, how many sixes ? 9. In twenty how many fives ? EXERCISES. 10. In twenty, how many fours ? 11. In twenty, how many threes ? 12. In twenty, how many twos ? 13. In twenty, how many ones ? 14. What is one tenth of twenty ? (Solution : tV of 10 is 1, . •. jt^ of 20 is twice 1 = 2.) 15. What is one fifth of twenty ? 16. What is two fifths of twenty ? 17. What is seven tenths of twenty ? (/>) 1. What number with twenty makes thirty 2. Thirty, take away ten, leaves what ? 3. Thirty, take away two tens, leaves what ? 4. In thirty, how many twenties ? 5. How many tens in thirty ? 6. Three tens make what ? 7. In thirty, how many nines ? 8. In thirty, how many eights ? 9. In thirty, how many sevens ? 4. Notation. 1. From two tens take away nine ones. («) How many whole tens are left? 113 '-3 ■■ w Vufl l!.; V 3* ■;'■ 114 THE GROUND- WORK OF NUMBER. * (6) How many separate ones are left ? (c) What does the term eleven mean exactly ? 2. From two tens take away one ten and one. How many are left ? 3. From two tens take away eight ones. How many tens and how many ones are left ? What does the term twelve mean ? 4. From two tens take away one ten and two ones ? How many are left ? 5. From twenty take away seven. How many are left? What does the term thirteen mean ? Proceed similarly with the others. 5. Miscellaneous Exercises. 1. How many exercise books worth ten cents each can be bought for twenty cents ? 2. A boy has twenty cents. He spends five cents for a lead pencil. How many copy books at five cents each can he buy with the remainder ? EXERCISES. 115 3. It is twenty miles from Boissevain to Kil- lamey. A man rides ten miles of the distance to-day. How many miles has he still to go, and what part of the dis- tance has he travelled ? 4. There are nine square feet in a square yard. How many square yards in twenty square feet? 5. There are three feet in a yard. How many yards in twenty feet ? 6. There are twelve inches in a foot. How many feet in twenty inches ? 7. There are sixteen ounces in a pound. How many pounds in twenty ounces ? 8. What part of a yard is a foot ? 9. What part of a foot is one inch ? three inches ? five inches ? eight inches ? 10. There are twelve ounces in a pound Troy. How many pounds Troy in twenty ounces? 11. There are twenty shillings in a pound. What part of a pound is fifteen shillings ? 12. How many pounds of rice at eight cents can be bought for twenty cents ? 13. How many weeks in twenty days ? 116 THE GROUND-WORK OF NUMBER. fillet;; |! ' ■'i \i 14. How many days in two weeks and six days ? 15. There are six working days in a weok. How many working weeks in twenty days ? 16. A plot of ground is four feet by five feet How many square feet does it contain ? 17. Five square feet of the plot is in grass. What part of the whole is in grass ? 18. How many pounds of oatmeal at four cents a pound can be bought for twenty cents ? 19. Lawn is twenty cents a yard. Find the cost of half a yard — of a quarter of a yard ? 20. Four yards of cotton cost twenty cents. What will three yards cost? 21. Six pencils cost eight cents. What will three pencils cost ? What will two cost ? 22. Six hats cost nine dollars. What will four hats cost? 23. John has two cents. James has one cent. They put their money together and buy eighteen marbles. They then divide them fairly. How many should each get ? 24. John had ten cents. He spent two cents. What part of his money did he spend ? 'f*'h*\t part has he left ? 1 EXERCISES. 117 25. A boy caught ten fish, and gave away two fifths. How many had he left ? 26. John has twelve cents. He spent one third of his money for an exercise book, and one half of it for a lead pencil. What part of his money did he spend alto- gether ? The difference between what he spent for the pencil and what he spent for the book is what part of his money? 27. A boy has ten cents. He spends one fifth in marbles and one half of the remainder for an exercise book. What part of his money did he spend for the exercise book ? What part did he spend alto- gether ? What part has he left ? 28. Eight apples cost twelve cents. What will six apples cost ? 29. How many apples will three cents buy ? 30. One yard of tape costs sixteen cents. What will nine inches cost ? 31. There are eight gallons in a bushel. One bushel of oats costs twenty cents. What will six gallons cost ? i PI 118 THE GROUND-WORK OF NUMBER. m ii I 32. Oats are sixteen cents a bushel. Find cost of seven gallons. 33. Eggs are eighteen cents a dozen. How many eggs can you get for ten cents ? 34. Oranges are sold at the rate of eight for twenty cents. A boy has fifteen cents. How many can he buy ? 35. John caught two fish. James caught three. They sold their fish for fifteen centa How did they divide their money ? 36. A man had three horses. His brother had four. They hired them out together, and received fourteen dollars. How did they divide their money ? 37. Divide fifteen cents between John and James so that John will have twice as much as James. 38. Of twelve calves, five are black, one third are white, and the rest are red. What part are red ? 39. John and Mary have ten plums each. John eats six tenths of his, and Mary eats three fifths of hers. How many has each now? EXERCISES. 119 40. Three sheep are exchanged for five pigs. At the same rate how many sheep would you get for fifteen pigs ? 41. Ten men can stack a field of grain in three days. How many men would it take to do the work in five days ? 42. Divide eighteen apples between John and May, so that every time John gets four May will get five. 43. A boy had twelve cents. He spent one third of it in candy, and one fourth of the re- mainder in popcorn. How many cents had he left ? What part of the whole had he left? 44. Two fifths of a post is in the ground, and there are nine feet of it above ground. How long is the post ? 45. John has sixteen cents. He spends three eighths of it and loses one fourth. How many cents has he left? What part of the whole ? twelve dollars each. man sheep On one he gained one third and on the other he lost one third of what he paid. Did he gain or lose, and how much ? I''' ill 120 THE GROUND-WORK OF NUMBER. |:^ !i ' !.V 47. Two apples and two oranges cost fifteen cents. Two oranges cost as much as three apples. Find the cost of an apple. 48. John has five cents. Dick has three. They put their money together and buy six- teen apples. How should they divide the apples ? 49. Seven men do a piece of work in sixteen days. How long would it take eight men ? 50. How many men would do it in seven days ? 51. John can wheel three times as fast as James can walk. They start off" in opposite directions, and stop when they are twenty rods apart How far has James walked ? 52. In returning, James runs two thirds as fast as John rides. Where will they meet ? 53. How many ounces in one pound and one third Troy ? 54. What is the difference between two thirds of a pound and three fourths of a pound ? What part of a pound is it ? EXERCISES. 121 55. John can walk four miles an hour. How long will it take him to travel eighteen miles ? 56. How far will he walk in two hours and a half ? 67. How far will he walk in four hours and three quarters ? 68. Rice is six cents a pound. Find the cost of two pounds and one third. 59. Find the cost of one pound and two thirds. 60. John has six cents. William has four. They club together and buy newspapers. They sell, and gain a profit of twelve cents. How will they divide their profit ? 61. James has eighteen sheep, and sells fourteen. What part did he sell ? What part is left ? 62. There are ten hours in a working day. Wages are five dollars a day. What would a man earn who worked two hours a day for three days ? 63. There are twenty calves. Each stall will hold three calves. How many stalls will be used ? 122 THE GROUND-WORK OF NUMBER. U\ 64. What part of the whole number will be in the last stall ? 65. What part will be in the other stalls ? 66. Three men own a ship. One owns one third, another two fifths. What part does the third man own ? 67. A boy has eighteen cents. He buys five oranges at three cents each. What part of his money has he spent ? 68. There are five classes in a school of eighteen pupils. There are two pupils in the high- est class, and the other pupils are equally divided among the other classes. What part of the whole are in one of the other classes ? in three of the other classes ? 69. It takes John six minutes to write out six- teen words. How long will it take him to write out twelve words ? 70. There are fifteen minutes allowed for recess. John is kept in twelve minutes, and Mary ten minutes. What part of the usual time has each left for play ? 71. Fifteen men do a work in twelve days. How many men could do it in twenty days ? EXERCISES. 123 72. A railway train goes twenty miles an hour. How far would it go in three fifths of an hour ? 73. A man has eighteen horses. Two thirds are grey, two ninths are black, and the remainder white. What part are white ? What part are not black ? 74. Charlie had sixteen marbles. He gave three eighths to Tom, and one fourth to James. What part had he left ? How many more had Tom than James ? How much more ? 75. Divide eighteen cents between two boys, in the proportion of four to five. 76. Divide twenty cents among three boys, in the proportion of two, three, and five. It is unnecessary to suggest further exercises, as all other numbers are aggregations of tens and parts of ten.