Lectures on the logic of arithmetic

Front matter

LIBRARY

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Preface

Intropuctrory AppRESS TO YOUNG

TEACHERS How Men LEARNED IN OLD TIMES On Countine By TENS

Wuy WE DO NOT ALWAYS DO Sums THE WAY THAT COMES NATURAL ,

ARITHMETICAL SHORTHAND .

Keeping Accounts

Dis-MEMBERING AND RE-COLLECTING Weicuts AND MEasuReEs Muttipiying By Minus

Appine Minus ..

DivipING AND SHARING

In wHat consists Economy Economy or Minp-Force

EXERCISE IN ReLevance; Inrropucine THE IpEA oF PROBABILITY .

EXERCISE ON ZERO

eos

EQuIvVALENT FRAcTIONS Greatest Common MEASURE

STANDARD Wercguts AND MEraAsuRES

>. 0.41 XXIT.

XXITT.

CONTENTS

WHAT CAN BE SETTLED BY Human Law Parser Money |

Tue Doe's Patu

Tue Batu’s Pari

EXERCISE TO PREPARE FOR GENERAL ForMULAE .

APPENDIX: On Proportion.

PAGE 122

127 132 136

139 142

PREFACE

THE motto chosen for the title-page of this little work may seem unsuitable as an Introduction to a course of lessons on Arithmetic ; a subject which to many persons seems so eminently un-heavenly and dull. But then, the main reason why it seems so is that their teachers failed to put them in possession of that ‘ Recovered Past, the bearing of which on the Present forms the great clue to that knowledge of subtle forces which gives‘ its possessor the key of the Future.

The present little volume is not intended to interfere with ordinary methods of teaching Arithmetic, or to supersede the books already in use. It can be used under any School system and in conjunction with any text-book. Not more than one chapter is intended for use in any one term; the earlier chapters are suited to little children, the later ones for children of fourteen or fifteen. - The lesson for the term may be repeated each week with varying illustrations.

Teachers of such subjects as Electricity complain of the difficulty of getting pupils to apply what they know of Mathematics (at whatever level) to the analysis and manipulation of real forces. It is not that the pupil does not know enough (of Arithmetic, or Algebra, or the Calculus, as the case may be), but he too often does not see, and cannot be got to see, how to apply what he knows. Some faculty has been paralysed during his school-life; he lacks something of what should constitute a living mathematical intelligence. In truth he usually lacks several things.

In the first place, though he knows a good deal about antithesis of operations (e.g. he knows that subtraction is the opposite of addition, division of multiplication, movement in the direction minus — x of movement in the direction x, and so on), he has not the habit of observing in what respects antithetic operations neutralize each other, and in what respects they are cumulative; and surely no habit is more needed than this as preparation for making calculations in electricity or mechanics.

In the next place, he too often knows, about the idea of relevance, only enough to be foggy about it. The reason for this is that his study of the idea of relevance itself began where it ought to have ended ; his attention was never called to it till the stage was reached when it would have been right that he should direct his action in regard to it sub-consciously from long habit, leaving conscious attention free for dealing - _ with the actual elements of some question which is difficult enough to need thinking about. He was not made to grasp the fundamental idea that a statement may be relevant to one question and irrelevant to another, till some knotty problem occurred involving consideration of which statements are relevant to the special question in hand. So he had to try to grasp at once the idea of relevance and the question, what is relevant to what, in a special problem. Such thrusting on the young brain of two difficulties — of different kinds at once, is contrary to all accepted canons of Psychology. Examples are here suggested (Lesson XIII) in which there can be no doubt as to what is relevant to the question at issue; the child’s attention is therefore free to focus itself on the idea that there can be facts concerning a thing which in no way concern the particular question which is just now being asked about the thing.

Then again, whatever skill he may have acquired in the manipulation of those notations and formulae which he has been taught to use, he knows hardly anything about the manner in which such things come into being. Now an applier of Mathematics to real forces should be able, when occasion requires, to modify his notation, or invent a new formula, for himself.

He cannot begin to learn how to do this, straight away, while his mind is struggling with problems of electricity or mechanics; he should have had, from the first, the habit of seeing through formulae and notations ; of watching them coming into being, of helping to construct them.

The special difficulties found in interesting pupils in Arithmetic are in no way inherent in the nature of the subject ; they belong rather to the domain of medical psychology; their cause can be fitly described only in the language of that science. However logical may be the course of a teacher’s demonstration, gwen his premises, those premises are usually introduced to the pupil under conditions extraordinarily unfavourable to concentration of the attention-power. One can only describe them by saying that if analogous conditions were set up in the stomach-nerves at the beginning of dinner, the result would be lessened appetite and digestionpower; if in the muscles at the outset of a ericket-match, the running would in consequence be slower, the batting, bowling, and fielding less accurate, and the heart-action more irregular ; if in the eye- and finger-nerves before a drawinglesson, the result would be dazzled sight and trembling fingers.

It may be objected that if this temporary partial paralysis is set.up, in the Arithmetic class, only on the occasion of the first presentation of some new idea or principle, i.e. perhaps once in a term or two, it is not frequent enough to induce disease or permanent weakness of the attention-power. But if any condition of nerve disturbance or lowered vitality accompanies the first presentation of an idea or object, there is a strong probability that each subsequent contact with that object or idea will produce, by _association, a recurrence of similar disturbed action. Thus is set up a habit of scattered | attention which retards the development of the faculty of concentration, and makes any real grasp of principles difficult, especially in regard to Mathematics. One main object of the present volume is to present the premises involved in various arithmetical operations in a manner which concentrates instead of scattering the power of attention, and thus leaves the pupils fit to attend to the teacher's reasoning.

My own experience, however, is that a child who has been helped to grasp the premises of a mathematical argument, in any hygienic manner, seldom needs much assistance in making out for himself the logical consequences.

The Preface to a recently issued text-book of Arithmetic! asserts that ‘the young are always eager for a rule or formula which will save

1 Arithmetic, Kirkman and Field.

them the trouble of thinking for themselves.’ The young who are reeling under a blow on the head from a bludgeon may not unnaturally be eager to be taken home to bed in the easiest vehicle at hand; and there is no telling what habits of lethargy might not be generated by a systematic course of such treatment ; but the young who have been left in possession of their normal faculties prefer using those faculties m active personal exercise.

Another teacher says ! :—

‘In stocks and shares the pupil is usually dealing with things the nature of which. he does not understand. A little political economy must be drilled into him. Bring the meaning of the abstractions of the subject home to him.... The great difficulty is in problems. Nothing but stern practice can avail here. Discourage all formulae. Anybody who knows anything about boys knows that if, by any means under heaven, they can avoid using their brains and attain the same result by a mechanical application of a formula they will do so. They cling to a formula as to a straw, in the ocean of new ideas in which they are drowning.’ Exactly so; children’s minds are too often left to clutch at straws ‘in an ocean of new ideas.’

7 Workman, ‘ Oxford Junior Local,’ in Educational Review.

PREFACE 4a

If, two or three terms before a pupil has to do with fluctuating values, he had any clear idea why a Bank of England note has a fixed value, it would be easier than it usually is found to make him grasp the meaning of stocks ‘and shares.

The mechanism by means of which man learns is intended to play among the conceptions Unity, Negation, and Fraction ; Number ts less cognate to it. The kind of fractions dealt with in Arithmetic books is, it 1s true, too recondite to be thrust on a brain at an early stage. But if mathematical faculty is to develop normally in the youth, the child should have early practice in dealing with the idea of a unt broken into bits, which fit together to make up a whole.

Nature provides savages with opportunities for dealing with this idea, to an extent of which it is difficult for us to form any adequate conception. Something fas or has not been caught for dinner; and when it has been caught it has to be shared and is entirely eaten. Next day the child again sees an animal of the same kind whole. The child’s mind plays constantly between the idea of rabbit (or whatever animal may be used for food) whole, rabbit. shared, no rabbit; and again, whole rabbit. This has a very different effect from what is induced by the sight of a loaf or joint appearing as a matter of course; a loaf from which every one is helped, the rest being taken away when no longer interesting. This is one of the reasons why an existence uniformly, uninterruptedly prosperous is not always the most favourable to intellectual development.

In all ranks of life, it is true, infants provide themselves with opportunities for exercising the faculty of reconstructing a unit from its fractions, by tearing or breaking things and fitting the bits together again. Our Kindergarten systems to some extent provide for systematic exercise of the same kind by means of various toys. Even there, however, though much practice is given in constructing, too little is given in re-constructing a unit out of its fractions. But in schools the whole provision for this practice, which is the essential fibre of all mental growth, seems left to chance, and too much crowded out.

In Arithmetic, where it is specially important, hardly any opportunity is afforded for practice in swinging the mind between the conceptions Unity, Negation, and Fraction. The mass material of Arithmetic itself (i.e. the art of dealing with numbers) is packed into the mind artificially ; little or no possibility is provided for it to build itself up by natural accretion round its organic supporting fibre. This tends to induce a condition of what I have called mental rickets. ,

It is desirable to fix children’s minds on the act of Negation as a positive act of mind, to a far larger extent than is usually done. Opportunities for this might be taken in connexion with sums in multiplication and long division ; the zeros should be written in, as statements of fact, for some time after the pupil has grasped the idea that their omission does not actually affect the ultimate answer.

Use might also be made of such exercises as those of Chapter XIII.

Again, the teaching of Arithmetic is much facilitated, if, besides actual exercises on.o and I and very simple fractions, logical exercises parallel to the operations of Arithmetic and constructed on the same models, are carried on in that borderland region where mental operations parallel to those involved in Arithmetic deal with questions of Art, or such simple portions of History, Ethics, or social relations, as come fairly within the scope of a child’s intelligence. Hach operation of Arithmetic may find its analogue in one of these borderland exercises’ in Logic.

It is of primary importance that children should clearly distinguish between what they do or recite because they are told to do so, and what they themselves grasp or see. They must do certain things because they are told, and commit to memory facts which they will need to know; but they should always be taught to sg Snvsavtilh between :—

zt repeat pois because I was tlds and ‘I see that this is so.’

Some of the worst mental habits are induced by the practice of teachers making a statement as if ex cathedra, and then proceeding to bring forward proofs of its truth; this is Euclid’s method, but Euclid apparently wrote for grown-up men, perfectly well able to take care of their own minds; he wrote probably for the best intellects of his time. His method is unsuited to children. If a teacher has anything to say to children as a statement, he should say it, not exactly as dogma which they are bound to believe, but as working hypothesis which they are to assume as a basis for the present. Anything which he intends to prove should never be stated ; children should be led up to find it out for themselves by successive questions. No pains should be spared to keep the two sets of statements apart.

To return to Mr. Workman’s complaint of the abuse of formulae, surely the remedy lies not in discouraging the use of them, but in encouraging children to use them intelligently ~and to make them for themselves. They should begin early the practice of entering certain kinds of results in a book. This book may be divided into two parts: anything specially needful to remember, if found out by the children, should be entered in one part; if told them by the teacher, in the other.

‘The sentimental people who assert that everything in Arithmetic can be ‘ proved’ to children have, usually, no idea of what rigid proof means ; it is not necessary that the child should see the evidence for every hypothesis on which he works; what is necessary to mental health is a clear understanding of what constitutes evidence, and the power to distinguish between what is, and what is not, proved.

The children should be encouraged to use freely, for reference, any tables or formulae constructed by themselves. This should be permitted as an indulgence, a labour-saving luxury, the reward for accuracy in recording the results of previous investigation. In private teaching and in very small classes, no child should be allowed to enter any formula into the form-book till he is individually able to work it out unaided. In ordinary class-teaching this may be impossible. In that case the process should be gone over at intervals, till

BOOLE B the class as a whole can, amongst them, evolve every step. The formula should then be entered by each child in his own form-book; and the impression should be conveyed that it is the communal work of the class, the expression of its conjoint logical investigation.

The formula-book should begin with a blank form of multiplication-table, which is to be filled in by the child as practice in. addition. The operation of multiplying should be introduced to the child’s mind in some such way as this :—‘ Add together four and four and four and four and four. When the addition sum has been done correctly, say :—‘ You will have to add five fours together very often. You may enter the result in your table-book ; and next time I give you five fours to add together, I will allow you to look for the answer in your book instead of adding.’

All the modern higher Mathematics is based on a Calculus of Operations, on Laws of Thought. All Mathematies,-from the first, was so in reality; but the evolvers of the modern higher Calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about Algebra and Arithmetic as dealing with Laws of Number, and about Geometry as dealing with Laws of Surface and Solid Content, are doing the best that in them lies to put their pupils on the wrong track for reaching in future any true understanding of the higher Algebras. Algebra deals not with Laws of Number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on number. Plane Geometry deals with such Laws of Thought as have been discovered by men intent on finding out how to measure surface; and. solid Geometry with such additional Laws of Thought as were discovered when men began to extend Geometry into three dimensions. The branch of Mathematics called Quaternions deals with such Laws of Thought as reveal themselves during the process of investigating the relations between x and 2 + 1 dimensions. The sooner pupils are made to see all Laws of Arithmetic as Laws of Thought, not of things, the simpler and more satisfactory will their future course be.

This is especially the case with regard to the sion — ; it should never be interpreted in any such way as to convey the impression that it “indicates negative quantity, subtraction, or diminution of numbers or of things; but always as indicating something about the point of view from which the things or numbers are considered. Neglect of this caution lands one B 2 in endless metaphysical contradictions and absurdities. Such for instance as that a rectangular surface becomes = 0 if we happen to place the origin of co-ordinates in the middle of it; for it then divides naturally into the four rectangles X x Y, X x (— Y), (— X) x (= Y), (— X) x Y. The amount of confusion and hindrance to progress thus induced is sickening to think of.

Teachers seldom observe how early many children are affected by such anomalies as these, or how profoundly they are affected by them. If the same boys who are thus painfully impressed became teachers in their turn, they would no doubt remember their own past experience, and a better mode of presenting the primary conceptions of Arithmetic and Geometry might soon be evolved. But, unfortunately, these logical thinkers are usually repelled by the impossibility of extricating themselves from the network of insincerity, or what seems to them such; give up the study of Mathematics, and betake themselves to some other pursuit. Very much of the waste. and confusion which gather round the sign — might be obviated by making children go through such exercises as those given in Chapters VIII and IX.

Much needless difficulty is caused by the attitude of Mathematical teachers towards the symbol « , which they often interpret as somehow connected with endless length or exhaustless quantity. The effort to conceive of either endless length or exhaustless quantity is a metaphysical gymnastic very unhealthy for a young mind, It sets up also a set of ideas and connotations quite irrelevant to any use of the symbol « in the higher Mathematics. © has reference not to length or quantity, whether great or small; but to release from certain restrictions to which the values specified as finite have been subject. (See Lesson XV.) The whole terminology connected with the operations called in Arithmetic books ‘Multiplication’ and ‘Division’ is excessively misleading. The words ‘multiply’ and ‘divide’ might have been invented on purpose to create confusion in passing from integral to fractional Arithmetic. The continued use of the word ‘ dividend ’ especially, in Arithmetic, is a symptom of the carelessness of teachers in the matter of avoiding causes of confusion. In old days, when most people who had money hoarded it in the house or deposited it in a bank, till they could use it in some business of their own, the word ‘dividend’ had no connotation for a child’s ear, till it was explained to him as ‘an amount to be divided.’ But in these days of limited liability companies, the word ‘dividend’ is incorrectly but very commonly used, at home in the hearing of children, to indicate a sum of money which is the result, or quotient, of the division and distribution of the year’s profits among’ the shareholders: ‘I will do so-and-so when my dividends come in, &c. Even if the teacher is careful to explain that the word in the text-book does not mean what it means in common parlance, which not all teachers remember to do, that explanation in itself causes distraction ; some word should be chosen which has no misleading associations. Of course children ‘ outgrow’ any misapprehension which may be thus caused, and learn that ‘dividend’ has two meanings. But one wonders what teachers suppose is the effect on the mental habits, on the brain tissue, of struggling, for even a few days, to understand division before this discovery has been made! The present writer has not authority to devise a nomenclature ; and therefore in this introduction employs that in common use, though under protest. It should be avoided in speaking to the class. |

The teacher should, in dealing with any branch of the operation called ‘ multiplication,’ have. in his mind this idea :—‘ Multiplication’ means doing to the operand (multiplicand) what, if done to unity (1), would produce the operator (multiplier). ies

In dealing with ‘division’ he should keep in his mind the idea :—‘ division’ means doing to the operand (dividend) what, if done to the operator (divisor), would produce unity (1).

The operation of division should be introduced at first, not as the inverse of multiplication but as a succession of subtractions, thus :—

If five boys are to share twenty apples among them, how many can each boy have? Let us give each boy one apple first ; that takes off? Five apples. And leaves? Fifteen. Now we give each boy another apple: that takes off? Five more. And leaves? Ten. Another to each boy leaves? Five. Another to each leaves? None. So now we have finished. How many times did we go the round and give each boy an apple? Then each boy has? Four apples.

A. little practice in such successive sub tractions should be gained, before the idea is introduced of doing long division straight off. Division should not be treated as the mere inverse of multiplication, but as a way of cutting short a laborious series of subtractions... The child should be led to see that the multiplication table can be used to cut short not only series of similar additions, but also series of similar subtractions. A great deal of the dithculty afterwards found in dealing with such operations as G.C.M. is traceable to neglect of this simple precaution.

The manner in which that most logical of all text-books, Euclid, has been misused to induce illogical habits of mind, repeats itself in most departments of elementary Mathematics. <A good instance of such misuse can be shown in connexion with the subject of equivalent fractions (Lesson XVI). Let us suppose that the — teacher wishes to prove to the class that three-quarters of one is equal to one-quarter of three. He takes as his unit of thought the concept apple. He too often begins by stating a thing which is not true in itself: viz. that three-quarters of an apple is the same thing as a quarter of three apples. For the purpose with which the teacher's mind is occupied at the moment (the equivalence of fractions), the statement is true; it is as true as that, from the shop-keeper’s point of view, twelve pence are a shilling. But the child, who has not yet been introduced to the conception of equivalence of fractions, is at a point of view quite different. To the child’s imagination ‘great lots of grub’ have a fascination quite independent of the size of his own share. To suggest (for instance) that a bun for one child is the same as a hundred buns for a hundred children would be to insult the pupil’s understanding and his feelings— feelings which affect him all the more keenly that as yet he does not know how to express them. If he could believe you, what would be the meaning of school treats and social picnics? In proportion as you ultimately induce him to feel that one bun for a child is the same as a hundred buns for a hundred children, he will be, in future, the worse man, citizen, political economist ; and the less fit to apply mathematics to problems in real forces.

But we are speaking now of the effect of such statements on his intellectual processes, at the age when his sensations are as yet unwarped. The statement that a quarter of three apples is ‘the same as’ or ‘equal to’ three-quarters of an apple has brought prominently into the field of his imagination two pictures :—an apple, and a group of three apples. The natural way of making an apple equal to three apples is to bring forward two more apples. It is the way things used to be made equal when he was learning addition ; why not now? The teacher has therefore started him looking out for the two more apples ; and as what is said next does not seem to be tending in that direction, his attention is distracted; he gives only half his mind to what is said, Thus he fails to get hold of one or more links in the chain of argument. Some children soon recover themselves, and attend to the rest of what is said. And, as the teacher seems to be satisfied with the reasoning and to expect them to be so, they imagine they are so. This helps to form a habit of intellectual dishonesty, of confusing sham proof with real proof. The children who are either less sharp or more logical and thorough, look longer for the non-appearing two apples, and miss more links of the chain of proof. These children feel that the whole demonstration has passed in that region where grown-ups conduct a self-satisfied mental life in which children cannot share. Thus are formed habits of ‘hopeless non-comprehension,’ perhaps even of ‘self-protecting and contemptuous non-atten-— tion. The chapter on equivalent fractions (Lesson XVI) suggests ways of attacking one 7 problem not open to this objection.

It may occur to some that too much use is made of examples relating to food. But we cannot make mathematicians by insisting upen a non-existing superiority to physical facts. Apple or bun forms the natural unit for a child; the sharing of a cake or fruit is the natural fraction as well as the true introduction to the higher ethical life. As a matter of fact the Arithmetic of grown people is largely occupied over questions of food supply and of personal or family interest. To place children’s ideas about such subjects on an honest basis and give them a social direction would surely tend more towards ethicalizing the community than the setting up of topics of fictitious interest, and then teaching children to be satisfied with reasoning which is either imperfect in itself or but partially understood by them. To let the children be thoroughly logical about their own sphere of action among things which they do care about, and deal practically with duties which they must perform, is far better training than to encourage them to express opinions about subjects the data of which lie beyond their personal ken.

The pauses of sitting with slack muscles and taking slow quiet breaths, to make mind-pictures, is not intended to supersede the usual permitted ‘intervals’ between lessons; they are an integral part of the Logic lesson itself. They, as well as the rest of the scheme here suggested, form, it is believed, the first attempt to adapt to elementary education here the magnificent method of study described in Gratry’s Logique. A certain element of normal mental life, essential to sound intellectual development, and which can only be received during suspension of the active perceptions of intellect, is being driven out by modern educational grind. This tends to produce a sort of mental and moral rickets, analogous to the physical disease induced by a diet deficient in the bone-forming element. Teachers who have studied any sort of medical psychology bitterly lament that the requirements of existing systems force them to inflict this grievous wrong upon the rising generation. In the present volume, an attempt is made to re-introduce at least a small share of the skeleton-forming elements of mental life, in a manner which does not necessitate disturbing existing arrangements. The author has been kindly assisted in this matter by Mrs. Archer and her pupil Miss Cross, of Coombe Hill School.

English children have now to be taught to relax, to breathe, to see. It is probable that the cultivated art of relaxation will ultimately prove more satisfactory than the mere following of natural impulse to relax.

But it is certain that the teachers who have weeded out the wild growths must see about planting the cultivated ones. Such exercises as those here recommended tend to promote sane and healthy mental action, and to make the discipline of school more organic and harmonious. They have a great fascination for children, as my young friends at Coombe Hill School know. I am sure they will be glad that I am attempting to give to other boys and girls a share in the fun that we had over our little sums.