Arithmetic

“ How far every boy ought to be made to do mathematics (beyond simple arithmetic) I cannot say. I know that I am extremely grateful for the small amount of mathematics forced into me. I am even thankful for a very mechanical side of the subject, namely, the use of mathematical tables in general, and for being compelled to work out innumerable sums by logarithms, which we had to do in a “neat tabular form” to quote our precise master’s words. ”

“ When we enter into the spirit of the methods of arithmetic, we perceive that they all flow clearly and simply from the very principles of numeration, from some precise definitions, and from certain ideas of relations between numbers, which all minds easily perceive, and which they even possessed in advance, before their teacher made them recognize them and taught them to class them in a methodical and fruitful order. ”

“ The transformation of functions belongs strictly to algebra. Teachers who love arithmetic, and who have themselves success in working out the so-called numerical conundrums, defend with much earnestness the current practice which uses so much time for arithmetic. They see in it a valuable training for ingenuity and logical analysis, and believe that the industry which discovers arithmetical ways of transforming the functions given in such problems into plain numerical operations of adding, subtracting, multiplying, or dividing is well bestowed. ”

“ And I believe that arithmetic, if it deserves the high place that it conventionally holds in our educational system, deserves it mainly on the ground that it is to be treated as a logical exercise. It is the only branch of mathematics which has found its way into primary and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favor of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to schoolboys. ”

“ Should you teach any one or all of the first eight years of school work the teaching of arithmetic will be of practical interest to you. Even the most modern of school courses has not eliminated arithmetic nor curtailed to any great extent the proportionate share of attention given to it. Most teachers will admit that the subject has been over-estimated, and that it is not of the great practical value popularly attributed to it. The man in business does not use the principles of arithmetic half so often as many believe. ”

“ On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science. It is no more than an interesting subdivision in a general theory. ”

“ I have said that all questions in arithmetic require nothing but addition and subtraction. I do not mean by this that no rule should ever be used except those given in the last section, but that all other rules only shew shorter ways of finding what might be found, if we pleased, by the methods there deduced. Even the last two rules themselves are only short and convenient ways of doing what may be done with a number of pebbles or counters. ”

“ The inductions of arithmetic are of two sorts: first, those which we have just expounded, such as One and one are two, Two and one are three, etc., which may be called the definitions of the various numbers, in the improper or geometrical sense of the word Definition; and secondly, the two following axioms: The sums of equals are equal, The differences of equals are equal. ”

“ IN our day arithmetic is looked upon as a science of which every boy at fourteen ought to be master. Such was not the case a century or so back. In England, as well as upon the Continent, arithmetic was long considered a higher branch of science, and a university study, like geometry. In part, this is accounted for by the strong conviction which has always possessed mankind until within the last two hundred years, that numbers have about them very potent and mystical properties. ”

“ Sometimes your teachers set you more complicated problems than:—What is the price of six pounds of sugar? For instance:—In what proportion must one mix tea bought at 1s. 4d. a pound with tea bought at 1s. 10d. a pound so as to make 5 per cent. profit by selling the mixture at 1s. 9d. a pound?
Arithmetic, then, means dealing logically with certain facts that we know, about number, with a view to arriving at knowledge which as yet we do not possess.
When people had only arithmetic and not algebra, they found out a surprising amount of things about numbers and quantities. ”

“ As counting is the fundamental operation of arithmetic, and all other arithmetical operations are simply devices for speed by using remembered countings instead of going through the detailed work again each time, the hint is furnished 19 the teacher for the first lessons in arithmetic. This hint has been generally followed out and the child set at work at first upon the counting of objects so much alike that the qualitative difference is not suggested to him. He constructs gradually his tables of addition, subtraction, and multiplication, and fixes them in his memory. ”

“ The £1 is termed the unit. A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times 1s. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced. ”

“ It is possible, as one knows from experience, for a boy to have mastered all the arithmetical rules and tables with which his master has supplied him, and to have all his measures and weights at his fingers' ends, and yet to be so destitute of the arithmetical sense as to give without a moment's misgiving an entirely nonsensical answer to a simple arithmetical problem,—to say, for example, as I have known half a class of boys say, that a room is five shillings and sixpence wide. Such a boy, though his head may be stuffed with arithmetical information, has no knowledge of arithmetic. ”

“ The sciences classed as pure mathematics are Arithmetic, Geometry, Algebra, Analytical Geometry and Calculus. Arithmetic is eminently the science of numbers, and treats of, or practically illustrates their nature and uses. ”

“ But mathematics uses these facts to deduce more and more complicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. The question, “What is a number?” is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. ”

“ In the ordinary arithmetic the numbers or figures employed, taken separately, have always the same value, and the result, when, sometimes by a tedious process, obtained, is applicable only to the particular question proposed, but in solving the problem by algebra, since we employ letters to which any values may be attributed at pleasure, the result obtained is largely applicable to all questions of a particular class. ”

“ They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year.
When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. ”

“ IN this age the knowledge of arithmetic is so widespread that it is difficult to conceive that it is comparatively young. The three R's, Reading, 'Riting and 'Rithmetic, seems to our mind almost axiomatic and fundamental in regard to education. It is, therefore, very interesting to read in histories of arithmetic that the earliest known book which contains a systematic exposition of the decimal system of numeration was written in the seventh century, and that our familiar arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, can not be traced to an earlier century than the seventh. ”

“ The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. ”

“ In mathematics, too, we should be looking in vain, at that stage where the real instruction in mathematics begins, for anything new and more easy, based on the whole previous instruction of the exercises of the second [Pg 272] year up to 20. Where in arithmetic the real difficulties are met with, where it becomes necessary to explain the subject from all its sides to the pupil, as in numeration, in addition, subtraction, division, in the division and multiplication of fractions, you will not find even a shadow of anything easier, any new explanation, but only quotations from old arithmetics. ”

“ Since the abilities which together constitute arithmetic ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. ”

“ When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers. ”

“ Arithmetic is the art or science of computing by numbers. It is national, political, military, and commercial. It is of the highest importance to the community; because it pre-eminently teaches us to take care of Number I. Our ministers succeed according to their knowledge of the science of numbers. Witness the skilful management of majorities of the lower house.
He who understands the true art of Addition, Subtraction, Multiplication, and Division, as here laid down, will not be considered a mere cipher in [4] the world ”

“ In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic ”

“ Efforts to arrive at a much more accurate definition of the term arithmetic are apt to meet with disappointment. On the one hand, we meet with contradictory classifications among works of the highest authority. The great mathematical encyclopedia which is being published almost simultaneously in German and French includes determinants under arithmetic, while the International catalogue of scientific literature places determinants under algebra. ”

“ This is expressed in the following, which is tacitly accepted in algebra, and should be accepted equally in arithmetic:
'In all computations and operations in arithmetic, all numbers are essentially abstract and should be so treated. They are concrete only in the thought process that attends the operation and interprets the result.'"
(4) Least common multiple.—The whole set of bonds involved in learning 'least common multiple' should be left out. ”

“ These proposed measures of ability to apply arithmetic illustrate very nicely the differences of opinion concerning what applied arithmetic and arithmetical reasoning should be. The thinker who emphasizes the fact that in life out of school the situation demanding quantitative treatment is usually real rather than described, will condemn a test all of whose constituents are described problems. ”

“ The measures which are used in this country are not those which would have been chosen had they been made all at one time, and by a people well acquainted with arithmetic and natural philosophy. We proceed to shew how the results of the latter science are made useful in our system of measures. ”

“ The principle is obvious enough and would probably be admitted in theory by any intelligent teacher, but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied. For example, we inherit the convention of studying addition of integers thoroughly, and then subtraction, and then multiplication, and then division, and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”