The Parents' Review

A Monthly Magazine of Home-Training and Culture

First Stage in Arithmetic

[Robert Hebert Quick, 1831-1891, graduated from Trinity College, Cambridge, and was assistant master at Cranleigh School, and Harrow School (both near London). He lectured on the history of education, wrote about Froebel, and edited John Locke's "Some Thoughts Concerning Education."]

A Lecture given at the College of Preceptors [May 16, 1888]
[From pamphlet printed for the Author by C. F. Hodgson & Son, I, Gough Square, London]

In addressing you to-night, for the first time in this building, I naturally go back in memory to other lectures I have given at the College Preceptors. It is now twenty years since the College first asked me to lecture, and great changes have taken place since then. Now we have this splendid building instead of the "adapted" dwelling-house in Queen's Square. And I am glad to think that the change in premises is a material expression of the change in the position of the College itself, and that not only has the College prospered more and more during these twenty years, but the great object with which it was founded, the improvement of education in this country, has also been greatly advanced.

Now that I am getting very near the time of life which Horace describes as beset by disadvantages, I ask myself if there are any compensations. According to his forecast, I shall soon, if I live, be

"Difficilis, querulus, laudator temporis acti
Me puero, castigator censorque minorum."

[Google translate:
More difficult, given to praising the act
My child, chiding and criticizing.]

But I cannot say that I feel all this coming upon me. On the contrary, the longer I live the less I am tempted to praise the past time of my boyhood. Difficilis, querulus [More difficult, complaining]. Yes, I may be; I am not easily satisfied, and I am by no means satisfied with things as they are. But laudlor temporis acti [praising the years] I am not; and if I am ever aridus futuri [the dry future], it is chiefly because I should like to see the great changes which I believe to be impending.

So to me, one of the commoda [advantages] of old age, which we may set against the incommoda [disadvantages], is the thought that we have seen vast changes for the better, and from them may derive good hope for the time to come. To-night I wish to avail myself of this opportunity of addressing you to urge another change, a change in the method of giving children their first notions of number.

We practical English people are, of course, very suspicious of theory and theorists. We like to boast with the man in Poponilla, "Whatever theorists may say against our system, nobody can deny that it works well." But, unfortunately, we can't do this of our system of teaching arithmetic. Here is an art that has been practised ever since the schoolmaster came into existence, an art dependent not at all on fleeting fashions, but resting on the immutable laws of things and "not liable to any ruin." The practical man has had his fling, and surely he ought by this time to have brought the art to perfection. But even he must admit that he has not. The failures in arithmetic are notorious, and there seems hardly any art in a more deplorable state in this country than the art of teaching arithmetic. So, as the practical man has come to grief, he should not be above asking theorists to give him a helping hand.

A candid friend of mine, a schoolmaster, is fond of assuring me (with or without justice I, of course, cannot decide) that I am "too theoretical." However this may be, I have too much of the British Philistine in me to care very much about theory for its own sake. I wish to use it as a lamp, and, instead of staring at the flame, to pick my way by the light it gives.

As I think, theory may be of immense use to us in teaching arithmetic; and it is on this account that the subject seems interesting to me, and that I have ventured to try to interest you in it this evening.

When we consider such a subject as algebra, we distinguish between two parts of it. First, we have a number of principles which must be grasped by the intellect. Next, we have a system of conventional symbols, which we manipulate according to certain laws drawn from the principles. In this way we work out results on which we can entirely depend, although we do not with our mind's eye see the process as it goes along, any more than the miller with his bodily eye sees his wheat turning into flour. We observe, by the way, that many of these processes may be conducted by those who have no notion of the principles on which they depend. Most schoolboy algebra is mere dodging about with letters according to certain rules. The whole thing is meaningless.

What I have said about algebra is true also of arithmetic. There are certain truths of number which exist quite independently of our conventions. These truths we have to apprehend with our minds; and they, having adopted a system of symbols, we work out results by their means. In arithmetic, to be sure, we can see through the operation much farther than in algebra, but not very far after all.

How far, we never inquire. According to the doctrine of payment by results, we have only to give our lads a set of rules and to see that with them they can work out the right answer. In this way, and in this way only, we can "pass" our pupils. But this sort of arithmetic has two drawbacks, one from the theorical and one from the practical side. First, it does very little for the boy's thinking powers--indeed, it is in some ways injurious, for it gets him into habit of going along with the eyes of his mind closed; and, secondly, even the practical people find that arithmetic by rule breaks down directly there is the smallest variation which the rule has not provided for; so even they are in favour of building an intellectual basis on which the practical skill might rest.

But then comes the difficulty. There is this constant demand for results. In my schoolboy days, over 40 years ago, when the reign of the classics was unassailed, there was a measure by which the progress--intellectual, if not moral and physical--of the schoolboy was always gauged; "What book is he doing?" "Oh, he is in the first book of the Aeneid." "How old is he?" "He was 11 last March." "That is very good, if he has got into Virgil before he is 12!" It never seemed to occur to anybody that even the boy's knowledge of the Latin language, let alone his mental development, could not be fixed by the particular Latin author he was boggling over. I hope there has been some light let in on this point since those days. But, even now, a child's progress in arithmetic is measured by "what sums he can do"; and, while this is so, it seems almost a stern necessity, enforced by the law of payment on results, that a child should be early taught to do a number of things that he can by no possibility understand. The examination question is too large for us to discuss to-night. In my mind, there is no doubt that, generally speaking, examinations have, in some respects, a good effect, in others a bad effect. But I am now concerned with the earliest stages of learning only, and here I am convinced that examinations by an external authority make proper teaching impossible. Children themselves are apt to adopt the examination system, and when they have sown seeds, they every day or two remove the earth to inspect how their seeds are getting on. The answer made by the seeds to their inquiries is virtually of Miss Mary Hopkins, familiarly addressed by her admirer as Polly Hopkins:

"None the better, Mr. Tomkins.
For seeing of you,
For seeing of you."

The seeds, in fact, get on very much worse for the examination system; indeed, it is often fatal to them. And, if ideas are to germinate and develop in the minds of the young, a good deal must go on before they are ready for examination. At present, the inspector comes on a fixed day, and "passes" such of the young plants as his measure shows to be the regulation height above the ground. So the object of the teaching is not to produce correct ideas, all alive and growing, but to present the required appearance at the appointed time. As the inspector, by the rules of the game, must consider nothing but height, it is found the simplest plan to stick into the ground rootless twigs, which, for examination purposes, are just as good as plants, and give everyone much less trouble.

Our present system has got such a firm hold, and from use and wont seems to us so "natural," that I almost despair of a radical change, and yet there is little hope of good arithmetic without it. We must give up the examination system, and, instead of expecting children to do sums in this rule and that rule, we must gradually develop their conceptions of number.

As I hold, the most important stage of all is the first stage. If this is so, the beginners require the most skilful teaching. Mulcaster, the first headmaster of Merchant Taylors' School, whose "Positions" I have just republished, ["Positions," by Richard Mulcaster, 1551? (Republished 1655. Longman-103. 6d.?] said this of all subjects as early as the reign of Elizabeth, and he even recommended that those who taught beginners should get the highest salaries. As far as I am aware, his advice has not as yet been followed--in this country, at all events. Pupil-teachers who teach beginners arithmetic are, I suppose, not the best teachers, and they are certainly not the best paid; and the same thing is true--as to pay, at least--of nursery-governesses.

How children should get their first notions of number hardly anybody in this country knows, and, but for the goodly band of ladies who have now begun to study education scientifically, we might add hardly anybody in this country seems to care.

On the Continent, the problem was examined by Pestalozzi, whose celebrated threefold division of elementary instructions was Form, Number, and Speech. In this country Horace Grant's "Arithmetic for Children" [Published by Bell & Sons.] was a great stride in the right direction. Messrs. Sonnenschein and Nesbitt's "A B C of Arithmetic" [Swan Sonnenschein & Co.] also deserves honourable mention. [All teachers should, I think, know De Morgan's "Arithmetic"] The Grube method, well known in Germany, is now spreading in America, and there are at least two American books about it. [Grube's Method of Teaching Arithmetic by Levi Seeley, 1891; Grube's Method of Teaching Arithmetic Explained by Frank Louis Soldan, 1878] From such books as these, teachers may now, if they will, get some notion of how children should be taught.

On one occasion Pestalozzi went to Paris in the hope of interesting "the great Napoleon," as he is called, in education; but Napoleon treated him with contempt. He couldn't trouble himself, he said, with questions of A B C. Well, those who profess the art of destruction have always been honored, and those who profess the art of instruction have not. But in this case, instruction proved itself to be the greater force. Little did Napoleon dream that the down-trodden Germans would learn of Pestalozzi, and that when their day of triumph came, they would see in his ideas one of the main causes of the regeneration of their country. (See Address quoted in preface to Guimps' "Pestalozzi.") He who rules by ideas is also a potentate, and a higher kind of potentate than he who rules by the sword; moreover his power outlives him.

We see, then, that those who influence the young have their importance as well as those who make laws for the grown-up. This being admitted, I wish to go a step further and get you to agree with me that those who teach the beginners have greater influence than those who build on their foundation. All who have come to-night show by their presence that they are alive to the importance of the first stage in arithmetic at least; and although you may be well qualified to take the highest classes, you will not, like Napoleon, resent an attempt to bring before you questions of A B C. One of the most valuable results of intellectual education is the power to discriminate between facts and conventions. The teacher should always be extremely careful to exercise this power himself, and to develop it in his pupils. In the first stage of arithmetic, the great point is to keep to facts. I began by saying that algebra is a system of conventions made in agreement with certain principles of facts. Arithmetic in its first stage consists of facts, but these are soon used in connection with conventions.

The grand object in teaching children should be to get the facts firmly settled and incorporated with their thoughts before we go on to use the conventions. Now everything that the child uses without understanding is, to that child, a convention. To us it is not a convention that twice 12 are 24, but it is a convention to the child. In the first stage we must keep clear of conventions. But perhaps, some one may object in limine, "You must teach the child the names of the numbers, and words are nothing but conventions." In reply I might throw up the aegis provided me by two eminent men, Max Müller and the present Sir James Stephen, [See Nineteenth Century for April, 1888. pg 569, 743] and say that thought and language are identical. But neither their high authority nor their arguments, as far as I have examined them, have enabled me to adopt this position. I must admit, then, that directly we associate the sound one with a particular conception of the mind, and two with another, we have started a convention. But in all cases, let the fact come first, the conception before the sign. And the truth that we cannot live without food makes a poor defence for gluttony; similarly, though we must use a convention in sounds, this is no excuse for our present method or no-method, according to which children are plied with conventions, and are not taught (except as conventions) the simplest facts.

I hope, then, we shall all agree that to begin with we must settle firmly in the child's mind the facts about numbers up to ten; and make these facts thoroughly familiar, so that the child may be able to work with them. So far I trust I shall carry you all with.

Next, in the first stage keep as free as possible from conventions. Here again I feel so certain that this is right, that I shall be sorry if anyone dissents.

I proceed to some details, most of them in accordance with the Grube method. These I consider less axiomatic.

The Germans, as a rule, keep children employed for the whole of the first year on numbers up to 10, and for the whole of the second year on numbers up to 100. This seems to me entirely right; and in order, as far as possible, to exclude conventions, I would during these two years banish notations of every kind. In the second year the children might indeed be allowed to know the signs for the nine digits, but they should never work with them. They should learn to write them neatly, and both in upright and lateral columns, but this they should practice on paper as copies, and should never connect this writing with any computation.

Now for the first year, the Germans are most careful to make the instruction anschaulich, or as we might say, sensuous. Grube advances regularly a unit at a time and the children always have the units in some form in their hands. I think cubes of wood are the commonest form, and Tillich's box of bricks (sold by Birmingham Midland Educational Supply Company, Corporation Street, Birmingham, and by Mr. Arnold, Briggate, Leeds) are excellent for the purpose, but at present seem too dear for general use. [The large size cost 10s. 6d. the box. To be sure, they would last almost as long as the properties of numbers! W. Shepherd, 30, Paternoster Row, sells a nice box of 8 cubes ("3rd Gift.") for 3d.]

Another good form of unit would be the round coloured counters made of bone, which would be very inexpensive but, on the other hand, more easily lost.

With such things, the children find out experimentally all possible connections of the units and operations with them. That children should be kept for a whole year, say from five to six years old, over the first ten units, will seem to most English teachers an absurdity, if not an impossibility. I should not insist rigorously on the exact time, but I think if the child got thoroughly familiar with the ten units in the year, the time would have been well spent; and teachers should examine the plan as worked out by F. Louis Soldan, of St. Louis, ["Grube's Method of Teaching Arithmetic, explained by F. L. Soldan." Chicago. Interstate Publishing Company.] and on a larger scale by Dr. Levi Seeley, of Lake Forest, Illinois. ["Grube's Method by Levi Seeley." New York. E.]

Those who are new to the subject will be surprised to find how much there is to learn about the ten units. Many of us have not examined the notions of number in our own minds, still less in the minds of children. On examination, a very common confusion will be found between the conception of succession or order and the conception of co-existence. The two notions should be kept apart. Suppose, e.g., I clap my hands five times and count 1, 2, 3, 4, 5. In that case only one clap can be the fifth. But if I count out five shillings we get the notion of five co-existent things; the notion of order is lost. I may take them in any order, and any one may be the fifth. Now the notion of succession is sure to come to the child's mind as soon as he begins to count, but the notion of co-existent units of the same kind has to be developed. This is done by means of the things in the child's hands. By degrees he must learn to recognize their number up to 10 without counting. The notion of any of the odd numbers as a central unit with the same number of units on each side of it is a conception very different to that of succession, and children should be familiarised with it.

When the child has clearly grasped the notion of a number as of a collection of co-existing units, he will readily learn to see how the units may be divided into two or more groups. In 7, e.g., he will see 4 and 3, 5 and 2, and 6 and 1. He will thus learn to perform without conscious effort any separation of these groups for subtraction, or union of them for addition.

I have said that I am strongly in favour of teaching, for at least the first two years, numeration without notation. The reason I gave was that in this way we should teach facts before giving the conventions with which we connect those facts. If we consider how numeration arose, we find in it a beautiful simplicity. The primitive man when he counted up the fish he had caught, or the nuts he had gathered, put together one for each finger on each hand, thus making a heap of the first ten; then he began again till he was once more at the end of his fingers. The second heap of ten he put by the side of the first, and so went on till he had a third, fourth, fifth heap, &c. He then counted the number of heaps, and so long as he had not more than ten of them he managed easily enough. Unfortunately, however, the primitive man was not so successful in his language. When he set out to count his second ten, he distinguished it from the first by calling it 1 and 10, 2 and 10, 3 and 10, &c., &c. But having completed his second pile he got muddled in his nomenclature. Sometimes he said, "one and twain-ten," or one and twenty, "two and twenty," &c., but sometimes he named the piles of ten first, and said "twenty-one," twenty-two," &c. As he got further on, he adopted more and more the plan of naming the tens first, until the method he set out with was lost entirely. So we have to name the units first, up to 20; then we have a choice of tens or units for a little while, but this liberty leaves us again about 50. If I ask the age of a man under 30, I am told, say, that he is "about three and twenty." "About 23" is allowable, but only just allowable. If the man were 10 years older, I should be told, either "about three and thirty" or "thirty-three," and here one form would be as natural as the other. If we raise the age another to years, it would be spoken of as about 43, rarely three and forty: another 10, and 53 is almost the only form possible; and still more of the higher decades. If we heard anyone talking of "three and eighty," we should say he could not be an Englishman.

It seems to me a great pity that the symmetry of our numeration should be obscured by freaks of language. I should therefore insist on the tens being names first through out. After ten I should go on "ten-one, ten-two, &c.," or, if one might take such a liberty with the language, I should use onety-one, onety-two. [I am glad to find that these suggestions for the beginner's nomenclature have been anticipated by Mr. Wm. Wooding, Assistant Master in the City of London School. The plan also for teaching notation with vertical wires is carried on in his "Decimal" Abacus, which seems to me admirably adapted to its purpose. For particulars apply to the Agent, Mr. Thomas Mole, 72, Downs Park Head, Hackney, London.]

In adding, it might be well to complete one set of ten, and then append the odd units, e.g., if 7 has to be added to 8, we might look at 7 as 2 and 5, add the 2 to the 8 for the 10, and append the 5 to it.

To come at last to notation, I should be sorry indeed to speak evil of the beautiful convention which, as it came into Europe through the Arabs, we call the Arabic notation.

But, beautiful as it is, it has a danger which we ought to guard against. As we saw, our numeration is simply by a series of complete tens. This, however, is slightly obscured by our notation, which has no sign for ten. So the break seems to come, not after ten but after nine, and this often leads to confusion of thought. Some at least, many I hope who are here to-night will see the beginning of the 20th century. Those who do will find that the newspapers and people in general will be deceived by our notation, and will declare that the 20th century has begun when they reach 1st Jan, 1900. The truth is, the year 1900 belongs to the present century. On any other supposition, the first year of the Christian era was the year "0," which, as Euclid would say, is absurd. If I myself do not live long enough to bring this absurdity home to the newspaper people at the proper time, I trust some of my younger friends will prove themselves equal to the occasion. It is worth observing how often we are the slaves to symbols. It seems strange to most people that 21 is fixed on the year of coming of age. Why not 20? The truth is that at 20 we are not "out of our teens," i.e., we have not completed our second decade. We enter on our third decade with our 22nd year. This confusion of thought which is apt to follow from the Arabic notation, should be carefully guarded against.

The notation itself might be explained sensuously with a row of upright wires or files in a horizontal bar of wood. [Ibid.] On to these wires we should be able to fit pierced cubes of wood of such a size that each wire would take nine and no more. By such a device the Arabic notation might be made comprehensible "by the meanest capacity."

But remember that, if you take my advice, you will, for at least two years, keep entirely clear of any sort of notation. I am confident your pupils will be the gainers in the end.

(To be continued)

Typed by Dawn Taylor, "excelsiorwarriors," Oct, 2006, Proofread by LNL, Apr. 2021