The work of the Mathematical Association, in connection with its activity in promoting the reform of Mathematical teaching in our schools, necessarily involves the expenditure of much time and thought upon the detailed discussion of specific schemes for the improvement of the teaching in special departments of Mathematical education. It is, however, well that we should sometimes reflect upon the more general aspects of our work; and perhaps a Presidential Address affords the most suitable occasion for reducing some such reflections to an explicit form, even though nothing essentially new can be said upon the matter. To attempt to offer opinions on many of the important questions of detail which come up before the Association and its Committees for discussion would be an impertinence on my part. Much of the experience requisite for the formation of considered opinions on points of detail is lacking to me; and I consequently regard myself as in the position of a learner in such matters. Indeed I consider myself fortunate in being allowed to bear the opinions of the many experts, of the greatest experience in all matters appertaining to the school teaching of Mathematics, who are to be found amongst the members of our Association.
In making a few brief remarks upon the general character of the reform movement, I propose to emphasize one or two governing principles which I regard as of fundamental importance in relation to Mathematical teaching. If I venture, in the course of my remarks, to make some suggestions on less general matters, the adoption of such suggestions as parts of the policy of the Association would only be possible after much detailed discussion of the manifold points which would have to reach some degree of settlement before the suggestions could be translated into the domain of practice.
The modern tendency which has exhibited itself in our time in greater or less degree in all countries, in educational policy in general, may be described as the tending towards the democratization of Education. This term, or some synonymous one, has frequently been used to denote the extension of Education to wider classes of the population; but it is not in this quite general sense that I intend here to employ the expression. I mean by it rather the progressive adaptation of educational methods to the intellectual democracy; the transformation of the methods of teaching and of the matter of instruction so as to meet the needs of those who are lacking in exceptional capacity, at least in relation to the particular branch of study in question; in other words, the concentration of the attention of the Educator, in a much greater degree than formerly, on the work of developing the minds of the average many, and not solely of those of the exceptionally gifted few. The progress of democratization of Education, in this sense, has been perhaps more marked in the case of Mathematical instruction than in other departments. In our own country the Mathematical Association has been conspicuous as an agent in furthering the democratization of Mathematical Education. It is very certain that no such democratization could be effected without more or less radical changes being made both in the methods of teaching and in the selection of the matter taught. It would be of but little avail that the attention of the teacher should be concentrated in a greater degree than formerly on the average many, if the methods of teaching, and the material taught remained unreformed. With a view to the formation of some estimate of the profit and loss due to the changes which have taken place of late years in the teaching of Mathematics in many of our schools, let me briefly glance at some of the differences, both in theory and in practice, which distinguish the older and the newer methods from one another. Any exaggeration of which I may be thought guilty must find its excuse in the fact that I am attempting to indicate only the more salient features in a continuously progressive movement.
In accordance with the older and traditional treatment of Mathematical instruction in our schools, Geometry was treated in a purely abstract manner; the idea being that Euclid, as a supposed model of purely deductive logic, should be studied entirely with a view to the development of the logical faculty. Any knowledge of space relations which might have been imparted by this study was reduced to a minimum by the excessive insistence on all the details of the syllogistic form; the whole attention of the pupils being engrossed by the effort to commit to memory a long chain of propositions in which the actual geometrical content was exceedingly small. On the other hand, Algebra, and to a great extent Arithmetic, were taught without any regard to their logical aspects, but mainly as affording discipline in the purely formal manipulation of symbols in accordance with prescribed rules; little or nothing being said as to the origin of such rules. The teaching of Mechanics was assimilated as far as possible to that of Geometry, the true position of the subject as a fundamental part of Physical Science being almost wholly obscured. That the average boy or girl is not by nature appreciative of formal logic, or of the interest and meaning of abstract symbols, was thought to be a reason why the subjects so treated should be especially insisted on. In fact, the notion of Mathematical teaching was that it should be in the main medicinal and corrective. Its advantages consisted largely in calling forth the use of faculties which are the rarest in the average boy or girl, and were therefore thought to be in special need of development. It was thought to be by no means wholly a disadvantage that these subjects, so treated, were found hard and repulsive by the majority. It was thought that the hard discipline involved in the attempt to assimilate them developed a kind of mental grit, and involved a certain species of moral training, even when the intellectual results were small. A certain strengthening of faith, to be acquired in the process of hard work spent on subjects of which neither the aim nor the utility was obvious to the pupil, was thought to be highly beneficial.
It is unnecessary for me to enlarge upon the defects of this system, and on the inadequacy of the ideals underlying it. The existence of the Mathematical Association is a warrant of the widespread dissatisfaction with these methods, both in their results and their aims. The system as it existed in our schools was condemned by its failure. It failed to attain even its own narrow ideals, except in the case of a very select few among the pupils. The many rejected the material which was for them wholly indigestible mental food. The system was, in the sense in which I have used the term, undemocratic. The results obtained in the case of the vast majority were deplorable; and it needs indeed a strong faith in the anti-democratic principle to imagine that this failure was compensated by the effect of a hard and bracing training on the few who, by mental constitution, were enabled in some degree to profit by it. Even the chosen few suffered severely from the effects of the narrow conception of education which lay at the base of the methods of instruction. For the purely abstract treatment failed to disclose the close relations of Mathematical ideas with the physical experience in which the abstractions took their origin. That Euclid has any relation to the problems of actual space was seen by the majority of those who suffered under this system only at a later time, if at all. The relations of symbols with the concrete, and the economy of thought involved in their use, remained for the most part unappreciated; such appreciation came, if at all, as the product of later reflection on the part of a very few even of those who had attained to some facility in the manipulation of the symbols.
Mais, nous avons changé tout cela. The modern methods of teaching appeal in the first stages to those interests which are strongest in the majority, instead of running atilt against the most undeveloped sides of the minds of the pupils. Geometry, the science of spatial relations, is introduced by the observational and experimental study of the simplest spatial relations, verification by actual measurement playing an important part; the abstract treatment in accordance with the deductive method being relegated to a later stage. The interests of the average boy are rather practical than theoretical; therefore, it is thought, he must be interested with space relations on their practical side. He is not interested in formal logic, therefore he must not be bored with learning a chain of theorems of which the object is not apparent to him. He is not usually ingenious, therefore, it is thought, no demands must be made upon him which require ingenuity. He does not readily move in the region of abstract symbolism, therefore be must be introduced to the use of symbols only in an arithmetic manner, in which the concrete implications are prominent. Laborious exercises in Algebra, in which expertness in the manipulation of symbols is the object to be attained, should, it is thought, be for the most part omitted.
Owing in large measure to the activities of the Mathematical Association, a considerable transformation in the methods and in the spirit of Mathematical teaching has already taken place in many of our schools, and the changes in the direction indicated by the newer ideals are no doubt destined to have even more far-reaching effects than at present. However, the old mechanical methods of teaching still linger on in many of our schools, in which conservative traditions are notoriously difficult to eradicate. The detailed discussion, both in print and viva voce, which arise in connection with the work of our Association, may be of inestimable value in directing aright the detailed development of the reformed methods of teaching. I hope, also, they may prove useful in the direction of checking those one-sided exaggerations which are always apt to arise in connection with activities in which the objects to be attained are various; as they must be in the case of so many-sided a branch of education as the one with which we are concerned. Some degree of compromise, without undue sacrifice of principles, may often reasonably be made, in adapting the teaching so as to take account of the widely diverging future careers in prospect for different classes of pupils.
It may, I think, be safely maintained that, the better the theory underlying the method of instruction may be, the more exacting will be the demands made upon the skill, the knowledge, and the energy of the teacher. My own early recollections of learning Mathematics call up memories of the Classical Master, without any real knowledge of, or real interest in, the subjects, hearing the repetition of propositions of Euclid, or setting a long row of sums in Algebra, monotonous in their sameness. Somehow, a few of us managed to learn something, but I tremble to think what would have been the result, had the said Classical Master attempted to teach in accordance with the newer methods. For the success of the teaching in accordance with the reformed methods, a high degree of efficiency on the part of the teacher is essential, if the results hoped for are to be attained; and even if those results are not in some respects to fall short of what was reached under the older system. The teacher must possess a high degree of skill in presenting his material; he must have a broad knowledge of the subject, reaching much beyond the range which he has directly to teach; he must have skill and alertness in handling a class, that skill having been developed by definite training, but of course presupposing a natural capacity for the kind of work. Some of the failures, of which one hears, of the newer methods to produce satisfactory results, may probably be traced to a falling short on the part of the teaching, in one or more of the points I have indicated.
At the present time it is not possible to form any precise estimate of the actual effects of the recent reforms in Mathematical teaching. It will only become possible to do so when the confusion incident to a state of transition has passed away. That in many quarters the gain has already been considerable, I have no doubt. Nor do I doubt that the principles underlying the newer methods are sounder than those which formerly held sway. I have no doubt that it is right to proceed from the practical and concrete side of the subject, rising only gradually to the more abstract and theoretical side. But the adoption of more correct principles is only one step; their actual translation into practice gives rise to many difficulties, and to many dangers, some of which have most certainly not been altogether avoided. The process of change has as yet not been one involving pure gain.
A perusal of some of the current treatises on "practical Mathematics" has led me to think that in some quarters the purely practical side of Mathematics is unduly emphasized. The teaching should, without doubt, commence with this side, and should never lose touch with it, but the study of Mathematics must be pronounced to be a relative failure as an educational instrument if it fails to rise beyond the purely practical aspect of the subject to the domain of principle. Purely numerical work, calculation with graphs, problems in which the data are taken from practical life - all these are excellent, up to a certain point, and they form the right avenue of introduction to scientific conceptions. But if this kind of work is unduly prolonged, and too exclusively practised, it tends to develop a one-sided mechanical view of the capabilities of Mathematical methods, and the study ceases to be in any real sense educational. Such practical work is only educational when it precedes and leads up to a grasp of general principles, and when it is employed to illustrate such principles. I do not wish in the least to depreciate the importance of Mathematics as providing the tools for a vast variety of applications, useful in various professions. This side should never be lost sight of in school work. But the most important educational aspect of the subject, is as an instrument for training boys and girls to think accurately and independently; and with this in view, the more general and theoretical parts of the subject should not be entirely sacrificed either to the exigency of providing useful tools for application in after life, or to the supposed need of sustaining interest in the subject by a too anxious adherence to its concrete and practical side.
I gather that, in some of the current teaching of practical Mathematics, a kind of perverse ingenuity is exhibited in evading all discussion of fundamental ideas, and in the elimination of reference to general principles. Instead of a skilful use being made of practical methods to lead up to general methods and illuminating ideas, practical rules seem sometimes to be made the end of all things. I have been told, for example, that the use of logarithms is sometimes taught to students who at no time attain to a comprehension of what a logarithm really is, or of the grounds upon which the rules for the use of logarithmic tables rest. Students who are in the habit of employing for purposes of calculation formulae the origin of which they do not understand have entered upon a path which will inevitably lead to disaster, not only as regards their mental culture, but also in the practical domain. If Mathematics is degraded to the level of a set of practical rules, of which the grounds are not understood, for dealing with practical problems of special types, the unscientific character of such a study will avenge itself even on the practical side of life. A student who proceeds on these lines will fail to arrive at those points of view that. are not only the most stimulating mentally, but of which the attainment is really essential for success in applying Mathematics to practical matters. The practical applications of Mathematics are much too varied to be capable of being confined within the range of any number of prescribed rules and formulae. Practical problems will be found constantly to arise in connection with professional work which are not quite on the lines of the rules that have been taught, and these problems can be effectively dealt with only by persons who possess some real grasp of Mathematical principles, as distinct from a mere knowledge of certain practical rules and methods. Whilst maintaining that a student should thoroughly understand the grounds upon which the formulae and rules which be employs are based, I do not believe that he ought to be expected to commit to memory, and to be able to reproduce at any time, formal proofs of all such formulae and rules. Much precious time and energy has been unprofitably employed in the past in attempting to satisfy the unreasonable demands made by Examiners in some branches of Mathematics that formal proofs should be forthcoming of everything that the candidates are supposed to have learned. The burden thus thrown on the memories of the candidates is far too heavy; and much time and energy which should have been employed in an endeavour to grasp and realize principles has thus been diverted to a far less profitable use.
It appears to me to be eminently desirable that the time saved by the diminution, in school work, of the amount of time spent on unessential details and on unnecessarily prolonged drill in the manipulation of symbols should be employed in introducing the pupils to a considerably greater range of Mathematical thinking than has hitherto been usual; and in particular in endeavouring to make them acquainted with more of the fundamental and fruitful ideas which make Mathematical Science what it is. In the higher classes some time might profitably be spent on the principles, as distinct from the practice, of Arithmetic. It would be of great educational value if the principles which underlie the practice with which all the pupils have become familiar were brought explicitly to their consciousness. For example, they should understand the principle of our arithmetic notation, so that they may have an adequate appreciation of its beautiful simplicity, and of the fact that it embodies a great time-saving invention. In order to attain this object it is necessary to deal with the theory of scales of notation and radix-fractions, so that the arbitrary element involved in the adoption of the scale of ten may be clearly appreciated. I do not of course contemplate the introduction into such a course of artificial problems on scales of notation; only the fundamental principles should be explained, with such quite simple illustrations as may be found necessary for their complete elucidation.
I do not know to what extent some rudimentary and informal treatment of the properties of simple figures in three-dimensional space has at the present time become part of the normal instruction in Geometry in our schools. I am quite sure of the urgent necessity for finding time for a small modicum of study of this part of Geometry. I remember, a few years ago, in a paper on Mathematics for Candidates for a College Scholarship in Physics, the candidates were asked to construct the shortest distance between two given non-intersecting straight lines. One of the candidates, who showed a considerable knowledge of plane Geometry, informed me that two non-intersecting straight lines are necessarily parallels. It is unnecessary to insist upon the importance of an endeavour to uproot ignorance of this kind, due as it is to lack of stimulation of the power of observing simple spatial properties.
In considering the various directions in which Mathematical teaching may be made to extend beyond the domain that consists of that drill in the employment of processes which up to a certain point is undoubtedly necessary, one question of great importance arises. That is the very important question as to the possibility of making a rudimentary treatment of the ideas and processes of the Calculus part of the normal course of Mathematics in the higher classes of schools. In the hands of a really skilful teacher, the purely formal element in the treatment of the Calculus could be reduced to very small dimensions; all the leading notions and processes could be sufficiently illustrated by means of functions of the very simplest types. I believe that some of the time saved by lightening the matter in such subjects as Algebra might be more profitably employed in this manner than in any other. The Calculus, as embodying and utilizing the fundamental notion of a "limit," is the gate to a Mathematical world of incomparably greater dimensions than the one in which the student has moved during the earlier part of his course. Any method of presentment which evades the notion of a "limit," as it appears in the differential coefficient, or in kinematics as a "velocity," is much to be deprecated. The possession of this notion is the most valuable result of the study, both for educational and for practical purposes. By means of carefully chosen examples, in both the arithmetic and the geometric domains, a pupil may be led up to this fundamental notion, so that it may in the end become really his own. To this end it is wholly unnecessary that any treatment of the subject should be employed which would satisfy the logician or the professional mathematician. The important point in connection with this idea, as with many others, is that the student should really have the notion as part of his permanent mental furniture; and not that he should be able to give a complete description of it, or of its philosophy, in conceptual language. I do not propose to indicate now, even in outline, a schedule of those parts of the Calculus which would be suitable as part of a general education. This is a matter which might with much advantage be fully discussed by the Association, when the views of practical teachers as to the possibilities in this direction would receive the fullest attention.
There is a danger which arises in connection with the democratization of education, that less than justice may be done to the minority, who, by natural aptitude, are capable of making much more rapid progress than the rank and file. This danger is probably not so great in this country as in some others; with us, the old leaven which impels teachers to make the most of their more gifted pupils still works strongly enough, and the questionable stimulus provided by Scholarship Examinations and other competitions exercises an influence in the same direction which is very powerful, and perhaps indeed too powerful. In some countries the rigid system by which every pupil in a school is taken in a general class, in a certain number of years, through prescribed portions of a subject, acts detrimentally upon those pupils who are capable of learning much more rapidly than the average. In America, I was told that it would be regarded as undemocratic to make any special provision in a school for the more rapid advance of gifted pupils. This view seems about as reasonable as it would be to prescribe, as a thoroughly democratic arrangement, that all the pupils should be supplied with boots of the same size. The general good demands that, as far as possible, equality of opportunity should be afforded to all for their mental development in accordance with their enormously varying abilities; it does not demand a mechanical equality of treatment represented by forcing all students to move at the pace of the less gifted or of the average. Although, however, this danger may be a real one in some quarters in this country, the opposite fault, of sacrificing to some extent the needs of the average to those of the abler students, is probably still the more prevalent one.
The movement which I have spoken of as the Democratization of Mathematical Education is a progressive development. Something not inconsiderable has been accomplished in our time; very much more remains to be done. The difficulties which arise in this connection are largely those of finding the true coordination between the practical and the theoretical sides of the subject. An undue emphasis placed on either side is apt to have disastrous results. The perfect mean is in all such cases probably an unattainable ideal; a certain degree of compromise, depending upon a variety of circumstances, is usually the practicable course; but the most earnest endeavours should be made to prevent such compromise going too far. Whilst recognizing to the full the importance of the practical side of Mathematics, both as affording the right approach to the subject, in view of sound psychological principles, and also on account of its importance as an equipment for various departments of practical life, let us never lose sight of the paramount importance of Mathematics as part of a real education of the intellect. Such education is incomplete unless a few at least of the many illuminating notions which our race has achieved in its long struggle to attain clearness in the domain of Mathematical thinking are made the common property of our intellectual democracy.
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