The Parents' Review

A Monthly Magazine of Home-Training and Culture

Teaching Arithmetic

Teaching Arithmetic: Some weaknesses I have found and their remedy.

While here and there one finds teachers of exceptional originality or, of what is equally important and almost so featureful as to be identical with originality, viz., teachers possessed of common-sense methods applicable to varying conditions alike of the local requirements and of the special natural variety of attainment of the pupils; yet, on the whole, arithmetic is taught along uniformly monotonous and unrealistic lines, and with varying objects and aims in view, most of which aims are at heart uneducational, valueless, and therefore futile. The result is that we get not only inaccuracy, but incompetency as the goal finally reached, while all along the road the work has been unintelligent if not unintelligible, and therefore uninteresting. A boy will beam with joy if he finds that he has the key which will give him easily the right method, and accordingly (if he is careful) the right result, just in the same way as he will delight if the method on which he has been taught spelling is such that he is able to master with ease the pronunciation of some extraordinarily large word without the help of others older than himself.

The child likes to exercise his own initiative and to find in its freedom of exercise the power of successfully accomplishing of himself that which older heads can do readily. That this is the general effect of present methods of teaching arithmetic I think few can claim. In most schools, more time is devoted to this subject than to any other, and with all the time spent, the result is more uncertain and the strain of teaching it (both to teacher and scholar) greater than is the case of almost any other subject.

Arithmetic is regarded by the child largely as an abstraction, and he thinks that all one is doing is to work the figures together in varying relationships and after multiple methods, so that in some way or other (to the child, apparently arbitrary) you get an answer which in his mind represents no definite actuality.

My first point, then, is that the first thing necessary is to give the child to understand that figures stand for some reality of value, in the same way as the letters t-a-b-l-e in reading books stand for an article of furniture with which the child is acquainted. In the first instance, this can only be done in the very simplest form, but it is a feature of the teaching which should dominate the teacher all the way along from Standard I. in an elementary school right on to University matriculation.

It cannot be too much impressed that the student, young or old, is not so much dealing with the manipulation of abstract figures, as that he is dealing with certain values whose results are utilized for the purpose of calculating accurately matters of first importance, whether it be the time and rate of travelling on a railway line, for drawing up a railway time table; or whether it be a matter of determining certain relative weights or distances for the purpose of exactitude in scientific research.

Now, the child referred to in Standard I. cannot grasp or appreciate this idea of values and their utilization, but when he gets to Standard IV. he will get to know somewhat of how the railway time table is worked out when he finds by proportion what trains travelling certain distances at varying rates of speed will do. Then the same child in the fourth standard will require the same feature of value elucidated and made still clearer step by step as he approaches the still more mathematical side and touches eventually on astronomic distances and sizes and weights. They tell me in the technical schools that if you set a boy to work in the laboratory, he will get on with the experiment very well until he comes to tabulate and work out results, but when he comes to these, if they are matters requiring calculations, he stops to know what he has to do next in order to work them. That is to say, he cannot state the proposition that has to be worked out. If you state the proposition for him, he will work out the mechanical part fairly well. Why is this? It should not be so. I think he has never been made to realize my first point thoroughly, and that is one reason. Another reason is that he has had too much done for him. The boy is not sufficiently stimulated to do his own work for himself, and further, the underlying principles are not sufficiently elaborated and recapitulated. In my Point II. let us deal with these questions of underlying principles and their actions as a stimulant to the scholar.

There are various reasons given for this want of more effectual and thorough attention to this matter of underlying principles.

Reason I. The authorities, in some districts at any rate, seem to aim to have all children of a given age in a given class or standard. This determination is arrived at with great disregard to allowing for the difference in the attainments of the children. Especially is this so in elementary schools. If a discreet head master classifies according to attainment, and, as a result, has children left behind in the same class a second year, he is open to all sorts of difficulty. The Government Inspector may regard him with suspicion. The Board or other authority press for speedy promotion. The parents are dissatisfied if the child does not go up regularly and continuously, notwithstanding that illness or other causes may have caused him to be absent a considerable portion of his school time. All these interested onlookers of the teacher's work forget that the slowest progress is always made in laying foundations, and that, if more time were devoted to the work of the child at the start, less time would be required at the end, and the child would do his work with pleasure instead of it being irksome. The difference, in so far as the child is concerned, is merely one of understanding and appreciating what he is doing, instead of his muddling along in incertitude without having any very definite realization of what all this labour with figures means.

Reason II. The mania for examinations. In my opinion, no child should be examined in any other way than by his form master or the head master, until he is about 13 or 14 years of age. The schoolmaster must examine if the classes are large, for the purpose of discovering what progress each pupil has made, and to enable him annually to classify the scholars on a fairly even basis of attainment. But, in addition to these examinations, we have now various public or semi-public examinations for prizes, scholarships, etc., and these are fatal to all thorough teaching of young children. In teaching for these examinations, the chief object of student and teacher (especially the latter) is to get as much of some subjects crammed into the child as will give him the greatest likelihood of being the distinguished—because successful—candidate. What is the result? If the scholar is successful, in most instances his connection with the old school ceases, and the teacher's connection with him is at an end. The cramming for the examination is soon obvious, and not very helpful to him in his new surroundings. As far as the teacher is concerned, mechanical cramming rather than thorough groundwork is the principal effort to which he devotes himself. Of course, the fact that scholars of one master or from one school are successful in annexing some greater numbers of these prizes than are those of other masters and other schools, at once marks that master for advancement and that school for the patronage of parents; whereas if they only knew the real facts they would realize that that is the master of all others who should not be advanced, and that is the school which should be avoided. Hence it follows in natural sequence that a master who, by cramming, gets his reputation, cannot give more than sparse attention to underlying principles. He cannot afford the time to fully develop then and recapitulate them, else the boy will be past the age required for, say, an Elementary County Council Scholarship; while the actual result is that the boy, on the other hand, will have an indistinct idea of his work, and probably fall away when put alongside others more carefully trained.

Reason III. More than a few teachers are untrained, and many of the trained ones are badly trained. Others again know how to get their required answers, but have no natural gift for making clear to a child what principles should be applied to some certain problem, nor why another principle would not give the same result equally as well as the method he has selected. Hence the child is not clear either as to what he is doing, or why he is doing it. What wonder then if the boy is not stimulated to do his work for himself. Some people think that the cane is a good stimulus for boys. It may be, and sometimes is for some boys, but it is not the one solitary stimulus available; it is really an added stimulus when it is a stimulus at all. If a boy has no idea of what he is doing, or of what is required, or for what result to look, then the stick will not stimulate, it will rather cause revolt. If, however, he knows what to do, and why, and how to do it, and what the use of doing it is, then the stick will help him if he is naturally lazy. But the first stimulus —the antecedent of all other stimuli and the only one of abiding value —is to give a boy such an appreciation of what is required, and the methods by which he will secure the required result, that this knowledge itself will be its own stimulus. A boy delights in letting you know how clever he is, how much he knows, how much he can show you, and how difficult it is for you to put a question or piece of work before him which he cannot do readily, quickly and well.

Under the present system of teaching arithmetic, this feature of teaching, viz., Continuity, in the teachers and their methods is most notable for not existing. This is a very difficult matter to get over. For the purpose of removing this difficulty to some extent and to deal with the weaknesses to which I have been drawing attention, I welcome most heartily the growing appreciation of Scheme B under the Government code for Elementary Education, and I would like to see its principles advanced much further than I have seen them. Here and there, portions of what I would like to see done I have viewed and recognised with pleasure, but nowhere have I seen a completely developed scheme and arrangement that would cover the whole of arithmetic.

Referring to the difficulties to be faced in overcoming want of continuity, I am bound to remember that only those methods are successful which have begotten enthusiasm in the teacher. A bad method well and enthusiastically taught will probably on the whole give better results than a good method badly taught. I do not know that one is justified in adducing that as a difficulty, and yet it is. The real fact is that in either of the above cases, the success (if there be any) will not be owing to the method, but rather in spite of the method.

Teachers are taught different methods at the different colleges they attend. In one school you may have seven or eight teachers all from different colleges and having different methods of their own. How are you to get continuity even in one school throughout, and how much less is the possibility of continuity where the child passes from one school to another?

Hence it follows that a headmaster takes up a method after having examined, and it may be after having tried several others. He clings to it because he gets what he considers better results. The results, however, are determined on the basis of a system now, I hope and believe, becoming more or less obsolete, viz., examination for all ages of children at every year's stage. It follows that if you show the head teacher a better method, he will naturally look askant. He sees at once all the difficulties it would incur, and does not see that the advantages will compensate for the disadvantages. First of all, it will take time to introduce it, and his time generally is fully occupied. Then he would have to make himself familiar with the method. Next he would have to get assistant teachers to relinquish methods he has insisted on in the past. Having done this, he would be uncertain whether or not something newer still would turn up as soon as this system was introduced throughout his school, and finally, he would have to fight all sorts of Inspectors and others who might view all his attempts with suspicion. It needs immense enthusiasm to do this. Still, a good man, if he is shown a good thing, is glad to try it, and if he finds it valuable will succeed in adopting it and in getting it adopted. He wants to be up-to-date at whatever cost.

I am rather inclined to favour the way in which one headmaster has minimised this difficulty by adopting a given method for a subject throughout the school. He has gone through his staff and ascertained what subject each man has the greatest fondness for and which of them he best likes teaching. He has then allotted to each the subject for which he is most suitable. What a man likes most he will teach best. He then makes each man responsible for his allotted subject throughout the school. He keeps the class teacher in his own class for all other subjects, and the class teacher goes through with that class from Standard I. to Standard V. (I do not consider this necessary.) When the boys go down for recreation in the morning and in the afternoon, they leave their class master and return to their class room under the master who is responsible for arithmetic in the morning, and under, say, the geography master in the afternoon, but only for one lesson. Hence every boy is taught arithmetic at least once a week by a master responsible for the arithmetic of the school, and the class master does the arithmetic another day on the other master's lines. The same method is adopted for every subject. It ensures that every master gets a change of class twice every day and every boy has a change of master. It ensures a continuous method of teaching each subject and that once during the week, at least, a boy is taught by the master responsible for that subject and for that method of teaching it. In such a case as this, a headmaster has only to get the co-operation of the arithmetic master in order to get a fresh method of teaching that subject tried. He will, of course, begin at the youngest children, and gradually they will introduce it until the method through the whole school is uniform.

Continuity suffers through want of co-ordination amongst different schools, and even amongst departments of the same school. Hence, if, in the former case, a boy gets a chance to attend what is supposed to be a better school, he finds that some subjects have been neglected which he ought to have started long ago, and other subjects he has been well informed in are not now encouraged much, and they may be left out of his curriculum altogether. As between departments it happens that the teacher in the junior school rarely or never goes down to find out the methods along which the children from the infant school coming up to him have been trained. As a result, some things they have been taught are allowed to lie dormant and forgotten; moreover, fresh lines of teaching are adopted and different teachers from those who have previously taught the children deal with them (in many instances, men taking the place of women teachers), and the teaching is on different lines and in a different way from what they have been accustomed to. It is also done under a different roof and in quite a new atmosphere altogether. The result is that the change makes the child's work more uncongenial and difficult, and much of their previous work lies unused and forgotten, and eventually, through not being continued, it dies gradually. Some of it has to be done over again at a later stage in the child's school life. The senior school is not quite so bad, perhaps, in this respect as is the junior school, but even in the senior there remains much to be desired. For example, at the youngest years, fractions can be taught, and after these are taught decimals can be allowed to follow on in close succession instead of being deferred until the child is 12 or 13 years of age, and these can be so taught as to help the work of the younger part of the school in arithmetic. I have seen fractions well and intelligently introduced to children of 6 years of age. These same children going into the elementary junior school from the infant school the next year would hear no more of fractions till they got to Standard IV. or V. at any rate.

In the public elementary schools little children are taught in some simple form the rudiments of addition, subtraction, and multiplication; but the multiplication is taught more in the form of "multiplication tables." The child at this stage may regard it as so many of a number added one to the other, or he may not. In any case, at the end of his sixth year of age he has to know the tables up to 6 times 12. The remaining tables up to 12 times 12 he learns during the next year in Standard I. In Standard I. the children take up short division as well, so that by the time the child leaves Standard I. he is supposed to have a fair knowledge of these four rules. Under the B scheme, to which I have referred earlier, these rules are taught simultaneously side by side with one another and the sums are set in problem form, leaving the child to determine which rule he has to follow in order to get the answer. In Standard II. they go on to long division, and perhaps they do some revision as well. But fractions are dropped, and as for alluding to decimals, it would be regarded as ridiculous by many. But the fractions taught in the infant school could easily be carried on side by side with these rules, and a good insight be given to the child thereby. It could be brought to realize the power of multiplying by ten, or dividing by ten, and hence of the meaning of one-tenth = 1/10 or one over ten. Here, or earlier, you could introduce decimals in some simple if not very thorough form. Addition of decimals is quite as good practice in the mechanics of adding as any other form of addition.

Of course, you would not start the year in Standard I. with advanced decimals, but you would lay yourself out to work up to and do something with simple decimals all the way along throughout the year, as I shall show later. If they got to know very simply something of the principle behind all decimal calculations by the end of the year, then the development of the principle and its more thorough meaning and value would be made clear in all the higher classes as the child goes along, and he would become familiarized with the value of decimals and their uses, while at the same time you would be keeping them in touch with fractions and the ordinary rules of arithmetic. It would give them a twofold method of working, and when they got to realize the great advantage of decimals, they would naturally give up old methods. By this means, a strong factor in educating the country to a decimal system would be established.

The same interest and affection for this the coming system is not engendered when, for nearly all his school career, the boy is soaked in the old system, and for his last year, decimals are introduced as a new subject to be mastered, and which, while valuable for some few scientific and international purposes, are for the most part not of value for the average boy because they are not current in his every-day life. If he were fed on them and made to utilize them from infancy, then when he came out into every-day life, he would be restless with our old methods until they were changed to the decimal system. Thus one advantage would be the breeding throughout the length and breadth of the land a new form of public opinion. I find that boys and girls going up for Pupil Teachers' Examinations, and what is known in Wales as Intermediate Examinations, who have been taught the decimal system thoroughly and made to apply it, break away from their teachers in examination. So much more at home do they feel with their old system in which they were grounded in their earliest years, that in examination, they actually abandon the later and more correct way of working, and, to the disgust of their teachers, the examiners give them marks and pass them, which of course makes the child glory in the idea that she knows as much or more than her teacher. This is particularly the case with girls. Sometimes they fail and are afterwards more amenable.

(To be continued.)

* Mr. Wilkinson, of Adelaide, has been in England for some years studying educational methods, and is shortly taking his report back to Australia,

Proofread by Leslie Noelani Laurio, November 2008