£)&?)$ Preprinted from the January, February, March and April numbers of Yol. XVIII, 1918, School Science and Mathematics THE STATUS OF MATHEMATICS IN SECONDARY SCHOOLS 1 . r By Alfred Davis, Francis W. Parker School, 330 Webster Ave., Chicago. Chairman of a Committee Appointed by the Mathematics Club of Chicago. ■ . One might judge by the criticisms that come from certain quarters that algebra and geometry are in a precarious situa¬ tion. Some would-be reformers would have us abandon the teaching of these subjects in high school in favor of what are said to be more useful subjects. Others would make them optional studies, or they would devitalize them by making them an easy and pleasant pastime for all who are required to take them. Still others would have us believe that the public is de¬ manding such changes as these, and that it is only a matter of time until algebra and geometry as cultural studies are relegated to the scrap heap. It Is a matter of considerable interest to know what persons outside the mathematical world, and who are qualified to speak of the subject, have to say as to the impor¬ tance of mathematics in our scheme of education. Our committee, representing the Chicago Mathematics Club, sent the following letter to prominent doctors, lawyers, mer¬ chants, bankers, etc., in the city of Chicago. We have avoided for the most part members of the teaching profession: “The undersigned committee of the Chicago Mathematics ^Club, in cooperation with a National Committee of the Mathe- L matical Association of America, is investigating the reasons for teaching mathematics in our secondary schools. We re- 2 spectfully submit the following questions to you in order that _ we may have the benefit of your practical experience. Further¬ ed more, your name will give to your answers a weight not attached to the opinions of persons less well known: “1. Was the study of high school mathematics worth while to you? If so, why? “2. Did its study contribute to your development anything which could not have been __ secured in equal degree from some other study which might have been substituted for it? “3. In employing a young person or in advising him to follow your profession, what impor- * tance would you attach to a good school record in mathematics? ^ ‘ ‘4. In planning the secondary education of your son or daughter, would you include algebra —and geometry? Why? “5. Is a knowledge of algebra or geometry of any practical value in your business? If so, - in what way? ( 3 — “6. Do you think algebra and geometry should be retained in our secondary schools? “Your answers to these questions will be of great value to those who are striving to secure for our schools the best possible 'This part of the report was read before the Club at its regular October meeting, October 6, 1917. 26 SCHOOL SCIENCE AND MATHEMATICS course of study, and your assistance will be greatly appreciated by our committee.” Fifty-five replies have been received as follows: Physicians and surgeons.. 7 Lawyers. 12 Bankers.. 7 Merchants. 4 Engineers... 1 Clergymen..... 9 Manufacturers.. 2 Authors and newspaper writers—. 2 Social workers.... 1 Business managers.. 1 Economist—.. 1 Real estate men.. 1 Broker___ 1 Typefounder_:. 1 Railroad men—. 2 Secretaries. 2 Printers.. 1 Answers to question 1: Affirmative... 43 Very emphatic—. 7 Reasons assigned: Mental training.... 27* Training in accuracy....—. 5 Training in concentration.. 3 Pleasure... 3 Practical value__ 2 Preparation for engineering—. 2 Preparation for physicians.. 3 Preparation for college... 3 Negative..1. 6 Reasons: Never used it.. 1 Of little value... 1 Answers to question 2: Affirmative.. 36 Very emphatic_ 9 Uncertain—. 8 Negative.... 6 (No reasons given, no comment.) Answers to question 3: Great importance.. 19 Banker. 3 Engineers. 1 Physicians. 2 Lawyers. 3 Merchants. 2 Clergymen.. 2 Brokers.. 1 Printers.. 1 Manufacturers.. 2 Railroad men_ 1 Considerable importance. 21 Bankers._ 3 Physicians. 2 Economists_ 1 Lawyers. 7 Clergymen.. 4 Railroad men.. 1 Real estate men.. 1 Secretaries. 1 Very little importance... 2 Lawyer and social worker. No importance.. 3 Physicians____....___ 2 Typefounder.. 1 Answers to question 4: Both.!_ 42 Algebra at least. 3 Neither.. 2 Only as a prerequisite for professonal work. 3 Answers to question 5: Affirmative_’.___23 Physicians........ 3 1, Biological chemistry, microscopic study. 1, Absolutely essential for science involved in medicine. 1, Necessary in vital statistics. Lawyers. 5 Engineers—. 1 Bankers.. 4 1, Investment mathematics. 1, Refers to the graph as important. Writers—. 1 Mentions its value in allusion. Ministers. 2 1, Logical help. 1, Argument and persuasion. Real estate man.. 1 Manufacturers.... 2 Economists.. 1 Statistics and city planning. Printer.___ 1 Railroad man.. 1 Negative.. 26 Bankers.. 3 Lawyers. 5 Clergymen...— 5 Merchants. 3 Physicians. 4 Social worker Broker Typefounder Railroad man. Answers to question 6: Affirmative... 46 Very emphatic.. 14 For professional preparation only—.. 2 Optional—. 2 Negative.. 2 The following additional information contained in the replies is of considerable interest: Mr. John G. Shedd, President, Marshall Field and Com¬ pany: “I firmly believe in the value of mental training derived from the study of mathematics, both the elementary and the higher branches. I know that training has been of especially great value to me, and the principles are constantly used in everyday business transactions. It seems to me that the aim of all branches of a school curriculum should be the development MATHEMATICS IN SECONDARY SCHOOLS 27 of analytical and logical thinking by the student. Nothing in my judgment tends more to this than thorough training in mental and written mathematics.” Mr. C. F. Hoerr, President, Home Bank and Trust Company: “I should certainly consider it a serious mistake if algebra and geometry were not a part of the education of our children. I should want them to have algebra and geometry for a number of reasons: “1. For the development of their reasoning and logical powers, and the concentration these subjects will demand. "2. In case the children should want to major in any of the sciences. “3. As a matter of general culture and training whether they desire to so major or not. “Statistics are coming into their own, and the question of graphs is very important. We are making use of graphs in our business, and I feel that through this medium of presenting statistics we are better posted in our business.” Mr. G. M. Reynolds, President of the Continental and Com¬ mercial National Bankj “In planning the education of a child of my own I should insist upon algebra and geometry; for while only a few cases might occur in which the business man would make practical use of either, the mental training acquired in gaining a knowledge of algebra and geometry is of very great help in directing modern business, which is constantly becoming more complex by reason of increasing burdens—a condition which calls for the highest degree of preparation that the schools can give.” Mr. James B. Forgan, Ex-President, First National Bank, answers the questions as follows: “1. Fifty years have passed since I studied mathematics in school. I have always thought that it, better than any other study developed my mind along the line of concentration of thought on problems confronting me. It' was a study I liked and in which I excelled. “2. Yes, the power of concentration, and I know of no other subject that if substituted for it, in my case, could have at¬ tained as good results. “3. I would regard a good school record in mathematics as indicating a mind trained to concentration of thought on prob¬ lems until they are solved. “4. Yes. “5. Arithmetic sufficient. “6. I do.” Rabbi Emil G. Hirsch answers as follows: “1. It was. I had, even as a lad, a knack for languages. Be- 28 SCHOOL SCIENCE AND MATHEMATICS fore I was twelve years of age I spoke four and read eight. Mathematics provided for me the balance wheel. “2. It did. It taught me to reason in abstract relations. “3. Soundness of judgment, I hold,is apt to be better devel¬ oped, and sequence of thought as well. “4. I should include both. Educationally both are effective. Practical advantages—perhaps the discovery of mathematical bent in him. “ 5 . Not directly but indirectly it is—reasons given above. “6. I do most emphatically. High schools are not factories turning out, one tracked minds.” Mr. H. H. Kennedy, lawyer, Moses, Rosenthal and Kennedy, answers as follows: “1. Unqualifiedly yes. It is my judgment that the study of mathematics very materially assists in developing the reasoning faculties of the mind, and powers of anaylsis. “ 2 . Yes, because I do not believe there is any study taught in the high school that would tend to develop in an equal degree the faculties of the mind referred to. Logic, in my judgment, is merely a higher form of mathematics; a person who is a good mathematician is invariably a good logician. “3. In advising a young man to enter upon the profession of the law, I would urge him to take up mathematics above all other subjects in the high school, since I believe that the primary essential to qualify a person to successfully practice law is the mental faculty of reasoning and the power of analysis. It also tends to develop the powers of abstract thought to a much higher degree than any other subject taught in the high school and especially the powers of concentration essential to success in any profession, and more particularly the law. Should a young man come to me with an exceptionally good record in mathemat cs, I should conclude therefrom that he had a well developed mind, and was especially fitted to practice law, for the ability to reason well and the power to concentrate are evi¬ dence of a good legal mind. “ 4 . Yes, I would most certainly include these two studies in planning for the secondary education of my son; indeed, I pursued this very course, and in connection with his studies at the University High School and Princeton University I urged upon him the importance of these subjects, and he pursued this course, his intention being to enter later upon a legal education. “5. A lawyer enjoys a particular advantage if he is well MATHEMATICS IN SECONDARY SCHOOLS 29 equipped in algebra and geometry; he finds himself brought into all forms of intellectual inquiry which require abstract thinking; familiarity with these subjects comes into play every single day; and the principles of geometry are the foundation of logic. “6. By all means. The science of algebra deals with repre¬ sentations of facts and permits the easier handling of them by the mind. A person well versed in algebra is especially qualified to detach himself from the facts of life in such a way as to enable him to reach conclusions free from bias or personal interest. It is unquestionably true that a lawyer who is a good mathema¬ tician is almost invariably a lawyer of ability, because he has a trained mind that enables him to stand upon his feet and reason well.” Thomas E. Donnelley, President, R. R. Donnelley and Sons Company, Director of the Chicago Directory Company, answers as follows: “1. Yes, it trained me to think along logical lines and it gave me a clear idea of the necessity of be ng 100 per cent accurate. “2. Yes, there is no study that could possibly take its place. “3. I would not employ him unless he was good at mathe¬ matics. “ 4 . Yes, both, and trigonometry as well. It is the science of facts as they are. Anyone who doesn’t understand these studies lacks an appreciation that things happen because they must. “ 5 . Yes, it gives one the ability to solve a problem. “6. Yes. Who on earth thinks otherwise? It is the basis of our whole industrial life.” Judge E. 0. Brown of the Appellate Court answers as follows: “1. It was. I cannot for myself conceive why mathematical truths (which seem to me to be the thoughts of God before time was) should not be worth any man’s study. “2. It did. “3. I should advise any young person to study mathematics and to obtain as good a school record as he could in them. “ 4 . I would—and did with my three sons and two daughters. Because I consider them very important branches of knowledge and very stimulating in intellectual pursuits. “5. I hardly know what meaning to attach to ‘practical.’ I should be less competent in every way without such knowledge. “6. I certainly do. If they are to be eliminated I do not know what should be retained.” 30 SCHOOL SCIENCE AND MATHEMATICS Dr. James B. Herrick, physician: “A progressive physician today must know physics, chemistry, physiology, physical chemistry, etc. Without the elements of mathematics such knowledge is impossible.” Mr. Marquis Eaton, attorney: “These studies (algebra and geometry) tend, in my opinion, to develop the power of analysis, which power lies at the threshold of success in every profession, notably the legal profession.” Mr. James E. Otis, President, Western Trust and Savings Bank, “I consider the study of high school mathematics as a necessary part of a high school education. In so far as I am concerned, my knowledge of algebra, geometry and higher mathematics frequently enables me to analyze problems which come before me. It is the training of the young mind in the study of algebra which I consider of greatest importance and not the practical use he can make of the science in later life. .In so far as geometry is concerned, questions are constantly arising where I find my knowledge of geometry most helpful. I feel so strongly on the subject that I should consider the drop¬ ping of algebra and gometry in our secondary schools as a very serious mistake” Mr. S. M. Hastings, President, Computing Scales Com¬ pany of America: “My personal business experience has taught me that mathematics, algebra and geometry are impor¬ tant, and in my opinion they should be retained in our schools. With a knowledge of these, one is better fitted to cope with the business problems. “As we become a greater world power and our foreign business expands, the need of every possible mental equipment will be more and more in evidence.” Mr. C. A. Creider, Secretary to Edward B. Butler of Butler Brothers: “Anyone who has been able to compass the subjects of algebra and geometry is by very nature of his training equipped to comprehend a problem or to arrive at a solution thereof much more readily than the fellow who has not had the training referred to. In other words, even a private secretary, who might be expected to have very little to do with the principles of these two studies, finds it much easier to fill his position having gone through the mental exercise of study of these two subjects.” Mr. C. S. Cutting, attorney, Holt, Cutting and Sidley: “The study of high school mathematics was worth while to me for this reason, among others, that the cultivation of the habit of MATHEMATICS IN SECONDARY SCHOOLS 31 mathematical methods of thought not only fixed my knowledge of the so-called practical mathematics of the grammar school, but made all operations of like nature, whether strictly mathe¬ matical or otherwise, easier. . . It is not usually possible to shirk work (as a student) in mathematics (it is therefore de¬ sirable in the course of study). . . . We use geometry in the examination of titles to real estate, in the investigation of plats and in many other relations where mensuration of both lines and angles must be considered. In my opinion, however, this is the smallest part o the benefit derived from the study of these subjects. ... I believe that the emasculation of a literary college course by omitting the classics is an educational mistake which would be equaled by omitting the mathematics. IVty experience teaches me that the man who knows a few things well is better educated than the one who knows many things indifferently.” Mr. James K. Ingalls, President, Western Heater Despatch: “High school mathematics taught me system, to be accurate, to have confidence in my conclusions, and accustomed me to abstract reasoning.” Mr. Granger Farwell broker: “I employ algebraic and geometrical calculations in business and many times without knowing that I specifically do so.” Miss Harriet Vittum, social worker, says: “2. Yes, I disliked it so heartily I needed to have to do it. “6. Yes, as general mental discipline if not to fill specific needs.” Rev. James G. K. McClure President, McCormick Theo¬ logical Seminary, “I am accustomed to say that the classics have done more to prepare me for my life than any other study or studies. The drill in Euclid at Yale was very disciplinary.” Rev. M. P. Boynton, Woodlawn Baptist Church: “The practical geometry that I learned in my father’s carpenter shop, and which was based on the steel square of the workman, has been of greatest practical use to me. Though I had no occasion to use these rules for many years, I was delighted when in build¬ ing my summer home, and needing the rules for cutting and fit¬ ting the rafters and other timbers to find that these principles wh ch I actually worked out as a boy freely came back to me. I believe that this is nature’s great law and that we retain only such knowledge as we are able to make real use of.” Dr. J. Clarence Webster, physician, says that in planning 32 SCHOOL SCIENCE AND MATHEMATICS the education of a son or daughter he would not include algebra or geometry, “unless these subjects were directly related to their professional work. ... I think they are as useless as Latin and Greek and should not be taught to the great majority.” A prominent clergyman of the city writes: “Even the ad¬ mitted value of geometry in training the logical faculties has to be discounted because the faculty is there developed in a field entirely apart from any of the practical problems of life, and its general cultural value is therefore indirect and remote. This is not so much a matter of judgment on my part as a matter of practical experience. My judgment is that algebra and geometry are quite unnecessary as a part of general education and that for purposes of general culture other subjects are much more useful in the curriculum of secondary education. Yet it is per¬ fectly clear that engineering and architecture, not to mention certain branches of physics and astronomy, absolutely require _a knowledge of algebra and geometry. “My daughter, now twenty-one years of age, had a distinct gift for languages and history. Mathematics of any kind were exceedingly difficult for her and even untiring industry and ex¬ cessive efforts barely sufficed to secure a passing mark. Yet algebra was a required study and graduation from high school impossible without a reasonably satisfactory ^record in this study. She will never make any use of algebra and she has bitter memories connected with its study, together with a rank¬ ling sense of injustice in a system which requires a given standard in this useless subject and allows no substitutes in the require¬ ments for graduation.” This is probably as representative a group of persons as could be selected. A larger group would not be likely to change results greatly. They are all busy men of affairs. That they should answer so fully is significant of the importance they attach to such matters. About 90 per cent answer favorably to the importance 6f mathematics in secondary schools, some of them emphatically so. This is an overwhelming majority. An election with such a majority would be called “a landslide.” While there are many factors which may enter to make the opinion of a single individual, or even of a small minority, of doubtful value, great importance must be attached to the opinion of such a majority. The members of such a group cannot all be biased or prejudiced; they cannot all be out of date in such matters, or be unacquainted with the trend of affairs in MATHEMATICS IN SECONDARY SCHOOLS 33 the educational world. The fact that they are successful men precludes any such notion. Some of them are giants in the busi¬ ness world and in other fields—men to whom we look for leader¬ ship. However, the opinion of a minority, or even of a single in¬ dividual, is not a negligible matter. These frequently suggest important lines for improvement, or they raise an issue and put to the test things that have come to us through tradition. When we recognize a fallacy in the argument of an opponent we are reassured in the position we hold. Those who have opposed the teaching of mathematics in high schools—where they have ex¬ pressed themselves beyond the mere answer “no”—seem to think of mathematics only as a tool. To them its only value seems to be the direct application in the sciences or in the doing of the world’s work. This is a mistake, as is shown by the answers to questions one and five. Question one, which is considered as referring to cultural values, has an overwhelming majority in the affirmative; while question five, which repre¬ sents the utility values, has a small negative majority. Evidently the direct application of mathematics to other fields is of secon¬ dary importance. The chief values are culture and mental dis¬ cipline, training in logical and abstract thinking, the cultiva¬ tion of the power of concentration, and the acquiring of speed and accuracy in our mental activities. However, this matter will be treated at greater length in the more complete report to be made later by this committee. President Butler of Columbia University says in Educational Review, September, 1917: “No educational instruments have yet been found that, in disciplinary value, are equal to Greek, Latin, and mathematics. The descriptive and experimental sciences cannot do it—or at least they have not done it—and the same is true of the newer subjects of study that are humor¬ ously if roughly, classified together as ‘unnatural sciences’— economics, sociology, and the like.” An investigation, as reported by Mr. Harris Hancock in School and Society, June 19, 1915, was carried out recently in Cincinnati. Of 105 prominent persons in the city, and 99 outside the city, consulted with reference to the question, “What course of study should be taken by a boy who is enter¬ ing high school?” 167 answered that they would require mathe¬ matics, 18 would require either mathematics or the classics, and 12 would make mathematics elective. This experiment carried 34 SCHOOL SCIENCE AND MATHEMATICS on with a larger group more widely scattered gives approxi¬ mately the same majority in favor of mathematics as is shown in our experiment. The following came with the replies: Hon. Chas. Theodore Greve, of the Cincinnati Law School: “I have specialized in history and economics, but they can never take the place of classics or mathematics. Both are essential and there can be no possible substitute for either. . . . These two subjects are the essential groundwork of education that is to be of value for any subsequent career, professional, scientific, business or mechanical.” Hon. Roscoe Pound, of the Harvard Law School: “The two things which appear to me to be required of secondary education are, linguistic training . . . and some sort of train¬ ing which will form settled mental habits of accuracy and thor¬ oughness during the student’s formative period. I believe mathematics will achieve the latter result as nothing else may. Personally I never liked mathematics; but they were valuable, not for what I remembered, but because I was com¬ pelled to see that two and two make four, instead of presenting plausible arguments that it might be otherwise.” Mathematics is, however, an important study not for the boy alone but for the girl as well. Since the girls of today will be the mothers of tomorrow, and so will bear the greater share of responsibility in the education of the coming generation, for this reason, if for no other, algebra and geometry ought to be a part of the education of every girl. Margaretta Tuttle, an author, and also of Cincinnati, says in Good Housekeeping, September, 1917, that the old line edu¬ cation, even though requiring effort that is not always pleasant, is best. “Will-power continues to be the great need of all who hope to grow and be of use. And it continues to develop only by use. “Marianne hates solid geometry. Marianne’s mother does not see why her daughter has to waste a whole year on some¬ thing she can never by any possibility use. Marianne is liter¬ ary. Oh, no! Literary ability and mathematical ability rarely go together. Marianne, who is eighteen, and who knows a little about the lives of writers and nothing at all about engineers, will tell you so. But all the same, after Marianne, by the sweat of her brow, has worked out these different problems, she will have formed the habit of not flinching from thinking things out—one of the most necessary habits for a literary person and for any woman. MATHEMATICS IN SECONDARY SCHOOLS 35 “Women do not always do it, nor men either. It is the think¬ ing of things out that steadies the world. It is the confronting of a problem with the intent of solving it after you have seen both sides of it that makes you a real help in a world of unsolved problems. You cannot write a good story, short or long, with¬ out a problem in it that has to be thought out and then demon¬ strated. Many a literary person living a meager life on small success, unable to solve the problem of why it is so meager, needs a little mathematical training. Many a mother struggling with the problem of her child’s food, and wishing she had taken dietetics instead of piano lessons, needs the mathematical touch —the ability first to recognize a problem when she sees it, then to attack it without worry, to bring to its solution all the like problems that have been previously solved, all the axioms and theorems that actually apply to it, and then to come to a conclusion and to stick to it because it has been demonstrated. Quod erat demonstrandum! The woman who has learned the full power of that phrase has learned to handle facts.” These testimonies and the results of these investigations make it evident that algebra and geometry are not so much rub¬ bish to be thrown out at the back door as quickly as possible, as some would have us think. The number of persons who ob¬ ject to the requiring of mathematics in our high schools is rela¬ tively a small minority, which, in its efforts to be heard, some¬ times makes a noise all out of proportion to its numbers. The professional, the business, and the industrial worlds, at least, demand for the present, and will demand for years to come, that algebra and geometry be retained as required studies in the secondary schools of our country; and this notwithstanding the efforts of educational theorists and would-be reformers to the contrary. Would it not be time and effort better spent if the destructionists were to aim at the perfecting of the sub¬ ject matter, and to the improvement of its teaching? The Committee: ALFRED DAVIS, Chairman, Francis W. Parker School, 330 Webster Ave., Chicago, Ill. A. M. ALLISON, Lake View High School, Chicago, Ill. J. A. FOBERG, Crane Technical High School, Chicago, Ill. M. J. NEWELL, Evanston High School, Evanston, Ill. C. M. AUSTIN, Oak Park High School, Oak Park, Ill. J. R. CLARK, President (Ex-Officio) Parker High School, Chicago, Ill. 112 SCHOOL SCIENCE AND MATHEMATICS VALID AIMS AND PURPOSES FOR THE STUDY OF MATHEMATICS IN SECONDARY SCHOOLS. By Alfred Davis, Francis W. Parker School, 330 Webster Ave., Chicago. Chairman of a Committee of the Mathematics Club of Chicago Appointed to Investigate This Topic. It is our purpose in making this report to aid the National Committee of the Mathematical Association of America in investigating the reasons for the teaching of mathematics in our secondary schools. The topic is a vital one, since its con¬ sideration must determine the place and the nature of the mathematics taught in our high schools. Its discussion is timely, since the challenge has come from various sources to us, as teachers of mathematics, to defend our subjects, espe¬ cially algebra and geometry. Educational inertia will no longer protect us. A passive attitude is no longer tenable; we must make our position clear. Even though we have little that is new by way of argument to offer; and even though we prove nothing beyond the possibility of a question, a statement of what we are convinced is true cannot fail to be of value both to teachers and to laymen whom it may reach. Since it is desirable, if possible, to avoid every generalization not supported by the results of scientific investigation or by the highest philo¬ sophic authority, we shall, throughout the discussion, make generous use of the investigations of others and shall quote freely from those who can speak with authority. We make bold to do this because important statements and valuable work increase in importance with emphasis and use. In the words of Dr. Smith, of Teachers College, when speaking of philosophers of standing and of those who are masters in their respective fields of effort, “These after all are the men who are our leaders.” Part I. The study and the teaching of mathematics in our secondary schools may be justified under the following heads, and it is to these that we may turn to establish the validity of our aims and purposes: 1. Philosophy. 2. Psychology and Its Experiments. 3. Experience. 4. The Utility oj Mathematics / MATHEMATICS IN SECONDARY SCHOOLS 113 1. Philosophers from Plato to the present time have been almost unanimous in their approval of mathematical studies. They believe that culture and mental discipline result from their pursuit. These are men to whom we must listen. They are masters in the art of reasoning and arrive at conclusions free from sentiment or prejudice. Where exceptions occur, it is usually due to a lack of knowledge of mathematics and a con¬ sequent inability to judge. The famous criticism of Sir William Hamilton has no weight, since John Stuart Mill (“Examination of Sir Wm. Hamilton’s Philosophy,” p. 607) shows that Hamil¬ ton did not know mathematics; and Professor C. J. Keyser (in an address at Columbia University, October 16, 1907) shows that he was prompted by unworthy motives. Oliver Wendell Holmes and Thomas Huxley were sincere in their doubts as to the value of the study of mathematics, but these have been ably answered by J. J. Sylvester and C. J. Keyser. Professor A. N. Whitehead (“Introduction to Mathematics,” p. 113) says, “Philosophers, when they have possessed a thor¬ ough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have possessed a slight or hasty and late-acquired knowl¬ edge of it are entirely worthless, being either trivial or wrong.” Philosophy’s support of mathematics stands like the rock of Gibralter, secure amid the passing assaults made against it. Plato, “Let no one ignorant of geometry enter my door.” August Comte, “No irrational exaggeration of the claims of mathematics can ever deprive that part of philosophy of the property of being the natural basis of all logical education, through its simplicity, abstractness, generality, and freedom from disturbance by human passion. There, and there alone, we find in full development the art of reasoning, all the resources of which, from the most spontaneous to the most sublime, are continually applied with far more variety and fruitfulness than elsewhere. . ' . . The more abstract portions of mathematics may in fact be regarded as an immense repository of logical resources, ready for use in scientific deduction and coordination.” G. Stanley Hall, “Mathematics is the ideal and norm of all careful thinking.” 2. It was but yesterday that psychologists denied the trans¬ fer of training, and so to them the ability acquired in the study 114 SCHOOL SCIENCE AND MATHEMATICS of mathematics was of value only in the further study of mathe¬ matics. This position has not been successfully defended. Professor Smith, of Columbia, says, “The attack upon mathe¬ matics, that it has no general disciplinary value, has thus far been abortive scientifically. We have only to note how di¬ vergent are the results of various investigations to see the truth of the assertion.’’ Professor G. M. Stratton, of the University of California (“School and Society”) says; “It is a grave mis¬ take to suppose that the experimental work has proved that the idea of mental discipline is no longer tenable.” Today all psychologists of standing concede transfer. The only questions are as to how much, and by what agencies it is accomplished. Is it not probable that tomorrow all doubts regarding the mental transfer of training will have run their course and will be dead, at least for a time? It would be of interest to show that questionable results have been obtained, and that undue importance has been attached to experiments worked out by psychologists and others, in at¬ tempting to show that the value of the study of mathematics has been exaggerated. Indeed, when we consider that these experiments are likely to be limited in scope; that the material to be experimented with (human minds) is almost infinite in possible variety; and that the interpretation of results is largely an individual matter; we must concede that such efforts are of doubtful value. However, psychology and its experiments in relation to the teaching of mathematics is to be considered by another committee. 3. The experience of the race justifies the teaching of mathe¬ matics. This is almost too obvious to need mention. Civiliza¬ tion may be measured in terms of mathematical progress. Mathematics is the ladder, giving the sure footing, by which we have climbed steadily to higher levels of achievment. It is through mathematics that man has been able to strip mystery from the forces in nature and to harness them for his service. Colonel F. W. Parker (“Talks on Pedagogics,” p. 92) says, “The lower the grade of development in the human race the less there is known of number.” Mathematics is then an intrinsic element in human- progress. Scientific progress is impossible without it. It is interwoven in the fabric of our commercial and industrial life. Since this is true it must change as the race advances—perhaps slowly—but ever to increase the importance of its study. MATHEMATICS IN SECONDARY SCHOOLS 115 4. The direct practical utility of mathematics alone will not justify our present high school courses in the subject. To many earnest and sincere critics mathematics is only a tool invented by man for the mastery of other fields. In fact, this is the rock on which most of those who object to mathematics shipwreck. Mathematics is vastly more than a tool; it is a type of thought, and even high school students gain indispensable training in mental activities from it—such training as cannot be gained so well from any other study. Professor C. J. Keyser (“The New Infinite and the Old Theology”) says, “Mathematics is indeed a humble servant—a drudge, if you please; an unsurpassed drudge—in the sense that nothing else does a larger share of humble and homely work. To imagine, however, that her place in the hierarchy of knowledges is thereby defined is hardly the beginning of wisdom in the matter. It is necessary to look much higher. Her rank in the ascending scale is not that of a useful drudge, immeasurable as is her service in that capacity; it is not merely the rank of a metric and computatory art, in¬ valuable as the latter is, as well in science as in the affairs of a workaday world; it is not even that of a servant to other sciences in their fields of experimental and observational research, indispensable as mathematics is in that regard; over and above these things, she is charged with a sacred guardianship—in her keeping are certain ideals, the ideal forms of science and the standards of perfect thinking; she is concerned not with the vagaries, but with the verities of thought, with select matters independent of opinion, passion, accident, and will; it is thus peculiarly hers to release human faculties from the dominion of sense by winning allegiance to things that abide; her medita¬ tions transcend the accidents of time and place; it is their idiosyncrasy to have for subject proper, not the fickle and transi¬ tory elements in the stream of a flowing world, but those aspects of being that present themselves under the forms of the infinita^— and eternal.”^ The tendency today is to overemphasize the utility of all high school studies. Professor John Dewey (“How We Think,” p. 138) says, “There is such a thing, even from the commonsense standpoint, as being Too practical/ as being so intent upon the immediately practical as not to see beyond the end of one’s nose or as cutting off the limb upon which one is sitting. . . . Exclusive preoccupation, with matters of use and application, so narrows the horizon as in the long run to defeat itself. It does not pay to tether one’s thoughts to the post of 116 SCHOOL SCIENCE AND MATHEMATICS use with too short a rope. Power in action requires some large¬ ness and imaginativeness of vision. Men must at least have enough interest in thinking to escape the limits of routine and custom. Interest in knowledge for the sake of knowledge, in thinking for the sake of the free play of thought, is necessary then to the emancipation of practical life—to make it rich and progressive.” Lord Beaconsfield’s definition of a practical man is too often the truth: “A practical man is one who prac¬ tices the errors of his forefathers.” Many today make the mistake of measuring educational, values in terms of money. They would prepare the child merely to get the necessaries of life. These are the enemies of liberal education; for if such are our aims we are, as a people, in the process of decay. The aim of education should be to make lives worth preserving—lives that will, at least in some small measure, make the world a better place in which to live. But multitudes of men and women “like dumb driven cattle” go wearily to toil each day. They use the last measure of strength in earning a livelihood—beyond this they have no vision. If they seem satisfied it is because they lack the outlook on life which an education ought to give. There is danger that edu¬ cators will assume that people need training for efficiency in this sort of life, but such is not the vision of leadership. These people need the best possible education, as much of it as they can get; not an education in any sense inferior, or suited to an inferior station in life. Professor Keyser (Educational Review, April, 1917) says, “I desire to warn you, as a friend, against the enemies of liberal education. These are very numerous, being easy to produce, springing up like weeds along the dusty highway, almost under the very hoof of travel. I desire to warn you against the insidious and baleful influence of omnipresent, well-meaning, wingless-minded educators who unconsciously conceive young men and women as more or less sublimated beasts; and who regard colleges and universities as agencies for teaching the animals the arts of getting shelter and raiment and food.” The motto of the Pythagorean Brotherhood should be the motto of today, “A figure and a step onward: Not a figure and a florin.” Yet the study of mathematics will ever be important from the standpoint of utility. Geometry originated in Egypt from the need to survey the farms in the valley of the Nile, and to replace the landmarks swept away by the periodic overflow of the river. MATHEMATICS IN SECONDARY SCHOOLS 117 Any schoolboy knows that algebra enables us to solve problems which are practically impossible of solution by arithmetic. The invention of the calculus enabled Newton to apply his law of gravitation to the motions of the planets. As needs have arisen in various fields, mathematics have been invented to relate theory to fact. Discoveries have been made in pure mathe¬ matics, when studied for its own sake, and later these have been applied to practical ends. There are many fields in which a knowledge of mathematics is absolutely essential to their mas¬ tery; indeed, it seems to be necessary to all branches of knowl¬ edge as these become more complete and so more scientific. Dr. O. J. Lee, of Yerkes Observatory, calls attention to the fact that many astronomers, while in the opinion of the outside world successful, have failed from lack of a sufficient founda¬ tion in mathematics. The same is true of many other lines of effort. Professor J. W. A. Young (“The Teaching of Mathe¬ matics”) says, “For direct practical usefulness, mathematics is second only to the mother tongue.” Regarding the possible future of mathematics in this line, G. St. L. Carson (“Mathe¬ matical Education,” p. 51) says, “I believe that the modern theories of pure mathematics are destined to illumine our understanding of the human mind and of cities and nations, just as the pure mathematics of fifty years ago has already illumined the previously dark and chaotic field of physical science; that modern mathematics is or will be to psychology, history, sociology, and economics as has been the older mathe¬ matics to electricity, heat, light, and other branches of physical science.” It is evident that the study of mathematics is amply justified by the testimony of philosophers who know; by its importance as an aid in mental development; by the weight of human ex¬ perience; and because of its increasing utility. Mathematics is an essential part of any scheme of education that pretends to be well balanced or complete. Since the education of a rapidly increasing number of people ends with the high school, it must be taught there. What, then, may we hope to accomplish by its study? Part II. The aims and purposes which may be realized in the study of mathematics are determined by its values to the one who studies it. These possible values are almost without number. We shall consider some of the more important. 118 SCHOOL SCIENCE AND MATHEMATICS 1. Mathematics Teaches Logical Thinking. It is the most effective means for teaching logical thinking aside from the actual study of logic; in this it is unique. Every¬ one needs this training and no other high school study can give it so well. John S. Mill (“System of Logic,” Bk. 3, chap. 24, sec. 9) says, “The value of mathematical instruction as a prep¬ aration for those more difficult investigations consists in the applicability not of its doctrines but of its methods. Mathe¬ matics will ever remain the past perfect type of the deductive method in general.” Benjamin Pierce (American Journal of Mathematics , vol. 4, p. 97) says, “Mathematics is the science which draws necessary conclusions.” The type of reasoning most emphasized in mathematics is the deductive. This sort of reasoning is used in other subjects and is applicable to all sorts of situations in life. Deduction is the process of arriving at a logical inference based on accepted premises. Psycholo¬ gists claim that all thinking is problem solving. The process as outlined by Professor Thorndike, of Teachers College, is as follows: 1. A clear statement of the goal aimed at. 2. The selection of enough and representative individual facts. 3. Their arrangement in such a way as to make the general idea or judgment to which they lead obvious. 4. The verification of the conclusion by an appeal to known facts. 5. Its reinforcement and clarification by exercises in applying it to new individual facts. It is evident that the study of mathematics gives training in this process. The committee on Secondary Mathematics, appointed by the New England Association of Teachers of Mathematics, says, “Whatever one’s occupation, it will be funda¬ mentally important to have acquired in youth the habit of exact, orderly and logical thinking, which, if the experience of many centuries of teaching can be trusted, is best acquired by most high school students in mathematics well taught.” Mathe¬ matics is, then, of first rank among high school studies in teach¬ ing pupils how to think. Its fundamental concepts are few in number, very simple, and lie close to the experience of the pupil. It should therefore be an easy study for any normal mind. John Locke (“Conduct of the Understanding”) says, “Would you have a man reason well, you must use him to it MATHEMATICS IN SECONDARY SCHOOLS 119 betimes. Exercise his mind in observing the connection between ideas and following them in train. Nothing does this better than mathematics, which, therefore, I think should be taught to all who have the time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures. ... In all sorts of reasoning, every simple argument should be managed as a mathematical demonstration.” 2. Mathematics Creates Self-Confidence. This is not the confidence that makes “fools step in where angels fear to tread”; but a confidence based on that self-knowl¬ edge, and on that self-command which give poise and power. The gaining of this confidence is a necessary part of the early education of every individual, and it can be accomplished better by mathematics than by any other study. Other sub¬ jects depend on authority, and there is usually a difference between authorities and a conflict of opinions. The student is frequently bewildered. He wonders which authority to ac¬ cept, and he doubts his ability to think out a conclusion of his own. He finds mathematics different. In mathematics there is no such thing as an outside authority: given certain premises, there can be no doubt about the results of his reasoning. Results are either right or wrong and they can be checked: algebra by arithmetical calculation, geometry by actual measurement. The reasoning of a student who lacks ability to study mathe¬ matics, or who lacks training in it, is likely to be of doubtful value—he is likely to be uncertain of it himself. If mathematics has been properly taught and thoroughly mastered, one will not place too much dependence on the opin¬ ions of others. In this way its study will make for a better citizenship. The mastery of mathematics must insure the pres¬ ence in a man of those qualities, at least some measure of them, which make leaders. The success of a democratic govern¬ ment depends on the power of its people to think intelligently and to act wisely. Sir James Bryce says, “It is by the best minds that nations win and retain leadership. No pains can be too great that are spent on developing such minds to the finest point of efficiency.” It .’is equally true that no pains are too great that will raise the intelligence of the mass of the people to greater efficiency. It is economy to use mathematics as a means to these ends. The life of Lincoln illustrates this.^ In the “Life of Lincoln,” Nicolay and Hay, vol. 1, p. 229, we read: “It was at this time that he gave notable proof of his unusual 120 SCHOOL SCIENCE AND MATHEMATICS powers of mental discipline. His wider knowledge of men and things, acquired by contact with the great world, had shown him a certain lack in himself of the power of close and sustained reasoning. To remedy this defect, he applied himself, after his return from Congress, to such works upon logic and mathematics as he fancied would be serviceable. Devoting himself with dogged energy to the task in hand, he soon learned by heart six books of the propositions of Euclid, and he retained through¬ out life an intimate knowledge of the principles they contain.” This is not an isolated case; others give testimony to the value of the study of mathematics in giving power to direct men on important issues. 3. Mathematics Cultivates the Power of Concentration. The success of a student depends on his early gaining of the power to concentrate his mind on a given problem. To work with the highest efficiency the mind must be wholly absorbed with the task in hand, the nerves tense, and the body in an attitude that suggests attention and alertness. The easy- chair loafing method will not lead to a trained and developed mind. The pupil whose mind goes “wool-gathering” in the midst of important work, or who is easily distracted by his surroundings, is a failure as a student. This power to study is developed by mathematics as by few other studies. While apparent progress might be made in some studies without it, concentration is vital in this study. The New England Report says, “The real development of mankind lies in the growth of voluntary attention, which is not passively attracted, but turns actively to that which is important, significant, and valuable in itself. No one is born with such power. It has to be trained and educated. This great function of education is too often neglected.” We are inclined to forget that real progress is directly proportional to the conscious effort of the individual. Prof. W. C. Bagley, of Teachers College, says, “Bricks cannot be made without straw, nor can mental growth be achieved without individual effort and individual sacrifice.” Work is not play and the child must distinguish between the two. Something must be done besides amusing our pupils in class. There is something more important than catering to the child’s wishes. Sometimes he may have duties to perform that he will not like. He needs the training that such effort will give, since he must learn to adjust himself to environment, and to the comfort and wishes of others as well as of himself. MATHEMATICS IN SECONDARY SCHOOLS 121 Spencer defined the educated being as one who did what he ought, when he ought, whether he wanted to or not. The pupil must learn that the world does not revolve about himself; and we as teachers should recognize the fact that a pupil does not find himself educated without knowing how it happened. Mathematics has the power to develop concentration; and it is our duty to furnish adequate motive for the effort that will accomplish this result. Professor R. E. Moritz (University of Washington) says, “That mathematics is the most efficient agency for acquiring the power of quick attention and prolonged concentration of mind has never been seriously questioned by competent critics.” Prof. Gonzales Lodge, of Teachers College (Address before the faculty of Teachers College, February 8, 1917) says, “It is apparently becoming more and more a cardinal doctrine of the shallow thinkers of the present day and genera¬ tion that the mind of man cannot be trained; that the only thing that can be trained is the hand. ... It seems to me axiomatic that mental training to be valuable must involve effort. ... I mean that kind of effort, necessary in the devel¬ opment of the attitude of mind that faces a problem squarely, which goes to work at it in detail, which analyzes it with care and exactness; which expresses the results of this analysis with the same care and exactness. The mind that can do that is cer¬ tainly a trained mind, and the benefits of such training are available in every walk of life.” 4. Mathematics Demands Originality in Its Study. It is probable that much of our school work stifles initiative and originality, if it does not kill these outright, in attempting to fashion large class groups according to one mould. This deplorable condition is aggravated by the excessive dependence on the text and the memory work so common in all studies. If the pupil can make a show of learning by repeating other people’s ideas he is passed on and the teacher is counted a suc¬ cess. Mathematics, properly taught, and especially in its application to well chosen practical problems, requires inde¬ pendent thought and judgment. The pupil is able to realize all the joy of individual discovery and achievement, which is in itself a sufficient motive for the study of any subject. His new consciousness of independence and of power is an inspiration and a delight to a reasonably keen-minded student and is fre¬ quently a stimulus to a slower mind. Mathematics is usually the child’s first introduction to the possibility of independent 122 SCHOOL SCIENCi1 AND MATHEMATICS thinking; and at a time when the independence of maturity- should be developing, it is especially suited to his needs. How¬ ever, it sometimes happens that a child’s ability to remember and to repeat glibly is mistaken for ability to think. Conse¬ quently a 1 ‘bright” pupil fails in mathematics and the subject or the teacher is condemned by disappointed parents. To say that the study of mathematics does not develop originality, or to teach it as a thing to be learned by rote and to be applied by machine methods, is to debase it to the rank of a mere tool and create the possibility that a show of learning may be made by a mere exercise of the memory. There can be little gain for the pupil in either case. J. J. Sylvester (“Mathematical Papers”) says, “As the prerogative of natural science is to cultivate a taste for observation, so that of mathematics is, almost from the starting point, to stimulate the faculty of inven¬ tion.” Again, some students may possess unusual abilities in mathe¬ matics or in related fields. A lack of a knowledge of algebra and geometry in early years may mean that these will remain forever buried and that the race may be deprived of an important possible contribution to its progress. Even unusual difficulties with these subjects is not always a sufficient reason for the abandonment of them. Dr. Smith (Mathematics Teacher , March, 1913) speaks of some of the world’s greatest mathe¬ maticians as being unpromising in early years. Florian Cajori (“A History of Mathematics,” p. 201) says of Newton, “At first he seems to have been very inattentive to his studies and of very low rank in school.” The study of mathematics will, then, draw out the individual powers of the pupil and enable him to “find” himself. It furnishes the best measuring rod for a pupil’s abilities and needs. 5. Mathematics Trains in the Precise Use of English. Teachers of algebra and geometry know that students have the greatest difficulties with translation problems in algebra and with originals in geometry. Most of the trouble is due to inability to read intelligently. Reading, of course, means getting the thought. Much of the pupil’s reading up to this point has been so simple as to require little effort to get the meaning, or it may have been the mere repetition of words. The pupil must be taught to dig below the surface, to properly balance statements and to get their real meaning. The pupil must also learn to express himself in a concise and forceful manner. MATHEMATICS IN SECONDARY SCHOOLS 123 Definitions must be clearly and accurately stated, not a word too many, not a word lacking, just the right word for each idea. Hypotheses must be stated exactly and kept distinct from conclusions to be reached* All of this is hard work and requires much patience, but it is a worthy effort for both teacher and student. English teachers who have had the opportunity to observe are emphatic in their approval of mathematics as an aid in the mastery of English, correcting slovenliness and inaccuracy. Much useless discussion and controversy would be avoided if the precise use of English were made a more direct aim in the study of mathematics. Professor Moritz says, “In mathe¬ matics, therefore, the student can be brought to recognize the absolute necessity of mastering the meaning of words pre¬ liminary to their use as a vehicle of thought. Half of the mis¬ understandings and futile controversies in active fife arise from ambiguity in the use of words. . . . Mathematics has come to be accepted as the synonym for exactness, clearness, certainty.” Perhaps in no other field is this training more essential than in the practice of law. Demonstrating before a class and answer¬ ing questions from teacher and fellow pupils gives self-possession, ability to think on one’s feet, and the power to adjust in proper order the important parts of any problem, giving to each its proper weight, so that desired ends may be attained. The pupil thus acquires ability to discuss a problem intelligently before a critical audience. Thomas Jefferson was of the opinion that, “mathematical reasoning and deductions are a fine prepara¬ tion for investigating the abstruse speculations of the law.” 6. Mathematics Trains in Accuracy. Probably no other subject demands this quality to so great a degree. There is no opportunity to cloak errors with results “nearly” right. If approximate results are sought the limits for the errors are known. Pupils can judge for themselves and make their own corrections. Again, carelessness in securing data or in the drawing of a figure may vitiate the whole prob¬ lem, no matter how perfect the reasoning. Furthermore, neat¬ ness is an essential part of accuracy and both should be conscious aims of both teacher and pupil. They should be emphasized more than at present in our teaching of mathematics. (To be continued) 208 SCHOOL SCIENCE AND MATHEMATICS VALID AIMS AND PURPOSES FOR THE STUDY OF MATHEMATICS IN SECONDARY SCHOOLS. By Alfred Davis, Francis W. Parker School, 330 Webster Ave., Chicago. Chairman of a Committee of the Mathematics Club of Chicago Appointed to Investigate This Topic. (Continued from the February number.) 7. Mathematics Gives Ability to Handle a Tool, Essential in Much of Life’s Work. Failure to recognize this often makes trouble for the student later. Prof. A. It. Crathorne, of the University of Illinois, says, “we have in the University of Illinois graduate students in agriculture who find themselves under the necessity of delaying the work in which they are directly interested in order to study the freshman algebra that they find essential to the study of their problems” (School Science and Mathematics, vol. 16, p. 420). And further, “The utility of algebra as a medium of expression is on the increase as surely as we are gaining more exact scientific knowledge” (Ibid.). The list of courses for which mathematics is an essential prerequisite, as given by a committee of this club (School Science and Mathematics, vol. 16, pp. 610-611) is as follows: scientific agriculture, engi¬ neering, physics, chemistry, art (drawing, designing, architec¬ ture, modeling, life and still life drawing, handicraft), pharmacy, dentistry, navigation, astronomy, naval and military engineer¬ ing, domestic science, insurance, forestry, commerce and ad¬ ministration, railway administration, political economy, sociology, hygiene, sanitation, education, medicine, and journalism. Law and theology are included with some reservations. It is not difficult to see that a knowledge of algebra and geometry is absolutely essential for some of these courses, while in the others it has important uses, particularly the formula, the graph, and the equation of elementary algebra. The man of ordinary education needs a knowledge of mathe¬ matics to appreciate everyday literature on many topics. Read¬ ing the many books and magazines relating to the automobile, the aeroplane, progress in science, or the war, without this as a foundation, is like reading about various places of interest without a knowledge of place geography. Prof. S. G. Barton, of the Flower Observatory, University of Pennsylvania ( Science, vol. 40, p. 697) says that in the Encyclopedia Brittanica (11th MATHEMATICS IN SECONDARY SCHOOLS 209 ed.) there are 104 articles which use the calculus, of which about one-fourth are pure mathematics. Some of these articles are: clock, heat, lubrication, map, power transmission, ship building, sky, steam engine, etc. There is much greater need for a knowledge of algebra and geometry in everyday reading. Prof. D. E. Smith, of Columbia, says, “Of the necessity for knowing number relations there can be no question, but fifty years ago one might well have cried the slogan abroad from the housetops, ‘Will anyone tell me why the girl should study algebra?’ Today a person would sadly feel his ignorance if he or she had to look with lack-lustre eyes upon a simple formula such as may be found in Popular Mechanics, Motor , the Scientific American, an everyday article on astronomy, a boy’s manual on the airplane, or any one of hundreds of articles in our popular encyclopedias. These needs come not only within the purview of the boy; they are even more apparent in the case of the girl, she who is to have the direction of the education of the generation next to come upon the stage of action. Each must know the shorthand of the formula, and the meaning of a simple graph, of a simple equation, and of a negative number, or else must feel the stigma of ignorance of the common things that the educated world talks about and reads about.” T. C. Record May, 1917.) It is not enough that we aim at the application of the results which others have worked out. We must go deeper than that. An intelligent use of mathematics demands a knowledge of the subject. It is the testimony of a teacher in a correspondence school that a student can in a few lessons learn the application of a formula to a particular situation; but that when a new situation arises there is no resourcefulness to meet the new need. The student has not mastered the subject. His place in the industrial world must be that of a machine. A. R. Forsyth, President of the British Association for the Advancement of Science, Sec. A, says regarding the Perry movement, “Some¬ thing has been said about the use of mathematics in physical science, the mathematics being regarded as a weapon forged by others, and the study of the weapon being completely set aside. I can only say that there is danger of obtaining untrustworthy results in physical science if only the results of mathematics are used: for the person so using the weapon can remain unac¬ quainted with the conditions under which it can be rightly applied. . . . The results are often correct, sometimes incorrect; 210 SCHOOL SCIENCE AND MATHEMATICS the consequence of the latter class of cases is to throw doubt upon all the applications of such a worker until a result has been otherwise tested. Moreover, such a practice in the use of mathematics leads a worker to a mere repetition in the use of familiar weapons; he is unable to adapt them with any confidence when some new set of conditions arises with a demand for a new method: for want of adequate instruction in the forging of the weapon, he may find himself, sooner or later in the progress of his subject, without any weapon worth having.” . . . “The witness of history shows that, in the field of natural philosophy, mathematics will furnish the more effective assistance if, in its systematic development, its courses can freely pass beyond the ever-shifting domain of use and application.” (Perry’s “Teach¬ ing of Mathematics,” p. 36, and Nature , vol. 56, p. 377.) The study of mathematics is useful in giving set and balance to one’s life. The more widespread its study and the resulting mathematical sense the less opportunity there will be for dis¬ honest practices by the unscrupulous man of affairs; a sort of intuitive sense of the correctness of business transactions will often prevent errors and losses; the elements of chance and luck will play a less important part in our affairs and superstition, the stronghold of ignorance blocking the way of progress, will be demolished. 8. Mathematics Gives Training in the Use of a Symbolic Lan¬ guage. Much of the world’s work is done by the use of symbols. They are the tools for rapid thinking and writing. The progress mankind has made would be impossible without them. We are convinced of this when we think of carrying on the simple opera¬ tions of multiplication and division without the use of figures. Newton’s law of gravitation: F = GMm/D 2 , where F is the force acting between two bodies, M and m are the masses of the bodies, D the distance between their centers, and G the constant of gravitation, would be cumbersome indeed if it were necessary to say it the long way in order to use it. By the use of the algebraic symbols the whole story is gotten at a glance, and in a form convenient for application to other prob¬ lems. There is economy of mental effort as well as of time. The importance of symbols is further illustrated in the fact that Newton chose a clumsy system of notation for the calculus while Leibnitz chose the present system, which is much better. The English adopted Newton’s method and so fell far behind the MATHEMATICS IN SECONDARY SCHOOLS 211 mathematicians of the continent in development and applica¬ tion of the calculus. A reprint from the Engineering Supple¬ ment of the London Times ,. June 19, 1910, says, “The extent to which mathematics is capable of exact prediction depends on expressing the problem in mathematical language. The greater ability of engineers of today to translate problems into this language has led to an increasing number of successful inven¬ tions.The interpretation and application of formulae is nec¬ essary in the reading of all sorts of current literature, and in the application of general principles to all sorts of human effort. Algebra and geometry offer unequaled opportunity for the mastery and use of symbols, since the solution of problems requires constant translation back and forth. Dr. Smith ( T . C. Record , May, 1917), “One merit of mathematics no one can deny—it says more in fewer words than any other science in the world.” {The Nation, vol. 33, p. 426), “The human mind has never invented a labor-saving machine equal to algebra.” Lack of training in the use of symbols fixes very narrow limita¬ tions to one’s life. The graph has become one of the most important symbols in modern life. The meaning and use of the graph cannot be properly taught apart from algebra. We see at a glance the re¬ lationship of two interdependent variables at any stage. Prof. Crathorne (School Science and Mathematics, vol. 16, pp. 423-4) says, “The world is full of variables which depend on other variables, presenting to us the problem of finding out and exhibiting the manner of dependence. The office of the graphical methods of algebra is to exhibit this dependence to the eye, and not, as many textbooks would imply, merely to aid in the solution of equations. ... In a recent book for work¬ ingmen in shop mechanics, a full page is devoted to an explana¬ tion of the stretch of copper wire for different loads. The author, no doubt realizing the vagueness of his explanation, then clear¬ ly sums up by a graph with three lines of English under it.” The graph has become well-nigh indispensable in presenting statistics. Much labor is necessary to get facts from numerical data, while the graph gives the same information almost without effort and presents a picture to the mind that is easy to recall. The graph makes the function concept clear to the pupil. He is able to understand the meaning and something of the impor¬ tance of “x is a function of y.” When the graph is subject to a known law, as represented by an equation, it is a necessary step to the study of advanced mathematics. 212 SCHOOL SCIENCE AND MATHEMATICS Regarding the saving of mental energy by a knowledge of mathematics, and by a mastery of the use of its symbols, Prof. Chas. H. Judd, of the University of Chicago (“The Psychology of High School Subjects,” p. 131) says, “No student will know what mathematics is until he realizes the great economy of mental energy which this form of experience makes possible.” 9. Mathematics Develops the Imagination. G. St. L. Carson, in “Mathematical Education,” p. 41, says, “The operations and processes of mathematics are in practice concerned at least as much with creations of the imagination as with the evidence of the senses.” F. J. Herbart says, “The great science (mathematics) occupies itself at least just as much with the power of imagination as with the power of logical con¬ clusion.” Prof. H. H. Horne, of Dartmouth College, says, “Apart from its manifold applications, mathematics is the inevitable disciplinary element in the curriculum. It trains in the habit of logical and symbolic thinking, of precision and con¬ centration, and it develops the imagination.” The concepts of space and number relationships are fundamental in educa¬ tion. Col. F. W. Parker (“Talks on Pedagogics,” p. 50, etc.) says, “I think we can truthfully say that form is the supreme manifestation of energy, and without a knowledge of form and without the power to judge form with some degree of accuracy, there can be no such thing as educative knowledge. . . . Form and number are modes of judging and are necessary to a knowl¬ edge of the external world. . . . The study of form and geometry are of fundamental, intrinsic importance in education.” In the study of mathematics, images of one, two, and three d mensions are constantly before the mind. At first, objects and drawings are used to give clear pictures, but the student soon learns to depend on the imagination to reproduce the images and to frame new relationships. This is especially true in the study of geometry. In the discussion of a problem, all the possibilities of a given case must pass in order before the mind. Surely the imagination is used and cultivated in the study of mathematics. A few well-selected problems in physics and astronomy, taught with appreciation by the teacher, will aid in the cultiva¬ tion of the imagination and broaden the life of the pupil. These may relate to the velocity of light and the length of its wave; the relative sizes of the planets and stars and how these are measured; the distance of the planets and stars and the meaning of “light years”; etc. Experience shows that problems relating MATHEMATICS IN SECONDARY SCHOOLS 213 to such topics are fascinating to the student. Carson (“Math. Ed.,” p. 10), says, “One of the few really certain facts about the / juvenile mind is that it revels in the exploration of the un¬ known.” Astronomy, in particular, leads the mind to the thresh¬ old of the unknown, and exposes it to the Infinite. Who can be little, or narrow, or prejudiced, if his imagination has been inspired by mathesis! 10. Mathematics Leads to a Knowledge and an Appreciation of the Foundations of Science. All science is ultimately mathematical in its methods; the more completely it is developed the more mathematical a science becomes. Mathematics enables us to apply accepted laws to Nature’s problems. In this way Newton’s and Kepler’s laws have been established and have been made to extend to almost infinite‘reaches into space and to unfold the mysteries of a universe of which we are an infinitesimal part. Astronomy was astrology until mathematics released it and it became a science. But even astrology depended somewhat on mathe¬ matics. Sir John Herschel (“Outlines of Astronomy,” Intro¬ duction, Sec. 7) says, “Admission to its sanctuary (astronomy) and to the privileges of a votary is only to be gained by one means, sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry , without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range.” It is through mathematics that we are gaining knowledge of molecules, atoms, electrons, ions; of the wonderful changes that are occur¬ ring in these and of the laws that govern them. Mathematics has well-nigh unlocked the secret of matter itself. Indeed, we could know little of chemistry, physics, or of any other science were it not for the aid of mathematics; witness the fol¬ lowing testimony: Roger Bacon (“Opus Majus”) “Mathematics is the gate and the key of the sciences. . . . Neglect of mathe¬ matics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy.” A Kant, “A natural science is a science only in so far as it is mathe¬ matical.” Laplace, “All the effects of nature are only mathe¬ matical results of a small number of immutable laws.” W. W. R. Ball (“History of Mathematics,” p. 503) “The advance 214 SCHOOL SCIENCE AND MATHEMATICS in our knowledge of physics is largely due to the application to it of mathematics, and every year it becomes more difficult for an experimenter to make any mark in the subject unless he is also a mathematician.” Comte, “All scientific education that does not begin with mathematics is defective at its founda¬ tion. ... In mathematics we find the primitive source of rationality; and to mathematics must biologists resort for means to carry on their researches.” J. F. Herbart, “It is not only possible but necessary that mathematics be applied to psychol¬ ogy; the reason for this necessity lies briefly in this: that by no other means can be reached that which is the ultimate aim of all speculation, namely conviction .” Novalis, “All historic science tends to become mathematical. Mathematical power is classifying power.” Prof. A. Voss, of the University of Munich, in a lecture in 1903 (quoted by T. E. Mason of Purdue in Science , December 15, 1916) said, “Our entire present civilization, as far as it depends upon the intellectual penetration of nature, has its real foundation in the mathematical sciences.” Prof. Thos. E. Mason {Science, December 15, 1916), “Can you realize what would happen, just what stage of civilization we should be in, if all that is developed by the use of mathematics could be removed from the world by some magic gesture? Every branch of physics makes use of mathematics; chemistry is not free from it; engineering is based on its development; sociology, economics, and variation in biology make use of statistics and probability. Our skyscrapers must disappear; our great bridges and tunnels must be removed; our transportation systems, our banking systems, our whole civilization, indeed, must step back many centuries.” The student who leaves high school without a knowledge of the importance of mathematics in science has a serious lack. Problems in algebra and geometry should, when convenient, relate to the various sciences. If the text does not furnish such problems the teacher should provide them. 11. Mathematics Should Be Appreciated as One of the Great¬ est Achievements of the Human Intellect. In this respect there is the same reason for studying mathe¬ matics as for studying literature, language, art, or history, for it is only as we learn to appreciate the greatest in man’s efforts, achievements, and aims that we can have the proper ideals and purposes in our lives. It is only through this knowledge and appreciation that one is able to take an intelligent part in the MATHEMATICS IN SECONDARY SCHOOLS 215 world and in the age in which one lives; that one can be a live force and not an encumbrance in the world. The path by which men have traveled is shown by history; literature and art beau¬ tify its borders; language furnishes the bond of unity; but science, including mathematics, is the pavement and even provides the light by which they walk. Indeed, mathematics is the greatest of the sciences. Hermann Hankel says, “In most sciences one generation tears down what another has built and what one has established another undoes. In mathe¬ matics alone each generation builds a new story to the old structure.” The earth as the center of the universe, the cor¬ puscular theory of light, the indestructibility of the atom, these and many other theories, at one time considered funda¬ mental, are now fit only for the intellectual museum; but the contributions to mathematics endure. Dr. Smith (T. C. Record , May, 1917) says, in speaking of the theorem of Pythagoras, “Before Mars was, or the earth, or the sun, and long after each has ceased to exist, there and here and in the most remote regions of stellar space, the square on the hypotenuse was and is and ever shall be equivalent to the sum of the squares on the sides. All our little theories of life, all our childish specula¬ tions as to death, all our trivial bickerings of the schools—all these are but vanishing motes in the sunbeam compared with the double eternity, past and future, of such a truth as this.” Surely we can appreciate Laisant when he says, “Mathematics is the most marvelous instrument created by the human mind for the discovery of truth”; and Leibnitz, who says, “Mathe¬ matics is the glory of the human mind.” Do we aim, as teachers, to give this appreciation of mathe¬ matics? It can be accomplished only by the study of mathe¬ matics, and not by a course about mathematics, as some have suggested. As well expect nourishment from talking about food as to expect knowledge and appreciation of mathematics from talking about it. More importance ought to be given the history of mathematics in our teaching. Much inspiration and enthusiasm can be gained from the lives of great mathe¬ maticians. And at appropriate times there should be given the student a prospective view of the richness and beauties of the subject to be realized by advanced study. 12. Mathematics May Make An Important Contribution to the Aesthetic , Moral , and Religious Life of the Individual. Henri Poincar6 (Annual Report , Smithsonian Institution, 216 SCHOOL SCIENCE AND MATHEMATICS 1909) says that mathematics has aesthetic value in the feeling of elegance in a solution or demonstration; in the harmony among parts, their happy balancing, and their symmetry. The feeling of elegance may come from unexpected associations and kinships among things. The sense of beauty is bound up with the economy of thought. We have seen high school pupils fascinated by the application of the binomial theorem; by the power and elegance of an algebraic solution; or by the Golden Section and other geometrical constructions. They are de¬ lighted with beauties in nature and art that are revealed for the first time through the study of algebra and geometry. Prof. S. G. Bartoij ( Science , vol. 40) says, “Beauty is con¬ cealed by ignorance of the mathematics necessary for its inter¬ pretation. The student of mathematics will see that of which the untutored mind has no conception, because lying beyond its comprehension. . . . One of nature’s demands in which she is inexorable is a study of higher—the highest—mathematics. The interpretation of her laws requires it. “The massive bridge once wonderful because of its enormous size, when its principles of construction are understood, be¬ comes a thing of beauty, a wonderful monument to the intel¬ lects of the designer and the constructor. The great tunnels, turbines, subways, are changed to objects of wonder, to those capable of understanding the difficulties overcome in their construction. The stars in the universe above, which nightly dissipate some of their light upon the earth, bespeak their Creator’s glory in voices but faintly heard by those whose training does not enable them to comprehend the reign of law there prevailing. To such an one The heavens declare the glory of God’ in a more real and exalted sense.” Thus mathe¬ matics literally opens a new earth and a new heaven to us. It unlocks for us the “music of the spheres”; it reveals the thoughts of the Eternal. The study of mathematics leads to clear thinking; to honest and patient effort; to reverence for truth; and must, therefore, have a large place in the building of character. B. F. Finkel says, “Mathematics is the very embodiment of truth. No true devotee of mathematics can be dishonest, untruthful, unjust. Because, working with that which is true, how can one develop in himself that which is exactly opposite? Mathe¬ matics, therefore, has ethical as well as educational value.” Prof. H. E. Hawkes, of Columbia (Mathematics Teacher, March, MATHEMATICS IN SECONDARY SCHOOLS 217 1913) says, “What is simplest and most beautiful in the domain of pure mathematics too often corresponds to the facts of na¬ ture to be accidental. I contend that it is our privilege to point out, at every possible turn, this coordination of number and form, of formula and physical law, of unity between mind and nature. This is an experience of no mean moral value, to realize that our mathematical procedure is attuned to the harmony of the universe.” In mathematics, then, the human mind relates itself to the Supreme Intelligence, whose thought is manifest in the universe; and the contact must leave its imprint on our lives. This leads us to the consideration of the religious value in the study of mathematics. Man has an innate desire from early years to reach towards the Infinite and the Eternal. Clerk Maxwell, towards the end of his life, said, “I have looked into most philosophical systems, and I have found that none of them will work without God.” Mathematics reaches farther and with greater certainty than any other philosophy; towards a Supreme Being, a great First Cause. It looks down the vistas of the ages past; and into the dimness of the eons to come; and the human mind is awed by the sub¬ lime majesty of the Divine. Prof. D. E. Smith says, “The proper study of mathematics gives humanity a religious sense that cannot be fully developed without it. . . . In the history of the world, mathematics had its genesis in the yearning of the human soul to solve the mystery of the universe, in which it is a mere atom. ... It seems to have had its genesis as a science in the minds of those who followed the course of the stars, to have had its early applications in relation to religious formalism, and to have its first real development in the effort to grasp the Infinite. And even today, even after we have been pushing back the sable curtains for so many long centuries—- even today it is the search into the Infinite that leads us on.” Col. Parker (“Pedagogics,” p. 46) says, “I can assert that, from the beginning, man’s growth and development have utterly depended, without variation or shadow of turning, upon his search for God’s laws, and his application of them when found, and that there is no other study and no other work of man. We are made in His image, and through the knowledge of His laws and their application we become like unto Him, we ap¬ proach that image.” . . . “There is but one study in this world of ours, and I call it, in one breath, the study of law, and the study of God.” That the study of mathematics is the study of 218 SCHOOL SCIENCE AND MATHEMATICS God’s laws and so must lead to God there can be no question. Plato, “God eternally geometrizes.” C. J. Keyser says, “Again it is in the mathematical doctrine of invariance, the realm wherein are sought and found configurations and types of being that, amid the swirl and stress of countless hosts of transforma¬ tions, remain immutable, and the spirit dwells in contempla¬ tion of the serene and eternal region of the subtle law of Form —it is there that Theology may find, if she will, the clearest conceptions, the noblest symbols, the most inspiring intimations, the most illuminating illustrations, and the surest guarantees of the object of her teaching and her quest, an Eternal Being, unchanging in the midst of the universal flux.” “And reason now through number, time, and space Darts the keen lustre of her serious eye; And learns from facts compar’d the laws to trace Whose long procession leads to Deity.” —Jas. Beattie, “The Minstrel,” Bk. 3. No wonder the mind is fascinated by the fields opened through the study of mathematics. One is led to a spirit of reverence when he contemplates the human intellect as revealed in the ever unfolding and almost limitless range of mathematical achieve¬ ment. It reveals and inspires Godlikeness. Mathematics deals with the eternal verities: it is, if you will, the majestic mountain peak that rises in sublimity above the clouds of doubt and uncertainty and basks in the sunlight of eternal truth. These values in the study of mathematics will suggest the more important aims and purposes for its study and its teach¬ ing. The teacher will do well to have them in mind as a back¬ ground for his teaching; in this way they will permeate and vitalize his work. But this is not enough. The pupil must also appreciate their importance, to give sufficient motive for his work, to satisfy him that the study is worth his effort. Not all pupils will appreciate these to the same extent. The teacher should ascertain the interests of the student and make the appeal chiefly in accordance with these interests. Bertrand Russell says, “Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.” The same is true of all purposes; neither teacher nor student should work blindly without knowing what to expect as a result of his effort. We have made our defense for the teaching of mathematics in high schools when we have shown that some of these values MATHEMATICS IN SECONDARY SCHOOLS 219 can be realized alone by the study of mathematics, and that others can be realized better through mathematics than by the study of any other subject. We have shown that mathematics is most important in its culture values—that it is indispensable to everyone who would live his best. We might add the words of President Hadley of Yale, “The value of an education large¬ ly consists in studying facts that will not be used in after life, by methods that will be used.” Dr. Snedden, of Teachers Col¬ lege, recognizes the fact that the chief claim for the study of mathematics is on the cultural side. In “Problems of Secondary Education” (recently published), p. 223, he says, “I am con¬ vinced that the prominence of algebra (and geometry) in secon¬ dary education rests not so much upon faith in its usefulness as a tool of further learning as upon belief in its value as a means of ‘mental training’ and upon the faith that somehow a knowledge of algebra is essential to general culture.” Again (p. 229), speaking of a secondary school curriculum, he would “Seek to develop a ‘culture’ course in mathematics which should prove attractive to students seeking to inform themselves about the world in which they live; this to include some account of the evolution of mathematics as a human tool and as a means of interpretation, as well as a survey of modern applications of mathematics to the understanding of the universe and to the work of the world. Just as many of us can respond to operas, epics, and great paintings without being artists in these fields, so I think many could be led to appreciate the place of mathe¬ matics without becoming mathematicians.” In the light of what we have said, we would change “should prove attractive,” to “should be made attractive,” throwing responsibility upon the teacher. We would require a course in mathematics of everyone. We would make this course to include the actual study of mathematics; for while we would not seek to make mathematicians of all students, in the sense of making each a specialist, everyone needs to study mathematics as well as to study about it. And further, the cultural value of mathematics is of immeasurably greater importance than the mere getting of information about things. However, a knowledge of mathe¬ matics is essential in much of life’s work; and it is evident that it will become more and more important as a means of investiga¬ tion in practically every field of endeavor. Let us remember, withal, that the greatest values and aims in the study of any subject, like all the greatest things in life, 220 SCHOOL SCIENCE AND MATHEMATICS cannot be easily reduced to exact measurement. We have heard Dr. Smith say that he wished someone might measure the value of the early study of Greek in his life. So far no one has, and it is not likely that the psychologist ever will invent a measure for such values. Therefore, while we satisfy ourselves and seek to satisfy others by giving reasons for our faith in, and enthusiasm for, the subject we teach, we do not hope to close the door to the questions of doubters and critics. Nor do we think the door can ever be closed or ever should be. It is the glory of mathematics that, while it has had destructive critics through the centuries, it has survived them all and is marching steadily onward to a higher and a surer place in our civiliza¬ tion. Like everything worth while, it has its critics and enemies, but these ultimately contribute to its strength. They aid mathematics in leading its advocates to establish more firmly its claims, to adjust it better to changed conditions; and they are the means of heralding its worth to the multitude. This is the logical outcome of present criticism. (To be Continued) MATHEMATICS IN SECONDARY SCHOOLS 313 VALID AIMS AND PURPOSES FOR THE STUDY OF MATHEMATICS IN SECONDARY SCHOOLS. By Alfred Davis, Francis W. Parker School, 330 Webster Ave., Chicago. Chairman of a Committee of the Mathematics Club of Chicago Appointed to Investigate This Topic. (Continued from March number) Part III. If we are to realize our aims and purposes in the teaching of mathematics it may be necessary, as times and conditions change, to vary both subject matter and methods of teaching it. However, such change is evidence of life, not of death. The New England Report, already referred to, says, “Some teachers of mathematics in the high schools are inadequately prepared for their work, perhaps teaching it only incidentally. Conditions in normal and grammer schools are probably worse.” Matthew Arnold says, “The plan of employing teachers whose attainments do not rise far above the level of the attainments of their scholars has been tried and it has failed.” By far too much* emphasis is put on methods in schools of education for prospective teachers when these would-be teachers have not thoroughly mastered the subjects they seek to teach. Not that any teacher can know too well how to teach, but the study about education should never be at the expense of education itself. The scholarship requirements for teachers should be higher than they are; indeed, we can scarcely make them too high; for superior scholarship in the subject one is to teach is of supreme impor¬ tance, and all of us would be happier with greater efficiency in our chosen field. Much of the criticism of mathematics today is from those who are perfectly well informed as to how it ought to be taught, but they have little knowledge—and less apprecia¬ tion—of the subject. Prof. L. T. More ( Nation, May 3, 1917) says, “Afew months in a schoolroom acting as an assistant to an experienced teacher after a sound college course will prepare a person for teaching more effectively than pedagogical courses.” The plan is worth thinking about. On the other hand, granting adequate scholarship, all who enter the teaching profession need technical training in teaching, more than most of them get at present. Knowledge of a sub¬ ject does not necessarily imply ability to teach it; it is frequently assumed that it does, but some brilliant students have made 314 SCHOOL SCIENCE AND MATHEMATICS very poor teachers. It is a crime to make the classroom a labora¬ tory in which the would-be teacher shall try by hit-and-miss efforts to gain experience. Doctors of medicine do not gain all their preparation by experimenting on the helpless sick. Lawyers are not entrusted with our reputations before they have earned to practice law. The teacher in training should be under the direction of an expert. Our material, the plastic minds of boys and girls, is too valuable to be trifled with. Be¬ sides, we have too much at stake in the reputation of our subject to take such chances. The teacher must learn to see the sub¬ ject from the pupil’s angle and to present it effectively; to accom¬ plish this he must, while practicing under an expert, master the fundamentals of psychology and of pedagogy. Above all, the teacher’s preparation should draw out his individuality, the thing that makes him different from every other teacher; and it should enable him to avoid the ruts from the beginning. Much of the inability of students to master mathematics is due to poor teaching at some point in their career. Those who cannot master a reasonable amount of mathematics are probably no more numerous than physical defectives. Someone has said that algebra is “fool proof,” and this has been assumed true by many principals in assigning work to teachers. Dr. Snedden, (“Problems of Secondary Education,” p. 225) says, “Algebra is one of the easiest of secondary school subjects to teach with a certain degree of apparent effectiveness. Lessons and hard tasks can be assigned easily, and a very duffer of a teacher can make pupils work slavishly on this subject. In most small high schools today it will be found that the teacher with least special preparation for his work is usually teaching algebra. The case here is somewhat analogous to the practice of ‘electric’ and ‘magnetic’ healing by ‘near’ physicians.” This is a somewhat frank statement that the trouble with secondary mathematics is not with the subject but with the teaching of it. Poor teaching gives a subject the reputation of being diffi¬ cult. When pupils pass the word along that algebra or geometry is hard and uninteresting the battle is lost before it is begun. Even a good teacher will have difficulty in overcoming this prejudice. Another source of trouble is the assumption that when a pupil masters a principle this guarantees his ability to use it. Such is not the case. Many of our best students appear sadly inefficient when they leave high school. Application is quite MATHEMATICS IN SECONDARY SCHOOLS 315 as important a part of our work as the teaching of principles. Of what value is a tool unless we are taught the actual use of it? Prof. Judd (“The Psychology of High School Subjects/’ p. 130) says, “It is contrary to experience to assume that students can apply mathematics to the other sciences or to the practical affairs of life unless they are trained to see mathematical rela¬ tions in other forms than those in which they are commonly presented in the schools. The student who knows the abstract demonstrations of geometry, but does not realize that knowledge of space is involved in every manufacturing operation, in every adjustment of agriculture and practical mechanics, is only half trained. Application must be a phase, and an explicit phase, of school work. Application is as different from pure science as pure sciences are different from each other.” In addition to the inefficiency of teachers we have the ineffi¬ ciency of students as such. This is probably due to two causes. First, the idea is somewhat prevalent in the elementary schools that a pupil may gain an education by being amused. He is not trained to feel nor to know personal responsibility. His education is entirely a matter for the teacher to look after. He even resents any attempt to secure his initiative if the effort he is to make savors of drudgery or requires hard thinking. School work must be a pleasure and without responsibility. ^Second, there is a tendency on the part of college students to ignore study. This spirit finds its way into the high school, which so often tries to imitate the college. Consequently, many who could be good students fail; mathematics, requiring the sort of effort they know nothing about, is impossible with them. Ability in mathematics may, usually, be considered synonymous with ability to be a good student. Prof. Schultze (“Teaching of Mathematics in Secondary Schools,” p. 25) says, “It is a common experience to see a pupil in the upper grades suddenly wake up to the meaning of mathematics and thereby change his attitude towards study in general.” A good standing in high school is almost certain to guarantee a good college record and a successful life. This matter has been splendidly treated by Wm. T. Foster, President of Reed College, Portland, Oregon, in “Should Students Study” (Harper’s Month¬ ly, September, 1916). He raises the question, “Are good students in high school more likely than others to become good students in college?” Three colleges in as many states are considered. Of hundreds of students in the University of Wisconsin, above 316 SCHOOL SCIENCE AND MATHEMATICS 80 per cent of those in the first quarter in the high school re¬ mained in the upper half of their classes throughout the four years of their university course, and above 80 per cent of those who were in the lowest quarter in high school did not rise above mediocre scholarship in the university. Only one in five hun¬ dred of those in the lowest quarter reached highest rank in the university. The University of Chicago found that students that failed to receive in high school an average higher than p£ss by at least one-fourth of the difference between the pass¬ ing mark and 100 per cent failed in college; such students are, therefore, not admitted; where exceptions are made the record in college is seldom satisfactory. Reed College, at its fall opening five years ago, admitted only those students who ranked in the first third in the preparatory schools: about 20 per cent were exceptions to this rule and 2 per cent were below the median line; these exceptions were selected as the best below the first third. Of these exceptions, practically none rose above the lowest quarter in their college classes. The same results are shown to be true of those who go from college to the profes¬ sional schools. Surely “promise in high school becomes per¬ formance in college/’ and the mediocre in high school are out of the race. President Foster says, “If all these studies prove anything, they prove that there is a long chain of causal con¬ nections binding together the achievements of a man’s life and explaining the success of a given moment. . . . Luck is about as likely to strike a man as lightning and about as likely to do him any good. The best luck a young man can have is the firm conviction that there is no such thing as luck and that he will gain in life just about what he deserves and nor more. . . . At a convention of teachers not long ago a speaker ridiculed a German boy who, upon failing in a recitation, put his head upon his desk and cried. He said he had never seen such a boy in the schools of this country. . . . Nothing seems to promise failure in the tasks of tomorrow with greater certainty than failure in the studies of today. . . . Among teachers the greatest number of criminals are not those who kill their young charges with overwork, but those who allow them to form the habit of being satisfied with less than the very best there is in them.” From a study by President Lowell, of Harvard ( Educational Review, October, 1911), it appears that President Foster’s conclusions apply to Harvard students. Of 609 who graduated MATHEMATICS IN SECONDARY SCHOOLS 317 from college with A. B. plain, only 6.6 per cent obtained cum laudi in Law School. Students of mathematics attained high¬ est honors in Law School. Students of classics stood next. The qualities of diligence, perseverance, and intensity of appli¬ cation acquired in the study of mathematics secured a higher degree of success than was obtained by others. The pupil should be impressed with the importance of proper habits of study and he should be aided in forming these habits. If he finds the work very hard, let him profit by the determination of Robert Bruce as he watched the spider, after many futile efforts, finally reach the ceiling. If he is inclined to waste time let him get inspiration from the words of Hotspur before the battle of Shrewsbury: “Oh, gentlemen, the time of life is short; To spend that shortness basely were too long, If life did ride upon a dial’s point, Still ending at the arrival on an hour.” The New England Report says, “There is real danger that depreciation of mathematics by persons supposed to be experts in matters of secondary education may, unless vigorously met, exert an unfavorable and undue influence on public opinion.” It would seem desirable that teachers of mathematics assert their claims more energetically, for it is not always safe to assume that right will win without an advocate. It is not necessary to enumerate these criticisms; we are all familiar with them. Some have been justified and have already resulted in improved courses and methods in the better high schools; for no subject is so perfect as to need no improvement; and any subject will soon be out of date if it lacks the elasticity which enables it to fit new conditions. There is still room for much improvement. Dr. D. E. Smith says, “I think unquestionably there has been too high a pace set in the matter of abstract drudgery. I think there is no question but that'we must harmonize and vitalize our algebra.” Other criticisms have been answered time and time again, but like Banquo’s ghost they will not down. Prof. R. D. Carmichael, in Science , May 18, 1917, says, “It seems that no body of thought has been of more importance in human progress and at the same time been criticized more freely than the science of mathematics. Much of this criticism seems to be good natured and to amount to little more than a quasi - humorous way of expressing the critic’s unashamed ignorance. At first sight one might treat this as harmless; but from the 318 SCHOOL SCIENCE AND MATHEMATICS point of view of general interest it can hardly be passed over in such a way. How this ignorance is to be overcome I cannot say. Perhaps one of the first requisites is to find some means of over¬ coming the shamelessness with which individuals otherwise well trained contemplate their own ignorance of mathematics.” Prof. Chas. N. Moore, University of Cincinnati, in The Ameri¬ can Mathematical Monthly , February, 1916, says, “The move¬ ment against mathematics is, for the most part, confined to a group of theorists who feel that they must advocate something new in order to convince their readers that they are investigators. This group, however, has made up in volume of sound what it has lacked in numbers. . . . The statement that there is at the present time much uncertainty as to the educational value of algebra and geometry will not bear examination.” Dr. D. E. Smith, in T. C. Record , May, 1917, says, “It is by no means the ad¬ vanced educator who denies a disciplinary value to geometry; it is rather, I think, either the educator who is slipping behind in the race, or the one who has never been in the race at all. If you do not think so, ask a man like Professor Thorndike. If anyone says to me that we have statistics to show that young people spend a year on a subject whose chief purpose is the logical proving of statements and are not thereby made more logical in their other lines of mental activity, now or in years to come, then I tell him frankly that not only do I not believe it but that scientific men generally do not.” Our critics seem to be so fascinated by the weight and worth of their own ideas that no reply can reach them. Some seek to surprise us with the resurrection of issues that have been dead for ages. Others, in the face of irrefutable argument, do the seemingly impossi¬ ble; they absolutely ignore the other side of the issue; they continue their argument as though it were impossible for them to be mistaken or for anyone to make a reply worthy of notice. Prof. Paul Shorey, in “The Assault on Humanism” {The Alantic Monthly, June, 1917), says, “They either have not read the literature which they controvert, or they intentionally ignore it. They do not inform their readers of its existence, and they do not even tacitly amend their own arguments to meet its specific contentions. In controversy, this is what Lincoln .called Tush whacking.’ In the authors of the textbooks of the science or the history of education it is the abandonment of the scientific for the frankly partisan attitude. . . . They not only argue as partisans against the classics but they systematical- 1 MATHEMATICS IN SECONDARY SCHOOLS 319 ly suppress both the arguments and the bibliography of the case 2 for the classics. . . . The principal effort of the classicist who aims at argument rather than eloquence must be to shame his opponents from their unfair tactics, their neglect of the evidence, their preposterous logic, and to urge the educated public to examine the matter for themselves. He must wearily repeat his old list of ‘must nots’ and ‘don’ts.’ ” These statements apply with equal force to many of the critics of mathematics. As teachers of mathematics, we are too often the victims of men officially higher up, who know little of mathematics or of its teaching, and whom we allow to do our talking and planning for us. Why should courses and curricula in mathematics be arranged and criticized by principals and superintendents who have no appreciation of the subject? It seems to be different in Europe. Prof. J. C. Brown, in a recent bulletin of the U. S. Bureau of Education, says, “European school men believe that a course in mathematics should be planned by those who know mathematics rather than by educators who are practically ignor¬ ant of the subject^ Educators not familiar with mathematics may aid in an advisory way; but, if the subject does what we * claim for the student, it must have done that in some measure for the teacher; who then can be so well fitted to mould the subject into better form or to adjust its courses? Some of the difficulties, and lack in interest of students in the study of algebra and geometry, are the result of preconceived notions, the echoes of what educational theorists have said. We have heard a teacher say that it is a pity that the active productive minds of young people should be burdened with so formal a study as algebra. A student who is looking for an easy time and who finds algebra hard work thinks he has sufficient authority in such a statement for loafing on the subject. This point is cleverly made by Prof. Crathorne in a parable given in “Algebra from the Utilitarian Standpoint” (School Science and Mathematics, vol. 16, pp. 418-431). After considerable discussion on the part of Dr. Highbrow, Dr. Practical Man, Dr. Brown, and an Average Parent regarding difficulties in algebra and dislike for the subject, which the Average High School Pupil seemed to be experiencing, the pupil broke in with, “But I don’t dislike algebra. I never did; but so many people argued that I disliked it that I began to think there was something wrong with it.” That much of the lack of interest on the part of students towards mathematics is merely assumed is shown 320 SCHOOL SCIENCE AND MATHEMATICS in a report by this club published in School Science and Mathematics, October, 1916. A test was given in three high schools—University High and Hyde Park, Chicago, and Oak Park, Ill.—to determine from the statements of students in these schools the measure of enjoyment they found in their various studies. Out of a total of 2,018 pupils reported as taking mathematics, 1,769, or 87.7 per cent, said they got some enjoyment from the study; 991, or 49.1 per cent, claimed to get very much enjoy¬ ment; and only 229, or 12.3 per cent, said they got no enjoy¬ ment. With one year of mathematics (at least) required of all students in these schools, and in view of present criticisms, these results are remarkable. Subjects which ranked higher were purely elective. It is the opinion of Prof. J. W. Johnson that 49 per cent of the pupils who study mathematics, includ¬ ing the high schools of the entire country, would vote that they got very much enjoyment from the study of mathematics. It would be a mistake to try to persuade ourselves that no criticisms are worth our attention. The destructive critic always defeats his own ends, and is unworthy of serious consid¬ eration; but the constructive critic is always welcome; providing he maintains a friendly spirit. We must have the assistance of the latter. G. St. L. Carson, in “Mathematical Education,” pp. 61 and 85, says, “The whole world is going through a trans¬ formation, due in part to scientific and mechanical invention and in part to the growth of separate nations, each with its own methods and ideals, of which no man can see the outcome. Our function, the function of all teachers, is to produce men and women competent to appreciate these changes and to take their part in guiding them so far as may be possible. Mathe¬ matical thought is one fundamental equipment for this pur¬ pose, but mathematical teaching has not hitherto been devoted to it because the need has but recently arisen. But now that it has arisen and is appreciated, we must meet it or sink, and sink deservedly. Neither the arid formalism of older days nor the workshop reckoning introduced of late will save us. The only hope lies in grasping that inner spirit of mathematics which has in recent years simplified and coordinated the whole struc¬ ture of mathematical thought, and in relating this spirit to the complex entities and laws of modern civilization. . . . Salva¬ tion from our present difficulties can come only from the efforts and experiments of teachers themselves. Educational matters are in a ferment. Men are asking more and more insistently MATHEMATICS IN SECONDARY SCHOOLS 321 why this and that are done, and they are right in their insistence. Unless fitting answers are ready, our work will stand condemned; the degradation of our subject to the domain of purely immedi¬ ate utility will surely follow, as also the loss of that higher mental training which is so essential to the formation of an efficient citizen.” To meet this situation we would make the following suggestion: Mathematics should be required of every secondary school pupil. This required work may consist of courses alloted to the junior high school, if that institution is to prevail; or it may, under the present arrangement, be reduced to one year and con¬ sist of algebra and geometry, with some reference to the use of the trigonometric functions. The purpose being, in any case, to give to all, early in their course, some command of mathe¬ matics as a tool; to give some knowledge of its historical develop¬ ment; to instill an appreciation of the importance, possibilities, and beauties of the subject; and, above all, to give the pupil a chance to find himself. Those who have little aptitude for mathe¬ matics would not elect additional courses, and yet would realize some of the benefits from its study. This would probably reduce pupil mortality in our h gh schools and remove much of the grounds for present criticism. On the other hand, such an arrangement would throw responsibility on the teacher to interest his pupils in mathematics while taking the required courses so that they would elect other courses in the subject. This might aid in securing better teaching and better teachers. This club has already expressed its approval of reducing the required mathematics in high school to one year. Prof. Paul H. Hanus, of Harvard, in his book, “A Modern School,” pp. 76 and 83, says, “Permanent lack of interest in a given field of work is an indication of corresponding incapacity; for growing interest and capacity always go together. ... No pupil should be required to pursue a study after it is clear that it does not appeal to him. Under most circumstances one year is enough— and it is not too much—to ascertain whether a study does, or does not, challenge a youth’s interest and capacity. . . . vOne year of elementary algebra and geometry may open the pupil’s mind to one of the most useful, the most profound, and to some minds most fascinating systems of thought which man has de¬ veloped—a result which can never be expected to follow from what the pupil has learned in the narrow field covered by arith¬ metic.” Regarding the nature of elementary courses in mathe- 322 SCHOOL SCIENCE AND MATHEMATICS matics, Prof. G. A. Miller, in the preface to his “Historical Introducton to Mathematics” (1916), says, “Early mathematical courses should be more informational, especially along historical lines, on the ground that knowledge begets interest in knowl¬ edge.” That a well-planned and interesting course should be required there can be no question. No one can know whether or not he has aptitude for the subject until he knows some¬ thing about it. One cannot know anything about mathematics, to such an end, until one has studied the subject. The Lincoln School, established this fall as a part of Teach¬ ers College, is an effort to determine the value of some of the modern theories of education, and to work out a better program of study than we have at present. We shall await with interest and shall welcome the results of this effort. It announces that it will attempt “in the subject of mathematics to develop a course which connects the study of mathematics with its use, adequate provision being made for those whose special abilities or future interests relate to mathematics. And in all subjects, whenever feasible, effort will be made to base schoolwork upon real situations, to the end that schoolwork may not only seem real to the pupil, but be so.” It would seem that our work in algebra and geometry might be more completely related to the interests and activities of the pupil than at present. It would seem that this result might be accomplished by a greater num¬ ber and variety of practical problems, even though these had little direct relation to the utility of the subject in later life. To say that only those problems which are directly related to the doing of the world’s work are legitimate is to place mathe¬ matics on a utilitarian basis, and this we have shown to be wrong. The values we have suggested can be realized when the pupil is thoroughly interested in his work regardless of its actual use in later life. However, as often as possible problems should be useful as well as interesting. Considerable progress is being made in this line in various parts of the country. Surely all important departures from the established order of things should be made slowly, and only after these have been carefully tested. Experiments with high school mathematics should be made in schools adapted to that use, and by those who are expert in mathematics as well as in the teaching of the subject. This, of course, would not preclude the possibility of important discoveries and contributions by any teacher situated in any school. MATHEMATICS IN SECONDARY SCHOOLS 323 ■ The high school is not the place to specialize. The required ■ courses in mathematics must be adapted at the same time to Bthe pupil who does not and to the one who does go to college. ■ Diversified courses may have a place as electives and in voca- B tional and technical schools. Only a very small number of high I school pupils can have any adequate notion of what their life’s I work should be. It is a great mistake to force—or even to per- I mit—them to follow courses that may exclude them from the 1 vocation for which they are best fitted. President Hadley, of Yale, says that only about 8 per cent of pupils previous to seventeen years indicate inclinations towards any vocation. Dr. Joseph Rausohoff, an eminent surgeon of Cincinnati says, (School and Society, June 19,1915) “Up to the fifteenth or sixteenth year the average boy who goes to a high school can have no idea as to the work he expects to follow in later life. A general course will give the boy a general knowledge which will later permit him to develop along certain lines, as his bent or neces¬ sity may indicate. Such a course makes the possibility, at least, of a general culture which will permit him to indulge in one or another intellectual hobby later in life. I would above all things not exclude mathematics, but make it compulsory in every high school curriculum, because it is, after all, the only study which will inculcate into the young mind that absolute- precision is among the human possibilities.” Opportunity for the broadest possible education must be given to everyone, i This is an essential thing in a democracy where equal opportunity should be enjoyed by all. We cannot say to one pupil, “The door to the highest attainments in education is open to you”; and even suggest to another, by limiting his field, that the same door is practically closed to him. This should be our ideal in curriculum making and in teaching, and it should be kept constantly before both teacher and student. The way to the highest achievement must not be closed to anyone. We have quoted freely in defense of our arguments, for the words of a thinker carry their own weight. We are reminded of the words of Emerson, in “Letters and Social Aims, Quotation and Originality”: “A great man quotes bravely, and will not draw on his invention when his memory serves him with a word as good.” The value of a well-expressed thought or of a good piece of work depends on the use made of it—the passing of it along. We have quoted from specialists in mathematics, for who but those who know and appreciate a subject can speak 324 SCHOOL SCIENCE AND MATE cs with authority about it. Lest this should seem a biased attitude, | we have also quoted specialists in other lines, and have e*en given the opinion of students in high school. Some may object that mathematics has changed since some of these statements were made concerning it, and that they are therefore of doubtful value. While mathematics is not today what it was yesterday, it is not less but more important. That mathematics is advancing today as never before is shown by the following quotations: G. A. Miller, “Historical Introduction to Mathematical Literature,” p. 22, “It would be very conservative to state that the first decade and a half of the present century (twentieth) produced at least one-fifth as much (mathematical literature) as all the preceding centuries combined. Hence it appears likely that the twentieth will produce, as the nineteenth century has done, much more new mathematical literature than the total existing mathematical literature at the beginning.’’ C. J. Keyser, in a lecture ten years ago, “2,000 books and memoirs drop from the mathematical press of the world in a single year, the estimated number amounting to 50,000 in the last generation. . . . “Indeed, the modern developments of mathematics consti¬ tutes not only one of the most impressive, but one of the most characteristic, phenomena of our age. It is a phenomenon, however, of which the boasted intelligence of ‘universalized’ daily press seems strangely unaware; and there is no other great human interest, whether of science or of art, regarding which the mind of the educated public is permitted to hold so many fallacious opinions and inferior estimates. The golden age of mathematics—that was not the age of Euclid, it is ours.” President N. M. Butler, of Columbia University, “Modern mathematics, that most astonishing of intellectual creation, has projected the mind’s eye through infinite time and the mind’s hand into boundless space.” Jas. Pierpont, “Surely this is the golden age of mathematics.” ALFRED DAVIS, Chairman, Francis W. Parker School, 330 Webster Ave., Chicago, Ill. J. A. FOBERG, Crane Technical High School, Chicago, Ill. A M. ALLISON, Lakeview High School, Chicago, Ill. M. J. NEWELL, Evanston High School, Evanston, Ill. C. M. AUSTIN, Oak Park High School, Oak Park, Ill. J. R. CLARK, . President Ex-Officio, Parker High School, Chicago, ill.