The Parents' Review
A Monthly Magazine of Home-Training and Culture
"Education is an atmosphere, a discipline, a life."
Mathematical Teaching and Its Place in General Education.
by T. James Garstang, M.A.,
Putting aside for the moment the more abstract question, "What is mathematics?" let me say at once that whatever method of teaching is adopted, it must have regard to four factors:--(1) it must be interesting; (2) it must appeal to the imagination; (3) it must be logical in any development introduced; (4) and as far as possible it must be up to date, i.e., it must attempt to provide the most powerful intellectual technique yet devised.
In order to awaken interest, the subject-matter must have some connection with every-day experience. So the first stage is undoubtedly concrete; counting must needs involve objects to count. Later on, problems involving money, surface areas, etc., can be introduced, carefully graded to any required standard. But in all these concrete questions too much attention cannot be given to one point, viz.:--the necessity of children having first-hand experience of the concrete subject-matter referred to in the question. As personal experience gets wider, possibly some advantage may be gained by departure from this principle; but in the earlier stages, I feel that it is impossible to carry it out too strictly, if it is desired that the mathematical teaching should be directed towards producing real power of mind.
Secondly, it must appeal to the imagination. I do not expect anyone will question that this ought to be true of all teaching; but possibly many may ask, "With such a dull subject as mathematics how is it possible?"
Thirdly, it must be logical in any of its developments. By this, I do not mean that the whole subject must be built up into a vicious imitation of Euclid; but that when any, even the smallest, deduction from given premises is attempted, the conclusion of course must follow from them, but also pupils must be allowed time enough to derive it personally, and must not take it for granted because printed in a text-book.
Fourthly, it must be up to date; by this I mean that if any system can be found admittedly superior to all others in providing the mind with the means of reasoning about any collection of observations presented to it, then that system ought, if possible, to be taught at school, and all the rest of the teaching in mathematical subjects ought to be directed to lead up to this system, as shortly as a thorough grasp of necessary principles allows.
All these factors are effective, if the method adopted reproduces more or less in the minds of the pupils during their school life the discoveries made by men of genius from the earliest times to the present day; or shortly, if the subject is taught with one eye on the lines of historical development. Wherever possible, each new principle, law, or general proposition ought to be introduced to a child through an attempt to pick up the traces of the inspiration of a former age; and when this is done, interest and imagination are preserved during lesson times without difficulty.
Allow me a moment or two to explain more fully what is meant by the words, "to pick up the traces of the inspiration of a former age." And for this purpose let us consider the well-known theorem in Euclid, Book I., Prop. 47.
Imagine a noble Greek of sage and reverent countenance, forsooth Pythagoras, at his meal in a hall whose floor in keeping with the age is paved with tiles. Well-skilled geometer that he is, his eyes pick out the lines and squares in answer to that wondering tension of his mind--"What relation holds between the adjacent sides of an angle of any triangle, and that third completing side which lies in opposition?" "I see," says he, "that with a smaller angle, smaller is the side, and longer when the angle too is large: but there is that within me draws me further on, and refuses more delay."
And then at length the picture, ever present to his eyes, the lines, the squares, rise up in sight and yield their mighty tale. "Behold! the square on this side the longest side contains parts equal to those which fill these two. Is this the secret of my long-tried search?" With smaller tiles he puts all to the test--as one might play with children's bricks; and pressing on, he plucks at length the fruit, the heritage he has left to all.
Here I should like to add a word of caution on a practical point. Because the method of teaching may conveniently be analysed into the four factors already referred to, beware lest hasty attempts should be made to upset any actual system of teaching, possibly quite efficient from the point of view of discipline, in order to adopt a system containing these new elements of stimulus, but omitting all the steadying influence of the old. In practice a teacher must find a way of balancing the two; personally I have found long or short periods of alternation give satisfactory results.
No single part of any system can at one and the same time contain all four requirements. Therefore, practically, the method resolves itself into two parts. One, accurate, plodding, strictly logical, and up to date. The other, in which free rein is given to the imagination of both teacher and pupils, to run wild among the data afforded by both history and legend, in order to pick up clues of connection with past inspirations. In the one, the bonds of discipline may be braced up and work exacted; in the other, the muscles should be relaxed, and discipline, never perhaps quite absent, must be quite unconscious and merely a natural offering to the joy of discovery.
In my own work I have assumed that the most powerful mental instrument yet devised is the Differential and Integral Calculus. This then has been the goal towards which my own teaching has been directed. At the outset I was confronted with a serious doubt. The Calculus as ordinarily introduced presents so much difficulty to mature minds that perhaps children have not the necessary power. Such a point could not be settled off-hand; but applying the historical method for the purpose of finding an answer, I was soon convinced that the difficulty felt by older students was due to the method of presentation and not to any feature or property of elementary Calculus itself. My conclusion was reached on these grounds:--most, if not all mathematical subjects, Arithmetic, Algebra, Geometry, etc., are taught (or were until quite recently) in watertight compartments, with a wet sponge most carefully drawn over all the remaining traces of original discover; and the Calculus in particular was approached by methods involving no attempt to reproduce a mental atmosphere similar to that which existed after the discovery of Co-ordinate Geometry by Descartes. That the step preliminary to the discovery of the Calculus was the synthesis of Algebra and Geometry by this philosopher is a statement easily verified from mathematical history; the step preliminary to teaching the Calculus to young children may be to break down the barriers, artificially created for examination purposes, between Arithmetic, Algebra, Geometry, Trigonometry and the inductive experimental sciences. This barrier-breaking I have systematically carried out for some years at Bedales School; and it may be not without interest to describe how. The instruments which are essential are fortunately not expensive--merely blackboard ruled into squares for oral work with the class as a whole, and squared paper for each child for individual practice. Straight lines and curves of many kinds are easily traced; while the work requires the accurate use of decimals, and is self-corrective, but ultimately repays all the care it demands.
And here let me pause to pay a tribute of respect and appreciation to one who has endeavored so long, and may we add, not unsuccessfully, to lead the reform of mathematical teaching along the only possible lines--I refer to your fellow citizen, Professor [George] Chrystal. How valuable his works on Algebra have been to me personally, is perhaps impossible to say; but if anyone is desirous of grasping the fundamental principles of Algebra and of acquiring a knowledge of curve tracing capable immediately of extension in many fruitful directions, I cannot do better service than refer such an one to Professor Chrystal's Introduction to Algebra. But returning to Bedales, we find the seed has grown until the idea of a limit is common knowledge in four classes of girls and boys ranging from twelve to eighteen years of age. Systematic work in both Differential and Integral Calculus is taken by the older pupils; and the practical course of Mechanics and the courses of Physics are carried out in ways involving the application of easy Calculus.
Lately we have attempted to arrange the work of the lower and preparatory schools on lines parallel to those already outlined. Our old friend the tiled pavement takes on a new dress, and in the form of a squared blackboard does excellent service. But for the youngest children, let me call attention to the method of tracing curves in silk originated many years ago by Mrs. Boole, and recently described in the chapter on the "Dog's Path" in her Logic of Arithmetic. Such work, of which I am fortunately able to show you some specimens developed after her idea, is thoroughly enjoyed by the children without strain. Indeed they are quite eager to do all, and many of these cards are the original designs as well as executions of the children themselves. To those who may not be familiar with this work, I may point out that the curves are not drawn with pencil and then merely stitched round and the rest filled in; but that the curves are evoked by the alteration of merely straight stitches according to a rhythmic law. We have then, rest for the nerves during work, and rest for the eyes in seeing curves of such delicate shape. In addition the children must obviously get practice in choosing colours; and since the curves in technical mathematical language are really envelopes, the whole is an unconscious preparation for the Differential Calculus.
I have already referred to the work of Descartes as having been the condition necessary and preliminary to the discovery of the Calculus. In order to give children a living knowledge of mathematics, we must attempt to reproduce somehow or other the atmosphere of the seventeenth century--of that wonderful age in which Kepler, Cavalieri, Fermat, Pascal, Barrow, Wallis, Newton, Leibnitz, and many others lived, and under whose stimulus and inspiration they produced their work. Great indeed were our own mathematicians until, yielding to a narrow national prejudice, they drove away the spirit of inspiration through refusing to co-operate in the development of an universal notation. And English mathematics had reached a low ebb in comparison with the brilliant work abroad, when the foundation of the Analytical Society at Cambridge in 1812 by Herschel, Peacock, Babbage, and a few others proved to be the dawn of a new era. The sympathetic co-operation of these mathematicians inspired the younger men with brighter hope; and the whole movement may perhaps be considered to have been crowned by the work of De Morgan, George Boole, and their friends, who strove hard and successfully to effect that synthesis between logic and mathematics proper, which through them laid bare for all the way to a fuller vision. The real nature of mathematics can be found admirably portrayed in a series of sectional addresses to the British Association during the late sixties, by Spottiswoode, Tyndall, Sylvester, and Clerk Maxwell; while in the eighties Chrystal and Henrici voiced the urgent need of the reform of elementary teaching in no uncertain tones. Where then may we find the path of true reform? Who is to lend a guiding hand to the coming generations deprived of most, if not all, support from those social customs and religious conventions which helped so much to steady growth and upbringing in the past? To answer is to name George Boole; and though we have him not to lead us here on earth, yet his spirit still speaks through his work on the Laws of Thought. What shortly does he ask us to perceive? The break in the Roman digits at V. and X., or the change in our modern arithmetical symbols at the number 10, connect numerical notation with the structure of the human body, viz., the number of fingers on one or both hands. Again, fill in the C line between the two musical staffs and the eleven lines are then sufficient to indicate the notes throughout the medium range of the human voice--in this way connecting musical notation with a human trait--the limits of our vocal capacity. Is it surprising then that the intellectual notations by and through which man brought order where chaos reigned before--is it surprising that, to the inspired seer, these notations proved the means which led him to perceive the general laws of thought? The games of children--their hide-and-seek or simple see-saw--the rhythmic movement of the dance, the rhythmic movement of the heart, the varied movements of a complex theme in music, all point to a natural alternation. Is mind alone to be exempt? It is not so; and thus Boole's Laws of Thought provide that foundation on which our future work must all be built.
Of notation in its international and universal aspects little need be said. But notation is an instrument of thought, enabling the human mind to look forward into the unknown future, and almost to focus it into the present. Poisson, Hamilton, Leverrier, and Couch Adams, and not least Clerk Maxwell, have proved this more than once. The moral is obvious; if any group of people desire to think about any particular subject, let them first invent or rediscover an ad hoc notation to assist the human mind in its search for truth. Without notation, there is self-deception and exhaustion; with adequate notation, there is self-correction and control.
Finally, I should like to add that the result of some years' mathematical teaching, for the most part in complete ignorance of Mrs. Boole's ideas, but at the same time under a system allowing complete freedom of choice, has convinced me not only of the value of her methods, but also that until it is more generally recognised that mathematics is the science of the Laws of Thought applied to number, size and form, we cannot get the best out of mathematical teaching. Only under such a recognition will mathematics take that place in general education to which it is entitled, viz., the safest and surest method yet devised by man for regulating inspiration from the unknown.
(1) The Logic of Arithmetic. Mrs. [Mary Everest] Boole. (Clarendon Press.) (To be used in conjunction with any ordinary Arithmetic.)
(2) An Introduction to Algebra. Professor Chrystal. (A. & C. Black.) (Especially Chapters II.--VI. inclusive, and Chapter XXV.)
(3) Symbolical Methods of Study. [Mary Everest Boole]
(4) The Mathematical Psychology of Gratry and Boole. (Both by Mrs. Boole; to be obtained from Benham & Co., Colchester.)
(5) Brotherhood. Bound volume, May, 1903--April, 1904. (F. R. Henderson, 26, Paternoster Square, E.C.)
(6) British Association Reports for 1865, 1868, 1869, 1870, 1883 and 1885. Presidential Addresses to Section A.
(7) A book, now in proof, by Mrs. Boole, and to be issued shortly by the Clarendon Press, describes the application of this method in the education of young children.
(8) The Laws of Thought. Professor G. Boole.
(9) The IXth Bridgewater Treatise. Charles. Babbage.
Typed by LovieWie, August, 2023; Proofread by LNL, August, 2023
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