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 MODERN TENDENCIES IN 
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MODERN TENDENCIES IN THE 
 
 TEACHING OF MATHEMATICS 
 
 By Professor ISAAC J. SCHWATT 
 
 READ AT THE MEETING OF THE 
 
 New York Section of the Association of Teachers of Mathematics 
 December ii, 1908 
 
 Reprinted from 
 
 Mathematics Teacher, published by the Association of Teachers of Mathematics 
 in the Middle States and Maryland 
 

 f 
 
 'i 
 
 4 
 
 
MATHEMATICS- 
 ' LIBRARY 
 
 : /O' i 
 
 s oL 
 
 
 
 MODERN TENDENCIES IN THE TEACHING OF 
 MATHEMATICS. 
 
 By Professor Isaac J. Schwatt. 
 
 Until the middle of the last century, mathematics had been 
 developed only from the top, so to speak; but during the last 
 decades, the efforts of some of the ablest mathematicians have 
 been directed towards obtaining clearer conceptions of the foun- 
 dations of mathematics. 
 
 Cantor, Weierstrass, Dini, Dedekind and many others have 
 studied the fundamental conceptions of number and space, and 
 their work has resulted in more accurate ideas about these 
 conceptions. 
 
 With all the advance in the knowledge of mathematics, with 
 our more thorough conceptions of the foundations of the sci- 
 ence, and with the resulting tendencies to change the methods 
 of presenting its various branches, the effect on the student’s 
 knowledge of the subject is far from encouraging. 
 
 During the last few years, quite a little unfavorable criticism 
 has been made on the efficiency of our schools. Many of these 
 criticisms refer to the results we now obtain as compared with 
 those obtained before. We do not concern ourselves with 
 the criticisms made by skeptics ; they have been, and they always 
 will be with us. Nor ought we to take seriously the statement 
 by George Bernard Shaw : “ Those who can, do ; and those who 
 cannot, teach.” But there have been criticisms made by members 
 of our own craft. Professor Charles W. Earned, of the United 
 
 41 
 
42 
 
 THE MATHEMATICS TEACHER. 
 
 States Military Academy, in an article entitled, “The Ineffi- 
 ciency of onr Public Schools,” North American Revieiv, Sep- 
 tember, 1908, tells us that out of 314 candidates who submitted 
 in June, 1908, to the examination for entering West Point, 
 154 failed in algebra, and 237 in geometry. The examination 
 questions were not difficult. They were free from what are 
 often termed ‘ catch questions,’ and have impressed me as 
 being eminently fair. I understand that during the examina- 
 tion the candidates are very closely watched, and that they have 
 to depend entirely on their own resources. 
 
 The results of the examinations in other subjects are corre- 
 spondingly lamentable. This is the more surprising since, as a 
 rule, only the ablest candidates are chosen, and many win their 
 appointment by competitive examination. The candidates rep- 
 resent a class of earnest men, 206 out of the 314 having either 
 wholly or in part earned their own living while at school and 
 while preparing themselves for the Academy. 
 
 In 1905 the Association of Mathematical Teachers in New 
 England addressed letters to teachers of physics and chemistry 
 throughout the schools and colleges of New England, inquiring 
 as to their students’ ability to perform arithmetical and alge- 
 braical operations.* One of the questions of this inquiry reads : 
 “If you have taught for some time, please state whether you 
 notice any difference in the pupil’s preparation in mathematics 
 now and formerly.” 
 
 Some of the answers would indicate that the pupils have 
 deteriorated under the modern methods of teaching, or what- 
 ever the cause might be. 
 
 Professor Barns, of the Department of Physics, Brown Uni- 
 versity, writes : “ There is a change for the worse. Coming 
 from the high school of Cincinnati to Columbia College thirty- 
 one years ago, we were decidedly better prepared in mathemati- 
 cal subjects than the men I teach now within forty miles from 
 the ‘Hub of the Universe.’” 
 
 Professor Bartlett, of the Department of Chemistry, Dart- 
 mouth College, reports : “ I refuse to teach arithmetic. . . . 
 Most of the students are helpless. Incorrect calculations are so 
 common that the first attention is always directed to the calcula- 
 
 Third rc‘i)orl of the Association of Mathematical Teachers in New 
 Knfj[lan(l, ](}o6. 
 
at 
 
 5 \ 0.7 
 
 CL.V\ 
 
 MODERN TENDENCIES IN TEACHING MATHEMATICS. 43 
 
 tions in case of erroneous reports. . . . Problems involving a 
 little algebra usually throw down the whole division .... I 
 have been teaching here for twenty-six years and had four 
 years experience in secondary schools previously. I think stu- 
 dents are much less competent in arithmetic now than formerly.” 
 
 As a member of the teaching profession, I feel keenly the 
 humiliation and the sting of the criticisms on the results of our 
 efforts. If the president of one of our leading universities can 
 give expression to the following: “We all know that the chil- 
 dren of the last two decades in our schools have not been 
 educated. With all of our training, we have trained nobody. 
 With all of our instructing, we have instructed nobody” — there 
 must be something decidedly wrong with the teaching of the 
 subjects of the curriculum and among them with the teaching of 
 mathematics, which is acknowledged to be one of its strongest 
 educational mediums. I am sometimes surprised at the apparent 
 indifference and complacency with which we teachers in general 
 seem to take these criticisms. If the criticism on the efficiency 
 of our schools is justified, it is our duty by single and united 
 effort to remove the cause of it. If education is good at all, 
 it must be effective, and its results must be commensurate with 
 the high importance we attach to it, and the large amount of 
 money we expend on it. 
 
 Whether the fault lies with us teachers, or with those who 
 are charged with the administration of our schools, or with the 
 object of our efforts, the pupil, upon us devolves the duty to 
 study the conditions earnestly, carefully and persistently, and to 
 bring about such changes as will improve the existing conditions. 
 
 It is pathetic to think that our efforts have not been crowned 
 with better success. On all other questions affecting the wel- 
 fare of the community and the state, every one of its mem- 
 bers has his say. But the average man or woman pays very 
 little attention to the work of the teacher. We teachers have 
 it all our own way. We can make up our curriculum and 
 choose our methods of teaching without interference from 
 the public. The public judges us by the results of our labors 
 only. It is therefore the more our duty to do our best and to 
 see that our efforts shall be most effective. 
 
 In a paper entitled “ Our Duty as Teachers,” I have con- 
 sidered one of the causes for the apparent failure of our efforts. 
 
 9 
 
44 
 
 THE MATHEMATICS TEACHER. 
 
 This cause is the tendency of the student to conceal his ignor- 
 ance from his teacher, instead of frankly trying to obtain all 
 possible help in his difficulties. 
 
 There was a time when the peculiarities of the teacher were 
 of such a nature as often to enable a pupil, who was not true 
 to his own interests, to pass a subject on insufficient knowledge. 
 The pupil took advantage of these peculiarities of the teacher 
 who was easily led to have a higher opinion of the pupil’s dili- 
 gence and knowledge than he deserved. P)Ut the day of such 
 teachers has passed. They survive only in the comic papers. 
 The teacher of to-day ought to be well able to cope with the 
 pupil who hides his ignorance intentionally or otherwise. 
 
 In this connection I am reminded of a little story which is 
 quite characteristic of the old-time professor. It is told of a 
 professor in one of our largest universities, that while walking 
 on the campus and meditating upon a problem, he met a student. 
 Linking arms with the student the professor proceeded to ex- 
 plain the problem to him and to demonstrate its solution. When 
 he was at the end of the demonstration, he said to the student, 
 “Don’t you see that the result is (let Is say) a?” The stu- 
 dent, who had not paid much attention and was, in fact, unable 
 to follow the demonstration replied with surprise, “Is it?” 
 The professor, believing that the student while following the 
 demonstration had obtained a different result, went over the 
 demonstration, and found that his first result was incorrect. The 
 apparent knowledge and ability of the student so impressed the 
 professor that from that time on, the student ranked in the 
 mind of the professor as one of the ablest mathematicians in 
 the university, always receiving the highest marks in the subject. 
 
 The failure of the pupil to derive from his studies the mental 
 development, for which they are intended, renders futile, at 
 least in part, the aims and the purposes of education. 
 
 What then is the purpose of sending children to school? 
 What, in short, is the puprose of education? On the answer 
 to tliis (jucstion depend the subjects to be taught in the school, 
 the metliod of instruction, and the entire attitude of the teacher 
 towards the student. 
 
 In answering tliis (piestion, we shall exclude from our con- 
 sideration the professional schools, the industrial and trade 
 schools, etc., and shall consider the elementary and such sccon- 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 45 
 
 dary schools and colleges as give what we term liberal courses 
 only. 
 
 Now, as I understand it, the purpose and aim of education 
 should be the physical, moral and intellectual development of the 
 young, whose fitness shall determine what the world is to be in 
 time to come. 
 
 It is true beyond discussion, that in order to do right, to be 
 useful to ourselves and to our fellowmen, we must first and 
 above all be healthy and strong. Our bodies must have the 
 power of resistance to enable us to discharge the manifold 
 duties incumbent upon us. '* 
 
 The school should make provision for the physical develop- 
 ment of the child. The school ought to teach the laws of 
 hygiene and the rules of right living. The child must be taught 
 that the proper care of the body is as important as the care of 
 the mind, that one of the means for keeping the body in perfect 
 health is the use of proper and well prepared foods, of pure 
 water, pure air, etc. 
 
 The second function of the school ought to be the formation 
 of character, the inculcation of moral habits, of habits of right 
 acting, and of all that is true and noble and just. 
 
 Most schools seem to consider the intellectual development of 
 the youth as their most important, if not their only function ; 
 but as the motto of the University of Pennsylvania, “ Literse 
 sine-moribus vanae ” (learning without morals is vain), em- 
 phasizes the fact that, one of the most important functions of the 
 school ought to be the development of character, the inculcation 
 of habits of sincerity and frankness, of diligence and perse- 
 verance, of honesty of purpose and application to duty, of rever- 
 ence and kindness and consideration of others. 
 
 A person possessed of these qualities and only moderate in- 
 telligence is a more desirable member of society and is bound to 
 succeed better in life, in the real sense of the word, than a 
 person who is lacking in some of them, no matter how great 
 his intelligence or how strong his mental powers might be. 
 
 We must take into consideration that there are many parents 
 who have not the time to supervise the moral education of their 
 children. There are persons who have assumed the responsi- 
 bilities of parenthood without being well fitted for this great 
 task. On that account, very often they do not enjoy the rever- 
 
46 
 
 THE MATHEMATICS TEACHER. 
 
 ence of their children. The latter, as a rule, show more respect- 
 for their teacher than for their parents. With such children 
 especially lies the great power of the teacher, and his opportuni- 
 ties and possibilities to develop and influence character. More- 
 over since the child spends the greater part of the day in school 
 and in the preparation of his lessons, there is not much time left 
 for even the intelligent parents, with high moral ideas, to con- 
 duct the moral education of their children. The duty of mould- 
 ing the character of the child therefore devolves, in part at least, 
 upon the school and the teacher. 
 
 It is perhaps a fortunate circumstance that in most schools 
 no formal provision is made for the moral education of the 
 child and no dogmatic teaching of morals is included in the 
 curriculum. If without preaching to the child, we use every 
 opportunity to draw a lesson in morals and character building, 
 the purpose will be much better accomplished. 
 
 We now come to the third function of the school — the intel- 
 lectual development of the young. We believe in the goodness 
 of human nature, and in the instinctive desire of every indi- 
 vidual to do right, if he can only think right. I am in full 
 accord with Madame de Stael, that all bad actions are only the 
 result of thoughtlessness. Mental development, power of mind, 
 the ability to think right, are the first requisites for character. 
 If we could only think right in all matters pertaining to our- 
 selves and to our relations with others, we would also act right. 
 
 There was never a time when people were so easily convinced 
 by arguments without submitting them to the careful scrutiny 
 of tlieir reason and understanding. People seem now more 
 willing than ever to have their thinking done for them. There 
 was never a time when the power of the self-constituted leader 
 who takes it upon himself to do the thinking for the people, was 
 so great as it is to-day, when all kinds of sects of religion and all 
 kinds of social theories were so prevalent ; when the influence 
 of the “ walking delegate,” whose office is to hand out ready- 
 made ideas to his constituents, helpful or baneful, was as strong 
 as it is to-day. 
 
 Just as we must submit all our actions to the test of the 
 highest standards of morals, so we must submit everything we 
 learn, read, or hear to the test of our understanding. We very 
 often think that we understand an idea, but when we come to 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 4/ 
 
 express it, or carry it into execution, we find that the idea is 
 not entirely clear to us. 
 
 One of the most pernicious habits of the mind, against which 
 we must constantly and most vigilantly guard our students, is 
 that of superficiality; of accepting an idea before submitting 
 it to the test of his understanding; of being satisfied to take up 
 another idea before the first one is entirely clear and assimilated. 
 
 This habit of superficiality is only a prelude to a slip-shod 
 way of doing things, to be satisfied when a task has been done 
 so that the result will look right, yet not stand close scrutiny 
 or careful examination. To do every task right and to the 
 best of one’s ability is an important and essential quality in 
 every occupation in life. The time to acquire this all-important 
 habit is in childhood, and the place to begin such a preparation 
 is in the classroom, whatever the subject taught. 
 
 Our number system, which is so well adapted to the perform- 
 ing of arithmetical operations in a more or less mechanical way, 
 and in this lies the claim for the superiority of our decimal sys- 
 tem over all other systems, has contributed a great deal towards 
 the pernicious tendency of doing things mechanically, without 
 much exercise of the power of mind. 
 
 The pupil carries this tendency to his work in all the other 
 mathematical subjects. Since the young find concentration of 
 the mind the hardest task, they easily give way and do things 
 mechanically wherever possible. 
 
 While the method of the ancients, like the Egyptians and the 
 Romans, for carrying out arithmetical calculations were rather 
 cumbersome, one cannot help thinking that these methods were 
 more conducive to the acquiring of mental power, than the re- 
 fined methods we apply in our calculations to-day. 
 
 A child who recites a poem without understanding its con- 
 tents receives very little benefit from memorizing it. The per- 
 nicious habit which the child acquires of saying things without 
 understanding them, is a danger to its moral and intellectual 
 development. We must, above all, educate the children to think 
 and to reason. While thinking is the hardest work, we can so 
 cultivate this all-important habit of the mind in the child that it 
 will enjoy reasoning, and find thinking an agreeable task. No 
 person can discharge his duties of citizenship unless he is able 
 to reason correctly, and to think right. 
 
48 
 
 THE MATHEMATICS TEACHER. 
 
 It is the duty of the teacher to see that the student has a clear 
 understanding of every idea presented to him, and that he 
 acquires the habit of submitting every idea to the test of his 
 reasoning before accepting it and calling it his own. Every idea 
 introduced must be of such a nature that the student can gain a 
 clear understanding of it, his age and mental development being 
 taken into consideration. 
 
 It is the duty of the school and the college to pay special 
 attention to those students who are less gifted, or who are lack- 
 ing in the sense of duty, and have not the strength of character 
 to apply themselves to their studies. If schools and colleges 
 drop young men from their rolls because they do not show 
 aptitude and inclination for work, or because they are unruly, 
 such youths without our help and encouragement may become 
 a burden to themselves and to those around them. If they do 
 not do their duty in their work in life, there are other men and 
 women eager to take up their positions, and with little senti- 
 ment they are simply dismissed. 
 
 It is the special duty of the school to improve such youths, 
 to strengthen their character, to build them up mentally, and if 
 possible, make useful men and women of them. In the long 
 run, we shall pay the penalty if we neglect to help those who 
 need our help most. 
 
 The student who leaves school or college because he is not 
 successful in his studies often does well when he engages in 
 manual work or in an occupation which requires little mental 
 effort and little continuous exercise of his power of thinking. 
 If the failure in their studies is due to lack of ability, it might 
 be better for such youths to continue their studies, and by 
 proper training and direction acquire the power of mind, with- 
 out which their usefulness as citizens must necessarily be 
 impaired. 
 
 The function of the school is to prepare the child and the 
 youth for life. The pupil ought therefore to be taught that all 
 of liis actions, while at work or at play, must conform to the 
 highest moral standards. He must understand that to go to 
 school is an occu])ation, that it is as important for him to be 
 successful in this occu])ation, as for his father in that which he 
 ])nrsiK‘S, and he will be the better fitted for his work in life the 
 better he a])plies himself to all phases of his school life. 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 
 
 49 
 
 The satisfaction resulting from having done work right is 
 invigorating and is conducive to true happiness. Some of us 
 perhaps remember the worry we experienced when we were 
 not sufficiently prepared in our lessons, and the anxiety we have 
 felt lest we should be called upon to recite. Those children 
 who are diligent and have all other good qualities, and are there- 
 fore free from anxiety and fear of punishment, are as a rule the 
 healthiest and happiest. What is true of adults is true of them. 
 If we are successful in our efforts, we are happy and as a rule 
 healthy, however hard we may work. It is worry and not work 
 that impairs our health. If we all think right, we shall act right, 
 and we shall be less subject to worry, and more endured to work. 
 
 All the functions of the school are so interrelated that if 
 one is neglected the others are bound to suffer. To do right, we 
 must think right, and to think right we must be healthy in body 
 and mind. Mens sana in corpora sano. To be healthy we 
 must have an easy conscience, we must be able to decide and 
 choose what is good for our mind and body. 
 
 To acquire knowledge, the student must possess certain traits 
 of character. He must have the qualities of diligence, sincerity 
 and perseverance. If he is lacking in these, it is the duty of the 
 school to develop them, not only because they are all-important 
 for every occupation in life, but because without them his 
 mental development is not possible. Any failure to awaken and 
 develop those qualities in the pupil and the consequent failure to 
 derive the mental benefits from his studies, are bound to cripple 
 his character for life. 
 
 The character of the child which is still passive, with no 
 expressed tendency for good or evil, can be formed in such a 
 manner as to acquire habits which will influence his life for 
 good. If he acquires while at school the habits of diligence, 
 of being punctual, of being sincere, of being perseverant in his 
 work, he is most likely to carry these habits to his work in life. 
 If he is, however, negligent in performing his duties while at 
 school, he is likely to shirk his duties when he is a man. If he 
 pretends to know what he really does not understand, he will 
 acquire the habit of insincerity. If he is quieting his conscience 
 and making himself believe that he understands an idea without 
 giving his understanding a searching and scrutinizing test, this 
 will tend to create in him the disastrous habit of superficialty. 
 
50 
 
 THE MATHEMATICS TEACHER. 
 
 If he is inclined to be irregular in attendance while at school, 
 he is likely to show this same tendency later as a man. What an 
 injustice to the child who will have to pay the ])enalty for any 
 failure in his character due to the neglect of the teacher! 
 
 We must teach our students that diligence, perseverance and 
 sincerity are as desirable qualities in the youth as they are in the 
 grown-up person ; that there is only one step between school life 
 and what we call practical life; that since the habits acquired 
 in youth are most lasting, he must acquire habits of right action, 
 and practice them in all his activities while at school. 
 
 The methods of education ought to be the same whether the 
 child is to become a laborer, a mechanic, or a professional man. 
 The standard of morals and the ability to think right and to 
 act right ought to be the same whatever may be the station or 
 the occupation of a person in life. Just as there is one legal 
 code, there is only one moral code, and we all have to conform 
 to its teachings. Whatever advantages outside of the school 
 the child of the well-to-do may enjoy over the child of the 
 poorer classes, each must have the same advantages for his 
 physical, moral and mental development, if he is to be able to 
 make use in the same measure of the opportunities, which our 
 country offers to all its citizens. 
 
 The stability of this republic depends on the ability of the in- 
 dividual to thing and to act right and on his proper conception of 
 happiness and in what ways true happiness and enjoyment in life 
 can be attained. 
 
 The unsatisfactory results of our efforts have caused teach- 
 ers of mathematics, and especially associations like this to look 
 for remedies. This dissatisfaction has made itself felt in cer- 
 tain tendencies in the teaching of our subject. 
 
 In the following we shall discuss some of the aspects of these 
 tendencies and make a few suggestions, in the hope that those 
 who do not agree with the views herein presented, will yet 
 consider them of sufficient weight to continue their discussion, 
 and that some conclusions may be reached which each and every 
 one of us will endeavor to apply for the betterment of the teach- 
 ing of our subject. 
 
 There is a demand for better text-books, more thorough and 
 more lucid, for syllabi to serve as guides in determining what 
 to choose in the teaching of a subject and what to omit. I say 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 5 I 
 
 it with a great deal of hesitancy that in most cases the prepara- 
 tion of a syllabus is a misdirected effort, and that the use of a 
 syllabus may have an ill effect on the true purposes of teaching. 
 With the increase in the number and in the variety of text- 
 books for all grades in the same subject, a syllabus can be dis- 
 pensed with. 
 
 Moreover I feel that, as a rule, the purpose of a syllabus is to 
 serve as a guide as to what topics some of the colleges expect 
 candidates for admission to be most efficient in. But since the 
 great majority of secondary school pupils do not enter college, 
 a syllabus has no place in such schools. 
 
 There is a tendency to represent the subjects of the curriculum, 
 including mathematics, in such a manner as to incite the interest 
 of the learner. But if the young are brought up with the idea 
 that everything they do must incite their eagerness or curiosity, 
 there is no wonder that those who fail to secure work which is in 
 accord with their tastes are augmenting the ranks of those whose 
 theories may mean ruin to civilization and to the progress of 
 mankind. 
 
 Those are blessed who have found their work. The large 
 majority of men and women, however, must find the inspiration 
 for their work, and for doing it right, in the consciousness that 
 they are doing something to help the common weal ; that it gives 
 them the means to provide for their own needs and for the needs 
 of those who are dependent on their support, that if they do their 
 duty to the best of their ability, they may in time find work 
 which is more congenial to them. 
 
 The children and the youth ought to be taught to do their 
 duty. The teacher should insist upon such a course on the part 
 of the student with firmness and strictness. 
 
 The methods by which children are made to acquire the habit 
 of doing their work and doing it promptly and well, will differ 
 with the individual child. _Some children will give heed to a 
 word, others will be influenced by persuasive talk, again others 
 will do right, only after more or less severe punishment. 
 
 What an injustice to the pupil, and what harm is done to his 
 entire future, if the teacher does not insist that he be accurate, 
 punctual, and diligent, and that he devote himself with all his 
 energy to his work. The pupil when he leaves school to-day and 
 finds employment to-morrow will be judged by his conduct and 
 
 
52 
 
 THE MATHEMATICS TEACHER. 
 
 by his diligence. If he fails to comply with the requirements 
 that govern his employment, he may lose it, which may mean 
 humiliation and disappointment, and may result in sad con- 
 sequences for him, and for those nearest to him. 
 
 Education ought to be a preparation for life, as life is, or as 
 it ought to be. The pupil and the student of to-day are the 
 employer and employe of to-morrow, and what they will be, the 
 world will be. 
 
 The best preparation for life is to make the young do that 
 which requires effort — effort of thought, effort to overcome any 
 possible disinclination to work; to derive pleasure from doing 
 one’s very best and from the satisfaction of overcoming diffi- 
 culties. 
 
 Life is a continuous rendering of service. Service in order to 
 earn a livelihood ; service in bringing up our children and in 
 caring for those around us ; service in taking part in the higher 
 purposes of life — moral, social and intellectual. 
 
 Wherever we turn, there are conditions to improve and work 
 to be done, which call for service, hard and unremitting service, 
 and very often with only the reward that comes from the con- 
 sciousness of having done something worth while. 
 
 There is a tendency, even in the secondary schools, to present 
 the various subjects of mathemathics with a view to their possi- 
 ble application in life, believing that education in order to be 
 effective, ought to be practical. 
 
 It is, however, a matter of experience that we soon forget 
 whatever we learn at school, and it is true beyond question that 
 in life, we need more than anything else, health, character and 
 power of mind, for on these our success will largely depend. 
 We can get along with very little mathematics, very little 
 geography and science, very little of the other subjects in the 
 curriculum, if we have only character and the ability to think 
 right. 
 
 Even professional schools can give their students only such 
 knowledge as forms the foundation of their special science 
 and of what they will meet in the practice of their professions, 
 d'he school ought to imbue its students with the scientific spirit 
 and with the thirst for knowledge. When they enter their pro- 
 fession they will have continually to study and to learn. Any 
 change in the hy])()t]iesis of their science, every new discovery 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 53 
 
 and invention in it they will have to apply, if they do their full 
 duty. 
 
 The great majority of men and women will have little use in 
 life for mathematics, beyond the operations with integers and 
 simple fractions. It is greatly to be regretted that even in the 
 engineering profession, mathematics is being used less and less, 
 in the actual solving of practical problems. Some of the larger 
 manufacturing establishments, such as the Cambria Steel Com- 
 pany and the Carnegie Steel Company are publishing handbooks 
 for engineers which contain formulae and all kinds of data 
 which an engineer is likely to need in his work. These books 
 have created a class of engineers called “ handbook engineers ” 
 who use the formulae and the tabulated data very often without 
 a knowledge and understanding of the methods by which they 
 have been obtained. The use of such books tends to lessen the 
 mental activity of those who use them. 
 
 A large engineering company has lately provided its engineer 
 with slide rules, expecting them to be used in making computa- 
 tions, and thus to save time. 
 
 But just as it is impossible to prepare the student of engineer- 
 ing in all problems which he may meet in practice, so it is im- 
 possible to include in a handbook all the data he may need in 
 his profession. In some cases he will have to fall back on his 
 mathematical knowledge and the more thorough it is the better 
 fitted he will be for the work requiring mathematical theories, 
 the results of which are not to be found in a handbook. The 
 world is a vast store of undiscovered energies and opportunities. 
 We must only sharpen our minds to discover them, and acquire 
 the knowledge and ability to exploit them for the benefit and 
 comfort of mankind. 
 
 The greater usefulness of the college-trained engineer com- 
 pared with the engineer who has learned his profession in the 
 factory and in the field, the so-called practical engineer, lies in 
 •the higher development of power of mind in the former and in 
 his ability to use such methods and results as his knowledge of 
 mathematics and mechanics affords. 
 
 In manufacturing the aim is to produce an article in the short- 
 est possible time, with the least expenditure of material, but 
 with due regard to quality and efficiency. Now the practical 
 experience and his natural instinct will often tell the engineer 
 
54 
 
 THE MATHEMATICS TEACHER. 
 
 in what way a saving of material, etc., can be effected, but in 
 many cases, such a knowledge can be obtained by mathematical 
 theory only. 
 
 Take for instance, the manufacture of reservoirs, of basins, 
 or simply of tin cans. We can construct tin cans of different 
 shapes, and therefore from different amounts of material, but 
 all of one and the same content, a quart, say. Now, unless there 
 is a reason that a can should be of a certain shape (as a can used 
 for baking powder, or for spices, etc.) it ought to be constructed, 
 for a given volume, with the least amount of material, i. e., with 
 the view to economy. Mathematical calculations lead to the 
 result that to construct a can having a top and bottom, the least 
 amount of material will be required if the diameter of the base 
 is equal to the height of the can, and the larger the diameter of 
 the base of a cylinder with neither a top nor a bottom, the less 
 material will be necessary to construct it. 
 
 As a result of the tendency to make the work in arithmetic 
 more interesting and practical, the child often learns how to use 
 a commutation ticket or a mileage book, and the names of parts 
 of cattle in the language of the butcher, but does not get the 
 mental training which can be obtained from a proper study of 
 the subject, nor the ability to perform with accuracy some of its 
 simplest operations. Such a course is as much to be con- 
 demned as the tendency of some people to provide themselves 
 with luxuries, without being able to satisfy the more substan- 
 tial necessities for their body and mind. 
 
 There are so many faculties and abilities to be developed in 
 the young, that the school cannot accomplish even the smallest 
 part of them. It would certainly be most desirable to teach the 
 children to model, and to paint, and to play a musical instru- 
 ment, and to sing. 
 
 It would be highly desirable for the school to discover any 
 latent talents and abilities of the pupil. Some persons have not 
 been aware of possessing high talents, which would have re- 
 mained latent if they had not been discovered by accident. But 
 until the school can fulfill well the great and difficult tasks of 
 developing the body, the character, and the mind of the young, 
 no other functions should he included in its work. 
 
 ddie scliools cannot teach all that is worth knowing, all that 
 we ought to know in order to understand social and economic 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 55 
 
 conditions and changes, the workings of nature and to enjoy 
 its beauties. But the school ought to awaken the desire for 
 knowledge. It should train the young in power of observation, 
 in power of discrimination, power of concentration, and in all 
 those abilities and qualities of the mind which are developed by 
 the study of the different branches of the school curriculum, 
 and in no small measure by the study of mathematics. 
 
 There is a tendency to correlate the different branches of 
 elementary mathematics, especially algebra, with some of the 
 branches of the natural sciences, particularly with physics, and 
 mechanics. There are a number of causes which have led to this 
 tendency, one of them, I feel, has as its base the dssatisfaction 
 of teachers of physics and engineering with the average stu- 
 dent’s ability to perform with facility and accuracy algebraical 
 and arithmetical operations, and to solve simple equations or 
 the different letters which occur in them. Also, the equations 
 in most of the books on algebra have integral coefficients, and 
 give for the value of the unknown, integers or simple fractions, 
 but many of the problems in physics and engineering lead to 
 equations whose coefficients are decimals and the results are 
 very often complicated fractions which must be expressed in 
 decimals. I claim that the student who is well trained in per- 
 forming arithmetical and algebraical operations and who has a 
 clear conception of the principles underlying these operations, 
 ought to be able to handle algebraical expressions, and to solve 
 the equations to which many of the problems in physics or 
 engineering may lead. 
 
 There are three kinds of problems from physics and mechanics 
 that might be used to illustrate mathematical operations. To 
 understand the conditions of one kind of these problems, a 
 thorough knowledge of the principles of the physical and me- 
 chanical phenomena involved is necessary, but to effect their 
 solution only the simplest arithmetical operations need be used. 
 The solution of such problems belongs in a course of physics. 
 The age and mental development of the pupil when he studies 
 arithmetic will as a rule not permit of the thorough understand- 
 ing of the principles of physics and mechanics necessary to 
 translate the conditions of these problems into the language of 
 arithmetic. 
 
 There are problems from physics and mechanics in the solu- 
 
56 
 
 THE MATHEMATICS TEACHER. 
 
 tion of which the methods of algebra may be used to advantage, 
 and in which the translation of the given conditions leads to 
 very simple algebraical expressions. The solution of this class 
 of problems also belongs more properly in a course in physics. 
 By introducing into such problems conditions which are arti- 
 ficial, the translation of the conditions and the resulting alge- 
 braical expressions may be increased to any degree of difficulty. 
 
 The great majority of problems from physics and engineer- 
 ing, some even of the simplest kind, need for their solution a 
 knowledge of calculus, and many of them, a knowledge of the 
 more advanced branches of mathematics, like differential equa- 
 tions, definite integrals, etc. 
 
 To solve these problems, perfect facility in handling alge- 
 braical and trigonometrical expressions, as well as a thorough 
 knowledge of the conceptions and operations of calculus is 
 necessary. 
 
 I feel that the correlation between mathematical subjects 
 and those of some of the natural and applied sciences has at 
 its base also the desire of the teacher to satisfy the tendency to 
 make the study of mathematics practical and interesting to the 
 student. But most of the problems from physics and mechanics 
 which I have seen or have been able to compose, and the solution 
 of which can be effected by algebraical operations, are of 
 little practical use. There is not much more practical infor- 
 mation in figuring out how long it takes a boat to cross a 
 river under certain conditions of wind and current, than there 
 is in figuring out how old Ann is. To calculate how long it 
 would take a boat to actually cross a river, it would be neces- 
 sary first to determine the velocity of the wind, the velocity of 
 the current, the weight of the boat and so on. To obtain some 
 of these data, a knowledge of meteorology, hydrodynamics, etc., 
 and skill in handling delicate apparatus are necessary. 
 
 The entire question as it appears to me is whether the formal 
 study of mathematics, especially of algebra, has its place in the 
 curriculum, and whether the formal study of this subject helps 
 to develop qualities of mind which are desirable. 
 
 Just as it recjuircs different ingredients to make up the diet 
 for the body to kee|) all the organisms healthy and strong, and 
 in a haniKmions state of dcvel()])nicnt, so the mind needs for 
 its development, a curriculum embracing different subjects. 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 5/ 
 
 The study of formal algebra, without any regard to its appli- 
 cation, teaches continuous concentration of attention, uncon- 
 scious following of a number of rules in more or less rapid 
 succession, and is an excellent discipline for acquiring habits 
 of precision, accuracy and power of mind. 
 
 Facility in handling algebraical expressions is a necessary 
 preparation for all other branches of mathematics. Without a 
 thorough knowledge of the principles of algebra and perfect 
 facility in performing arithmetical and algebraical operations, 
 the student will not be able to gain a knowledge of the prin- 
 ciples of trigonometry, solid geometry, calculus and other 
 branches of mathematics, and to solve the problems of these 
 subjects. The student will need all the time which is ordinarily 
 given in the secondary schools to the study of algebra, to acquire 
 facility in this subject, and not until he has attained perfect 
 facility in performing algebraical operations, should other mat- 
 ter be introduced. 
 
 No student can hope to acquire a thorough mastery of a 
 branch of mathematics unless he is able to perform with little 
 effort such operations from other branches as may be needed 
 in this one. 
 
 When he takes up the study of algebra, the student must be 
 able to perform with ease such arithmetical operations as he 
 may need in algebraic work. In the study of trigonometry he 
 must have perfect facility in such algebraical and geometrical 
 work as may be required in the solution of the trigonometrical 
 work. Again, if he is to successfully acquire a thorough knowl- 
 edge of the principles and operations of calculus, he must have 
 perfect mastery of such arithmetical, algebraical, geometrical 
 and trigonometrical work as he may need in the work in 
 calculus. 
 
 Just as when we have acquired facility in spelling and writing, 
 we need pay little attention to these, but many concentrate our 
 efforts on the formulation of our ideas. 
 
 Unless the results of a problem are accurate, they are of 
 little use. To carry a mathematical operation through all of 
 its stages to an accurate result, requires a thorough knowl- 
 edge of the subject and a facility which can only be acquired by 
 constant practice and by hard work. It is a great injustice to 
 the student who chooses a profession like engineering, to use 
 
58 
 
 THE MATHEMATICS TEACHER. 
 
 the time needed to acquire facility in performing algebraical 
 operations, in the solutions of problems to arouse his interest, 
 or which appear to him to he practical. 
 
 The educational value and the mental development which is 
 derived from the study of physics is in a sense different from 
 the educational benefit to he derived from the study of mathe- 
 matics. There can be no objection to the use of occasional 
 illustrations from physics or mechanics, or from any of the 
 sciences, to show the student the power of mathematical study, 
 outside of its value as an educational medium, but the student 
 must have a thorough knowledge of the principles and laws of 
 the sciences from which the illustrations are drawn in order 
 to translate intelligently the conditions of such problems in the 
 language of mathematics. 
 
 In the teaching of mathematics, we must be very careful not 
 to give way to fads or to apply new methods and tendencies, 
 unless we are sure that they will lead to more effective results. 
 
 We must not experiment on the child that has but a single 
 opportunity to gain or to lose. In the interest of the children 
 who are to make the world what it will be during the next 
 generation, and in a way, during all generations to come ; in the 
 interest of the larger purposes of true education, I wish to 
 sound a warning that we exercise, as far as the teaching of 
 mathematics is concerned, the greatest care and caution before 
 applying a “modern tendency’’ to the minds of the young. For 
 years we have been following in some of the elementary branches 
 of mathematics, the German methods of teaching. There is 
 now a decided tendency to follow Professor John Perry, of the 
 Royal College of Sciences of London, who terms the German 
 method the “ intellectual Strassburg goose-stufhng,” and in an- 
 other place, the “ intellect by order of the authorities.” He calls 
 the English system “ antiquated,” and has a method all his own.* 
 
 This country has proven the most fertile soil for all kinds of 
 experiments in the teaching of mathematics. We have been 
 quick to adopt methods, especially if advocated by foreign 
 ])edagogues. 
 
 Our course in this respect has been similar to that of a person 
 who attaches a higher appreciation to a thing, even if of in- 
 
 * School Mai/icmdtics, Marcli, 1004, p. 195. 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 59 
 
 ferior make and quality, simply because the label states that 
 it is made in a foreign country. 
 
 The Grube system, long after it has been discarded in the 
 land where it originated, is still in use in many schools here. 
 While the method of teaching and the selection of text-books 
 are in some foreign countries in part regulated by the Govern- 
 ment, and therefore more or less uniform, it is left here, as a 
 rule, to every city, and in some places to every school, to choose 
 its text-books and the method of presenting the subject. 
 
 It is true that we are a young nation, but we have already 
 reached maturity. We ought not to follow the utilitarian ideas 
 in education of one nation, nor the impractical ones of another. 
 We ought to develop a system of teaching mathematics — if we 
 find the present system wanting — which is peculiarly our own, 
 which is in accordance with our ideals and inspirations, our 
 temperament and environments, in short, which is American. 
 
 While Germany was, and in a measure still is, the nursery for 
 the world in some of the sciences and arts, other nations of 
 Europe have used the German ideas only to reach their own 
 maturity. They have then developed their sciences and arts 
 in accordance with their own national tendencies and peculi- 
 arities. 
 
 The Italians have developed a distinct school of music, and 
 also of some of the sciences — among them mathematics. 
 
 The French have a distinct school of the art of painting, and 
 have methods and tendencies in some of the sciences which 
 are peculiar to them and to their national characteristics. 
 
 No matter how experienced the teacher may be, he must be 
 sure that he has a thorough mastery of an idea before present- 
 ing it to his students, and he must plan and lay out in every 
 detail the work which he is to take up with the class. In sup- 
 port of this argument, I may be permitted to state an incident 
 of my own experience. After having taught a certain subject 
 for a number of years, both in the classroom and in private 
 tutoring, I felt that I was no longer in need of going over the 
 details of the lesson before meeting the class. It happened, 
 however, that during one year I had two sections in one and the 
 same subject, on the same days of the week, but at different 
 hours. The sections did not follow each other on the different 
 days in the same order; on some days one section and on other 
 
6o 
 
 THE MATHEMATICS TEACHER. 
 
 clays the other section came first. I found that after teaching 
 one section, I could do the work with the next section more 
 effectively, and go over the ground in a shorter time than with 
 the first. To give both sections as equal advantages as possible, 
 I made it a rule to go over the lesson before meeting the first 
 section just as though I were a novice in teaching the subject. 
 
 The abstruse manner in which a subject is presented must not 
 be confused with scientific treatment. The more difficult a 
 mathematical idea, the more lucidly and clearly it ought to be 
 presented to save the student time, strength, and often dis- 
 couragement. 
 
 Mathematics is advancing at such a rapid rate that it is becom- 
 ing quite impossible even for those who have high ability and 
 much leisure to devote to its study, to obtain a knowledge of all 
 its different branches. If this immense amount of subject matter 
 is not presented lucidly, as is the case in some of the works of 
 the foremost writers on mathematics, the task of acquiring 
 mathematical knowledge is quite discouraging, even to the ablest 
 students. 
 
 The great masters in their flights of genius perhaps ought not 
 to be expected to descend to the level of average mortals ; but 
 the writers of text-books, and those who record the results of 
 their research but who do not belong to this very exclusive class, 
 ought to present their ideas and their work in a clear and lucid 
 manner. The late Professor Tait must have assumed that all 
 the readers of his rather abstruse “ Elementary Treatise on 
 Quaternions,” Cambridge, 1890, would bring to it the same 
 ability, experience and knowledge in the subject which he has 
 possessed when he wrote the book. The following quotation 
 from this book page no, shows his sad lack of sympathy with 
 those who are perhaps, less gifted but v/ho are striving for 
 knowledge. “ [The words above, ‘it is evident,’ have been ob- 
 jected to by more than one correspondent. But, on full con- 
 sideration, I not only leave them where they are, but put them 
 in Italics. For they are, of course, addressed to the reader only; 
 and it is to be presumed that, before he reaches them, he has 
 mastered the contents of at least the previous sections which 
 bear on tliis (juestion. If, with these sections in his mind, he 
 does not see tlie ‘ evidence,’ he has begun the study of qua- 
 ternions too soon. I ” 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 
 
 6l 
 
 There is among teachers a great deal of difference of opinion 
 as to the method of conducting the work in the classroom, 
 especially in the advanced classes of the high school, and in the 
 college. Some teachers favor exclusively the so-called recita- 
 tion method ; others believe that the best results will be obtained 
 by explaining the subject to the class for the entire period. 
 This method most frequently results in a formal lecture. By 
 means of an examination in the subject at the end of the term 
 or of the year, or by occasional tests during the term, the teacher 
 gains information as to the standing in the subject of the 
 members of the class. But neither of these methods of conduct- 
 ing the work with a class does justice to the pupil. It is the 
 duty of the teacher to find out the state of knowledge of the 
 student from day to day. In this way the teacher will see in 
 what particular the student’s understanding of the work is 
 wanting. By giving the necessary explanation he will make the 
 ideas and the work clear to the student. This method with some 
 modifications ought to be used even in the work of the graduate 
 and the professional school. 
 
 We can never be sure whether or' not we know a thing, or 
 understand an idea, unless we come to practice it, or to express 
 it or to apply it. One of the means of acquiring clear ideas is to 
 speak the ideas out or to write them out. We must have the 
 student talk the ideas over with us. In this way only will he 
 find out whether or not they are perfectly clear to him. Human 
 nature is such that we are apt to do promptly what we must 
 give an immediate account of, and postpone such work for 
 which we are not at once accountable. 
 
 While I thoroughly believe that a person cannot do as good 
 work at one time as he can at another, and that we must in a 
 certain sense be disposed to do our work, yet practical life does 
 not tolerate such tendencies and variances of the mind. In life, 
 we have to do our day’s work, whatever our personal inclina- 
 tions may be. We ought therefore to train the youth to do his 
 work from day to day, and not to give way to slight indisposi- 
 tions and moods, unless the mind and body are in such a state 
 that it would be wiser to submit than to try to conquer them. 
 
 Besides getting clear ideas of the conceptions and laws of 
 mathematics, the student must also acquire facility in perform- 
 ing its various operations. The acquiring of this facility is an 
 
62 
 
 THE MATHEMATICS TEACHER. 
 
 excellent discipline, and is necessary if the student is to continue 
 the study of mathematics. Part of this facility the student must 
 acquire by means of home work. He will have to work at the 
 examples assigned to him, until he has entire mastery of all of 
 the operations involved and can with little effort carry them 
 through to correct and accurate results. 
 
 The teacher must indicate the mistakes which the student has 
 made in such work and supplement the work of the student in 
 such a manner as to make it entirely clear to him. The teacher 
 must continually encourage the student to ask pertinent ques- 
 tions and not rest content until he has entirely mastered each 
 point of the lesson. 
 
 Very often a student refrains from asking a question because 
 of a somewhat indefinite idea which seems to exist in the mind 
 of some of them, that they have, so to speak, no claim on the 
 services of the teacher, but that these services are rendered, in 
 a sense, as a matter of favor. If such an opinion is prevalent, 
 the teacher must make it clear to the student that it is his duty 
 to give the student all the information and help he may need, 
 and to answer such questions as will make the ideas presented 
 entirely clear to him. If the teacher is strong in character and 
 in mind — and only such ought to be employed — the clear under- 
 standing which the students will have of the teacher’s duties 
 and functions will not lessen their respect for him and not 
 minimize his importance. 
 
 Since the acquisition of knowledge is, as we have stated, 
 a matter of evolution, we must do our work from day to day. 
 We must digest and assimilate the ideas we learn one by one, 
 if we are to acquire clear conceptions of them, and if they are 
 to help our mental development and increase our mental power. 
 
 In most of the foreign universities attendance at classes is 
 not compulsory. The student is not held accountable for his 
 work from day to day. The work in the classroom consists of 
 more or less formal lectures delivered by the teacher. 
 
 It is })rcposterous to think that a young man of eighteen or 
 nineteen, with very little experience of the world and the things 
 in it, and often with little maturity of mind, should consider 
 it an affront if he made to attend lectures and submit to 
 being (jiiestioned, unless he does it from his own volition. 
 To he com])elled to attend lectures and recite, would, in the 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 63 
 
 opinion of these students be a violation of their so much cher- 
 ished “ Akademische Freiheit ” (academic freedom). 
 
 In some of these universities the student, whatever his age, 
 considers himself as belonging to an exclusive and privileged 
 class, and I cannot help but feel that he is often encouraged in 
 this belief by the authorities of the university. If he commits 
 an offence which would make him liable to punishment, in ac- 
 cordance with the law as applied to any other person, he is in 
 some of the university towns, not judged by the properly con- 
 stituted courts, but is left by them to be judged by the officers of 
 the university. The punishment inflicted by these authorities 
 generally consists in incarceration in the university prison 
 (Karcer). The jolly hours spent by the student in this 
 “ Karcer ” have been described in prose and verse. 
 
 The general conviction in this country that all men are equal 
 before the law, and the consequent respect for the law, has a 
 beneficial effect on the morals of the people. 
 
 In spite of the fact that in nearly every country in Europe, 
 an accurate record is kept of the occupation of every inhabitant, 
 of his previous residence, etc., I feel that the personal safety 
 and the moral status here, where all these regulations and pre- 
 cautions do not exist, are much greater than in most European 
 countries. 
 
 The system of education as it is carried on in some of the 
 foreign universities is bound to affect the intellectual and moral 
 status of the entire nation. If we can judge the signs of the 
 times and predict the future, the center of gravity of intellectual 
 powers is bound to shift from such nations to those where edu- 
 cation is conducted in a more effective manner. 
 
 While I am not able to support my arguments by statistics, it 
 is my own observation and experience that the number of stu- 
 dents who leave some of the foreign universities without finish- 
 ing their studies is growing larger and larger. 
 
 Such men, as a rule, become a burden to society, and are a 
 curse to civilization. Having been accustomed to the “ Aka- 
 demische Freiheit,” many of them are not fit to take employment 
 and to be subservient. They augment the ranks of those who 
 are dissatisfied with the world and the things in it. Some of 
 them become “ walking delegates,” or — I rather hesitate to ex- 
 press it — missionaries to foreign countries. 
 
64 
 
 THE MATHEMATICS TEACHER. 
 
 This shall not he construed as showing on my ])art a lack of 
 appreciation of the self-sacrificing men and women who are 
 willing to go to far-ofif lands and administer to those who need 
 their help. There are men and women who have felt the calling 
 to be missionaries, and many already while in youth have pre- 
 pared themselves for their chosen life work. 
 
 The teachers in some of the foreign universities consider 
 research in their special branch of learning, as their principal 
 duty, and the instruction of the students a matter of secondary 
 importance. 
 
 They pursue this course not as a result of any lack of a sense 
 of duty, but as a matter of custom and tradition, dating from 
 the time when only a few, and those the ablest, attended a 
 university. These students came to the university more for the 
 sake of getting the stimulus and the inspiration which comes 
 from contact with great minds, than for the purpose of instruc- 
 tion. But a great injustice is done to the student who attends a 
 university in order to acquire knowledge and obtain informa- 
 tion, if he does not receive instruction in his studies from those 
 whose duty it is to give it. 
 
 To many of these investigators mankind owes a debt of grati- 
 tude for the unselfish manner in which they have devoted them- 
 selves to the advancement of science. They have by their labors, 
 not only increased knowledge, but also have added to the com- 
 forts and conveniences of life, and often their discoveries and 
 inventions have been the means of producing wealth. The only 
 reward for their efiforts is the pleasure and satisfaction which 
 comes from searching for truth and finding it. 
 
 The man who makes the discovery or the invention, especially 
 if it is along scientific lines, does not, as a rule, reap any 
 material benefit from his abilities and labor. One of the most 
 touching incidents I know of is that of Heinrich Hertz, the in- 
 ventor of wireless telegraphy. He sacrificed his life for science, 
 and overwork led to a premature death. Science is enriched by 
 his labors. The comfort and convenience which his invention 
 has added to life is hard to estimate. It is pathetic to think that 
 his name is not even connected with the invention, and that he 
 is not even given j)opular credit for his labors. 
 
 It is true that there are men who, while carrying on research 
 in their chosen science, are still able to descend to the level of 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 65 
 
 the beginner, and maintain an intellectual companionship with 
 even the youngest student. Such teachers, I fear, are the ex- 
 ception. I feel that the majority of the teachers when engaged 
 in the all-absorbing work of scientific research, are not apt to 
 be in sympathy with the needs of the beginner in the subject. 
 
 Hegel is said to have calmly finished his “ Phaenomenologie 
 des Geistes ” at Jena, on October 14, 1806, not knowing anything 
 whatever of the battle that was raging around him. The carry- 
 ing on of original investigation, especially in the pure sciences, 
 requires the closest concentration of our mental powers which 
 very often absorbs our physical strength and endangers our 
 health, and may thus unfit us for the difficult task of teaching the 
 young. I also quote in this connection from a letter of the great 
 Jacobi : 
 
 “ It is a wearisome work which I have done, wearisome work 
 in which I am engaged. It is neither diligence nor memory 
 which leads here to the goal, for they are here merely the sub- 
 servient ministers of the moving pure thought. But tenacious, 
 brain-splitting pondering (Nachdenken) requires more strength 
 than the most persistent diligence. If therefore, by the con- 
 stant practice of this pondering I have acquired some power 
 in it, let it not be thought that it has been made easy for me, 
 perhaps by a fortunate gift of nature. Wearisome, wearisome 
 work I have had to undergo, and the anxiety of the pondering 
 has often seriously endangered my health. The consciousness, 
 however, of power attained, forms the best reward of the work, 
 and is also an encouragement to continue and not to relax.” 
 
 The young ought to be taught not only to husband their physi- 
 cal strength, but to be economical with their time as well. It 
 is most important that the student, whether in preparing his 
 lessons or in his reading, learn to accomplish the best results 
 with the least expenditure of time. In other words, he must 
 learn how to study and how to work. 
 
 The method of the spiral system which has been successfully 
 used in the teaching of arithmetic and which ought to be applied 
 to the teaching of each of the branches of mathematics, ought 
 also to be followed in reading and studying any subject, especi- 
 ally the more profound ones. It would be a waste of time, and 
 with some people, a source of discouragement, to attempt to 
 read a difficult work, say on philosophy, and endeavor to have 
 
66 
 
 THE MATHEMATICS TEACHER. 
 
 a thorough understanding of it as we go along. A good plan is 
 to read the whole book, or a part of it carefully, without at- 
 tempting to thoroughly understand its most difficult ])assagcs. 
 The second reading of the book will clarify some of these pas- 
 sages which are not clear at the first reading. Repeated readings 
 will make the book clearer and clearer until a thorough under- 
 standing of the whole work is reached. 
 
 In mathematics, the student must go over the lesson carefully, 
 concentrate his mind and his attention on it, without using an 
 undue amount of time on the more difficult parts. After going 
 over the lesson a second time, he will find that the ideas are 
 much clearer to him. He will have to go over the lesson again 
 and again until he has a thorough mastery of it. 
 
 The student must learn to concentrate all of his thoughts on 
 a piece of work in hand, and must not allow his attention to be 
 distracted by exterior influences. He ought to be trained to 
 think and to meditate. Meditation is a great help towards 
 clarifying our ideas on subjects we have learned, or we have 
 heard, or read or seen. Meditation develops and strengthens 
 the power and the habit of thinking. In the rush of life, we do 
 not meditate enough. The youth must be constantly reminded 
 and persistently urged to make the best use of his opportunities 
 and of his time. 
 
 If I am permitted to be again personal — and I feel that we 
 certainly can speak more authoritatively about our own experi- 
 ence than about the experience of others — there is nothing that 
 fills me with so much regret as to think of the days and the 
 hours of which I have not made proper use. We feel the regret 
 the more, as we grow older. O ! for the hours wasted which 
 never return ; for the opportunities lost which never come again. 
 
 I do not, however, wish it to be understood that the young 
 ought to be expected to employ their time and energy only in 
 study and in such a manner as will be of immediate use and 
 direct benefit to them. 
 
 I'ime si)ent in well directed out-door exercises, time spent in 
 reading a good book, or in the company of a stimulating friend, 
 in nursing good comradeship; time spent in listening to good 
 music, in seeing a good l)lay, or in watching manly sport, is time 
 well sj)ent. 
 
 It would ai)pear that the teacher in the college or in the 
 
MODERN' TENDENCIES IN TEACHING MATHEMATICS. 6/ 
 
 university is, so to speak, the court of the last resort, and is the 
 only judge of his efforts, yet it is far from being so. 
 
 While the work of the secondary and elementary schools has 
 been lately subject to a great deal of criticism, the work of the 
 college has been attacked by men of affairs on whose sympathy 
 with the college its maintenance depends, and its great mission 
 is made possible. 
 
 These attacks have been found to be worthy of comment, 
 even by some of the most conservative college presidents. 
 While the work in the college depends on the preparation which 
 the pupil receives in the elementary and secondary schools, yet 
 the college is judged irrespective of this preparation, and only 
 by what the finished product — the graduate of the college — is 
 able to do and by what he represents. 
 
 Those who have been successful in life, in spite of their lack 
 of college education, or, as in some cases without any school 
 education because of their superior qualities of character and 
 abilities of mind, and perhaps also because they were aided by 
 favorable conditions, are apt to be rather skeptical of the bene- 
 fits to be derived from higher education. There are some who 
 believe that a higher education even unfits a man or woman 
 for the work of life. I cannot help thinking that such criticisms 
 are merely the result of experience with those college men, who 
 have failed to make proper use of the opportunities and facilities 
 offered by the institutions which they have attended. A person 
 may go through college, and yet not be educated. 
 
 The amount of ground covered in a subject does not deter- 
 mine the mental status of the learner. The inventory of the 
 mental powers of a person cannot be measured by the number 
 of books read, by the number of topics studied, by the number 
 of facts learned, but only by the number of clear ideas acquired, 
 and to what extent he is able to supply them to his own moral 
 and physical betterment, and to the welfare of his fellowmen. 
 Such criticisms will be silenced when the young realize the true 
 purpose for which they go to school and college, when they 
 avail themselves of the opportunities, and make proper use of 
 the facilities offered for their physical, moral and intellectual 
 development. 
 
 True education ought to teach the youth that all work is 
 ennobling if it is done well and if it helps towards obtaining the 
 
68 
 
 THE MATHEMATICS TEACHER. 
 
 necessary comforts and conveniences of life. That experience 
 gives facility, and without facility the most effective results 
 cannot be obtained. To do our duty should be our highest 
 ambition. The greater our opportunities, the greater our re- 
 sponsibilities. A man’s worth ought to be measured by his 
 character only, and the man who has been fortunate enough to 
 enjoy the advantages of education, ought to become a missionary 
 for all that is good and noble. He should administer his ideas 
 by precept and example to those who have not enjoyed such 
 advantages as he has. 
 
 Idleness and vice go hand in hand, but neither ever remains 
 unpunished. Reward and punishment do not in general follow 
 our actions immediately, but will surely come some day, per- 
 haps not until the lapse of generations. Leisure is only to be 
 enjoyed by those who have done their life’s work, and who 
 know how to use it. 
 
 True arguments must not be supported by quotations, but 
 must be substantiated by reasoning, or by results which confirm 
 the experience of mankind. 
 
 Education must teach the student true modesty. He must 
 learn that our knowledge is little as compared with what we still 
 have to learn ; that we are but passing trifles when compared 
 with the eternity of matter and the immensity of the universe ; 
 that unless we contribute something towards helping the world 
 to become better than we find it, we have lived in vain. 
 
 It is the wish and the hope of parents that their children be 
 in every way better than they are. It must be our purpose to 
 leave the world better than we have found it. In this, and only 
 in this way, is the progress of the world morally, intellectually 
 and socially possible. 
 
 While a knowledge of the history of mathematics and of all 
 other kinds of information as to the development and the philo- 
 sophical aspects of the subject will broaden the teacher and make 
 him better fit for his work, yet all this will be of little avail if 
 he has not a thorough knowledge of the subject he is teaching, 
 ])erfect facility in solving any of the examples and problems in 
 connection with the work, and such a mastery and resourceful- 
 ness in the subject which is above the hook and syllabus and as 
 will enable him to adapt the instruction to the needs of the 
 students of different ability, and different environments. 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 69 
 
 Better text-books, a change in the number and the kind of 
 subject taught, a better distribution of the hours for work and 
 the hours for recreation, both for pupils and teachers, a change 
 in the ratio between the number of pupils and the number of 
 teachers, will help and help a great deal if the teacher has the 
 highest qualities of character and mind, and is able to impress 
 his pupils with them, and use them to make the pupils follow his 
 teachings and advice in the classroom and out. The most 
 efficient supervision, the wisest direction by school officers, all 
 advances in the physical equipment of the school will be of little 
 avail, unless the government of the school is able to attract and 
 retain men and women of the highest qualities of character and 
 mind, well equipped for their work. The plans of the architect 
 are of little use, unless the builder and sculptor are able to 
 execute them and to give form to the plans. 
 
 The teacher must understand the peculiarities of the student 
 and their difficulties, he must be able to descend to the level of 
 the student and to enter into an intellectual companionship even 
 with the beginner in a subject, and also with those of little 
 ability. If the teacher will do this, then such charges as are 
 often made by students, that while the teacher seems to have a 
 knowledge of the subject, he does not seem able to impart it to 
 his students, will no more be heard. 
 
 The mode of presenting a subject, the manner of handling a 
 class cannot very well be learned from books. The description 
 of an event is no match to actually seeing it happening. The 
 reading of instructions is only a weak substitute for the “ living 
 word,” and seeing these instructions actually carried out. No 
 one should have the right to teach unless he attends for a certain 
 period of time the classes in actual progress of men and women 
 who have reached a high proficiency of imparting knowledge and 
 conducting classes. The candidate must have studied and be 
 proficient in the subject of which such a class is treating, in 
 order that he may be able to concentrate his attention and efforts 
 on the mode of presentation, and the manner in which the class 
 is conducted. 
 
 What we need more than anything else is to attract men and 
 women to our profession who not only have perfect mastery 
 of the subject they are teaching, and the ability to impart the 
 knowledge to their pupils, but also who have the personality, the 
 
70 
 
 THE MATHEMATICS TEACHER. 
 
 enthusiasm for their high calling, and all those qualifications 
 of character and mind to make them an inspiration and a stimu- 
 lus to their students. There is no occupation in which (jualities 
 of sincerity and frankness and the most extreme straightfor- 
 wardness are more important than in teaching. If the teacher 
 is to accomplish his great task, he must have the complete con- 
 fidence of his pupils. The teacher exerts the most lasting in- 
 fluence during the formative period of the child’s character. The 
 character of the teacher, his habits, his entire life must be models 
 after which the student can most safely pattern his own life. 
 
 While the profession of the teacher is in a sense rather re- 
 spected, yet on the whole, the teacher does not enjoy the appre- 
 ciation which is accorded to members of other professions. I 
 feel that the public at large, and especially men of affairs, are 
 inclined to consider the teacher whose actions and activities 
 extend only to the young, as not possessing such qualities as 
 the men who are in daily contact with the practical world. 
 
 If the teacher is to discharge his important duties of develop- 
 ing the character and the mind of the young, and thus effecting 
 the amelioration of the race, he must be well informed about 
 the world and the things in it, about everything that is good and 
 noble and everything that needs to be improved and remedied. 
 
 The teacher with the opportunities for self-culture and medi- 
 tation, ought to be best equipped to take a leading part in every 
 movement that will make for moral and intellectual betterment. 
 
 While the profession of the teacher has the attraction, that 
 it gives more leisure than is enjoyed by the members of most 
 of the other professions, it does not, on the other hand, offer 
 such possibilities as other professions hold out to those who 
 enter them. The teacher on entering his profession knows 
 pretty well what limit he is able to reach; he cannot, on the 
 whole, aspire to reach such a goal as the average American boy 
 may set for himself when he enters other professions or business. 
 
 The tendency has prevailed at all times, and is stronger now 
 than ever before to offer in any of the activities of life, high 
 rewards for superior qualifications and high efficiency of service. 
 The same tendencies ought to apply to the teaching profession 
 also. Teaching is an art, and a high and a noble art. It is a 
 (iofl-given art to cut a statue out of a piece of marble, which 
 after all is only subject to the chisel of the sculptor. What 
 
MODERN TENDENCIES IN TEACHING MATHEMATICS. 
 
 71 
 
 nobler art it is to mould the character and the mind of the child 
 — with its environments, home influence and temperament — so 
 as to make it, morally, mentally and physically as nearly as 
 possible, a perfect man or woman. 
 
 There is a difference in the degree of ability necessary to solve 
 the various problems which the follower of most of the profes- 
 sions meets in his practice. It is not as difficult to diagnose 
 and cure some diseases as it to conquer others. Some engineer- 
 ing problems require for their solution a high degree of ability 
 and often great ingenuity, while a moderate degree of ability 
 is necessary to solve others. The same is true of the problems 
 in the other professions. As far as the teaching profession 
 is concerned, it takes as high a degree of ability, of character 
 and mind, and of all other high qualities, to mould the character 
 and mind of the child attending the lowest grades of the ele- 
 mentary schools as it does to influence the student in the highest 
 classes of the college. In all other professions and occupations 
 there is room for persons with moderate ability. If they per- 
 form their duties conscientiously, their services are as neces- 
 sary and important as are the services of those who are engaged 
 in the more difficult problems of their profession. In the teach- 
 ing profession there is room only for those of highest ability and 
 superior qualities of character, mind and heart, if education is 
 to accomplish its great mission, the amelioration of the race. 
 
 The public provides, on the whole, liberally for the necessities 
 of the community. If it could be made to see the great influence 
 and the mission of education, then the respect for education 
 and for the teacher will rise, and he will be accorded in every 
 way the credit which he deserves. 
 
 University of Pennsylvania, 
 
 Philadelphia, Pa. 
 
UNIVERSfTY OF ILLINOIS-URBANA 
 
 3 0112 114019281