Modern Teaching of Arithmetic
IN his book On the Education of an Orator, Quintilian gives an excellent series of reasons why the pleader should be taught mathematics. His doctrine is that geometry, as he calls it, is, in its two branches of " numbers ” and “ forms,” important for an orator on practical grounds, for example in cases concerning real estate or accounts; and he vividly pictures the embarrassment which the speaker will show if he is awkward at the problems of arithmetic which necessarily come into his oration. Quintilian does not stop here : he admits as a well-known principle that geometry is an admirable training for the reasoning powers; and the experience of later ages has fully confirmed this view.
It will be remembered that the Greeks and Romans had no algebra, and very troublesome systems of arithmetical notation. For both reasons their arithmetic furnishes an admirable mental training of the kind which is still much in favor with old-fashioned teachers, — a training now based upon the use of artificial obstacles. If any one who knows Greek or Latin will take the pains to read the seventh and following books of Euclid, especially the voluminous tenth, he will find ordinary arithmetic treated under difficulties of a kind quite analogous to those which have been artificially produced in our “ higher arithmetics.” Indeed, it must seem strange to any one who holds certain theories of mathematical teaching to discover that these books have for centuries been neglected in the schools, and replaced by vastly easier methods. For an English Euclid containing them we must look in editions published a century or more ago; and if mere difficulty supplies an excellent basis for mathematical training (as some people seem to think), this neglect of so much of the immortal author’s work appears disrespectful.
Archimedes was the discoverer of an approximation to the quadrature of the circle; the latest editions of this great writer’s books show very clearly how much he was hampered by the Greek method of calculation, and how much more he could have done had algebra and the Arabic notation been then invented. We now employ the infinitesimal calculus to gain with great ease the results which he obtained with enormous labor. No one at present thinks of using his method in instruction, although its difficulties are so considerable. After the Middle Ages the Arabic notation was introduced into Europe; and at the revival of learning great mathematicians restored the science and art of numbers to its old place in the schools, and began the scientific investigation of nature with the study of astronomy. These philosophers made it possible for the mariner to find his latitude at sea, and occasionally to guess at his longitude ; the ocean was no longer absolutely unknown, and the discovery of America was possible.
Such writers taught arithmetic more as a matter of rules than of reasoning. Very possibly the rules were supplemented, in some cases, by the abstract theories of Euclid ; but on the whole the rules prevailed in the schools, and the numerical part of mathematics was a practical subject, taught for the sake of mechanical facility. In England and her colonies this method was long retained, owing in part to the extremely artificial character of the denominations employed in money, weights, and measures, and the steady conservatism of the English people. Even now children in England gain much “ mental training ” (of the kind due to needless difficulties) from the use of pounds, shillings, and pence. There the method of “ Practice, Simple and Compound,” which with us has been long forgotten, is still in full vigor.
Our conservative instructors held on to the debased currency in English denominations used in this country much longer and more strenuously than was at all necessary; and I have no doubt that they delayed the final introduction of federal money more than a few years. The worst consequence of the old ways — the teaching by rule rather than by reasoning—has not entirely disappeared. The ordinary books give, it is true, a short course of reasoning preparatory to each rule; but the rules are many, and the reasoning is often so lightly indicated that many teachers lay no great stress upon it, and the children work by the mechanical process. So, at least, it appears when the methods are tested at a later stage of education.
The ideas of the celebrated Pestalozzi were translated into practice by his numerous disciples in all civilized countries. In arithmetic Warren Colburn was the most practical and successful American writer of this century. He emphasized the idea brought forward by Quintilian, that mathematics is especially valuable as means of mental training ; and it may be questioned whether, at first, some teachers did not pay too strict attention to this side of the matter. But it soon became the usual practice to combine the two methods : to employ Colburn’s First Lessons as a textbook for mental arithmetic, and some larger one for written. The consequence in many cases has been the retention of mechanical methods in written arithmetic, which has been sometimes kept quite separate from mental.
Since Colburn’s time graded schools have been established far and wide in this country. Their principles have taken up the German method of dividing numbers into so-called “ circles,” — 1 to 10, 10 to 20, 20 to 100, and so forth ; at first without definite uniform boundaries. The circles were, in fact, bounded differently for the various operations. Thus the English New Code of 1888 gives as the work of the First Standard: “ Notation and numeration up to 1000. Simple addition and subtraction of numbers of not more than three figures. In addition not more than five lines to be given. The multiplication table to 6 times 12.” A distinction is thus made between addition and multiplication. Similar programmes have been made in this country for many cities; but the latest tendency is to the general adoption of Grube’s method.
This is a method in which separate numbers, 1, 2, 3, 4, and so on, as far as 100, are taken up one by one and analyzed. The pupil learns the qualities by his small experience, first of the number 1, then of the number 2, both by itself and as compared with the preceding number. Then follows 3, which is already more complex ; its slight complexity is illustrated in every possible way by objects, and it is thoroughly mastered so far as the child’s mind can deal with it. The separate numbers are mental “ objects,” as one may technically express it; the mental objects are definitely presented before the mind by the comparatively small degree of abstraction required to separate the idea of 3 from the idea of 3 fingers, 3 cents, 3 pencils, or other small but familiar things.
The underlying theory is that otherwise the child is required to perform so great a degree of abstraction that the thought becomes mechanical; and the method is brought forward as a contribution from experience to the psychology of the growing mind. It is’very clear to those who have thought about it that this method of dealing with individual numbers in their orderly succession, one by one, is more natural than the older way of taking for granted, after a few trifling exercises, that the young pupil was thoroughly grounded in the necessary concepts, and could go on at once to more highly abstract notions. There are features of Grube’s method in which many good teachers think him extreme ; but his main principle is very widely recognized as a true one.
Colburn’s First Lessons, as we intimated, have been often improperly employed ; the mental work has been combined with written exercises upon larger numbers; and in England even the First Standard, usually passed by children when seven years old, requires the use of far too great numbers. In some places Grube’s method is used in the same improper manner, contrary to its author’s intention, along with exercises on much larger numbers with slate and pencil.
The gist of this method lies in the numerical analysis. We may, for instance, suppose that the numbers 1, 2. 3, 4, have been already taught; let us see what the lesson on 5 will contain. 5 is 4 and 1 ; 3 and 2; or 2x2 and 1 over. This will be illustrated on the hand. The children will then imagine ways of expending five cents ; will be taught the relation of the five-cent piece to the single cents ; will find out that no number of equal groups can be made of five things; and so that 5 is a prime number. In a word, the single number 5 is taught so as to review the earlier ones, and prepare concretely for later abstractions. But when 20 has been so passed, there comes to the instructor the temptation to hurry on and generalize ; the primes especially become less interesting. Thus 24 and 25 far surpass 23 or 26 in special properties. The objection ordinarily made to the thorough methods in arithmetic is that they are not rapid enough. For instance, Grube proposes to keep children a year upon the numbers 1 to 10, and at least a year more upon those from 10 to 100. The English First Standard goes far beyond this in one direction, but fails to attain it in another.
It must be remembered that Grube’s circles of numbers do not extend in any way beyond their boundaries. The first year’s work, as specified above, does not involve the addition of 6 and 5 or 9 and 2, but merely such additions, subtractions, multiplications, and divisions as can be carried out with data and results under 10, and accompanied by many little problems in the application of those numbers.
Shall we then consider the child of seven who knows only so much arithmetic as this a backward or neglected pupil ? On the contrary, it is quite clear that such a child has learned a great deal. He has practiced the four ground rules thoroughly, and has applied them to problems covering what is for him a great range of ideas. Where schools are poor, intelligent parents will have little confidence in them ; it may be that even a good school is unable to bear the criticism that its pupils know a small amount, even if that little is carefully taught.
The main difficulty of introducing the improved methods arises from the state of public opinion, just as it did in Colburn’s time. His First Lessons was adopted and used, but in combination with books of an older and less excellent type. In the same way, Grube’s method is only partially employed, and more or less side by side with superficial ones which give the appearance of progress. Our school work is very apt to be done in a. hurry, with the final result that our scholars do not finish their general education as soon by a year or two as those of other countries.
Up to the number 20 the way seems clear to adopt Grube’s analysis; but at this point teachers appear to desert, or at least to postpone it. If a child has acquired skill with the smaller numbers, why, it will be said, can he not proceed at once to the unlimited range of Arabic notation? Simply because important matters are overlooked. The range of numbers which is of most importance to every one is precisely that from 1 to 100. Any question of wages, for example, is settled by steps within this range ; no strike of workmen would, I fancy, take place for one per cent advance, but might well be undertaken for five per cent. We are all pleased to know that the diameter of the earth is nearly 7900 miles, but do not care about the odd miles ; or that the sun’s average distance from the earth is 93,000,000 miles, where a hundred thousand more or less makes no difference to us. Grube’s analysis to 100, number by number, seems to have a basis in the ordinary relation of the mind to numbers, and if faithfully carried out would make the four fundamental operations, especially division, much easier. What is called long division is a great stumbling-block in arithmetic ; and its difficulty arises largely from the uncertainty of the first figure of the quotient when the divisor has, we will say, 18 or 19 in its first two figures. Any one can see that a thorough study of every number to 100, with respect to its factors or those of other numbers near it, would be a very great and ready help.
The definite questions to be settled are these : Shall children of seven be taught numbers in general, Arabic notation as a system, the abstract ideas of addition. subtraction, multiplication, division, and with numbers larger than they can comprehend ? Or shall the simpler exercises of analysis, easy problems with numbers less than 100 and familiar applications of the four forms of calculation, be practiced with less abstraction ? It would be easy to choose between these two alternatives, were not all our habits formed in the first manner, so that the second and better one appears strange even to teachers.
A pedagogical maxim of some importance is that the matter taught must be mastered by the teacher, not only in scientific form, but also in the form in which it is to be taught. The common defect of textbooks is that they’ present the subject nearly as an expert would review it, and without much thought for immature minds. The teacher of arithmetic forgets his early struggles to master the art of numbers, and considers the real basis of the subject very simple, merely because he has been well trained in it; and it requires, in fact, much labor to know the first hundred numbers, as the analysis requires the teacher to do. Grube ’s method of employing this analysis is only sketched in his book, and its easy handling needs a great degree of pedagogic insight and experience. While interesting to the learner, it is not easy for the teacher ; yet it is very fascinating to those who study it.
We may say, then, that this method is gradually coming into use ; every advance in the qualifications of teachers makes it easier to take it up more thoroughly, and the training it gives children is more and more appreciated. An important question for primary-school teachers is how far to employ actual objects. It seems probable that the first year ought to give the conception of abstract number, of two, three, four, as distinguished from two hands, three pencils, four children, and so forth. The use of some contrivance for showing the collection of tens into a hundred, and of representing the intermediate numbers, is called for ; but if the single numbers are learned one by one, and abstraction is gradually introduced by small steps, the need of actual objects for exhibition diminishes little by little. When the numbers between 100 and 1000 are studied, it may be well to exemplify the Arabic principle by mechanical devices ; but the Grube analysis of the smaller single numbers now gives the qualities of the tens, and the material objects need be only sparingly introduced. In fact, the necessity of employing such devices as a main element of training beyond 100 would seem to indicate that the earlier work is not well done.
Grube takes up fractions one by one; and the proper fractions whose denominators are less than 10 furnish material for a half year’s lessons. Here, of course, objective illustration is needed. The somewhat stolid way in which he goes on, taking up the denominators 2 to 9 in their order, seems to displease teachers who consider other fractions, as tenths, twelfths, more important than sevenths or ninths, just as these same teachers would omit 23, 29, and other prime numbers, while paying individual attention to 24, 25, 28, or 30. But yet the numbers and fractions in regular order offer much more variety to the pupil from the alternation between odd and even, prime and composite ; the German method is after all that which gives the most solid foundation. As Grube suggests, prime numbers are to be studied more for the sake of the abstract number, composite for their applications.
Fractions in the abstract are so difficult that any method of lightening the work upon them is a great gain ; and there seems to be no doubt that the pupil can be led to understand, in their order, halves, thirds, quarters, and so forth, and to determine their relations to each other, much more easily than he can be led to form the abstract idea of a fraction and deduce the general rules of operation. Most grown people are easily puzzled by quite simple fractional questions, for the very reason here indicated ; they have attempted as children to learn general statements before particulars, if not to deal with the subject by mechanical rules.
The true method for learning elementary mathematics is the heuristic, the method of discovery. The pupil should be shown or taught the mathematical object. In arithmetic the objects are whole numbers or fractions, in algebra quantities not always discontinuous. These objects should be presented in the simplest manner when new ideas are to be developed. The more nearly spontaneous the pupil’s thought can be made, the better; the teacher keeps his own wider and more abstract thoughts about the object as far in the background as possible, and attempts to enter into his pupil’s mind. Suppose 28 is the number studied ; 27 was the last one. The child has by this time learned to expect 28 after 27, as 18 followed 17, and 8, 7, The teacher hardly needs to name it ; the symbol or actual counting gives the name. But 27 was 9x3, an odd number. 28 is even ; it is divisible by 4 ; it is 4 x 7 or 7 X 4 ; it can be divided by 2, 4, 7, 14, — no other numbers. How does it compare in this respect with previous numbers ? 24 has had six factors, 25 one, 26 two, 27 two. In actual life what is the importance of this number ? Four weeks are 28 days ; February has 28; in England 28 pounds make a quarter, and so on. Some of these things the teacher must state; others can be readily educed from the children’s thoughts, still others from their memories. But everywhere the method is heuristic, not dogmatic ; the pupil’s own faculties are briskly exercised. The single numbers seem to offer materials for such exercise almost spontaneously, in the greatest abundance.
At the next stage, where numbers of three figures are taken up, the objects become more abstract. Few numbers in the hundreds are very significant; the process is more a synthesis than an analysis. Having learned with numbers of two figures what the Arabic principle is, we inquire what will be produced when we carry it a step farther. We have several bundles of sticks or the like to represent hundreds, and we put them together; we count (synthetically) 100, 200, 300, 60, 5. We now have a larger number (represented) than any of which we have much experience. We compare this number so made up with other like numbers ; add, subtract; or we put two or more equal bundles together, and multiply. In a word, we are now extending our pupils’ ideas of number, not a single little step at a time, but on the large scale.
To return to the debatable point of the proper conclusion of the one-by-one analysis: is it not nearly certain that this can better go on to 100 than stop at 20 ? We are now generalizing, in one sense, but with limitations. We still hold to our three figures. It is a very good arithmetician indeed who knows all that is to be known about the first thousand numbers ; and our pupils of eight years do not need to go beyond them to be very thoroughly trained in the four fundamental processes as a matter of practice rather than of theory.
The next step is a still more abstract one, — numeration. The process itself is now the object to be presented. No material objects are to represent the things counted; hut the law of place in the Arabic system is explained with proper illustrations. So far three figures only have been used ; the principle is thus readily fixed and extended. The employment of an unlimited number of figures can be hinted at, and applied so far as need be; then should follow the study of the four fundamental operations, definitely separated and practiced apart. Up to this time numerical analysis and synthesis have used the operations quite freely as a means, and, so to speak, empirically; they are now to be studied for themselves. “ Carrying ” has been practiced, but instinctively, heuristically ; it is now taught as a distinct mode of operation. By the time the child is nine years old, he (or she) is able to perform the four fundamental operations in whole numbers, both pure and applied, without any special restriction of the magnitude of the numbers involved, save as common sense dictates. The operations, separately considered as objects, cannot be rationally taught to young children until they are familiar with many numbers and perform the calculations habitually; for the study of the object " multiplication " requires introspection together with interest in and power over the process.
A year ’s course in fractions, spent half on the individual fractions and half on the fundamental operations as such, concludes the arithmetic of the primary school. At the age of about ten years the pupils are ready to go on with the practical study of the subject, or that higher work which furnishes a base for algebra.
In this country taken as a whole, I fancy that no more than a third in number of the actual teachers are even partially in favor of this reform. Many are inexperienced; many others are looking forward to other professions or to marriage; a good many have no wish to be martyrs to principle or leaders of reform ; some, who would like to improve their work, are hampered by circumstances or public opinion, and perhaps grow discouraged and leave off teaching. A well-made speech, full of glittering generalities and commonplaces, will command much applause in a teachers’ meeting; and the forward movement among educators does not go on as smoothly as if all were professional, permanent, and fully interested in their work. But, in the long run, those of our teachers who are advancing will prevail.
Aside from the specialties of Grube’s method, there are certain well-recognized truths which no teacher can afford to forget. Mere calculation by rule should be abandoned ; in its place training in the use of small numbers, and consequent formation of right habits, should be introduced. All arithmetic is mental; written arithmetic, so called, is merely for the purpose of diminishing the strain oon the memory. All exercises in this subject should he predominantly mental, and deal by preference with small numbers ; taking up larger ones for practice only. The weight of teaching should be on the mental side, not the mechanical. When written arithmetic is practiced, the work should be neatly and systematically done.
Grube’s Leitfaden was first published in 1842. By some German teachers and writers of textbooks the analysis was restricted to 20, and the generalization begun at that point. They did, however, introduce one of his essential principles, which our teachers do not seem yet fully to approve, the separate treatment of the simpler fractions up to ninths and tenths.
The present article cannot be better concluded than by some extracts from Dr. Kellner’s Volksschulkunde, sixth edition, published at Essen in 1868. This work is quoted, as not at all a radical or venturesome one ; in fact, the author was then Catholic school counselor at Treves, with jurisdiction over the schools of a population of perhaps 400,000. The book may be compared with Emerson and Potter’s The School and the Schoolmaster, or later books of the kind ; and in it Grube is mentioned with approbation. Kellner says, in substance, that arithmetic has been too much employed for formal education, and that in consequence its true importance has been overlooked, and an artificial formality introduced; that the examples have lost relation to the life and business of the common man, while referring to all sorts of so-called business methods ; that the length and complexity of the road traversed are a special hindrance to the many-sided and thorough study of the separate portions ; and that the whole process of instruction has been crowded into the old mechanism from which teachers were trying to get free. It was not enough for the pupils to divide numbers of six figures by numbers of three, but the dividends must be billions, and the divisors hundreds and thousands of millions. The fraction 1/1 1/3 was not sufficiently complicated ; the pupil must reduce of a dollar to lesser denominations. Mental arithmetic was kept strictly apart from written, and by special devices carried beyond the powers of average pupils.
The scholar should, on the contrary, be taught to solve the moderate examples naturally appropriate to him in as independent a way as possible; not by mechanism and complicated formulæ, but by his intellect. Mental arithmetic should be introduced everywhere and accompany every exercise. The small practical result obtained from it, sometimes.urged, is due to the neglect to give regular hours to it in connection with written. Kellner advises finishing the common school course with fractions, and suggests that the rule of three and interest be taught as their applications.
From an educational periodical he quotes " six rules for teaching arithmetic badly,” which are here condensed.
First. Divide your hours for arithmetic into theory, mental arithmetic, and written. In each division pay no attention to either of the others.
Second. In theory, proceed from abstract ideas ; use foreign and high-sounding words ; spend the most time on what is of no practical use; give a detailed theory of proportion.
Third. Arrange your mental arithmetic so that the children shall not employ any processes of their own; make it as much an arithmetic of figures as possible ; if the scholar is to divide mentally, accustom him to write the dividend and divisor in the air with his finger.
Fourth. Have some special devices in mental arithmetic to throw dust in the eyes of the public.
Fifth. In written arithmetic, let each child do the sums from a h=book, imitating a process which has been shown him, but not explained. Let every one go on for himself ; if he gets the right answer (by the key, which you keep), say Right! if not, say Wrong! and leave him to find out for himself how to get a better result. This we may call training in independence.
Sixth. An especial means of hindering all progress in arithmetic lies in the examples. Large numbers, unintelligible denominations, matters which the children do not understand, — all these should be thoroughly employed.
By these six rules you will be pretty sure to attain your object of teaching without any result.
In thus quoting Kellner’s book I do not care to lay any stress upon the fact that it is German. The author, though a German and a Catholic, understands well the nature and capacities of such children as we find in American schools. He is, in fact, a practical teacher and superintendent, who has leisure enough to put into words the results of a long experience ; and American teachers -well know that the European boy, French, German, Italian, Slavic, Scandinavian, is after all very much like the young American in the growth of his mental processes.
We must carefully guard ourselves from the illusion that the average rate of progress of our sons and daughters is more rapid than that of European children. It is quite the contrary ; and that this is so is owing to many causes, very prominent among them the fact that the material development of this country has greatly taxed the mental energies of the race; and even in education theory has been looked upon askance, and the practical man, who can produce the tangible results called for by uninstructed public opinion in the quickest and cheapest manner, has been glorified to the disadvantage of such dreamers as Pestalozzi and Groebel. The great sums now devoted to the higher learning will, if we are wise in their application, give our scholars leisure to theorize in such a practical manner that our common schools shall in part reap the benefit.
Truman Henry Safford.