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IN MEMORIAM 
 FLOR1AN CAJORI 
 
JPROBLEMS IN 
 
 THE TEACHING OF 
 
 SECONDARY MATHEMATICS 
 
 AN ADDRESS DELIVERED BEFORE THE 
 
 NEW ENGLAND ASSOCIATION OF 
 
 TEACHERS OF MATHEMATICS 
 
 BY 
 
 DAVID EUGENE SMITH 
 
 GINN AND COMPANY 
 
 BOSTON NEW YORK CHICAGO LONDON 
 
 COPYRIGHT. 1913, BY GINN AND COMPANY 
 
CAJORI 
 
$41 
 
 PROBLEMS IN THE TEACHING OF 
 SECONDARY MATHEMATICS 
 
 It is not without considerable hesitancy that I venture 
 to address a body of teachers upon some of the great prob- 
 lems that confront us at the present time in the work in 
 secondary mathematics. This hesitancy arises from several 
 causes, prominent among them being the feeling that I 
 shall only be " carrying coals to Newcastle." For surely 
 these problems are already in your minds, and many of you 
 have pondered over their significance and their solution 
 quite as seriously as I, and no doubt with a more satisfactory 
 issue. I hesitate, also, because I can merely state them with 
 no attempt at solution, mindful all the time of the ancient 
 adage referring to questions which a wise man cannot 
 answer. But after all, there is a value in clearly stating 
 from time to time the large questions that confront our 
 guild, for if problems were never formulated they would 
 never be solved, and it is upon associations like this that 
 we must largely depend for the solution of those that I 
 shall venture to lay before you. 
 
 I. HAS THE DAY OF SECONDARY MATHEMATICS 
 PASSED? 
 
 The first of the great questions that confront us at the 
 present time relates to the very existence of secondary 
 mathematics in our curriculum. To many of us it may 
 
2 PROBLEMS IN TEACHING 
 
 seem preposterous that the question should seriously be 
 asked. We say to ourselves that if anything is to be blotted 
 out, let it be some language or one of the various manual 
 arts that are from time to time exploited only to find, in 
 most cases, an early resting place in the great educational 
 necropolis of forgotten graves. 
 
 But the question cannot be dismissed in any such way 
 as this. Other subjects have been seemingly as well in- 
 trenched as mathematics, and yet they have passed away. 
 Formal logic was at one time one of the great features of 
 a liberal education ; it gave place, in the secondary school, 
 to formal grammar ; but a university course in formal logic 
 is now a rarity, and a high-school course in formal gram- 
 mar, as it was conceived of a few years ago, is almost 
 unknown. The world seems to proceed as well without 
 these subjects as it did when they held prominent place, 
 and we have to face the question whether it would not 
 get along just as well if algebra and geometry followed 
 them into educational oblivion. The medieval Compotus 
 was once an essential feature in the education of a learned 
 man ; it was apparently intrenched in a position of secu- 
 rity ; and yet, as I mention it to-day, half of this audience 
 may be ignorant, and excusably so, of even the meaning 
 of the word. Somebody at some time asked the question, 
 "Why should an educated man need to study the Com- 
 potus ? " and the answer came in due time, " There is 
 no reason," and the subject was soon forgotten. Somebody 
 to-day raises the question, "Why should an educated man 
 need to study algebra ? " and we, the teachers of mathe- 
 matics, must answer. A high school in which I am inter- 
 ested holds its last class in Greek this year one of the 
 
SECONDARY MATHEMATICS 3 
 
 best-known high schools in the country, a school of five 
 hundred selected pupils, all of whom are hoping to enter 
 college. Shall it be, a few years hence, that this same 
 school shall be teaching its last class in geometry, com- 
 pelled to drop another ancient and honored subject because 
 it is no longer demanded ? 
 
 We may say to ourselves that high-school mathematics 
 has always existed, that it is not a college subject, and that 
 it is absurd to talk of abolishing it. But if we say this we 
 forget that the American high school is itself a new insti- 
 tution ; that it has no exact parallel in other countries ; 
 that other countries select their pupils for secondary work 
 while we seek to educate the mass ; that 95 per cent of 
 our high-school pupils do not go to college, and do not 
 hold the intellectual standards held by the boys of our old 
 academies ; and that it is comparatively a recent demand 
 of the American college that its candidates must offer 
 any mathematics beyond arithmetic. So when we speak of 
 high-school mathematics we should bear in mind that our 
 high school has not as yet proved its worth, and is even 
 now being weighed in the balance with rather unsatisfac- 
 tory results, and that precollegiate mathematics is only a 
 recent matter. If a pupil postpones his Greek until he 
 enters college to-day, why should he not postpone his 
 Latin, his algebra, and his geometry to-morrow ? 
 
 Now this is not a cry of alarm for the sake of temporary 
 effect ; it is a succinct statement of the arguments that we 
 frequently hear to-day from the general educator. Up and 
 down this country, before many gatherings of teachers, the 
 question is being vociferously propounded, " Why should 
 the girl ever study algebra ? " Even in associations of 
 
4 PROBLEMS IN TEACHING 
 
 teachers of mathematics the question is being asked, " Who 
 would stand to-day for the spirit of Euclidean geometry? " 
 And the men who ask these questions hold prominent 
 positions ; they are professors in universities, educators of 
 influence, men whom the mass of teachers naturally look 
 to as leaders. The problem is, therefore, a real one and 
 one that we have to face. 
 
 But we must not deceive ourselves by thinking that 
 we can successfully meet it by mere opprobrium. We not 
 infrequently hear it asserted that the general educator is 
 usually a man of a low degree of scholarship, that his range 
 of culture is limited, that he was taught Latin so poorly 
 that he believes it can never be taught in any other way, 
 and that he rarely stands for any intellectual ideals ; but 
 we must remember that, even if the assertion has some 
 truth, enough people of this type might easily create a 
 Zeitgeist that would not down by any such formula as 
 Weave a circle round him thrice." These men who 
 attack the ancient culture have been rather recklessly called 
 educational muckrakers, men who seek only the bad and 
 judge everything by that ; they have been hysterically de- 
 nominated pedagogical anarchists, men who destroy with- 
 out rebuilding ; and they have been frequently looked upon 
 as intellectual iconoclasts, those who in their zeal to destroy 
 idols are willing that all the beauties of art, which their 
 dull vision fails to see, should go to the scrap heap. In all 
 these assertions there may be some grain of truth, but no 
 one gains anything in an argument by giving expression 
 to such an attitude of mind. Such an expression merely 
 brings the countercharge that those who hold other views 
 are reactionaries, laudatorcs temporis acti, unprogressive, 
 
SECONDARY MATHEMATICS 5 
 
 and selfish clingers to their little jobs. The epithet of 
 "old fogy" is as weighty as that of " muckraker." 
 
 When we calmly consider the question, we find that it 
 relates to the value of algebra and geometry for the democ- 
 racy that America, in distinction from the rest of the 
 world, is trying to educate in the high school. What does 
 democracy want of mathematics ? And in our America of 
 the dollar we find that the question is often reduced to that 
 of the immediate utility of algebra and geometry. The 
 potential utility does not seem to enter into the considera- 
 tion of the type of reformer that seems to speak the loudest 
 upon the subject. And here appears to be the real point 
 at issue : one side demands the immediately useful, while 
 the other stands for that which it claims to be potentially 
 so. Can we, therefore, justify our secondary mathematics 
 on the potential side ? For surely no one would for a 
 moment claim that the teaching of the immediately practi- 
 cal part of algebra to a mechanic would require more than 
 a month, or that the immediately practical part of demon- 
 strative geometry exists at all, taking these words in their 
 usual popular significance. Such, then, is the first of the 
 large problems that seem to loom up before us. Call it 
 a claim for mental discipline if you please this is a mere 
 question of fashionable or unfashionable phraseology ; it 
 is a claim for serious attention to a vital issue in education. 
 
 II. WHY SHOULD NOT ALL MATHEMATICS BE 
 ELECTIVE? 
 
 The general educator is usually found to concede that 
 mathematics should be taught in our high schools, but he 
 is frequently heard to assert that it should be elective. 
 
6 PROBLEMS IN TEACHING 
 
 Many teachers of mathematics, perhaps most of them, 
 would personally welcome such a change, since the pleas- 
 ure of teaching is largely increased if the learner takes the 
 subject con amore. But of late a new type of educator has 
 appeared, the one who proposes to weigh in psychological 
 scale the intellect of youth and to guide it aright. You 
 have it in your own part of the country to-day in the phase 
 of vocational guidance, and in this work so many excellent 
 people are seriously engaged that we are certain to see it 
 become an important phase of modern education. Let the 
 boy who gives promise in science begin his specialization 
 early, say those who seek to guide the youth in a scientific 
 manner, and let the one who takes to Latin bend his 
 energies there. Let there be scientific tests to show whether 
 or not the particular individual can hope for success in the 
 particular vocation a worthy effort and one that will 
 produce good results. But there are not wanting those 
 who will be less scientific, and who will assert that one 
 who, by virtue of his surroundings and family, is destined 
 to be a hewer of wood, should early come to like to hew, 
 and should be taught chiefly the nobility of labor with the 
 hand. That we may realize some of the dangers that beset 
 those who seek to guide the youth aright, and who may 
 feel called upon to sidetrack all that is not immediately 
 practical, let me tell you some advice that I myself have 
 given within a few years past in cases like these, and lay 
 before you the problem that I had to face. 
 
 Not long ago there came to me a father who wished to 
 train his boy for trade in a seaport town, and who asked 
 my advice as to the proper education to give him. The 
 problem seemed simple. The community was not an 
 
SECONDARY MATHEMATICS 7 
 
 educated one ; it lived off its little shipping industry ; the 
 boy was destined to small business and to small reward ; 
 he gave no promise of anything better, and the advice was, 
 therefore, unhesitatingly offered that the only mathematics 
 he needed was arithmetic through the sixth grade. 
 "Another parent asked me a little later about his son. 
 The boy was of the ordinary type and would probably 
 follow his father's occupation, that of a sculptor. What 
 mathematics would it be well for him to take ? I suggested 
 a little study of curves, some geometric drawing, and the 
 modeling of the common solids a bit of vocational 
 guidance that seemed to me then and seems even yet 
 particularly happy. 
 
 A third boy happened to be with me on a steamer and 
 I took some interest in talking with him and with his 
 mother. They lived in a city of no particular note, at any 
 rate at that time, and the boy was going into the selling 
 of oil within a few years. The profits of the Standard Oil 
 Company appealed to the family, and I advised him to 
 learn his arithmetic well and get into business as soon 
 as he could. 
 
 Out of the store of my memory I recall a curious lad 
 whom I came to know through my sympathy with the 
 family. The mother was a poor woman, and she took the 
 boy, when little more than a baby, over to Riverside Park 
 one day when there was a naval parade. A drunken sailor, 
 having had a fight with a group of hoodlums, rushed 
 through the crowd of spectators and slashed right and 
 left with a knife. In the excitement the boy, in his mother's 
 arms, was horribly cut in the face. When I knew them 
 he was about ten years old, unable to speak plainly, and 
 
PROBLEMS IN TEACHING 
 
 already a misanthrope through his affliction. I advised the 
 mother to give the boy a vocational education, telling her 
 that through the use of the hands he would satisfy his 
 desire for motor activity, and that this would compensate 
 him for the loss of verbal fluency and would tend to make 
 him more contented with his lot. In this advice I feel 
 that I would have the approval of educational circles. 
 
 And finally, out of this series of experiences, let me 
 recall the case of a boy whom I came to know through 
 a noble priest who found him one morning, an infant a 
 few days old, on the steps of his church. We talked over 
 the best thing to do for such a foundling, one who, at the 
 time I knew him, was in the primary grades. He showed 
 no great promise, he was without family recognition, and 
 his only chance, apparently, was in the humbler walks of 
 life. I recommended a vocational school where he could 
 quickly prepare for the shop or the lower positions of 
 trade, and the good priest approved my plan at the time, 
 although he finally followed quite a different course. 
 
 It is apparent, however, that I have here spoken in 
 parables. Perhaps you already recognize the boys, and 
 perhaps you feel how sadly I blundered in my counsel. 
 For the first of these whose cases I have set before you 
 felt a surging of the soul a little later, and this was recog- 
 nized in time, and he became one of the Seven Wise Men 
 of Greece Thales the philosopher, he who introduced the 
 scientific study of geometry into Greece. The second felt 
 a similar struggle of the soul, and his parents recognized 
 my poor counsel in time to save him and to give to the 
 world the founder of its first university Pythagoras of 
 Samos. The third boy, for whom only .the path of commerce 
 
SECONDARY MATHEMATICS 9 
 
 seemed open, and this in a town only just beginning to 
 be known, was the man who finally set the world's first 
 college-entrance examination, the one who wrote over the 
 portal of the grove of Academos the words, " Let no one 
 ignorant of geometry enter here " - Plato, the greatest 
 thinker of all antiquity. The fourth, the hopeless son of 
 poverty, maimed, sickly, with no chance beyond that of 
 laboring in the shop for such wage as might by good for- 
 tune fall to his lot, became the greatest mathematician 
 of his day always the stammerer (Tartaglia), but one 
 whom Italy has delighted to honor for more than three 
 centuries. And the last one of the list, the poor foundling 
 on the steps of St. Jean-le-Rond in Paris, became D'Alem- 
 bert, one of the greatest mathematicians that France, a 
 mother of mathematicians, ever produced. 
 
 Shall we, then, advocate the selection of those who are 
 to study mathematics and close the door to all the rest ? 
 Are we so wise that we can foresee the one who is to like 
 the subject, or succeed in it ? Have we so adjusted the 
 scales of psychology that we can weigh the creases in the 
 brain, or is there yet invented an X-ray that will reveal to 
 us the fashioning of the cells that make up its convolutions ? 
 
 Of course it will at once be said that these illustrations 
 that I have given are interesting, but that they are unfairly 
 selected ; that those boys gave earlier promise in mathe- 
 matics than I have said. It will be asserted that I should 
 have taken the case of the stupid boy, the one who did 
 not like school, the one who liked to play with little wind 
 wheels, who liked to fight, who actually did run away 
 from school, and who stood near the bottom of the class 
 in mathematics. Such a case would be a fair one, one in 
 
10 PROBLEMS IN TEACHING 
 
 which we could safely say that prescribed algebra and 
 geometry are out of place. And I suppose we must 
 agree to this and confess that the argument from the histor- 
 ical incidents that I have mentioned was unsound. Let us 
 rather take this case that I have just described, and let us 
 see to whom the description applies. I need hardly tell 
 you who this boy is ; he is well known to you ; he is well 
 known to the world ; and long after every educational re- 
 former has passed into oblivion his name will stand forth as 
 one of England's greatest treasures, for it is the name of 
 Sir Isaac Newton. 
 
 But again I have been unfair, perhaps. I should have 
 taken positively hopeless cases, for such can surely be found. 
 I should have taken some illiterate man, one who does not 
 learn to read until he is nearly out of his teens, or else 
 some man who shows no promise in mathematics by the 
 time he reaches manhood, or some one who by the time he 
 is thirty is to show no aptitude in the science. It is so easy 
 to theorize ! But let us have care, for the men whom I have 
 now described are Eisenstein, Boole, and Fermat. Take 
 them away and wnere is your theory of invariants, your 
 modern logic of mathematics, and the greatest genius in 
 theory of numbers that the world has ever seen ? 
 
 But I am wandering afield, and I fear I may be inter- 
 preted to question the modernizing of our educational work. 
 I thoroughly favor scientific, vocational guidance as under- 
 taken by a small number of our best scholars in the field. 
 With equal zeal do I favor industrial education when it is 
 not so narrow as to condemn a boy to some particular 
 groove in life, and I earnestly hope that we shall so guide 
 our youth that every boy and girl will leave school fitted to 
 
SECONDARY MATHEMATICS 1 1 
 
 do something well. I do not believe that any thoughtful 
 educator wishes to guide a pupil in a narrow path nor keep 
 from him the chance that the world owes him. It seems 
 right, however, to set the problem clearly before us : Can 
 we safely say that we may close the door of mathematics 
 to any boy ? Should he not be given the chance ? If he 
 fails, that ends it, but if he succeeds, the world is the winner 
 in the lottery. Of course this does not answer the question 
 as to what this chance should be ; it is quite possible that 
 it should not be our present algebra ; it is even possible 
 that it should be merely some form of mensuration that 
 masks under the time-honored name of geometry ; it may 
 conceivably be some emasculated form of fused mathematics 
 that has none of the logic of geometry and none of the 
 beauty of algebra, although I do not believe it ; and it may 
 even be some form of technical shop mathematics that 
 appeals to but few pupils because of its very technicalities. 
 This is the part of the problem to be solved. But that the 
 door of mathematics of some substantial character shall 
 not be opened, and opened after arithmetic has been laid 
 aside as the leading topic, seems unthinkable. 
 
 III. DOES THE GIRL NEED TO STUDY ALGEBRA? 
 
 A third question that seems at the present time to agitate 
 .the educational interests in some parts of the country relates 
 to the study of algebra by the girl. This carries with it 
 the corollary that no mathematics whatever, beyond mere 
 computation, is to be required of at least half the force 
 that controls the world. 
 
 How this meets the views of the emancipated woman I 
 do not know. I assume that she would say, if asked, that 
 
12 PROBLEMS IN TEACHING 
 
 if algebra and geometry are good for the boy, save in the 
 narrowest technical sense, they are good for the girl also. 
 I should think she would say that if mathematics is the 
 one subject that makes us understand our infinitesimal 
 nature in the infinite about us ; if it is the one science that 
 has had the most to do with banishing the superstition that 
 comes from simply looking at the heavens with lackluster 
 eye ; if it brings a mental uplift that no other science brings 
 and lets us see what seems the nearest to exact truth of 
 anything that we meet ; and if we find at every turn the 
 mathematical invariant, this timely symbol of the un- 
 changeable that presents itself just as the youth is passing 
 into manhood, I should think that she would say that 
 if mathematics brought these things in its train it is worth 
 while for the girl if it is worth while for the boy. 
 
 But to me there is a more serious side to the problem. 
 Mathematics, in one form or another, is going to continue 
 to be taught in the schools. It will gradually change to 
 meet the future demands of the times as it has always done 
 in the past. So far as we can see there will be a mathe- 
 matics that is immediately practical for those who are not 
 hoping for any intellectual leadership, and there will be the 
 mathematics that I have described as potentially useful for 
 those who are not content with remaining in the lower 
 intellectual class. But in any case there will be mathematics, 
 and the boy will study it. The question as it relates to the 
 girl is, as I have intimated, a more far-reaching one. When 
 this was an agricultural country the father directed the 
 education of the children. In the long winter evenings he 
 had the time and inclination to help those of his household 
 who at that season of the year were taking what was termed 
 
SECONDARY MATHEMATICS 13 
 
 their schooling. The mother had other duties that filled 
 her time, and, moreover, was not. herself well enough edu- 
 cated to give much assistance in the matter of study. But 
 America has changed. With our urban population at least, 
 the father no longer has control of the education of the 
 children. In our manufacturing centers he is busy in the 
 shop, and his hours are no longer limited by the light of 
 day. On the other hand, the urban mother no longer 
 weaves and spins, no longer helps in the fields, no longer 
 preserves fruit, and has almost forgotten how to make 
 bread. She has more time for the higher life, and it is 
 she, rather than the husband, who gives the direction^ the 
 help, and the encouragement in the education of the son 
 as well as the daughter. The father may need some algebra 
 in his trade, for if he reads the artisans' journals he must 
 know how to manipulate formulas, but "the mother must 
 have a general knowledge of the subjects that children study 
 if she is to be the sympathetic director of their intellectual 
 activities. It is the woman quite as much as the man who 
 needs a broad education at the present time. 
 
 When I hear some man who is skilled in dialectics 
 attack the teaching of algebra to the girl because it does 
 not enter into her immediate life, I am led to wonder if 
 he would have women learn anything whatever except the 
 work of the household and the club. The classics of the 
 world appeal to her immediate needs and interests quite 
 as little as mathematics, and hence these would be dropped. 
 Indeed, we hear not infrequently of some school principal 
 who proposes the epoch-making plan of studying literature 
 only from current magazines. If we agree to drop all noble 
 literature, all knowledge of history except what comes from 
 
14 PROBLEMS IN TEACHING 
 
 the newspapers, all study of good music and art except by 
 those who expect to be musicians or painters or designers, 
 all science except such as relates to the simple chemistry 
 of foods, all astronomy, all physics, everything except what 
 concerns the science of good physical living, then we may 
 let mathematics go with the rest. Perhaps this is not what 
 these agitators mean one never knows, for they them- 
 selves never seem to have any exact ideas upon the subject. 
 Theirs it seems to be to agitate, hut never to formulate, 
 and yet they serve a purpose in the economy of education as 
 a yeast microbe does in the economy of nature. I refer here 
 not to the scientific psychologist who is seriously working on 
 the great problem of education, but to the man of brilliant 
 speech who prostitutes his powers of persuasion to cast 
 doubt on whatever fails to appeal to his immediate fancy. 
 Hence the problem seems to me not so much to decide 
 whether or not the girl should study algebra, as to decide 
 how we shall so teach the subject to her that she will 
 know of its beauties, of its purposes, and of the feeling 
 of mastery that comes from its pursuit. Such a problem 
 may well occupy the attention of associations like the one 
 I have the honor to address. 
 
 IV. WHAT SHALL BE THE MATHEMATICS OF 
 OUR TECHNICAL SCHOOLS? 
 
 A fourth problem that is thrust upon us by modern con- 
 ditions relates to the mathematics of our technical schools. 
 While we do not hear such vociferous assertions as we did 
 a short time ago about the fact that "the doctrine of formal 
 discipline has been exploded," a resonant but rather 
 uncertain phrase that was fashionable for a time, it is 
 
SECONDARY MATHEMATICS 15 
 
 neverthless quite axiomatic that the mathematics of a 
 technical school should aim at something in addition to 
 general culture or power. In a school of mechanics we 
 need the mathematics of mechanics, and so for other 
 special fields. And yet, as we look over courses of study 
 for our agricultural schools, for example, we find nothing 
 but arithmetic taught in one college, while mathematics 
 through the calculus is taught in another ; and in the high 
 schools the confusion is equally apparent. And what is 
 true for agricultural schools is quite as true for other 
 technical institutions. We have not even begun to think 
 seriously about solving the problem for our high schools, 
 and the same thing can be said for our industrial schools of 
 a more elementary character. Mere technical mathematics 
 alone has never succeeded, and the nature of the general 
 mathematics that is best suited to develop the power to 
 handle the problems that confront the foreman who is 
 erecting a skyscraper has never yet been determined. 
 Here, then, is another serious problem that meets us when 
 we try to settle upon the best course in mathematics for 
 the technical schools of our country. These schools are 
 hardly started in America as yet, although in Europe they 
 are well established ; and if we may judge from world expe- 
 rience, their success is to depend in no small degree upon 
 the quality of mathematics that will enter into their curricula. 
 
 V. THE PROBLEM OF OUR BACKWARD 
 AMERICAN MATHEMATICS 
 
 There were in the mind of those who initiated the 
 work of the International Commission on the Teaching of 
 Mathematics three large problems. The first was that of 
 
16 PROBLEMS IN TEACHING 
 
 leading each nation to take stock of \t& nwn work in 
 presenting mathematics to its youth, that it might have 
 before it a kind of moving picture of its teaching, from 
 the kindergarten through the university. The second was 
 that of informing other nations of this work, to the end 
 that all might profit by the success and failure of each. 
 And the third problem was that of looking beyond the 
 confines of one's own land and seeing what the rest of the 
 world is doing. In the United States upward of a dozen 
 reports were issued, telling the story of our own work. 
 But there remains the third problem, that of looking abroad 
 and seeing wherein other nations are surpassing us, and 
 then of finding the causes and the remedy. This is the 
 problem of the immediate future, and it is proper on this 
 occasion to set forth one of its phases. 
 
 When we compare, year for year, the work in mathe- 
 matics here and abroad, we are struck by the fact that we in 
 the United States are not only not the leaders, but in nearly 
 every respect we are distinctly behind the other prominent 
 countries of the world. At the end of our seventh grade 
 we are about a year behind, and at the end of our twelfth 
 school year we are about two years behind, other great 
 educational nations in the teaching of mathematics. You 
 say this to a professional pedagogue of the general type 
 and he will make all sorts of apologies. He will say, "Oh, 
 this is America, and we have different problems," as if 
 that were any excuse. He may lay it to climatic conditions, 
 to the necessity for assimilating a million immigrants a 
 year, to the paucity of teachers in a new country, to the 
 brief tenure of office of women teachers and even of men, 
 to our shifting population, to our democracy of education, 
 
SECONDARY MATHEMATICS 17 
 
 to the greater breadth of our curriculum, or to any one 
 of dozens of other causes. But he is a rare educator 
 who will come out and assert that we have a soft peda- 
 gogy that often dominates our elementary school, a sweet 
 but mushy pedagogy that brings a maximum of temporary 
 pleasure with a minimum of intellectual attainment. Per- 
 haps the reason that he does not say this is that it is not 
 true. Perhaps we are wiser than our European neighbors 
 in selecting topics for our schools that are better adapted 
 to train to good citizenship. If so, the results certainly 
 do not show it as yet. And when one sees the vigor, 
 contentment, good spirit, and comradeship that are found 
 in this generation in so many of the schools that represent 
 the influence of Pestalozzi in certain of the countries 
 abroad, and then sees the intellectual progress that these 
 schools foster, he must question, if he is open-minded, 
 the wisdom of those who have directed the American 
 policy. We are behind ; bad as the European arithme- 
 tic is, ours is worse ; backward as the European boy may 
 be in his algebra, ours is more so ; faulty as may be his 
 attainment in geometry, that of our boy or girl is still 
 more so. I know of plenty of schools in which boys at 
 the end of the twelfth school year have a good working 
 knowledge of trigonometry, a fair command of the basal 
 principles of analytics, and enough ability in the calculus 
 to meet the demands of a pretty good course in analytic 
 mechanics, but they are not American schools. I join 
 you in excusing ourselves ; I know the standard explana- 
 tions by heart ; I am even willing to join with every 
 American, of the type satirized by our foreign friends, 
 in asserting our claim to having the longest river, the 
 
1 8 PROBLEMS IN TEACHING 
 
 biggest lake, the tallest skyscraper, the wealthiest men, and 
 the most abject poverty to be found anywhere. But after 
 all this boasting is over, I have to confess to myself that, 
 in the teaching of the science in which this association 
 
 and I are interested, America is behind, definitely and un- 
 questionably behind, and I seek not for a dozen causes 
 so much as for one good remedy. 
 
 When we attempt to free ourselves from our insular 
 habit, and turn our attention to learning from the' expe- 
 rience of the rest of the world instead of from the theories 
 of the lecture room, we find that the work of the first two 
 years is generally more definite in other countries than in 
 ours. The idea that arithmetic shall be merely incidental 
 in these years is not held abroad, and, indeed, has little 
 scientific standing even here, although it has numerous 
 advocates. Sixj^ears are elsewhere generally deemed suf- 
 ficient to cover the essentials of arithmetic, the subject 
 thereafter being reviewed and applied along with algebra 
 and geometry. Instead of beginning formal algebra as a 
 new topic in the ninth school year as with us, the subject 
 is introduced by easy steps in the sixth and seventh grades, 
 so that the pupil is initiated by slow degrees into the ad- 
 vanced stage. Instead of withholding, geometry until the 
 tenth school year, and then suddenly springing the demon- 
 strative "pKase upon the pupil, this subject is introduced 
 along with algebra in the elementary grades. In other 
 words, instead of devoting the seventh and eighth grades 
 to a business arithmetic that is often too difficult for the 
 children, a simple initiation is given into the algebra of 
 the formula and equation, into geometric drawing and the 
 simpler demonstrations, and into such higher arithmetic as 
 
SECONDARY MATHEMATICS 19 
 
 is within the grasp of the pupils of these grades. There 
 is thus an intelligent preparation for formal algebra and 
 demonstrative geometry that is quite lacking with us. 
 Moreover, in the eleventh and twelfth school years there 
 are often opportunities to take courses in trigonometry, 
 analytics, the calculus, and mechanics, and sometimes 
 mathematical astronomy, that are almost never found in 
 our schools, and that have proved their worth by the 
 results attained. 
 
 It would be a sad error if we should conclude from such 
 a statement that our work is all bad and the foreign work 
 all good. The human tendencyjthat Horace satirized, of _. 
 looking on our neighbor's possessions as better than our 
 own, must always be recognized. The fact is that we have 
 "IrT America an excellent course in algebra and geometry, 
 better in some respects than those found abroad. Our 
 arithmetic work, too, has many features of superiority. But 
 our deficiency seems to lie in three features : our dawdling 
 over early primary arithmetic, our neglect of the initial 
 stage of algebra and geometry in the seventh and eighth 
 grades, and our failure to offer advancedjel_ectiyes in the . 
 last two years of our high school. 
 
 I am well aware of the difficulties to be met in applying 
 the remedy, and it is unnecessary to dwell upon them here. 
 For example, I know the gap between democracy and aris- 
 tocracy in education, but I don't believe it to be as wide as 
 many people think. I wish to set the problem before you 
 rather than to attempt to point out the slow steps by which 
 the solution may be effected. If an ideal is kept before our 
 people we will all gradually move toward it, and this gradual 
 trend is wiser and safer than any attempt suddenly to attain 
 
20 PROBLEMS IN TEACHING 
 
 results that seem to us desirable. The problem is to preserve 
 the serious, orderly mathematics that we have, while adopting 
 such good features as the rest of the world may suggest to us. 
 
 VI. THE QUESTION OF PARALLELISM 
 
 A glance at the foreign schools, such as has been given, 
 suggests another problem that we are reasonably certain to 
 meet in the near future the one of rearranging our cur- 
 riculum. America is one of the few countries in which 
 algebra and geometry are commonly taught in tandem 
 fashion. The required work in most of our high schools 
 is one year of algebra followed by one year of geometry, 
 and this followed by an elective course in algebra, and this 
 by one in solid geometry. The rest of the world in general 
 does not pursue such a plan. It carries its algebra and 
 geometry separately, as we do, but "It carries them side by 
 side, say from the seventh grade through our high-school 
 ^period. _lt may definitely assign two days of the week to 
 algebra, two to geometry, and one to arithmetic ; or it may 
 make some other arrangement, but in any case the arrange- 
 ment is based upon the idea of parallelism. I do not believe 
 that America is ready for this at the present time. Indeed, 
 I think that it never will be ready unless it adopts the plan 
 of a six-year high school, beginning with the seventh grade, 
 or goes even farther in following the European arrangement. 
 But there are many arguments in favor of the scheme in 
 case such an administrative change is made. At any rate, 
 the plan succeeds everywhere else, and because it does not 
 succeed under our present conditions is no reason for believ- 
 ing that we shall not have to face the problem in the chang- 
 ing conditions that are likely to be met in the near future. 
 
SECONDARY MATHEMATICS 21 
 
 VII. ARE WE MAKING MATHEMATICS 
 INTERESTING? 
 
 Among all of the experiments that have been made in 
 teaching mathematics in this country, not much that is 
 strikingly new has been evolved, nor much that seems 
 permanent. We have had in general education a great 
 many bubbles to prick, as witness the failure of the type 
 of manual training that was a few years ago asserted to be 
 a panacea, the fading away of the doctrines of concentra- 
 tion and correlation, the semioblivion into which the cul- 
 ture epoch theory has passed, and the fate of the Grube 
 method. Likewise in the teaching of mathematics we have 
 our bubbles, bubbles blown with the enthusiasm of youth 
 and pricked with the experience of years. Sometimes the 
 bubble is a geometry syllabus, sometimes an impossible 
 fusion of mathematical topics as far apart _as_ Latin and~ 
 chemistry, sometimes a "ratio method," sometimes a wild 
 adoption of the graph, and sometimes a form of practical 
 mathematics that is so technical that it repels pupils and 
 teachers alike. And hence it is not without hesitancy that 
 I suggest what very likely may develop into another bubble 
 the problem of making mathematics more interesting in 
 and for itself. 
 
 Although no two sets of educational statistics ever seem 
 to prove exactly the same thing, all statistics that touch 
 upon the subject tend to show that mathematics ranks well 
 up in the scale of pupils' interests. The latest set that I 
 have happened to see placed it third or fourth from the 
 top on a scale of twelve, with the vocational training, that 
 was to have been our salvation, down at the bottom of the 
 whole list. However taught, mathematics always has in it 
 
22 PROBLEMS IN TEACHING 
 
 the game element. You play the game, and you win if you 
 really set about to do so, and when you win you have a 
 definite result. We start, then, in the teaching of mathe- 
 matics, with this great advantage. 
 
 But the question arises, Are we making enough of this 
 matter of interest ? If we give a couple of pages of dull, 
 dry, algebraic formalism, with nothing contributed by the 
 teacher to enliven it, are we doing our best ? If not, what 
 more can we do ? 
 
 The problem is not one to be solved in a moment. But 
 let me suggest that the history of mathematics is not being 
 used as wisely as it might be in our classes. I do not think 
 it is presented to best advantage in the form of discjorb_ 
 nected notes in a textbook, but as outside material to be 
 brought into class, to be the subject of a moment's inspir- 
 ing talk by the teacher, it has unquestionable value. And 
 so it is with the recreations of mathematics, a subject on 
 which we have considerable available literature. Are we 
 using this material as we should ? Is it feasible in our 
 schools to establish mathematical clubs, such as Mr. New- 
 hall describes in his" monograph in the Commission report 
 on mathematics in the secondary schools ? I have myself 
 ventured to suggest in a recent number of the Teachers 
 College Record, under the title " Number Games and 
 Number Rhymes," a few possibilities, and to incorporate 
 some material on the subject in my work on the Teaching 
 of Arithmetic. Certain it is that there is an opportunity 
 for serious work here, and that some good may be accom- 
 plished if we do not go to the extreme of making a mere 
 bubble out of the effort. There would not be so much 
 heard about the immediately practical in mathematics if we 
 
 I 
 
 ,- 
 
SECONDARY MATHEMATICS 23 
 
 would show our pupils the interest in the subject perse, 
 and the meaning of the science in the larger life about us. 
 I wonder if we ourselves ever stop to think of the effect 
 of blotting out of existence every book or manuscript, say 
 even on so important a subject as the general science of 
 education, and also every book or manuscript on mathe- 
 matics. In the former case the schools would open next 
 week as they opened this ; the^teaching would go on as 
 before ; the world would know no difference save in a 
 few institutions for the training of teachers I and even they 
 might conceivably be the betteroff . But in the case"~~tjf- 
 mathematics every great engineering project in the world 
 would be stayed ; the skyscraper would not be planned ; 
 the next_ocean leviathan of steel could not be begun ; the 
 banking of the country would halt ; all safe navigation 
 would cease ; the science of artillery would need to be 
 begun anew ; astronomy would stand aghast ; mechanics 
 would have again to frame its laws ; and civilization would 
 send out the hurry call for intellects to repair the damage. 
 The question, then, as to what would happen if mathe- 
 matics were taken away, might well be suggested to a 
 mathematics club in a high school, and this is a type of 
 dozens of other questions that would probably add to the 
 interest that the pupils take in the study. Imagine the joy 
 of a healthy-minded pupil when he first reads " Flatland.", 
 or " Another World," or Hill's " Geometry and Faith," or 
 when he comes to know Ball's "Mathematical Recrea- 
 tions," or White's " Scrap Hook of Mathematics " ! 
 
 Sometimes, however, I feel that we get little thanks for 
 such an attempt. I know a certain prominent educator 
 who in private conversation is always sane and thoughtful, 
 
24 PROBLEMS IN TEACHING 
 
 but who now and then seems to fall into grave error in 
 his public utterance. He is reported as having recently re- 
 marked, in one of his somewhat ill-considered outbursts 
 against mathematics, that no teacher had ever yet told his 
 daughter why she studied the subject. Of course it is 
 quite probable that no one ever told her why she studied 
 anything. He admitted, however, that she was very fond 
 of the subject, but he raised the question as to whether 
 this is any justification for teaching it to her or to any one 
 else. I know of no one who would make any such claim, 
 but I am equally certain that it is an advantage to have 
 the subjects that we teach made as attractive as possible 
 without taking from them the sterner qualities that they 
 may possess. And so, at the risk of being opposed because 
 I take such a commonplace position, I suggest as one of 
 the problems of our time the effort to make mathematics 
 still more interesting, independent of its higher purposes 
 or its narrow field of immediate applications. 
 
 VIII. THE FUNCTION PROBLEM 
 
 It is perhaps well to inject into this discussion a problem 
 that is a little more mathematical than the ones that have 
 been suggested. For this purpose I select one of which a 
 great deal has been said in Europe during the past five or 
 ten years, and one of which we are soon to hear. I refer 
 to the introduction of the function concept early in our 
 work in mathematics, making of it a kind of unifying 
 principle of the elementary science. I have not the time 
 to refer to its recent history ; to the rise of the idea of 
 introducing the concept into elementary work, due largely 
 to the efforts of the French engineers ; to its elaboration 
 
SECONDARY MATHEMATICS 25 
 
 by men like Tannery ; to its cool initial reception by the 
 French teachers ; to the subsequent efforts of Klein, and 
 to its recent success in Germany. Suffice it to say that 
 there seems to be good reason for bringing before the 
 pupil, as soon as he begins the study of algebraic symbols, 
 the idea of function. This idea naturally enters into the 
 study of the simplest formula, it is the essence of the 
 equation, it is the major part of mensuration and trigonom- 
 etry, and in the mathematics that follows it plays a part 
 of ever-increasing importance. The question is, What shall 
 be its fate in America ? Shall we receive it with Western 
 enthusiasm, exploit it as we did graphs, go to an extreme 
 from which we must recede as we did in that case, and 
 finally, after much waste of energy, come down to a sane 
 use of this valuable aid to the study of mathematics ? Or 
 shall we first ask ourselves the question as to why we study 
 the subject, what we propose to accomplish by it, where 
 we can really use it to advantage, and what are the extremes 
 to be avoided ? If we take the latter course, we shall reach 
 the sane stage much more quickly than by the former 
 route. Already the danger has appeared, and so it is well 
 that we consider this as one of the problems demanding 
 our serious attention. 
 
 IX. THE PROBLEM OF ALGEBRA AND GEOMETRY 
 
 It would be too much to expect you to listen to any 
 extended statement of the problem of our present algebra 
 and geometry. That such a problem exists is patent to us 
 all. That conditions are not ideal we would all agree. It 
 is not the best type of education that begins the work in 
 algebra with mere formalism, nor the work in geometry 
 
26 PROBLEMS IN TEACHING 
 
 with a difficult demonstration. But while we agree to this 
 statement, and object to the wretched plan that was at one 
 time followed, quite as strongly as our colleagues in the 
 general field of education can possibly object to it, we 
 must recognize the fact that we are already improving the 
 situation very rapidly, quite as rapidly, indeed, as seems 
 safe. Practically all of our better algebras to-day have in- 
 troductory chapters that set forth the uses of the science 
 in a simple and interesting manner, and the problems that 
 they offer are becoming more real from year to year. 
 Our better textbooks in geometry no longer begin with a 
 proposition to be demonstrated, but, like the algebras, have 
 their introductory chapters that show the significance of 
 the subject. This steady progress is accompanied, to be 
 sure, by many amateurish efforts and experiments that are 
 foredoomed to failure, but the great body of common-sense 
 teachers is not disturbed by these eccentricities and is 
 steadily progressing to a better understanding of the prob- 
 lem and to its certain solution. 
 
 X. THE PROBLEM OF SCIENTIFIC EDUCATION 
 
 It would not be right to close without some reference 
 to a problem that has recently come to the front in educa- 
 tional circles, fostered to no small extent by the labors of 
 my distinguished colleague Professor Thorndike, and car- 
 ried on by a number of his former students. I refer to 
 the attempt to find scientific sanction for what we do or 
 think we should do in the' classroom". We proceed at 
 present as a result of what we believe to be the best expe- 
 rience of the race, and this in itself is a scientific method. 
 But may we not accelerate this experience and improve 
 
SECONDARY MATHEMATICS 27 
 
 more rapidly ? It is the belief of most educators of schol- 
 arly rank that it is possible to find out the causes of any 
 unnecessary retardation in the advance of the mass of 
 pupils, and we should certainly encourage all scientific 
 attempts in this direction so far as they relate to the work 
 in our field. If the trained investigator can show us where 
 we can best begin informal algebra, informal geometry, 
 and formal mathematics ; if he can find a general norm 
 for the drill work that shall give the best results ; if he 
 can tell us what we may safely demand of the average boy 
 and girl ; and if, through the examination of a sufficient 
 number of cases, he can standardize the work in mathe- 
 matics in our high schools, we shall all be his debtors. 
 To solve this problem he must take account of a large 
 number of factors, and his efforts will not meet with suc- 
 cess until these factors are all considered ; but meanwhile 
 we should recognize the problem and give our encourage- 
 ment to its scientific solution. 
 
 XI. THE PROBLEM OF OUR DUTY 
 
 Addresses like the present one are ephemeral ; they 
 reach no conclusion, and they are not likely to provoke 
 very serious thought. And yet they serve a purpose if they 
 select from the wide range of questions of the day a few 
 that seem to demand special attention. As I said at the 
 outset, I have no definite solutions of any of these prob- 
 lems, or rather I do not believe that, by laying before you 
 such as I have, so much will be gained as by stating the prob- 
 lems themselves. The answer to all such questions will fol- 
 low in the general agreement of the large, silent, thoughtful 
 body of American teachers. But it would not be right that I 
 
28 PROBLEMS IN TEACHING 
 
 should close merely with the enunciation of a few questions, 
 accompanied by remarks that are more or less rambling. 
 I therefore wish to suggest one more problem, that of our 
 present duty, and to indicate what seems to me its solution. 
 I believe it to be our duty to stand solidly against the 
 lowering of the standard of mathematics that shall make 
 of it only a science that js imjne^iaHy ingtpaH nf poten- 
 ^tiallyjpractical. I believe that we should open the door of 
 the great field of algebra and geometry to every boy and 
 every girl in our country. If they fail, let them substitute 
 some other science lor this, but offer them the chance. If 
 we can extend our high school downward to include the 
 seventh grade, the offer may be made in the seventh and 
 eighth grades, and this may be better than to require the 
 algebra and geometry of to-day. But to close the door of 
 opportunity in any one of the great branches of knowledge 
 is a crime, whether it be in biological science, in music, in 
 language, in mathematics, in history, or in any other field of 
 world importance. I also believe it to be our duty to favor 
 the extension of high-school mathematics downward to the 
 seventh grade at least, taught by the high-school teachers, 
 which teachers should in the near future be trained in 
 physics as well as our own science. With this should come 
 the extension of electives upward, to include courses in an- 
 alytics, calculus, and mechanics. To say that this is impos- 
 sible is to say that the American youth and the American 
 teacher lack the abilities of their foreign colleagues. 
 
 I believe it to be our duty to encourage in every way 
 a proper industrial education, but to insist that serious, 
 thoughtful mathematics shall have its place, and shall not 
 become merely a collection of a few rules of thumb. I feel 
 
SECONDARY MATHEMATICS 29 
 
 that we should guard against making the function concept 
 ridiculous as we, in some quarters, made the graph an 
 absurdity ; but that we should accept it for what it is really 
 worth and use it wherever in our teaching, to use the jargon 
 of the pedagogue, it "functions." I conceive, furthermore, 
 that an association like this is unanimous in the belief that 
 we should " hold fast to that which is good," not allowing 
 ourselves to give up the serious study of algebra because 
 it has not always been well taught in the past, nor the 
 serious study of demonstrative geometry because some one 
 has, for the millionth time, slaughtered Euclid. The pos- 
 sible improvements that are suggested by the problems that 
 I have ventured to lay before you may be effected with 
 only a natural and gradual change of the science as we 
 have it ; they demand no cataclysm, least of all the cata- 
 clysm that would follow if some of our reformers had their 
 way. When we see a man claiming that he speaks for the 
 95 per cent of high-school pupils in demanding the aboli- 
 tion of real mathematics, we will do well to listen to his 
 objections, trying to remove any just cause for his com- 
 plaints. But if we find that he is merely the advocate of 
 the nonintellectual, and that he would take from us all the 
 idealism that we have, then we have a right to put forward 
 the claims of the boy and girl who really wish to learn and 
 in whose souls idealism is beginning to take a start. But 
 even more than this, we have the duty to do our best to 
 bring this idealism into the schools for the 95 per cent of 
 whom we hear so much, and this I believe we shall do 
 effectively only when our high-school work begins earlier 
 than at present, and we put some introductory and practical 
 work in the seventh and eighth grades. 
 
30 PROBLEMS IN TEACHING 
 
 And above all, it seems to me to be our duty to stand 
 for the interest of mathematics for its own sake, for setting 
 forth its beauty of symmetry, for voicing its poetry, for 
 living its religion, and for exalting it because of the truth 
 that it sets forth so clearly and because of the invariant 
 properties that characterize it in every branch. So far as 
 possible, all this should grow out of the child's experiences 
 and needs ; not merely out of his external experiences and 
 needs in the narrow sense of the workshop, but also out of 
 those which are more vital and of which he is conscious 
 within. It is only by being imbued with such feelings and 
 ambitions that we can bring our pupils to love the subject 
 and to feel the great mental uplift that comes from its 
 study. May the statement of the few problems that I have 
 set before you assist us all to maintain this spirit in the 
 future as I am sure we have tried to maintain it in the past.