II M3 UC-NRLF $B Ml? D^T Why Do We Study Mathematics: A Philosophical and Historical Retrospect By Thomas J. McGtrmack Why Do We Study Mathematics: A Philosophical and Historical Retrospect Address Delivered Before the Secondary Mathematics Section of the National Education Association Boston, July 8, 1910 By Thomas J. McCormack Principal of the La Salle-Peru Township High School La Salle, Illinois THE TORCH PRESS CEDAR RAPIDS, IOWA 1910 a- v\ ' i .' WHY DO WE STUDY MATHExM ATICS : A PHILOSOPHICAL AND HISTORICAL RETROSPECT Introduction The thought that impresses itself most forcibly upon one's mind as one approaches this subject of the part which the study of mathematics plays in education, is the extremely fortunate and exalted position that this discipline occupies in the hierarchy of the sciences that have been selected as especially designed to inform and cultivate the human mind. The oldest branch of human knowledge to be investigated, the first to take systematic and dogmatic form, it has, in the three thousand or more years of its development, rarely halted in its advances and never ceased to fructify either the fields of practical knowledge or the loftier realms of metaphysical thought. Men of affairs and philosophers, whether for ma- terial gain or pure intellectual enhancement, have, for ages, alike looked to it for succor and guidance; and wherever, in the history of human thought, it has appeared that the ideals of truth and certitude were destined to extinction, there in its despair thinking humanity has found in the irrefragable conclusions of mathematics an unfailing solace and support. Encircled with a halo of divinity by Pythagoras and Plato, with whom the laws of _naiura-ware but- the geometricaL t hought s oF"(jod, it ran serene and undisturbed the solitary gamut of its development through the entire history of phil- osophy, embracing, dominating, and even submerging in Des- cartes, Spinoza, and Kant the whole structure of human knowledge," and furnishing the prototype of all human inves- tigation of truth. And in the great scientific compeers of these last-named men, Galileo, Huygens, Newton, Euler, La- place, Monge, and Lagrange, it took that direct practical turn that made it the living groundwork of all the marvels of the 233109 material civilization of whicli we 6t today boast. On neither its theoretical nor its utilitarian side, therefore, does it need apology or justification. It needs simply, for our present pur- poses, analysis; and this, under one or two simple points of view, without any pretence of exhausting a well-worn subject, I purpose to offer, in the hope that the clarification which the process involves will present some important pedagogical im- plications. MATHEMATICS AN ECONOMIZATION OF THOUGHT, A CAPITALIZA- TION OF INTELLECTUAL LABOR-SAVING DEVICES The most salient practical feature of all scientific thought is its economic or labor-saving purpose. Even our every-day modes of thinking are impregnated with this trait. Education itself largely consists of the inculcating of ready-made, snap judgments on men and events, on sociology and economics, on politics and history. Prejudice is the supreme type of ready intellectual power; the mightiest weapon in our logical arm- ory : it is mental preparedness. # The laws of physics, of astronomy, of geology, and the rest are, on their practical side, mere shorthand or rather short- mind formulae or rules for recovering, with a minimum of mental labor, by means of little brains and the mechanical manipulations of a pencil, the past and future facts of nature, which, without these rules and laws, we should have indefinite labor and take indefinite time in recovering.^ Such are the so-called physical laws of refraction and of falling bodies, the prediction of eclipses, etc. Millions of cases reduced to a single case. 'All a saving of intellectual labor. But this economy of mental effort is most conspicuous in mathematics. Memorized addition-results save not only fric- tion of aboriginal fingers and toes, but save also brain-strain, 1 For the full development of this view of science as an economy of thought see the works of Ernst Mach, especially the "Science of Mechanics" and ** Popular Scientific Lectures," translated by T. J. McCormack and published by the Open Court Publishing Co., Chicago. The idea was contained in nuoe in the works of Adam Smith, Babbage, and several German writers quoted by Mach. It has also found sporadic expression in such quotations as that given in this section from De Morgan. an economy still further heightened and typified by the mod- ern adding-machines. The multiplication-table, the Arabic decimal machinery, determinants, integration are all thought- saving devices. But preeminent among all these economic mechanizations of human thinking are the tables of logarithms and products, powers, roots, interest and annuity tables, etc., in which the mathematical labors of generations have been capitalized and amassed for all time. In this view the development of mathematics becomes a continuously progressive abstract economization and petrifac- tion of quantitative thought, a permanent crystalization of quantitative architectonic thinking, a standardizing of the ma- chinery of the mind, which reaches the pinnacle of its refine- ment in the shorthand symbolism, or, as I prefer to call it, the shortmind or stenophrenic ^ (as distinguished from sten- ographic) symbolism of algebraic analysis. Algebra, in this view, to quote the lucid words of an old editor of Euclid, is that ''paradise of the mind, where it may enjoy the fruits of all its former labors, without the fatigue of thinking. " ^ It is the essential, paradoxical purpose of mathematics, in this con- ception, to get rid of thinking by very dint of thinking. This is the purely intellectual side. But this economization of mental and manual labor can be even more palpably traced in the domain of applied mathematics, in engineering, survey- ing, navigation and the rest, to develop which I refrain, from lack of time. PEDAGOGIC IMPLICATIONS OF THIS VIEV^ Now what is the lesson of all this ? I see in it two things. First, we have, in this process, in its purest and simplest form, the original and primordial type of all human intel- lectual activity, the incarnate essence of all human mental striving, which is intense economization, and progressive con- centration of mental effort. The student who has gained this point of view, by contact with mathematical and especially algebraic study, acquires from it a sense of intellectual power 1 From ffTev6i, short, and 0/ji}v, 4>pv6s^ mind. 1 Quoted by De Morgan in the Preface to his * ' Essay on Probabil- ities," London, 1838. and discipline, which can be brought home to him with equal force by the study of no other branch of knowledge, and draws from it an aesthetic inspiration that heightens his whole spirit- ual life. There is a moral sense of enlightenment in it that courts comparison with that of any other discipline; and I take it, that this view has, for its practical educational value, not yet received its proper emphasis in pedagogical theory. And the second lesson is, that this view, on its ultra-prac- tical side, connects our so-called formal theoretical study by a living link with our material industrial civilization, whicli sees the goal of life in the progressive minimization of all human labor and the saving of human resources for higher ends. Both the practical and ethical implications of this view are immediately apparent, and it is gratifying to see its main trend faintly emphasized in some of our recent text- books. No one who can grasp this point of view will ever have the hardihood to say that the science of mathematics is without a moral, let alone, an intellectually material, con- tent, a moral content for which alone it deserves to be studied, entirely apart from its applications, which so few students ever make, and entirely apart from the Jogical cul- ture which it gives and which has been its glory for centuries. This is one of the great benefits derived from mathematical study, which every thinking person will admit, which has not, in my opinion, been sufficiently recognized, but which alone would justify its pursuit as a branch of practical intellectual culture. PRACTICAL INTELLECTUAL CULTURE I would emphasize particularly this phrase I have used, practical intellectual culture. There are things intellectually and spiritually practical, as well as things materially practical. And the former have as much right to a place in education, from which they would seem to be now becoming displaced, as the former. And unless we recognize this, one-half of the studies of our present secondary curricula must go, and espe- cially mathematics must go; for upon its meagre quota of possible so-called practical, material applications, for the ma- jority of secondary students, it cannot rest. De Morgan, as early as 1838, in the preface to his "Essay on Probabilities," where he applied mathematics in the most practical possible manner to the questions of life-insurance and its related subjects, spoke with regret of the tendency which even in his day existed both in England and America with regard to opinion upon the end and use of knowledge, and the purpose of education; and his remarks have far great- er applicability now. He says: ''Of the thousands who, in each year, take their station in the different parts of busy life, by far the greater number have never known real mental exertion; and, in spite of the variety of subjects which are crowding upon each other in the daily business of our ele- mentary schools, a low standard of utility is gaining ground with the increase of the quantity of instruction, which deteri orates its quality. All information begins to be tested by its professional value; and the knowledge which is to open the mind of fourteen years old is decided upon by its fitness to manure the money-tree." FORM OR CONTENT But this leads to one of the celebrated moot points of edu- cational theory, whether mathematics should be studied for its intellectual or for its material content. And a glance at the history of our subject may throw some light on this con- troversy. Personally, I believe that both demands can be simultaneously satisfied ; but the proper sphere of each should first be clearly exhibited by analysis, and the rights and claims of each recognized and placed in their proper setting. I am the advocate of neither view; I am a believer in both. The truth is more important than either ; and the truth will make us free to teach according to reasoned and just convictions. KANT vs. SCHOPENHAUER The dominating and exclusive position, above referred to, which mathematics has occupied in the hierarchy of the sci- ences since Plato, whose utterances on its importance as a cultural discipline are now the commonplaces of pedagogy, has led not only to the sublimest flights of speculative thought and a consequent apotheosis of the value of mathematical 7 V training, but also to the most extravagant hyper-emphasis of mathematics as an engine of scientific and philosophic enquiry. Descartes saw truth only in geometry. Newton, by his en- thronement of the deductive method, retarded the advance of English science for a century. *And Kant went so far as to say that "in every branch of natural knowledge there is just so much and only so much genuine science as there can be mathematics applied in it. " ^ The world went mathematics mad. The calculus was as fashionable a fad in the 18th century almost as Christian Science is today. Ladies at Ver- sailles and the minor courts of Germany juggled with differ- ential coefficients and discussed the precession of the equinoxes at their toilettes, while Euler's letters on physics and deduc- tive logic to the Princess Elizabeth rivalled in continental pop- ularity the epistles of Lord Chesterfield. There were even mathelnatical schools of agriculture, and a mathematical school of forestry lasted in Germany even into the 19th century. The influences of the movement can be seen even in some modern phases of the science of psychology. The reaction was imperative and it was trenchant. There is a proverb of unknown origin to the effect that ''purus matliematicus, purus asinuSy'' or **Show me a pure mathema- tician and I will show you a simon-pure jackass." Frederick the Great called his mathematical colleagues of the Berlin Academy savages, hollow-pates and lack- wits.- Napoleon said that Laplace * * carried the spirit of his infinitesimal pettinesses into the business of state, ' ' and Schopenhauer bluntly assever- ated that "where mathematics begins, the comprehension of phenomena ceases. " ^ Such utterances I could quote ad libitum; as I could also their opposites. THE PLATONIC AND THE GASTRONOMIC POINTS OF VIEW How, now, for the purposes of practical pedagogy, and 1 * * Metaphysische Anf angsgriinde der Naturwissenschaf t. ' ' Vorwort. 2''0euvres de Frederic le Grand,'' tome 9 (6d. Decker, 1848), pp. 63-65, 68, 72-74; tome 19 (ed. 1852), p. 321; t. 22, p. 181; p. 199; t. 23, pp. 417-421. 3 ' ' Parerga und Paralipomena, ' ' Vol. II. 80. 8 this is our purpose here, are these strident contradictions to be reconciled? Why was it that Plato could say, as he did say in his ' ' Laws, ' ' ^ that the man who did not understand the theory of incommensurables was no better than a pig, that question-games on the ratios of incommensurable numbers were a far more graceful way of passing the time for educated men than playing checkers; while Savarin, the great French gastronomer, could proclaim, amid the applause of his con- temporaries, when Leverrier discovered by pure deductive mathematics, Neptune, that ''the cook who invented a new dish was a greater benefactor of humanity than the savant who discovered a new planet ! ' ' Here is trenchantly expressed the gaping hiatus that di- vides the camps of constructive educationists with regard to the value and the aims of mathematical study. I would call these two points of view (1) the Platonic point of view, and (2) the gastronomic point of view ; the point of view of pure in- tellectual culture, and the point of view of pure, practical, bread-and-butter utility. Which is to be predominant in the emphasis of topics and methods for secondary instruction in mathematics? Or, are the two reconcilable; and can both aims be adequately satisfied in our instruction ? I believe they can ; and that much of our worry and solicitude on this head is misplaced and supererogatory; that unconsciously and un- wittingly we always, despite ourselves, are fulfilling both pur- poses. ^^"^ THE IDEA ** practical'* EXTENDED ' ' Practical ' ' and ' ' utilitarian ' ' are current words of pop- v ular vogue that need sadly clarification and precise definition. Are not some things '/practical" and more eminently useful for purely intellectual, ethical and social purposes than are some other things that find ready and immediate application in the measurement of concrete realities^ I think we must seek a deeper meaning for the idea "practical" here. Not one-tenth of the graduates of our high schools ever enter pro- ^ fessions in which their algebra and geometry are applied to concrete realities ; not one day in three hundred and sixty-five 1 Plato, ''De Legibus,^' VII. is a high school graduate called upon to ''apply," as it is called, an algebraic or a geometrical proposition. Not one in ten of our high school students ever retain sufficient mathemat- ical knowledge or skill to solve even a tolerably difficult con- crete physical or mensurational problem after graduation. Why then do we teach these subjects, if this alone is the sense of the word "practical?" For abandon them, we are all agreed we should not. To me the solution of this paradox consists in boldly con- fronting the dilemma and in saying that our conception of the practical utility of these studies must be readjusted, and that we have frankly to face the truth that the "practical" ends we seek are in a sense ideal practical ends, yet such as have after all an eminently utilitarian value in the intellectual sphere. What practical profit can a girl, or nine out of ten high school boys, derive from a year and a half of algebra - unless it is the aesthetic joy and logical culture derived from participation in the upbuilding of a great abstract structure of symbolic quantitative thought, the sharing in the dramatic triumph of a great human intellectual conquest (which I referred to above in my references to mathematics as an econ- omization of human thought), or that invaluable, incalculably important mental and manual drill (which Chrystal has em- ^phasized), in accuracy, neatness and method that comes from the correct systematic manipulation of algebraic forms; or, can the same average student derive from geometry, save that magnificent training in logical deductive power, in capacity for intense concentration of the attention, in systematic un- erring pursuit of a goal, in intellectual self-restraint, and, last but not least, that intensifying and heightening of the \ power of precise English expression and English thinking that makes mathematics in our schools almost the substitute and the surrogate of a thorough legal training! Bent as I am personally and naturally to the making of all mathematical instruction practical and applied in its most popular sense, I must say that in summing up to myself the results of my own limited instruction I have always felt that the objects above enumerated have been attained in my classes^, 10 in tenfold greater degree than have the so-called concrete re- sults I have so ardently sought and wished for, but obtained only in a limited number of individual cases. We sometimes forget in our inordinate zeal to ''practical- ize" and popularize education, that our object is also to make men and women as well as engineers and "rope-stretchers," and that the former end is more commonly attained than the latter. Our trouble and indecision arise from the very im- portant psychological truth that it is impossible to weigh and measure psychical and ethical values, and that we have not always necessarily failed when we cannot palpably catalogue > our results. Education is a subtle process, and withdraws itself from quantitative observation. This is why Plato, with his divine tact, proposed for educated men the game of tit- tat-to with incommensurables, to replace the checkers of the porklings. It is the victory of the Platonic over the gastro- nomic point of view, the victory of Pythagoras of Croton over Mr. Crane of Chicago, and leads us to say with St. Paul that for the purposes of genuine education we ought sometimes to ' ' look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal. " It is these eternal impalpable things, which remain with us all from our study of algebra and geometry, that constitute the sole profit that ninety per cent of us ever derive from the study of mathemat- ics, and it is this that furnishes the foundation of truth to Emerson's paradoxical saying that "education is what remains to us after everything we have learned at school is forgotten. ' ' A CONCRETE ILLUSTRATION I will introduce here one concrete illustration at length, taken from a London Conference of Secondary Mathematics and Science teachers of last year, as exemplifying what I mean when I say with Chrystal that training in habits of accuracy, mental and manual, is the cardinal benefit derived from the study of secondary algebra for ninety per cent of us. The speaker, Mr. Jackson, asks : ' ' What, really, is our ob- ject in education; and why is elementary science going to do all that is expected of it? Mr. Searle, in his book on experi- 11 mental elasticity just published, observes: 'A demonstrator in physics spends much of his time in correcting student's mistakes. He has to discover, for instance, why a student ob- tains 537.86402 (no units stated) for Young's modulus for a brass wire. . . . The student has confused radius with diameter; has used a screw gauge in which one turn is equiv- alent to 1/50 of an inch, and treated one turn as equivalent to .5 of a millimetre. . . . has treated the millimetres as if they were centimetres. . . . and has used 32 for 'grav- ity' instead of 981.' ^ There is the enemy! The real enemy we have to fight against, whatever we teach, is carelessness, in- y accuracy, forgetfulness, and slovenliness. That battle has been fought and won with diverse weapons. It has, for in- stance, been fought with Latin grammar before now and won. I say that, because we must be very careful to guard against the notion that there is any one panacea for this sort of thing. It borders on quackery to say that elementary physics [and I should add, elementary mathematics] will cure everything. The personality and learning of the teacher are everything here. Nothing else matters very much. That ought to be re- iterated by every educational meeting that is held." I will revert to this question of the personality of the teacher under another head. We are concerned now with the benefits, non-material and unprofessional, yet none the less real and practical, that are derived from mathematical study by the majority of students. '^ And one of these is, as we have just seen, the conquest of ''carelessness, inaccuracy, forget- fulness, and slovenliness" in thinking and acting. USABLE KNOWLEDGE AND POTENTIALIZING KNOWLEDGE Let US assume, for the moment, however, that the other goal also is attainable, namely, the acquisition of concrete, usable mathematical knowledge. Let us admit that ten per cent of our high school graduates actually do acquire a mathematical knowledge, as distinguished from a mathematical culture, which they may be expected to apply to physical and men- surational problems; let us admit, too, that all graduates of technical schools and colleges possess that knowledge ; the ques- tion remains how long do they retain that knowledge and 12 whether they ever in the majority of cases possess it in a suffi- ciently powerful and ready form to apply it universally and successfully in practice. I personally know many successful engineers, but I know very few who know mathematics well and who can use it as a powerful auxiliary tool in the solution of new engineering and mechanical problems. Their knowl- edge of the differential and integral calculus is a faint dream of their collegiate career. They are devotees of elementary methods and rule-of-thumb procedures in the higher domain. A few formulae, a few tables of integrals and collections of mechanical rules and tabulated calculations are their chief stock-in-trade. Their sole pabulum is what I might call *' canned" mathematics, using that term in its best and most esoteric sense, and not in its present vulgar and figurative perversion, that capitalized stock of mathematical thought which is preserved in the economy-jars of logarithmic and other tables and in routine formulae, and which constitutes, in this capitalization and economization, a cardinal, essential feature of mathematical intellectual activity. This, ninety- ^ nine out of every one hundred engineers will admit. But they will not admit, nevertheless, that they have derived no ^ profit from their mathematical training; that they have not preserved from it that unalysable power of thinking and visualizing things mathematically and geometrically and o:B pre-constructing in symbols of thought and pictures of the imagination the projects which afterwards it is their business to create in space with physical materials. Now, what I contend under this point is, that, precisely as the engineer, who is trained for mathematics yet usually can- ^ not and does not use his mathematics, nevertheless derives in- estimable benefit and power from his mathematical education ; ^ so the average high school and university graduate wTio is never expected to use his mathematical knowledge at all, in- variably acquires from his mathematical instruction great moral and intellectual impressions and potential capacities of all sorts which will redound to his power and usefulness in every walk of life. And, in making this a cardinal benefit derived by the masses from mathematical study, I am not 13 slurring, in the slightest, the great material benefits of mathe- matical knowledge as aif auxiliary engine of research, control ^nd creation in every profession. I merely say that this lat- ter, important as it is, constitutes the least general practical acquisition from a mathematical training; while the former makes up in all cases, some nine-tenths of its value. For, if this were not so, then instruction in secondary mathematics should be limited, as the public would largely limit it, to p arithmetic and practical mensuration. And we should thus save at least one year of our time for the pure bread-and- butter studies. For, how otherwise could educators justify this economic fact, which I cannot prove statistically but which I believe to be true, that there is more human time and energy spent on writing text-books of mathematics, even ap- plied mathematics, and on teaching mathematics, than is ever after employed by all the people in the world in applying it to scientific and engineering problems! AESTHETIC CULTURE J should not leave this aspect of our subject without at least briefly referring to a subtle phase of purely aesthetic culture which occupation with mathematical studies inevitably imparts, although this consideration sinks behind the more practical intellectual benefits I have above referred to. There is in all mathematical research a genuinely aesthetic intel- lectual, as distinguished from sensuous, enjoyment, issuing from the contemplation of the unity, symmetry, and dramatic movement of mathematical creations, which is observable even in the logical upbuilding of the elementary parts of alegbra and geometry, and which should not be underrated in its educa- tional effects. This intellectually artistic side of mathematical study has found expression in all great thinkers from Pytha- goras to Pascal, Lagrange, and Kant, and has had its praises sounded by many lesser and later writers (Kummer, Ilelm- holtz, Boltzmann and Poincare). But I must abandon this theme for some more pressing topics. I have omitted in this paper all special discussion of the logical and disciplinary ad- vantages to be derived from mathematical study, as also the utilitarian benefits derived from its pursuit. These aspects 14 of the subject have been so often discussed as to be now commonplace and almost self-evident. PEDxlGOGrCAL COMPLICATIONS AND ECONOMY OF PRESENTATION The considerations adduced above lead now to an important practical pedagogical difficulty. We seem to be seeking in our high school curricula and text-books to devise a universal pedagogical machine for the instruction of everybody en masse for every possible end, cooks, dressmakers, and scribes, engineers and ''rope-stretchers," professional "ad- mirable Crichtons, ' ' and Jacks of all intellectual and utilitari- an trades. And as a result of this our manuals have become a veritable mathematical polyglot and Tohu-va-bohu. A glance at the problems of some recent admirable algebraic and geometrical text-books will amply show this. They seek, in their problems, to cover the entire universe of applied knowl- edge, and some additional domains besides. And this is a gratifying and refreshing symptom, growing out of a genuine desire to modernize and vitalize mathematical instruction. But we forget in our new-bom practicalizing zeal that all this has been done befoi > and that the different epochs of the long development of mathematical instruction have shown the same ideal and trend. Each period has had its practical prob- lems, some of them purely intellectual, but, according to my view, still practical. What we now regard in the older text- books as pure rubbish and antiquated intellectual ' ' survivals ' ' were once mostly real problems. But to show this, interesting as it is, would be to write a history of civilization, for which I have at present not the time. I merely wish to call to your attention the fact that in the past the same striving for mod- ernity and yitalization led, as it will likely lead now, to an almost uncontrollable emharras des richesses; and that this superfluity of material forced the old textbook writers, from sheer considerations of economy, to adopt the apparently harsh and arid, abstract method of presentation which we now so unanimously condemn, and to formulate general abstract problems for exercise, instead of a multitude of concrete ap- plied problems. For economy of presentation is of the very essence of our science and is as desirable in text-books from a 15 purely practical point of view as from a theoretic one. It was doubtless this consideration as much as the gibes of the Sophists that led Euclid to the formulation of his Elements; and any one who doubts this view has but to compare the beautiful, lucid, and concrete, but interminably prolix, treat- ises of the great Euler in the 18th century with the concise and condensed expositions of the generation that followed him, which we now so cordially anathematize. The pedagogic pendulum swings back and forth by a natural intellectual law. And in less than ten years, I prophesy, the same economic need will present itself in our American text-book world. ENDS AND MOTIVES Education, even mathematical education, is a class-problem, a caste-problem, an individual problem ; and what we need here is to differentiate. To make thinking men and women is one thing, to make pure mathematicians is another, and to make bookkeepers, engineers, mechanics, etc., is a third. Each end is legitimate in itself. Each requires its own pedagogical ma- chinery. But in the average text-book and in the average cur- riculum of the public school it is sought to realize all these ends at once. And hence results a necessary confusion and conflict of means and ends, which in a universal system of education it is difficult and, it may be, inexpedient, to disentangle. But, in undifferentiated curricula and school-systems such as ours, we should always bear this in mind, in our discussions and striv- ings for solutions. What part and amount of our work should be devoted to cultivating mathematics as a science in itself, as an independent body of natural knowledge having its own ends and methods ; what part, to it as a purely logical and cul- tural discipline; what part, to it as a science auxiliary to physics, mechanics, surveying, and engineering, or as a body of pure useful knowledge. These points of view should be considered as ends ; but they may also be considered in the role of their efficiency as at- tractions or stimuli to the study of mathematics, and a pro- portionately preponderant share accorded to each of them ac- cording to this aspect. And here the practical motives appear to be predominant, especially in the applications and the se- 16 lection of problems. For it is as essential in pedagogics as in dietetics, that we should ''first catch our hare before we cook it," or, as a writer whom I have before quoted, has phrased it, with some slight approximation to profanity, '' that what we want our boys and girls to believ e is that mathematics is indispensable in their daily life and not something they will have to do in hell. ' ' ^ But this aspect of the question will be adequately discussed by the gentlemen that are to follow me.^ Practical baits are legitimate, but they are not the end of educational psychology. PRACTICAL MATHEMATICS AND PRACTICAL PEDAGOGY 1 will now indicate the general character and trend of the so-called practical or Perry movement in the tekching of math- ematics, which constitutes in many of its features one of the most stimulating phases of recent educational thought. Some little discussion will probably place it in its true light as a branch of the practical psychology of teaching rather than as a new idea in educational science. " What is really meant by the term Practical Mathema- tics? " This question is asked by the speaker in the London Conference above referred to. "I am sure Prof. Perry's meaning has been misunderstood," he says. '' For instance, in the early days of the movement, he set a question about the relation between the capacity of a saucepan and its price. Some gentleman who took no interest in saucepans (I suppose he was a bachelor), took great exception to this question, and inferred that ' Practical Mathematics ' was a tin-pot business. " Now, what does Practical Mathematics mean? I doubt if Prof. Perry would be prepared to name a theorem that is certainly not ' practical. ' If I understand him rightly, he means the term to apply as in ' practical politics. ' You must consider not only what is right in the abstract, but what is right under given conditions. The fundamental theorem of Practical Mathematics is thisi There is not only the math- 1 ' ' Mathematical Gazette, ' ' Londt)n, Jan., 1909, p. 24. 2 See the program of the Mathematics Section of the Secondary Con- ferences of the National Educational Association for 1910, Boston meet- ing. 17 ematical proposition, but there is the boy into whose head that proposition is to be put, and the spirit in which the boy re- ceives the proposition. Prof. Perry's argument is that the mathematician is apt to deal with the wrong half of the prob- lem. He worries himself about Euclid I. 47, or about Fourier's Theorem. There is nothing wrong with them. The trouble is with the pupil who is to receive them. ' ' THE PROBLEM OF PRACTICAL PEDAGOGY This last remark brings us to the human, individual, psy- chological problem of practical pedagogy, to which we always have to revert in our discussions, with which all general solu- tions are entangled, from which no text-book can save us, the ever-recurring problem of the personality of the individual pupil and the personality of the individual teacher. This problem has been admirably sununarized by the great German mathematician Felix Klein in the following manner. He says : ''If I were to formulate the general problem of pedagogy mathematically, I should say that it consisted in marshalling the individual qualities and capacities of the teacher and his n students as so many unknown variables and in seeking the maximum value for a function of (1+n) variables, F (x^,x^, ...x;^ ), under given collateral conditions. If this problem, as a re- sult of the advances made by psychology, should some day ad- mit of direct mathematical solution, then from that day on- ward practical pedagogy would have become a science. Until then it must refaain an art.'' {" Deutsche Math.-Yerein. Jahresber.," 7, 1897-1898, p. 133.) We see, thus, that no text-book, no syllabus, no system how- ever admirable, is ever likely to free us from the necessity of the employment of individual skill and tact in instruction. The dominant element here is, and has always been, the per- sonality, humanity, and range of scholarship and sympathy of the teacher. Mathematics was taught as well one hundred and fifty years ago as it is today, and it was taught as poorly one hundred and fifty years ago as it is today. Wherever there exist in 18 the teacher's mind rich associations, a broad range of interests and didactic powers, there good teaching a lways exists. OLD AND NEW TEXT-BOOKS In this respect, and in respect to text-books, no other do- main can admit of comparison with mathematics. The bio- logical and descriptive sciences, chemistry and even physics are of relatively modern development, and the present text- books, syllabi, and schemes of instruction in these sciences are enormously superior to the books and schemes of twenty- five, fifty or seventy-five years ago, even where these exist; but the mathematics which is now taught in our secondary schools existed, part of it in its fullest development, as geo- metry, 2100 years ago, and part of it, in its algebraic devel- opment, two hundred years ago. In the years between 1700 and 1800 there were written text-books of algebra ^ in which our present high-school scholars would find as much satisfac- tion and from which they would derive as much profit by individual study as from some of the books published within the last ten years. It is a commonplace of educational history that geometry, as taught in our schools, is in its didactic form essentially the geometry of Euclid's Elements; and I make bold to say that, apart from the supplying of practical mo- tives in teaching, the only advance which has been made in the didactic presentation of the principles of geometry in 2000 years is, first^ the reduction of the verbiage of the old ele- ments and the economizing, by the use of concise language and shorthand symbols, of the linguistic form in which the proposi- tions were presented and proved ; and, secondly, the introduc- tion into the books of practical pedagogical, typographical and pictorial devices which have heightened the economy of sensual presentation and removed all physical sense-obstacles to the comprehension of the geometrical relations, an enormous advance in itself but one of which we should thoroughly understand the scope and which we should not overrate, recognizing that it is an advance which we owe nearly as much to the development of the arts of the printer and the illus- trator as we do to the ingenuity of the mathematical author. 1 By Clairaut, Euler, Lagrange, Laplace, et ceteris. 19 Our science, at least so far as we are concerned with it as secondary teachers, is not a new and growing science, but re- ceived its full development centuries ago; so much so, that one of the greatest mathematicians of the 18th century, La- grange, conscious of the exhaustion to which he was carrying his mathematical analysis, once remarked that it would not be many years before, for the purpose at least of pure inquiry and the discovery of new truth along the old lines, professors of mathematics would be as rare at the universities as pro- fessors of Arabic.^ This fortunate fact should be remembered in all our discussions. No mathematical author is now per- plexed with doubts as to the new subject-matter which he should introduce into his text-book as is the physicist for ex- ample when he comes to write about ions. This being the fact, our task, in my opinion, in both selecting and presenting the mathematical subject-matter in text-books is limited almost wholly to removing the sensual physical obstacles to the intel- lectual comprehension of geometrical and algehraic truths. It is a question of pedagogic economy. And how magnificently this is being done by color devices, photographs, models, sys- tematic and universal economic notations and designations in both our geometry and algebra text-books is known to every one who has seen some of the recent beautiful productions of our American text-book press. HISTORY OP THE PRACTICAL MOVEMENT This beauty, precision and economy of form which our mod- ern text-books are taking on, and this supplying of practical motives, have been the outgrowth of a long development, which may be unknown to some of you and of which the recent con- crete practical methods of teaching mathematics are themselves the outcome. The movement is not so new as one would infer from the prefaces to the American text-books of the last fif- teen years. The French Revolution gave the first universal impulse to the change, and in the Normal and Polytechnic 1 For a refreshing, but drastic and warped, characterization of the history of mathematical instruction and text-books see Eugen Diihring's masterful "Geschichte der Mechanik,'^ Leipsic, Fues's Verlag. Diihr- ing's work is little known in this country. 20 schools of Paris over one hundred years ago graphs, for ex- ample, were used by the greatest mathematicians of Europe to illustrate the principles of algebra although one would think from some recent writers that this device had been the pro- duct, if not of their own brains, at least of very recent years. But the great educational reformers of Germany hastened this process in the elementary schools. Trendelnburg was prominent in attacking the blind deductive form in which the Euclidean geometry was taught in the schools and to ask for a more concrete treatment. Schopenhauer's works are full of onslaughts on the sterile, formal, deductive methods of teach- ing geometry, among which the one making fun of Euclid's proof of the Pythagorean proposition, which Schopenhauer called the " mouse-trap proof," is the most classical. Study- ing geometry by Euclid's method, Schopenhauer said, " was like cutting off one 's legs in order to walk on crutches. ' ' ^ But it was the influence of Herbart that did most toward giving geometrical teaching its right bearings and toward starting the movement which has resulted in the methods of such men as Perry and of scores of other equally effective teachers and schools in our own country whose names do not happen to be known. ' ' This is the ideal that seeks simply to instill into the mind fruitful ideas, ideas which find a con- genial soil in the student's existing knowledge and interests, and which from a firm belief in the fertility of ideas has em- phasized the value of geometrical knowledge as opposed to the study of logical form. Let the boy, ' ' it says,^ ' * be thoroughly at home with a new fact or property before he begins to apply formal logic to it. To attain this familiarity, do not reject at any stage the help of [physics, mechanics and] experiment, and the recourse to common objects and experience. Geomet- rical experiment may use models, frameworks, machines; but there is a limit to the amount of apparatus that is convenient. We rely, therefore, in the main, upon figures : freehand sketch- 1 ' * W. als W. u. V. ^ ' I. 1. 15. Eousseau, ' * Confessions, ' ' has a similar remark. 2 Abstracted from a paper in the ' ' Mathematical Gazette, ' ' London, March, 1910. This excellent little magazine should be accessible to all secondary mathematics teachers. 21 es, where a sketch will reveal the fact that we are looking for ; accurate figures, where eye and hand alone are not clever enough. Hence the amalgamation of geometrical drawing with geometrical theory, subjects once divorced, to the great loss of both." THE HUMAN BRAIN ITSELF A MATHEMATICAL LABORATORY I will stop here a moment to consider how far experiment, physical illustrations, and models may be carried in the teach- ing of geometry. This tendency has been greatly exaggerated and overdone in some of our recent literature. I have seen in recent current American educational magazines several sug- gestions for the physical illustration of theorems of algebra and geometry which remind one of the proverbial procedure of using a steam-hammer to crack a nut, procedures which are altogether unnecessary and a sheer waste of time. Practical and laboratory methods in mathematics and mechanics have their limited sphere of application ; for it should be considered in this connection, and with special regard to our own science, that the human brain for our purpose is itself a store-house and laboratory of formal thought, where nearly all the ex- periences are collected that are needed for experiments in al- gebra and geometry, and which can be here conducted with far more dispatch and ease than in a special laboratory with levers and scale-pans. The human body and brain are a micro- cosm, a compact bundle of well-ordered space and mechanical experiences, which are always ready at hand. The chalk and the blackboard in our science are mightier than all the tin-can junketry of the physical laboratory. METHODIC AND GENETIC METHODS Several attempts also were made in the early part and mid- dle of the 19th century to write treatises in which the subject- matter and contents of geometry, entirely apart from its form, should be developed genetically, as would mechanics or elec- tricity, by the use of the principles of motion and continuity, and which discarded entirely the old arid and sterile deduct- ive form. But let me remark, parenthetically, that the re- jection of the old deductive form of presentation is not neces- sarily a rejection of the deductive form of reasoning. We are 22 not losing the value of the training in formal discipline when we study the contents of mathematics, purely for their con- tents ' sake. This is what I mean when I say that the two ideals which most people seem to separate in recommending the study of mathematics, namely, the formal ideal and the utilitarian ideal, are not necessarily contradictory and not necessarily exclusive. Some of these genetic attempts ^ have not yet been fully exploited in our American secondary text- books, which still halt between the two ideals, and, like the mediaeval donkey between the two bales of hay, find intellectual starvation while hesitating to decide which sustenance to take. But, rejecting both the formal and the genetic methods, and almost spurning mathematics as an independent science, the Perry movement takes the ultra-utilitarian bull by the horns, and seeks to convert mathematics into a mere auxiliary en- gine of physical and mensurational practice. This ideal is that the boy should possess such a ' ' command of mathematical methods that he can apply them in any new problems with ease and certainty. There should be no separate examinations in geometry, algebra, trigonometry, and mechanics. They are all one subject.^ Geometry should not be a study of logic, but should be a study of mathematical matter and method with experiment and common-sense reasoning. All parts of the elementary mathematics ought to be taught along with and through science and by the same master. [A rather difficult requirement for American schools.] Common-sense explana- tion, accompanying experiiiiBnt, should be the procedure. Our great difficulty is our being out of touch with our students. We must bring the teachers to see down to the level of the pupils and to think what the world needs. The traditional mode of teaching mathematics does not provide adequate mo- tives for the work from the point of view of the boy," we read,^ and the Perry reform is to take measures that such motives shall be supplied. Mathematics becomes simply an 1 For example, that of Snell, in a now almost unknown book, ' ' Lehr- buch der Geometric, " Leipsic, 1869. 2 Compare, as a variant of this plan, books like Holtzmiiller 's ' ' Meth- odisches Lehrbueh der Elementar-Mathematik, " Leipsic, Teubner. 3 *' Mathematical Gazette,'^ March, 1910. 23 auxiliary instrument for dealing with problems of a mechan- ical, physical, and mensurational character. Formal mathe- matical technique is a secondary consideration. The successful use of the instrument, no matter how clumsy the use, is the main end. I am tempted here to present some considerations on the relative value of the Perry method of teaching mathematics and the so-called '' methodic " methods of such foreign writ- ers as Holtzmiiller, as also of the pure ' ' genetic" method, where the entire subject-matter of mathematics is treated as an organ- ic body of knowledge which receives didactic presentation not in the separate, sundered form of arithmetic, algebra, geo- metry, trigonometry, etc., but as a unitary system of con- solidated facts, concepts, and ideas. This last method of pre- sentation, as supplemented by the Perry procedures, is com- ing generally to the fore in our text-book literature and un- doubtedly constitutes the ultimate goal toward which math- ematical didactics strive. But I can do nothing further here than to supply intimations. We are concerned merely with the philosophy and history which enter into these various con- ceptions, and I shall conclude merely with some remarks on how they have arisen. SUMMARY All the varying arguments, all the contradictory points of view above intimated, all this emphasis now of one side and now of the other side of the value of mathematical studies, are due to a lack of historical and philosophical comprehension. In its origin, mathematics was developed largely as an instru- ment for physical or mensurational research. After it was developed, and the logical, genetic interdependence of its truths and results was discovered, it became naturally an ob- ject of independent development in itself, and assumed neces- sarily an artificial, logical and purely conceptual character. It created a professional caste, like the grammarians of India, which cultivated it for its own sake ; it became an end in itself ; it was systematized, its principles economized and minimized, and its entire body of truth transformed and etherealized into a shadowy framework of pure logical forms; it lost its 24 foothold in reality and in the empirical soil from which it had sprung, became barren and sterile ; it ceased to produce fruit- ful and creative investigators, and was ultimately rejected in this form by practical minds. Hence the contradictory opin- ions and utterances on the value of the two phases of its de- velopment as an educational discipline ; and hence the varying hyper-emphasis that the one or the other aspect of it receives, according to the changing needs and ideals of each historical epoch. The truth, to which we must hold, lies in neither; it must be sought in an amalgamation of the two: mathematics arose from a physical, empirical soil ; it reverts in its applica- tions to that soil ; but in the transition it has passed through a purely intellectual domain, in which it has suffered a trans- formation that is of its very essence. And the study of its subject-matter, its logical and genetic development in this transformation is just as important an educational discipline for the intellectual needs of life as the study of its applications is for the material needs of life. And the former objects are, I think, more easily and more frequently obtained in practice than the latter. Two antithetic quotations in closing : Hankel, a Tvell-known German historian of mathematics, says: " Mathematics will never find its adequate appreciation among the general pub- lic until more than its A B C's are taught in the schools, and until the unfortunate opinion has been removed that its sole object in instruction is to impart formal culture to the mind. Mathematics finds its goal and purpose in its contents: its form is a secondary consideration and not necessarily that which it has come to be historically from the fact that it first took fixed shape under the influence of the Grecian logic. There is no more reason for studying mathematics for its formal culture than there is for studying history to strengthen the memory. ^^ ^ This is the utterance of the extreme technical party. It is attractive and forcible, even necessary in a certain stage of the development of educational practice, but does not contain all the truth. W. Kingdon Clifford has expressed the opposite 1 Hermann Hankel, * * Die Entwickelung der Mathematik in den letzten Jahrhunderten, " Tiibingen, 1869. 25 view, which holds to the more formal teaching of mathematics as a branch of pure scientific culture. " It seems to me," he says, '* that the difference between scientific and merely technical thought .... is just this: Both of them make use of experience to direct human action ; but while technical thought or skill enables a man to deal with the same circumstances that he has met with before, scientific thought enables him to deal with different circumstances that he has never met with before. ' ' I will conclude with these two quotations, leaving, for your consideration simply, the plenitude of implications they in- volve. I have endeavored, rather by way of intimation than by detail, to present a few central thoughts that this vast sub- ject adumbrates, in the hope that their mere suggestion will lead to some light and clarification. In magnis voluisse sat est. 1 "Lectures and Essays," Vol. I, p. 128. 26 OVERDUE- _-^====^^ OCT 28 193? St? ^^ '''' JUL It 19*^ 30Sep'49AM m %^% flBt^ ,^^4 29 V341 UOAK DEPT- i.n "1 ..S,3-i QiA M 3