Why Arithmetic should be taught.

297

spicuously one of those subjects of school instruction the purpose of which extends beyond itself. Its ideas and processes can be effectively applied to other regions of knowledge. You cannot measure its intellectual usefulness by looking only at its immediate aims. It is, in fact, both an Art and a Science :-an Art because it contemplates the doing of actual work, the attainment of definite and useful results; a Science because it investigates principles, because he who unearths the truths which underlie the rules of Arithmetic, is being exercised, not merely in the attainment of a particular kind of truth about numbers, but in the processes by which truth of many other kinds is to be investigated and attained.

an Art

Now it is unnecessary to remind you that of these Often rctwo aspects or uses of Arithmetic, the former is that which garded as we usually associate with the name. It is not reasoning merely. about numbers, but using figures for the purpose of calculation and working out sums, that we generally understand by the study of Arithmetic in schools. A text book of Arithmetic is often a book of exercises and problems, and nothing more. We all remember Goldsmith's schoolmaster, of whom it was said that

"Lands he could measure, terms and tides presage,
And even the story ran that he could gauge."

Such a pædagogue, who could do sums of surprising
length and intricacy, and set them down in beautiful figures
in a book duly garnished with flourishes, passed then
for the good arithmetician. The scholar who could work
out the largest number of problems by the shortest and
most dexterous methods was the winner of all the prizes,
and so long as he produced right answers, the extent to
which he had understood the processes he employed was
a matter of small concern.

Robert
Recorde's
Arith-
metick.

No doubt this notion of the place Arithmetic should hold in school-work, and of the object to be attained in teaching it, is still very prevalent. But it was not always so. Arithmetic, as taught in the schools of Athens or Alexandria; to the contemporaries of Socrates and Alcibiades; or later, when in the Middle Ages it shared with logic, geometry, grammar and rhetoric and music the distinction of forming one of the staple subjects of a liberal education, was taught in its principles, as a logical discipline; as something to be understood rather than as a series of devices for working out problems. It was however often mixed up with some wholly unsound and indefensible theories about the mystic properties of certain numbers; and numerical relations were supposed to furnish the key to certain moral and spiritual questions, with which we now think they have nothing to do.

It is interesting to turn to the oldest treatise in Arithmetic in our language and to see the spirit in which the subject was treated.

In Robert Recorde's Arithmetick, or the Grounde of Artes, dedicated to Edward VI., we have the first successful attempt to popularize the study of the 'Algorithmic science,' as it was then called, in England. It is written in the form of a dialogue, for, as the author quaintly says in his Preface, "I judge that to be the easiest way of instruction, when the scholar may aske any doubts orderly, and the master may answer to his question plainly." Accordingly, the book opens thus:

Scholar. "Sir, such is your authority in mine estimation, that I am content to consent to your saying, and to receive it as truth, though I see none other reason that doth lead me thereunto : whereas else in mine owne conceite it appeareth but value to

Robert Recorde's Arithmetick.

289

bestowe anie time privately on that which every childe may and doth learne at all times and hours.

Master. Lo, this is the fashion and chance of all them that seeke to defend their blind ignorance, that when they think they have made strong reason for themselves, then have they proved quite the contrary."

He goes on to vindicate his favourite study, and to shew its importance; and the docile pupil, whose function it is throughout the work to exhibit constant wonder and delight at the revelation of each new rule, soon expresses interest in the subject, and is conducted through the science in a spirit and temper which cannot. be too much admired, if we may take the following fragment as an example:

“Scholar. Truly, Sir, these excellent conclusions do wonderfully make me more and more in love with the art.

Master. It is an art, that the further you travell the more you thirst to goe on forward. Such a fountaine, that the more you draw the more it springes; and to speake absolutely in a word (excepting the study of divinity which is the salvation of our souls), there is no study in the world comparable to this, for delight in wonderfull and godly exercise: for the skill hereof is well known immediately to have flowed from the wisdom of God into the hearts of man, whom he hath created the chiefe image and instrument of his praise and glorie.

S. The desire of knowledge doth greatly incourage me to be studious herein, and therefore I pray you cease not to instruct me further in the use thereof.

M. With a good will, and now therefore for the further use of these two latter (multiplication and division) the seat of reduction."

In this way master and pupil proceeded amicably together through integral and fractional Arithmetic, only pausing now and then to congratulate one another, and to offer devout thanksgivings to God for the beauty of the science, and for its marvellous uses. Recorde subsequently published an advanced treatise, entitled

F. L.

19

The place of Arithmetic in a School

course.

the "Whetstone of Witte, containing the extraction of roots, the Cossike practice, with the rule of equations, and the woorks of surd numbers." This book contains an admirable summary for the period, of the chief rules for the manipulation of algebraic quantities; but throughout both books it is the intellectual exercise, not the useful application, which seems to the author to be of chief interest and importance.

It must be owned however that if early writers thought little of the practical usefulness of the applications of Arithmetic our immediate ancestors and many of our contemporaries have thought of these practical applications almost exclusively. Since Recorde's time the majority of authors-from Cocker, and Wingate, and Vyse, and Dilworth, to Walkinghame and Colenso-have treated Arithmetic from the utilitarian point of view exclusively. Their books give few or no demonstrations of the theory of numbers, but are filled with what are called commercial rules. There are tare and tret, alligation, foreign exchanges, partnership with time, partnership without time (whatever that may mean), bills of parcels, the chain rule, a new method of finding the cubic contents of a cask, and so forth. The goal to be reached in the teaching of arithmetic is very clearly defined, and all the progress towards it is regulated accordingly. The successful arithmetician is to be a good computer, a skilful tradesman, a land surveyor, or an exciseman; and the whole object of the art is to fit him to perform one or other of these important functions.

We are so accustomed to hear Arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction, that it seems like enquiring too curiously to ask why this should be. Reading, Writing and Arithmetic-these three are assumed to be of co

The place of Arithmetic among School Studies. 291 ordinate rank. Are they indeed co-ordinate, and if so on what ground?

In this modern "trivium" the art of Reading is put Two purfirst. Well, there is no doubt as to its right to the fore- roses to be served by most place. For reading is the instrument of all our it. acquisitions. It is indispensable. There is not an hour. in our lives in which it does not make a great difference to us whether we can read or not. And the art of Writing, too; that is the instrument of all communication, and it becomes, in one form or other, useful to us every day. But Counting-doing sums,-how often in life. does this accomplishment come into exercise? Beyond the simplest additions and the power to check the items of a bill, the arithmetical knowledge required of any well-informed person in private life is very limited. For all practical purposes, whatever I may have learned at school of fractions, or proportion, or decimals, is, unless I happen to be in business, far less frequently available to me in life than a knowledge, say, of the history of my own country, or of the elementary truths of physics. The truth is, that regarded as practical arts, reading, writing, and arithmetic have no right to be classed together as co-ordinate elements of education; for the last of these is considerably less useful to the average man or woman not only than the other two, but than many others which might be named. But reading, writing, and such mathematical or logical exercise as may be gained in connexion with the manipulation of numbers, have a right to constitute the primary elements of instruction. 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